Search Results for equation*

Biographies

1. Kuczma biography
   
   • For example in the student years he published papers such as: (with Stanislaw Golab and Z Opial) La courbure d'une courbe plane et l'existence d'une asymptote (1958), On convex solutions of the functional equation g[a(x)] - g(x) = j(x) (1959), On the functional equation j(x) + j [f (x)] = F(x) (1959), On linear differential geometric objects of the first class with one component (1959), Bemerkung zur vorhergehenden Arbeit von M Kucharzewski (1959), Note on convex functions (1959), and (with Jerzy Kordylewski) On some functional equations (1959).
     
   • On 1 December 1966, in addition to these position which he continued to hold, Kuczma became head of the Department of Functional Equations at Katowice.
     
   • From the founding of the new University, Kuczma became the head of the mathematics section and head of the department of functional equations.
     
   • In fact he published around 30 papers during the 1980s despite his severe disability and in [Selected topics in functional equations and iteration theory, Graz, 1991 (Karl-Franzens-Univ.
     
   • The first of these Functional equations in a single variable appeared in 1968 and was the first book to be written on this topic.
     
   • This is the first book ever published on functional equations in a single variable ..
     
   • The related questions of commuting functions, continuous iteration, and Schroder's and Abel's functional equations are also treated.
     
   • Kuczma's second book was An introduction to the theory of functional equations and inequalities.
     
   • Cauchy's equation and Jensen's inequality published in 1985.
     
   • Probably even the most devoted specialist would not have thought that about 300 pages can be written just about the Cauchy equation (and on some closely related equations and inequalities).
     
   • In the opinion and experience of this reviewer this is a very useful book and a primary reference not only for those working in functional equations, but mainly for those in other fields of mathematics and its applications who look for a result on the Cauchy equation and/or the Jensen inequality.
     
   • His final book Iterative functional equations was written jointly with Bogdan Choczewski and Roman Ger who had worked for his doctorate with Kuczma at the Silesian University of Katowice, graduating in 1971.
     
   • The book is a cohesive and exhaustive account of contemporary theory of iterative functional equations.
     
   • Fundamental notions such as existence and uniqueness of solutions of equations under consideration are treated throughout the book as well as a surprisingly wide scale of examples showing applications of the theory in dynamical systems, ergodic theory, functional analysis, functional equations in several variables, functional inequalities, geometry, iteration theory, ordinary differential equations, partial differential equations, probability theory and stochastic processes.
     
   • the paramount achievement of Professor Marek Kuczma was the creation and development of a systematic theory of iterative functional equations and founding a mathematical school centred around the seminar conducted by him since October 1964.
     
   • Kuczma was considered an outstanding mathematician highly esteemed by the international community of specialists; among the functional equationists he had commonly been treated as one of the informal leaders.
     

2. Bezout biography
   
   • As we have indicated Bezout is famed for being a writer of textbooks but he is famed also for his work on algebra, in particular on equations.
     
   • His first paper on the theory of equations Sur plusieurs classes d'equations de tous les degres qui admettent une solution algebrique examined how a single equation in a single unknown could be attacked by writing it as two equations in two unknowns.
     
   • It is known that a determinate equation can always be viewed as the result of two equations in two unknowns, when one of the unknowns is eliminated.
     
   • Of course on the face of it this does not help solve the equation but Bezout made the simplifying assumption that one of the two equations was of a particularly simple form.
     
   • For example he considered the case when one of the two equations had only two terms, the term of degree n and a constant term.
     
   • Already this paper had introduced the topic to which Bezout would make his most important contributions, namely methods of elimination to produce from a set of simultaneous equations, a single resultant equation in one of the unknowns.
     
   • He also did important work on the use of determinants in solving equations.
     
   • This appears in a paper Sur le degre des equations resultantes de l'evanouissement des inconnues which he published in 1764.
     
   • As a result of the ideas in this paper for solving systems of simultaneous equations, Sylvester, in 1853, called the determinant of the matrix of coefficients of the equations the Bezoutiant.
     
   • These and further papers published by Bezout in the theory of equations were gathered together in Theorie generale des equations algebraiques which was published in 1779.
     
   • The degree of the final equation resulting from any number of complete equations in the same number of unknowns, and of any degrees, is equal to the product of the degrees of the equations.
     
   • By a complete equation Bezout meant one defined by a polynomial which contains terms of all possible products of the unknowns whose degree does not exceed that of the polynomial.
     
   • nor could he even label his equations with a suffix notation.
     
   • History Topics: Quadratic cubic and quartic equations .
     

3. Ladyzhenskaya biography
   
   • It was here where she first started studying algebra, number theory and subsequently partial differential equations.
     
   • She became interested in the theory of partial differential equations due to the influence of Petrovsky as well as the book by Hilbert and Courant.
     
   • Being a talented student, the authorities often ignored absences at compulsory lectures while she attended research seminars including the algebra seminars of Kurosh and Delone and the seminar on differential equations headed by Stepanov, Petrovsky, Tikhonov, Vekua and their students and colleagues.
     
   • At the end of her fourth year she organized a youth seminar to study the theory of partial differential equations and persuaded Myshkis, a student of Petrovsky, to go with her to ask Petrovsky to chair the seminar.
     
   • Find the least restrictive conditions on the behaviour of parabolic equations under which the uniqueness theorem holds for the Cauchy problem.
     
   • For hyperbolic equations, construct convergent difference schemes for the Cauchy problem and for initial-boundary problems.
     
   • It was also here that she was strongly influenced to study the equations of mathematical physics.
     
   • In 1949 Olga defended her doctoral dissertation (comparable to an habilitation) which was on the development of finite differences methods for linear and quasilinear hyperbolic systems of partial differential equations, formally supervised by Sobolev though in practice it was Smirnov.
     
   • Her first book published in 1953 called Mixed Problems for a Hyperbolic Equation used the finite difference method to prove theoretical results, mainly the solvability of initial boundary-value problems for general second-order hyperbolic equations.
     
   • As in the previous decade, during the 1960s she continued obtaining results about existence and uniqueness of solutions of linear and quasilinear elliptic, parabolic, and hyperbolic partial differential equations.
     
   • She then studied the equations of elasticity, the Schrodinger equation, the linearized Navier-Stokes equations, and Maxwell's equations.
     
   • The Navier-Stokes equations were of great interest to her and continued to be so for the rest of her life.
     
   • Many papers written jointly by Olga and Nina Ural'tseva were devoted to the investigation of quasilinear elliptic and parabolic equations of the second order.
     
   • At the start of the last century Sergei Bernstein proposed an approach to the study of the classical solvability of boundary-value problems for equations based on a priori estimates for solutions as well as describing conditions that are necessary for such solvability.
     
   • From the mid-1950's Olga and her students made advances in the study of boundary-value problems for quasilinear elliptic and parabolic equations.
     
   • They developed a complete theory for the solvability of boundary-value problems for uniformly parabolic and uniformly elliptic quasilinear second-order equations and of the smoothness of generalized solutions.
     
   • One result gave the solution of Hilbert's 19th problem for one second-order equation.
     
   • When Olga first started to work on the Navier-Stokes equation, she was unaware of the work of Leray and Eberhard Hopf.
     

4. Li Zhi biography
   
   • This was a notation for an equation and, although the work of Li Zhi is the earliest source of the method, it must have been invented before his time.
     
   • Here the numbers which in our notation correspond to the coefficients of the equation are placed above each other so that the coefficient of x is placed above the constant, the coefficient of x2 is placed above the coefficient of x etc.
     
   • Unlike most western algebraists, Li Zhi never explains how to solve equations, but only how to construct them.
     
   • But he does not limit his reflections to equations of degree two or three; for him, the fact that polynomial equations of arbitrarily high degree are involved is of little importance.
     
   • Moreover, he never explains what he understands by an equation, an unknown, a negative number, etc., but only describes the manipulations which should be carried out in specific problems, without worrying about arranging his text in terms of definitions, rules and theorems.
     
   • To solve the above equation Li Zhi would bring the leading coefficient to -1 and then give the solution; in this case 20.
     
   • The type of problem which worried mathematicians in Islamic countries, and in Europe, concerning the solution of cubic, quartic, and higher order equations did not seem to arise in China.
     
   • Li Zhi seems happy with equations of any degree and, although methods to solve equations do not appear explicitly, one has to assume that he used methods similar to those Ruffini and Horner discovered over 600 years later.
     
   • If we examine Li Zhi's solution closely we see a remarkable depth of understanding of equations.
     
   • The problem leads to a quartic equation with a factor x + 16.
     
   • Li Zhi goes through the detailed, and quite hard, argument which leads to the quartic equation .
     
   • The central theme is the construction and formulation of quadratic equations.
     
   • Some of these equations are solved by the "coefficient array method" described above, but some are formulated using the tiao duan or "method of sections".
     
   • This older geometric style method of solving equations was used by Chinese mathematicians before Li Zhi and so the New steps in computation gives historians a unique opportunity to see the new coefficient array method beside the older method of sections.
     
   • This is the quadratic equation we wrote in Li Zhi's coefficient array method above.
     

5. Fredholm biography
   
   • As was always the case with all the deep mathematical results which Fredholm produced, this result was inspired by mathematical physics, in this case by the heat equation.
     
   • His 1898 doctoral dissertation involved a study of partial differential equations, the study of which was motivated by an equilibrium problem in elasticity.
     
   • He solved his operator equation in the particular cases which arise in the study of the physical problem in his thesis (and in the paper which appeared in 1900 based on that thesis) while the general case was solved by Fredholm somewhat later and not published until 1908.
     
   • Fredholm is best remembered for his work on integral equations and spectral theory.
     
   • Two years later in Stockholm a lecture about the 'principal solutions' of Roux and their connections with Volterra's equation led to a vivid discussion Finally, after a long silence Fredholm spoke and remarked in his usual slow drawl: in potential theory there is also such an equation.
     
   • In 1900 a preliminary report on his theory of Fredholm integral equations was published as Sur une nouvelle methode pour la resolution du probleme de Dirichlet.
     
   • Volterra had earlier studied some aspects of integral equations but before Fredholm little had been done.
     
   • Of course Riemann, Schwarz, Carl Neumann, and Poincare had all solved problems which now came under Fredholm's general case of an integral equation; this was an indication of how powerful his theory was.
     
   • Hilbert immediately saw the he importance of Fredholm's theory, and during the first quarter of the 20th century the theory of integral equations became a major research topic.
     
   • Fredholm published a fuller version of his theory of integral equations in Sur une classe d'equations fonctionelle which appeared in Acta Mathematica in 1903.
     
   • Hilbert extended Fredholm's work to include a complete eigenvalue theory for the Fredholm integral equation.
     
   • Fredholm's work on integral equations was met with great interest and boosted the morale and self-respect of Swedish mathematicians who so far had been working under the shadow of the continental cultural empires Germany and France.
     
   • Integral equations had now become a new mathematical tool not confined to symmetrical kernels.
     
   • Unlikely as it sounds, he built his first violin from half a coconut, while he also used his talents at building machines to make one to solve differential equations.
     
   • Fredholm received many honours for his mathematical contributions, including the V A Wallmarks Prize for the theory of differential equations in 1903, the Poncelet Prize from the French Academy of Sciences in 1908, and an honorary doctorate from the University of Leipzig in 1909.
     

6. Abel biography
   
   • While in his final year at school, however, Abel had begun working on the solution of quintic equations by radicals.
     
   • In 1823 Abel published papers on functional equations and integrals in a new scientific journal started up by Hansteen.
     
   • In Abel's third paper, Solutions of some problems by means of definite integrals he gave the first solution of an integral equation.
     
   • Abel began working again on quintic equations and, in 1824, he proved the impossibility of solving the general equation of the fifth degree in radicals.
     
   • Geometers have occupied themselves a great deal with the general solution of algebraic equations and several among them have sought to prove the impossibility.
     
   • The second of these explanations does seem the more likely, especially since Gauss had written in his thesis of 1801 that the algebraic solution of an equation was no better than devising a symbol for the root of the equation and then saying that the equation had a root equal to the symbol.
     
   • He had been working again on the algebraic solution of equations, with the aim of solving the problem of which equations were soluble by radicals (the problem which Galois solved a few years later).
     
   • Also after Abel's death unpublished work on the algebraic solution of equations was found.
     
   • If every three roots of an irreducible equation of prime degree are related to one another in such a way that one of them may be expressed rationally in terms of the other two, then the equation is soluble in radicals.
     
   • An extract from Abel's On the algebraic resolution of equations (1824) .
     

7. Al-Khwarizmi biography
   
   • Having introduced the natural numbers, al-Khwarizmi introduces the main topic of this first section of his book, namely the solution of equations.
     
   • His equations are linear or quadratic and are composed of units, roots and squares.
     
   • He first reduces an equation (linear or quadratic) to one of six standard forms: .
     
   • Here "al-jabr" means "completion" and is the process of removing negative terms from an equation.
     
   • The term "al-muqabala" means "balancing" and is the process of reducing positive terms of the same power when they occur on both sides of an equation.
     
   • Al-Khwarizmi then shows how to solve the six standard types of equations.
     
   • For example to solve the equation x2 + 10 x = 39 he writes [Muhammad ibn Musa Al-Khwarizmi : Algebra (London, 1831).',11)">11]:- .
     
   • The question therefore in this type of equation is about as follows: what is the square which combined with ten of its roots will give a sum total of 39? The manner of solving this type of equation is to take one-half of the roots just mentioned.
     
   • in his introductory section al-Khwarizmi uses geometrical figures to explain equations, which surely argues for a familiarity with Book II of Euclid's "Elements".
     
   • Al-Khwarizmi's concept of algebra can now be grasped with greater precision: it concerns the theory of linear and quadratic equations with a single unknown, and the elementary arithmetic of relative binomials and trinomials.
     
   • From its true emergence, algebra can be seen as a theory of equations solved by means of radicals, and of algebraic calculations on related expressions..
     
   • The final part of the book deals with the complicated Islamic rules for inheritance but require little from the earlier algebra beyond solving linear equations.
     
   • Al-Khwarizmi and quadratic equations .
     
   • History Topics: Quadratic, cubic and quartic equations .
     

8. Carleman biography
   
   • One reason was that many of his results, for instance the extension of Holmgren's uniqueness theorem, the analysis of the Schrodinger operator, and the existence theorem for Boltzmann's equation, were two decades ahead of their time and therefore not immediately appreciated.
     
   • As it is often the case with mathematicians who deal with differential or integral equations, Carleman carried a keen interest in the relationship between mathematics and applied sciences.
     
   • Before his professorship in Lund he published about thirty papers, the majority treating of the problems in the theory of integral equations, and the theory of real and complex functions, where he gave extraordinary evidence of originality, penetration and capacity to use various methods of analysis.
     
   • One of them is his fundamental contribution on singular integral equations and applications.
     
   • His first book Singular integral equations with real and symmetric kernel published in 1923 became fundamental.
     
   • Carleman is now remembered for remarkable results in integral equations (1923), quasi-analytic functions (1926), harmonic analysis (1944), trigonometric series (1918-23), approximation of functions (1922-27) and Boltzmann's equation (1944).
     
   • Names such as Carleman inequality, Carleman theorems (Denjoy-Carleman theorem on quasi-analytic classes of functions, Carleman theorem on conditions of well-definedness of moment problems, Carleman theorem on uniform approximation by entire functions, Carleman theorem on approximation of analytic functions by polynomials in the mean), Carleman singularity of orthogonal system, integral equation of Carleman type, Carleman operator, Carleman kernel, Carleman method of reducing an integral equation to a boundary value problem in the theory of analytic functions, Jensen-Carleman formula in complex analysis, Carleman continuum, Carleman linearization or Carleman embedding technique, Carleman polynomials, Carleman estimate in the unique continuation problem for solutions of partial differential equations and Carleman system in the kinetic theory of gas are well-known in mathematics (see [Encyclopaedia of Mathematics 2 (Kluwer 1988), 25-26.
     
   • In 1932 Carleman, following an idea of Poincare, showed that a finite dimensional system of nonlinear differential equations d u/dt = V(u), where Vk are polynomials in u, can be embedded in an infinite system of linear differential equations.
     
   • Results on unique continuation for solutions to partial differential equations are important in many areas of applied mathematics, in particular in control theory and inverse problems.
     
   • Carleman lectured at the Sorbonne in 1937 on Boltzmann's equation, which appears in the kinetic theory of gas, and published several papers on this subject.
     
   • Also his last book Mathematical problems of the kinetic theory of gas which deals with the mathematical aspects of the Boltzmann transport equation was published, after his death, in 1957 with some additional material submitted by L Carleson and O Frostman.
     

9. Euler biography
   
   • The core of his research program was now set in place: number theory; infinitary analysis including its emerging branches, differential equations and the calculus of variations; and rational mechanics.
     
   • Studies of number theory were vital to the foundations of calculus, and special functions and differential equations were essential to rational mechanics, which supplied concrete problems.
     
   • He introduced beta and gamma functions, and integrating factors for differential equations.
     
   • He discovered the Cauchy-Riemann equations in 1777, although d'Alembert had discovered them in 1752 while investigating hydrodynamics.
     
   • As well as investigating double integrals, Euler considered ordinary and partial differential equations in this work.
     
   • Problems in mathematical physics had led Euler to a wide study of differential equations.
     
   • He considered linear equations with constant coefficients, second order differential equations with variable coefficients, power series solutions of differential equations, a method of variation of constants, integrating factors, a method of approximating solutions, and many others.
     
   • When considering vibrating membranes, Euler was led to the Bessel equation which he solved by introducing Bessel functions.
     
   • Euler here also begins developing the kinematics and dynamics of rigid bodies, introducing in part the differential equations for their motion.
     
   • He published a number of major pieces of work through the 1750s setting up the main formulae for the topic, the continuity equation, the Laplace velocity potential equation, and the Euler equations for the motion of an inviscid incompressible fluid.
     
   • However sublime are the researches on fluids which we owe to Messrs Bernoulli, Clairaut and d'Alembert, they flow so naturally from my two general formulae that one cannot sufficiently admire this accord of their profound meditations with the simplicity of the principles from which I have drawn my two equations ..
     
   • History Topics: Quadratic, cubic and quartic equations .
     
   • History Topics: Pell's equation .
     

10. Bhaskara II biography
    
    • He reached an understanding of the number systems and solving equations which was not to be achieved in Europe for several centuries.
      
    • Bhaskaracharya studied Pell's equation px2 + 1 = y2 for p = 8, 11, 32, 61 and 67.
      
    • An example from Chapter 12 on the kuttaka method of solving indeterminate equations is the following:- .
      
    • The topics are: positive and negative numbers; zero; the unknown; surds; the kuttaka; indeterminate quadratic equations; simple equations; quadratic equations; equations with more than one unknown; quadratic equations with more than one unknown; operations with products of several unknowns; and the author and his work.
      
    • Equations leading to more than one solution are given by Bhaskaracharya:- .
      
    • The problem leads to a quadratic equation and Bhaskaracharya says that the two solutions, namely 16 and 48, are equally admissible.
      
    • The kuttaka method to solve indeterminate equations is applied to equations with three unknowns.
      
    • The problem is to find integer solutions to an equation of the form ax + by + cz = d.
      
    • Pell's equation .
      
    • History Topics: Pell's equation .
      

11. Lorenz Edward biography
    
    • The paper A generalization of the Dirac equations appeared in the Proceeding of the National Academy of Sciences in 1941.
      
    • in 1948 after submitting the dissertation A Method of Applying the Hydrodynamic and Thermodynamic Equations to Atmospheric Models.
      
    • A generalized vorticity equation for a two-dimensional spherical earth is obtained by eliminating pressure from the equations of horizontal motion including friction.
      
    • The generalized vorticity equation is satisfied by formal infinite series representing the density and wind fields.
      
    • An approximate differential equation is presented, relating the change in speed of the zonal westerly winds to the contemporary zonal wind-speed and the meridional flow of absolute angular momentum.
      
    • This equation is tested statistically by means of values of the momentum flow and the zonal wind-speed, computed with the aid of the geostrophic-wind approximation, from pressure and height data extracted from analyzed northern-hemisphere maps.
      
    • He was using a computer to investigate models of the atmosphere which he had devised involving twelve differential equations.
      
    • Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow.
      
    • Solutions of these equations can be identified with trajectories in phase space.
      
    • The set of equations and the attractors described by this set of equations are now called the 'Lorenz equations' and 'Lorenz attractors', respectively.
      
    • Another account of aperiodic behaviour in ordinary differential equations, and difference equations, in which Lorenz describes how he arrived, starting from the description of convection in meteorology, at the Lorenz equations is contained in his paper On the prevalence of aperiodicity in simple systems delivered at the Biennial Seminar of the Canadian Mathematical Congress in Calgary, Canada, in 1978.
      

12. Tartaglia biography
    
    • The first person known to have solved cubic equations algebraically was del Ferro but he told nobody of his achievement.
      
    • For mathematicians of this time there was more than one type of cubic equation and Fior had only been shown by del Ferro how to solve one type, namely 'unknowns and cubes equal to numbers' or (in modern notation) x3 + ax = b.
      
    • As negative numbers were not used this led to a number of other cases, even for equations without a square term.
      
    • In fact Tartaglia had also discovered how to solve one type of cubic equation since his friend Zuanne da Coi had set two problems which had led Tartaglia to a general solution of a different type from that which Fior could solve, namely 'squares and cubes equal to numbers' or (in modern notation) x3 + ax2 = b.
      
    • As public lecturer of mathematics at the Piatti Foundation in Milan, he was aware of the problem of solving cubic equations, but, until the contest, he had taken Pacioli at his word and assumed that, as Pacioli stated in the Suma published in 1494, solutions were impossible.
      
    • To Tartaglia's dismay, the governor was temporarily absent from Milan but Cardan attended to his guest's every need and soon the conversation turned to the problem of cubic equations.
      
    • Based on Tartaglia's formula, Cardan and Ferrari, his assistant, made remarkable progress finding proofs of all cases of the cubic and, even more impressively, solving the quartic equation.
      
    • Cardan and Ferrari travelled to Bologna in 1543 and learnt from della Nave that it had been del Ferro, not Tartaglia, who had been the first to solve the cubic equation.
      
    • In 1545 Cardan published Artis magnae sive de regulis algebraicis liber unus, or Ars magna as it is more commonly known, which contained solutions to both the cubic and quartic equations and all of the additional work he had completed on Tartaglia's formula.
      
    • For all the brilliance of his discovery of the solution to the cubic equation problem, Tartaglia was still a relatively poor mathematics teacher in Venice.
      
    • Ferrari clearly understood the cubic and quartic equations more thoroughly, and Tartaglia decided that he would leave Milan that night and thus leave the contest unresolved.
      
    • Fairly early in his career, before he became involved in the arguments about the cubic equation, he wrote Nova Scientia (1537) on the application of mathematics to artillery fire.
      
    • Quadratic, cubic and quartic equations .
      
    • History Topics: Quadratic, cubic and quartic equations .
      

13. Wang Xiaotong biography
    
    • The important innovation which is incorporated in most of these problems is that they reduce to a cubic equation which Wang solves numerically.
      
    • We do not know of any earlier Chinese work on cubic equations.
      
    • Of course one has to understand that when we say that the text is concerned with cubic equations, we do not see expressions with x, x2 and x3 in them.
      
    • Rather the equations are expressed in words and Wang thinks in a geometrical way.
      
    • For example where we might say "Let the height be x" and then produce an equation in x, Wang writes:- .
      
    • He then goes on to set up a cubic equation for the height.
      
    • when he is about to set up a cubic equation for the depth.
      
    • In setting up cubic equations Wang Xiaotong utilised a rule which is the same as the "tian yuan".
      
    • Data given for the work done by the workers allows the volume to be calculated, and a cubic equation is arrived at for x.
      
    • To be able to solve this problem Wang has not only to be able to set up a cubic equation and solve it, but he also needs to know a formula for the volume of his dyke with trapezoidal ends and varying cross-section.
      
    • Wang calls a the unknown and finds a cubic equation in terms of a.
      
    • Writing x for the unknown a, we have the cubic equation .
      
    • Try to set up the necessary equations in these two cases in a similar way to our solution to Problem 15 above.
      
    • Not only did Wang's work influence later Chinese mathematicians, but it is said that it was his ideas on cubic equations which Fibonacci learnt, probably first transmitted into the Islamic/Arabic world, and then brought to Europe.
      

14. Al-Tusi Sharaf biography
    
    • What is in this Treatise on equations by al-Tusi? Basically it is a treatise on cubic equations, but it does not follow the general development that came through al-Karaji's school of algebra.
      
    • it represents an essential contribution to another algebra which aimed to study curves by means of equations, thus inaugurating the beginning of algebraic geometry.
      
    • In the treatise equations of degree at most three are divided into 25 different types.
      
    • First al-Tusi discusses twelve types of equation of degree at most two.
      
    • He then looks at eight types of cubic equation which always have a positive solution, then five types which may have no positive solution.
      
    • We illustrate the method by showing how al-Tusi examined one of the five types of equation which under certain conditions has a solution, namely the equation x3 + a = bx, where a, b are positive.
      
    • Al-Tusi's first comment is that if t is a solution to this equation then t3 + a = bt and, since a > 0, t3 < bt so t < √b.
      
    • Thus the equation bx - x3 = a has a solution if a ≤ 2(b/3)3/2.
      
    • Then Al-Tusi deduces that the equation has a positive root if .
      
    • where D is the discriminant of the equation.
      
    • Al-Tusi then went on to give what we would essentially call the Ruffini-Horner method for approximating the root of the cubic equation.
      
    • Although this method had been used by earlier Arabic mathematicians to find approximations for the nth root of an integer, al-Tusi is the first that we know who applied the method to solve general equations of this type.
      

15. Lax Peter biography
    
    • He received his PhD in 1949, also from New York University, for his thesis Nonlinear System of Hyperbolic Partial Differential Equations in Two Independent Variables.
      
    • She was awarded a PhD in 1955 for her thesis Cauchy's Problem for a Partial Differential Equation with Real Multiple Characteristics.
      
    • for his groundbreaking contributions to the theory and application of partial differential equations and to the computation of their solutions.
      
    • The equations that arise in such fields as aerodynamics, meteorology and elasticity are nonlinear and much more complex: their solutions can develop singularities.
      
    • In the 1950s and 1960s, Lax laid the foundations for the modern theory of nonlinear equations of this type (hyperbolic systems).
      
    • Inspired by Richtmyer, Lax established with this theorem the conditions under which a numerical implementation gives a valid approximation to the solution of a differential equation.
      
    • Lax became fascinated by these mysterious solutions and found a unifying concept for understanding them, rewriting the equations in terms of what are now called "Lax pairs".
      
    • This phenomenon occurs not only for fluids, but also, for instance, in atomic physics (Schrodinger equation).
      
    • Their work also turned out to be important in fields of mathematics apparently very distant from differential equations, such as number theory.
      
    • In this monograph, written more than twenty years ago, we based our scattering theory on the wave equation rather than the Schrodinger equation.
      
    • Following up on a hint in Gelfand's address to the 1962 Stockholm International Congress, they showed that the Lax-Phillips scattering theory, applied to the wave equation appropriate to hyperbolic space, is a natural tool in the theory of automorphic functions.
      
    • Yet during the past five decades there has been an unprecedented outburst of new ideas about how to solve linear equations, carry out least square procedures, tackle systems of linear inequalities, and find eigenvalues of matrices.
      

16. Lions biography
    
    • Lions has made some of the most important contributions to the theory of nonlinear partial differential equations through the 1980s and 1990s.
      
    • Keep in mind that there is in truth no central core theory of nonlinear partial differential equations, nor can there be.
      
    • The sources of partial differential equations are so many - physical, probalistic, geometric etc.
      
    • - that the subject is a confederation of diverse subareas, each studying different phenomena for different nonlinear partial differential equation by utterly different methods.
      
    • 70 (2) (1996), 125-135.',3)">3] is his work on "viscosity solutions" for nonlinear partial differential equations.
      
    • The method was first introduced by Lions in joint work with M G Crandall in 1983 in which they studied Hamilton-Jacobi equations.
      
    • Lions and others have since applied the method to a wide class of partial differential equations, the so-called "fully nonlinear second order degenerate elliptic partial differential equations." The problem that arises is decribed in [Notices Amer.
      
    • such nonlinear partial differential equation simply do not have smooth or even C1 solutions existing after short times.
      
    • Another equally innovative piece of work by Lions was his work on the Boltzmann equation and other kinetic equations.
      
    • The Boltzmann equation keeps track of interactions between colliding particles, not individually but in terms of a density.
      
    • There are many nonlinear PDEs that are Euler equations for variational problems.
      
    • The first step in solving such equations by the variational method is to show that the extremum is attained.
      
    • The first volume covered his work on Partial differential equations and Interpolation, the second volume contained Control and Homogenization, and the third volume Numerical analysis, Scientific computation and Applications.
      
    • The models used in that field consist of complex sets of partial differential equations, including the Navier-Stokes equations and the equations of thermodynamics.
      
    • In spite of what Lions himself liked to call the 'truly diabolical' complexity of the set of partial differential equations, boundary conditions, transmission conditions, nonlinearities, physical hypotheses, etc., that appeared in those models, Lions, in collaboration with Roger Temam and Shou Hong Wang, was able to study the questions of the existence and uniqueness of solutions, to establish the existence of attractors, and to do a numerical analysis of these models.
      
    • Finally, with Vivette Girault, he worked until January 2001 on perfecting a finite element method using two meshes, one 'rough' and one 'fine', for the numerical simulation of the Navier-Stokes equations.
      

17. Diophantus biography
    
    • Diophantus, often known as the 'father of algebra', is best known for his Arithmetica, a work on the solution of algebraic equations and on the theory of numbers.
      
    • The Arithmetica is a collection of 130 problems giving numerical solutions of determinate equations (those with a unique solution), and indeterminate equations.
      
    • The work considers the solution of many problems concerning linear and quadratic equations, but considers only positive rational solutions to these problems.
      
    • Equations which would lead to solutions which are negative or irrational square roots, Diophantus considers as useless.
      
    • To give one specific example, he calls the equation 4 = 4x + 20 'absurd' because it would lead to a meaningless answer.
      
    • In other words how could a problem lead to the solution -4 books? There is no evidence to suggest that Diophantus realised that a quadratic equation could have two solutions.
      
    • Diophantus looked at three types of quadratic equations ax2 + bx = c, ax2 = bx + c and ax2 + c = bx.
      
    • He solved problems such as pairs of simultaneous quadratic equations.
      
    • Diophantus would solve this by creating a single quadratic equation in x.
      
    • The symbolism that Diophantus introduced for the first time, and undoubtedly devised himself, provided a short and readily comprehensible means of expressing an equation..
      
    • We began this article with the remark that Diophantus is often regarded as the 'father of algebra' but there is no doubt that many of the methods for solving linear and quadratic equations go back to Babylonian mathematics.
      
    • History Topics: Pell's equation .
      

18. Lopatynsky biography
    
    • His research interests then moved towards differential equations with his first paper on this topic Solution of the equation y ' = f (x, y) published in 1939, proving a general existence theorem.
      
    • He continued to undertake research on differential equations, but his interests now included differential operators.
      
    • The author studies linear (partial) differential equations from a formal algebraic point of view.
      
    • His treatment of this special case of the algebraic theory of algebraic differential equations yields a well-rounded ideal theory of linear differential operators; in many respects it differs essentially from the treatment due to Ritt (for instance, ideals and sums of integral manifolds are defined differently).
      
    • In 1945 Lopatynsky moved to Lvov where he was appointed to the chair of differential equations at Lvov University.
      
    • His seminar on differential equations at Lvov University attracted many mathematicians, both young men beginning their research activity and established researchers who found inspiration in the seminar.
      
    • Lopatynsky's research continued to impress as he continued to prove major results in the theory of systems of linear differential equations of the elliptic type.
      
    • In 1966 he became head of the partial differential equations Section of the Institute of Applied Mathematics and Mechanics of the Academy of Sciences of the Ukraine in Donetsk.
      
    • He was also appointed as Chairman of the Department of Differential Equations at Donetsk University.
      
    • Lopatynsky's contributions to the theory of differential equations are particularly important, with important contributions to the theory of linear and nonlinear partial differential equations.
      
    • He worked on the general theory of boundary value problems for linear systems of partial differential equations of elliptic type, finding general methods of solving boundary value problems.
      
    • he continued his studies of general boundary problems in differential equations using topological methods.
      
    • Recently he has obtained important results on solvability of the Cauchy problem for operator equations in Banach space and also on "almost everywhere" solvability of general linear and nonlinear boundary problems.
      
    • He also obtained some basic results in the solvability of the Cauchy problem for operator equations in Banach spaces.
      
    • In 1980 Lopatynsky published an important book Introduction to the Contemporary Theory of Partial Differential Equations.
      
    • This book makes the reader familiar with the basic notions and facts of algebra, topology, and functional analysis, and gives a general idea how to apply these notions to the theory of differential equations.
      
    • The book contains eight chapters: Sets; Basic algebraic notions; Algebraic equations; Topology; Differentiation and integration; Special linear spaces which are related to Euclidean spaces; Manifolds; and Elements of algebraic topology.
      
    • His next book, published in 1984 three years after his death, was entitled Ordinary differential equations.
      
    • We consider the basic methods of solving differential equations and methods of qualitative investigation of these solutions.
      
    • We emphasize the relation of the theory of differential equations to other areas of mathematics.
      
    • Finally let us mention that the Second International Conference for young mathematicians on Differential Equations and Applications held in November 2008 at the Donetsk National University was dedicated to Ya B Lopatinskii.
      
    • The conference is named after an outstanding mathematician, talented pedagogue and organiser, Academician of National Academy of Sciences of Ukraine Yaroslav Borisovich Lopatinskii who was a founder of both the Department of Differential Equations in Donetsk National University and the Department of Partial Differential Equations in the Institute of Applied Mathematics and Mechanics of National Academy of Sciences of Ukraine.
      
    • The conference is the continuation of the Conference on Differential Equations and Applications dedicated to the centenarian jubilee of Ya B Lopatinskii held in December 2006.
      

19. Ruffini biography
    
    • On the other hand it gave him the chance to work on what was one of the most original of projects, namely to prove that the quintic equation cannot be solved by radicals.
      
    • To solve a polynomial equation by radicals meant finding a formula for its roots in terms of the coefficients so that the formula only involves the operations of addition, subtraction, multiplication, division and taking roots.
      
    • Quadratic equations (of degree 2) had been known to be soluble by radicals from the time of the Babylonians.
      
    • The cubic equation had been solved by radicals by del Ferro, Tartaglia and Cardan.
      
    • Certainly no mathematician has published such a claim and even Lagrange in his famous paper Reflections on the resolution of algebraic equations says he will return to the question of the solution of the quintic and, clearly, he still hoped to solve it by radicals.
      
    • In 1799 Ruffini published a book on the theory of equations with his claim that quintics could not be solved by radicals as the title shows: General theory of equations in which it is shown that the algebraic solution of the general equation of degree greater than four is impossible.
      
    • The algebraic solution of general equations of degree greater than four is always impossible.
      
    • In writing this book, I had principally in mind to give a proof of the impossibility of solving equations of degree higher than four.
      
    • and recommend greatly the most important theorem which excludes the possibility of solving equations of degree greater than four.
      
    • your memoir on the general resolution of equations is a work which has always seemed to me worthy of the attention of mathematicians and which, in my judgement, proves completely the impossibility of solving algebraically equations of higher than the fourth degree.
      

20. Lewy biography
    
    • In this paper criteria are given for determining conditions which guarantee the stability of numerical solutions of certain classes of differential equations.
      
    • he published a series of fundamental papers on partial differential equations and the calculus of variations.
      
    • He solved completely the initial value problem for general non-linear hyperbolic equations in two independent variables.
      
    • On the basis of this, and using the daring idea of converting an elliptic equation into a hyperbolic one by penetrating into the complex domain, he developed a new proof of the analyticity of solutions of analytic elliptic equations in two independent variables, one which far exceeded the known proof in its elegance and simplicity.
      
    • He proved the well-posedness of the initial value problem for wave equations in what is now called Sobolev spaces two decades before these spaces became a common tool for specialists.
      
    • Nirenberg [D Kinderlehrer (ed.), Hans Lewy Selecta (Boston, MA, 2002).',6)">6] lists Lewy's mathematical papers under the following topics: (i) partial differential equations involving existence and regularity theory for elliptic and hyperbolic equations, geometric applications, approximation of solutions; (ii) existence and regularity of variational problems, free boundary problems, theory of minimal surfaces; (iii) partial differential equations connected with several complex variables; (iv) partial differential equations connected with water waves and fluid dynamics; (v) offbeat properties of solutions of partial differential equations.
      
    • Among the first papers he published after emigrating to the United States were A priori limitations for solutions of Monge-Ampere equations (two papers, the first in 1935, the second two years later), and On differential geometry in the large : Minkowski's problem (1938).
      
    • His paper An example of a smooth linear partial differential equation without solution (1957) gave a simple partial differential equation which has no solution, a result which had a substantial impact on the area.
      

21. Ince biography
    
    • Ince's research was mainly on differential equations.
      
    • Emile Mathieu discovered the Mathieu functions, which are special cases of hypergeometric functions, in 1868 while solving the wave equation for an elliptical membrane moving through a fluid.
      
    • By the use of convergent infinite determinants and continued fractions, with asymptotic formulae for large values, he succeeded in making computations practicable and after eight years' devotion to this task he published in 1932 tables of eigenvalues for Mathieu's equation, and zeros of Mathieu functions.
      
    • These tables were useful not only in the problems originally envisaged but also in more recent investigations such as quantum-mechanical problems leading to Mathieu's equation.
      
    • Ince published a major text Ordinary Differential Equations (Longmans, Green and Co., London, 1926).
      
    • He contributed Integration of Ordinary Differential Equations and set out his aims in the Preface dated May 1939:- .
      
    • The object of this book is to provide in a compact form an account of the methods of integrating explicitly the commoner types of ordinary differential equation, and in particular those equations that arise from problems in geometry and applied mathematics.
      
    • With this qualification, it will be found to contain all the material needed by students in our Universities who do not specialize in differential equations, as well as by students of mathematical physics and technology.
      
    • A C Aitken and D E Rutherford wrote the Preface to the second edition of Integration of Ordinary Differential Equations of April 1943: .
      
    • The nucleus of an integral equation for one of the periodic Lame functions is expanded in series of products of the characteristic functions ..
      
    • One further paper, Simultaneous linear partial differential equations of the second order, was edited by Erdelyi after Ince's death and published in the Proceedings of the Royal Society of Edinburgh in 1942.
      

22. Bateman biography
    
    • Bateman was awarded a Smith's prize in 1905 for an essay on differential equations.
      
    • Two further papers appeared in print in 1904, namely The solution of partial differential equations by means of definite integrals, and Certain definite integrals and expansions connected with the Legendre and Bessel functions.
      
    • It was during his visit to Gottingen that he learnt of work on integral equations being undertaken by Hilbert and his school.
      
    • One of these 1908 papers is his first publication on transformations of partial differential equations and their general solutions.
      
    • His 1908 paper was on the wave equation.
      
    • He is especially known for his work on special functions and partial differential equations.
      
    • In 1904 he extended Whittaker's solution of the potential and wave equation by definite integrals to more general partial differential equations.
      
    • Bateman was one of the first to apply Laplace transforms to integral equations in 1906.
      
    • In 1910 he solved systems of differential equations discovered by Rutherford which describe radio-active decay.
      
    • The finest contribution Bateman made to mathematics, however, was his work on transformations of partial differential equations, in particular his general solutions containing arbitrary functions.
      
    • In particular he applied his methods to equations resulting from electromagnetics, then later to those arising from hydrodynamics.
      
    • He wrote a number of texts that have been reprinted as classics: The mathematical analysis of electrical and optical wave-motion on the basis of Maxwell's equations (1915, reprinted 1955); Partial differential equations of mathematical physics (1932, reprinted 1944 and 1959); (written with H L Dryden and F D Murnaghan), Hydrodynamics, National Research Council, Washington, D.C.
      
    • (1932, reprinted 1956); and (written with A A Bennett and W E Milne), Numerical integration of differential equations (1933, reprinted 1956).
      
    • He only published five joint papers, one of those in 1924 being with Ehrenfest in which they looked at applications of partial differential equations to electromagnetic fields.
      

23. Plemelj biography
    
    • Plemelj undertook research under von Escherich's supervision and in May 1898 was awarded his doctorate for a thesis on linear homogeneous differential equations with uniform periodical coefficients (uber lineare homogene Differentialgleichungen mit eindeutigen periodischen Koeffizienten).
      
    • An important mathematical event occurred while he was at Gottingen, for that was the year in which Holmgren lectured on Fredholm's theory of integral equations at Gottingen.
      
    • The contributions he made to integral equations and potential theory were brought together in a work he published in 1911 for which he was awarded the Prince Jablonowski Prize.
      
    • Riemann's problem, concerning the existence of a linear differential equation of the Fuchsian class with prescribed regular singular points and monodromy group, had been reduced to the solution of an integral equation by Hilbert in 1905.
      
    • Plemelj discovered equations relating to boundary values of holomorphic functions which are now called the "Plemelj formulae" and shortly after this was able to solve Riemann's problem in his paper Riemannian classes of functions with given monodromy group published in Monatshefte fur Mathematik und Physik in 1908.
      
    • The equations are today important in a number of different fields, including neutron transport theory where a singular integral equation is encountered.
      
    • Plemelj's methods for solving the Riemann's problem were further developed by Nikolai Ivanovich Mushelisvili into the theory of singular integral equations.
      
    • Within the theory of differential equations he worked mostly on equations of the Fuchs type and on Klein's theorems.
      
    • He used to hold a general course of mathematics and a three-year cycle of lectures on differential equations, the theory of analytic functions, and algebra including number theory.
      
    • They were The theory of analytic functions (1953), Differential and integral equations.
      

24. Mazya biography
    
    • Solution of Dirichlet's problem for an equation of elliptic type (Russian) was published in 1959 and Classes of domains and imbedding theorems for function spaces (Russian) in 1960.
      
    • He published the two papers Some estimates of solutions of second-order elliptic equations (Russian) and p-conductivity and theorems on imbedding certain functional spaces into a C-space (Russian) in 1961, and then four further papers in 1962, the year in which he was awarded his Candidate degree (equivalent to a doctorate) from Moscow State University.
      
    • The Dirichlet problem for an arbitrary order elliptic equation in a domain with a cut off tubular neighbourhood of a smooth closed submanifold is considered in the second chapter.
      
    • The fourth chapter deals with asymptotic expansions of solutions to a quasilinear equation of the second order.
      
    • In 1997 (with Vladimir Kozlov) Maz'ya published Theory of a higher-order Sturm-Liouville equation which Eastham summarises by writing that:- .
      
    • the authors have identified a special type of higher-order analogue of the hyperbolic Sturm-Liouville equation and they have developed a coherent theory based on the Green's function.
      
    • One year later, in 1999, Maz'ya, together with Vladimir Kozlov, published Differential equations with operator coefficients with applications to boundary value problems for partial differential equations.
      
    • All the proofs are complete and rely on undergraduate university courses on real and complex analysis and some basic facts of functional analysis and of the theory of partial differential equations.
      
    • For example we list a few recent works without detailing the co-authors: Spectral problems associated with corner singularities of solutions to elliptic equations (2000); Asymptotic theory of elliptic boundary value problems in singularly perturbed domains (2000); Spectral problems associated with corner singularities of solutions to elliptic equations (2001); and Linear water waves (2002).
      
    • In addition the American Mathematical Society published Perspectives in Partial Differential Equations, Harmonic Analysis and Applications: A Volume in Honor of Vladimir G Maz'ya's 70th Birthday in their Proceedings of Symposia in Pure Mathematics series.
      
    • in recognition of his contributions to the theory of differential equations.
      

25. Petryshyn biography
    
    • from Columbia University for his thesis Linear Transformations Between Hilbert Spaces and the Application of the Theory to Linear Partial Differential Equations.
      
    • In 1962, Direct and iterative methods for the solution of linear operator equations in Hilbert space was published which does much toward developing a unified point of view toward a number of important methods of solving linear equations.
      
    • In the same year, The generalized overrelaxation method for the approximate solution of operator equations in Hilbert space appeared and in the following year the two papers On a general iterative method for the approximate solution of linear operator equations and On the generalized overrelaxation method for operation equations.
      
    • His major results include the development of the theory of iterative and projective methods for the constructive solution of linear and nonlinear abstract and differential equations.
      
    • He has shown that the theory of A-proper type maps not only extends and unifies the classical theory of compact maps with some recent theories of condensing and monotone-accretive maps, but also provides a new approach to the constructive solution of nonlinear abstract and differential equations.
      
    • The theory has been applied to ordinary and partial differential equations.
      
    • Approximation-solvability of Nonlinear Functional and Differential Equations appeared in December 1992:- .
      
    • This outstanding reference/text develops an essentially constructive theory of solvability on linear and nonlinear abstract and differential equations involving A-proper operator equations in separable Banach spaces, treats the problem of existence of a solution for equations involving pseudo-A-proper and weakly-A-proper mappings, and illustrates their applications.
      
    • Facilitating the understanding of the solvability of equations in infinite dimensional Banach space through finite dimensional approximations, Approximation - solvability of Nonlinear Functional and Differential Equations: offers an important elementary introduction to the general theory of A-proper and pseudo-A-proper maps; develops the linear theory of A-proper maps; furnishes the best possible results for linear equations; establishes the existence of fixed points and eigenvalues for P-gamma-compact maps, including classical results; provides surjectivity theorems for pseudo-A-proper and weakly-A-proper mappings that unify and extend earlier results on monotone and accretive mappings; shows how Friedrichs' linear extension theory can be generalized to the extensions of densely defined nonlinear operators in a Hilbert space; presents the generalized topological degree theory for A-proper mappings; and applies abstract results to boundary value problems and to bifurcation and asymptotic bifurcation problems.
      
    • In 1995 his second monograph Generalized Topological Degree and Semilinear Equations appeared in print, published by Cambridge University Press.
      
    • In this monograph we develop the generalised degree theory for densely defined A-proper mappings, and then use it to study the solubility (sometimes constructive) and the structure of the solution set of [an] important class of semilinear abstract and differential equations ..
      
    • A-proper mappings arise naturally in the solution to an equation in infinite dimensional space via the finite dimensional approximation..
      
    • Using these tools, the author defines the generalised topological degree for densely defined A-proper mappings, gives applications to the solubility of an important class of semilinear abstract and differential equations, and discusses global bifurcation results.
      

26. Jerrard biography
    
    • His most important work Mathematical Researches (1832-35) is on the theory of equations.
      
    • Viete and Cardan had shown how to transform an equation of degree n so that it had no term in xn-1.
      
    • These methods were, to a large extent, motivated by attempts to solve equations algebraically.
      
    • Abel and Ruffini showed this was impossible for general equations of degree greater than four.
      
    • In 1786 Bring reduced a general quintic to x5 + px + q = 0 while Jerrard generalised this to show that a transformation could be applied to an equation of degree n to remove the terms in xn-1, xn-2 and xn-3.
      
    • Hermite used Jerrard's result saying that it was the most important step in studying the quintic equation since Abel's results.
      
    • Jerrard wrote a further two volume work on the algebraic solution of equations An essay on the resolution of equations (1858).
      
    • Jerrard did not accept that the algebraic solution of the quintic equation was impossible.
      
    • In fact Jerrard had successfully shown that quintic equations could be solved but his error was to use methods which did not come under the precise definition of the 'method of radicals' which was required.
      

27. Zhu Shijie biography
    
    • In dealing with simultaneous equations, Zhu certainly presented improvements, giving a method essentially equivalent to Gauss's pivotal condensation.
      
    • He treats polynomial algebra, and polynomial equations, by the "coefficient array method" or "method of the celestial unknown" which had been developed in northern China by the earlier thirteenth century Chinese mathematicians, but up till that time had not spread to southern China.
      
    • Zhu, however, wants to illustrate something more advanced than solving a quadratic equation.
      
    • Although we cannot be certain that Zhu's methods are exactly what we have presented here, he certainly arrived at the equation (2).
      
    • He has illustrated how to work with the four unknowns x, y, z, t and he can now illustrate how to solve a quartic equation.
      
    • It is phrased in terms of a right angled triangle, but the conditions are so artificial that he is really simply giving a system of equations.
      
    • The following problem in the Siyuan yujian is reduced by Zhu to a polynomial equation of degree 5 (see [First Australian Conference on the History of Mathematics (Clayton, 1980) (Clayton, 1981), 103-134.',7)">7] for a detailed solution as given by Zhu):- .
      
    • The Siyuan yujian also contains a transformation method for the numerical solution of equations which is applied to equations up to degree 14.
      
    • This is based on the method to solve polynomial equations which was rediscovered by Horner and Ruffini.
      

28. Mordell biography
    
    • Rather remarkably, Mordell's future research interests were determined by these books, and his love of indeterminate equations came from this period.
      
    • For his Smith's Prize essay Mordell studied solutions of y2 = x3 + k, an equation which had been considered by Fermat.
      
    • Thue had already proved a result which, combined with Mordell's work showed that this equation had only finitely many solutions but Mordell only learned about Thue's work at a later date.
      
    • However he solved the equation for many values of k, giving complete solutions for some values.
      
    • Mordell was awarded the second Smith's Prize with his essay, and he went on to publish a long paper on this equation, now sometimes called Mordell's equation, in the Proceedings of the London Mathematical Society.
      
    • Mordell submitted his subsequent work on indeterminate equations of the third and fourth degree when he became a candidate for a Fellowship at St John's College, but he was not successful.
      
    • Indeterminate equations have never been very popular in England (except perhaps in the 17th and 18th centuries); though they have been the subject of many papers by most of the greatest mathematicians in the world: and hosts of lesser ones ..
      
    • marks the greatest advance in the theory of indeterminate equations of the 3rd and 4th degrees since the time of Fermat; and it is all the more remarkable that it can be proved by quite elementary methods.
      
    • He emphasised the fact that he was returning to Cambridge where he began his career by taking the equation y2 = x3 + k as the topic for his inaugural lecture to the Sadleirian Chair.
      

29. Klein Oskar biography
    
    • He defended his doctorate in 1921 at Stockholm Hogskola and was opposed by Erik Ivar Fredholm the mathematical physicist best known for his work on integral equations and spectral theory.
      
    • In a paper in which he determined the atomic transition probabilities (prior to Dirac), he introduced the initial form of what would become known as the Klein-Gordon equation.
      
    • The Klein-Gordon equation was the first relativistic wave equation.
      
    • The equation can be written: .
      
    • It is interesting to note that this equation appeared exactly as it has been written in David Bohm's 1951 book Quantum Theory but was not called the Klein-Gordon equation.
      
    • However, Bethe and Jackiw's Intermediate Quantum Mechanics, originally written in 1964, does refer to the same equation as the Klein-Gordon equation.
      
    • Klein and Walter Gordon were thus eventually honoured with having the equation named after them, though it seems to have taken over a quarter of a century to receive the honour.
      
    • Oddly enough, Schrodinger himself privately developed a relativistic wave equation from his original wave equation, which, in reality, was not that difficult to do, and did so prior to Klein and Gordon, though he never published his results.
      
    • The trouble came when the equation did not result in the correct fine structure of the hydrogen atom and when Pauli introduced the concept of spin a year later (1927).
      
    • The equation turned out to be incompatible with spin and, as a result, is only useful for calculations involving spinless particles.
      
    • He and Jordan showed that one can quantize the non-relativistic Schrodinger equation and, in honour of this work, he was the recipient of yet another named mathematical tool, the Jordan-Klein matrices.
      
    • Despite the so-called Klein paradox, that being that the positron was not completely understood by physicists, he was able to convince physicists of the soundness of Dirac's relativistic wave equation.
      
    • Of the many he helped, one included Walter Gordon who would later join Klein in being the beneficiaries of the named equation we have just discussed.
      

30. Nelson biography
    
    • The first of her two children, both daughters, was born shortly before she competed the work for her doctoral thesis The lattice of equational classes of commutative semigroups.
      
    • In addition to The lattice of equational classes of commutative semigroups referred to above, she published in 1971 the papers Embedding the dual of πm in the lattice of equational classes of commutative semigroups in the Proceedings of the American Mathematical Society and Embedding the dual of π∞ in the lattice of equational classes of semigroups in Algebra Universalis, both written jointly with Stanley Burris.
      
    • In the same year she published The lattice of equational classes of semigroups with zero in the Canadian Mathematical Bulletin.
      
    • Two which Nelson wrote jointly with Bernhard Banaschewski were On residual finiteness and finite embeddability and Equational compactness in equational classes of algebras both of which were published in Algebra Universalis.
      
    • In the following year she published Equational compactness in infinitary algebras again jointly with Bernhard Banaschewski.
      
    • W Taylor recently proved, among other results, that an equational class of finitary algebras contains enough equationally compact algebras if and only if the subdirectly irreducible algebras in the class constitute, up to isomorphism, a set.
      
    • This note provides a negative answer to the natural question whether the same equivalence holds for equational classes of infinitary algebras by exhibiting examples in which there are, up to isomorphism, only one subdirectly irreducible algebra in the class and no non-trivial equationally compact algebras at all.
      

31. Fuchs biography
    
    • Fuchs worked on differential equations and the theory of functions.
      
    • Fuchs was a gifted analyst whose works form a bridge between the fundamental researches od Cauchy, Riemann, Abel, and Gauss and the modern theory of differential equations discovered by Poincare, Painleve, and Emile Picard.
      
    • In 1865 Fuchs studied nth order linear ordinary differential equations with complex functions as coefficients.
      
    • Fuchs enriched the theory of linear differential equations with fundamental results.
      
    • He discussed problems of the following kind: What conditions must be placed on the coefficients of a differential equation so that all solutions have prescribed proberties (e.g.
      
    • This led him (1865, 1866) to introduce an important class of linear differential equations (and systems) in the complex domain with analytic coeffivcients, a class which today bears hios name (Fuchaian equations, equations of the Fuchsian class).
      
    • He succeeded in characterising those differential equations the solutions of which have no essential singularity in the extended complex plane.
      
    • Fuchs later also studied non-linear fifferential equations and moveable singularities.
      
    • In a series of papers (1880-81) Fuchs studied functions obtained by inverting the integrals of solutions to a second-order linear differential equation in a manner generalising Jacobi's inversion problem.
      
    • Fuchs also investigated how to find the matrix connecting two systems of solutions of differential equations near two different points.
      

32. Schmidt Harry biography
    
    • Another major publication during these years was his introduction to the theory of the wave equation Einfuhrung in die Theorie der Wellengleichung (1931).
      
    • While he was working at the laboratory Schmidt published, jointly with Kurt Schroder, a comprehensive report on the theory of laminar boundary layers deals with the basic conceptions and equations Laminare Grenzschichten.
      
    • The general equations of motion of hydrodynamics: .
      
    • The equation of continuity, the impulse theorem and the Navier-Stokes equations.
      
    • The equations of motion in orthogonal curvilinear coordinates.
      
    • Introduction of the boundary layer equation: .
      
    • A rigorous solution of the Navier-Stokes equations as an example.
      
    • The fundamental equations.
      
    • The equations of a steady two-dimensional motion with respect to the stream lines and their orthogonal trajectories.
      

33. Heaviside biography
    
    • Despite this hatred of rigour, Heaviside was able to greatly simplify Maxwell's 20 equations in 20 variables, replacing them by four equations in two variables.
      
    • Today we call these 'Maxwell's equations' forgetting that they are in fact 'Heaviside's equations'.
      
    • He introduced his operational calculus to enable him to solve the ordinary differential equations which came out of the theory of electrical circuits.
      
    • He replaced the differential operator d/dx by a variable p transforming a differential equation into an algebraic equation.
      
    • The solution of the algebraic equation could be transformed back using conversion tables to give the solution of the original differential equation.
      

34. Viete biography
    
    • In 1593 Roomen had proposed a problem which involved solving an equation of degree 45.
      
    • (If I asked for a solution to ax = b nobody asks: "For which quantity do I solve the equation ?") .
      
    • Viete made many improvements in the theory of equations.
      
    • However, if we are to be strictly accurate we should say that he did not solve equations as such but rather he solved problems of proportionals which he states quite explicitly is the same thing as solving equations.
      
    • Viete therefore looked for solutions of equations such as A3 + B2A = B2Z where, using his convention, A was unknown and B and Z were knowns.
      
    • He presented methods for solving equations of second, third and fourth degree.
      
    • He knew the connection between the positive roots of equations and the coefficients of the different powers of the unknown quantity.
      
    • When Viete applied numerical methods to solve equations as he did in De numerosa potestatum he gave methods which were similar to those given by earlier Arabic mathematicians.
      
    • Although this seems to make Harriot's dependence on Viete clear, one would have to say that the two men give very similar approaches to solving equations algebraically, yet Harriot shows deeper understanding than does Viete.
      
    • History Topics: Quadratic, cubic and quartic equations .
      

35. Ljunggren biography
    
    • Papers such as Fermat's problem by Oystein Ore and On the indeterminate equation x2 - Dy2 = 1 by Trygve Nagell were in the issue which contained the problems that he solved to win his prize and, through studying these and other papers, he was already interested in number theory before beginning his university course.
      
    • Almost all of Ljunggren's research was on Diophantine equations.
      
    • For example in A note on simultaneous Pell equations (1941) Ljunggren studied the simultaneous Pell equations .
      
    • One of Ljunggren's main interests was Diophantine equations of degree 4.
      
    • In 1923 Mordell showed that the Diophantine equation .
      
    • In the paper Ljunggren found bounds for the number of integer solutions for some special equations of this type.
      
    • In the first of these he proves that the equation in question has at most two positive integer solutions and gives an example of D = 1785 which does indeed have two solutions, namely x = 13, y = 4 and x = 239, y = 1352.
      
    • He proved that the equation .
      

36. Waring biography
    
    • We shall comment further below on this important work, covering topics in the theory of equations, number theory and geometry.
      
    • Meditationes Algebraicae, covering the theory of equations and number theory, appeared in 1770 with an expanded version in 1782.
      
    • In Meditationes Algebraicae Waring proves that all rational symmetric functions of the roots of an equation can be expressed as rational functions of the coefficients.
      
    • He derived a method for expressing symmetric polynomials and he investigated the cyclotomic equation xn - 1 = 0.
      
    • The most significant aspect of Waring's treatment of this example is the symmetric relation between the roots of the quartic equation and its resolvent cubic.
      
    • k equations in k unknowns can be reduced to one equation with one unknown.
      
    • His result that the product of the degrees of the original equations is the degree of the single reduced equation is known as the Generalised Theorem of Bezout.
      

37. Qin Jiushao biography
    
    • There is a remarkable formula given in this Chapter which expresses the area of a figure as the root of an equation of degree 4.
      
    • Again equations of high degree appear, one problem involving the solution of the equation of degree 10.
      
    • Qin obtains the equation (really an equation of degree 5 in x2, where x2 is the diameter of the city):- .
      
    • Throughout the text, in addition to the tenth degree equation above, Qin also reduces the solution of certain problems to a cubic or quartic equation which he solves by the standard Chinese method (namely that which today is called the Ruffini-Horner method).
      
    • For example the following two equations .
      
    • Qin also solves linear simultaneous equations, in particular the system .
      

38. Sneddon biography
    
    • It is a major text containing around 550 pages and is mainly concerned with applications which involve the solution of ordinary differential equations, and boundary value and initial value problems for partial differential equations.
      
    • Sneddon's next text Elements of partial differential equations appeared the following year in 1957.
      
    • The aim of this book is to present the elements of the theory of partial differential equations in a form suitable for the use of students and research workers whose main interest in the subject lies in finding solutions of particular equations rather than in the general theory.
      
    • The applications of the methods are again the strength of the book which considers the use of partial differential equations in thermodynamics, stochastic processes, and birth and death processes for bacteria.
      
    • The book deals with, among other topics, Laplace's equation, mixed boundary value problems, the wave equation, and the heat equation.
      

39. Littlewood biography
    
    • In the late 1930's the Department of Scientific and Industrial Research tried to interest pure mathematicians in nonlinear differential equations which were important for radio engineers and scientists because they described the behaviour of electric circuits.
      
    • The impending war motivated this interest and in 1938 the Radio Research Board asked British pure mathematicians for help in dealing with certain types of nonlinear differential equations arising in radio engineering.
      
    • Littlewood, working jointly with Mary Cartwright, spent 20 years working on equations of this type such as van der Pol's equation.
      
    • Monthly 103 (10) (1996), 833-845.',16)">16], and in particular the work on van der Pol's equation is discussed in [Harmonic analysis and nonlinear differential equations, Riverside, CA, 1995, Contemp.
      
    • Van der Pol's experiments with nonlinear oscillators during the 1920s and 1930s stimulated mathematical interest in nonlinear differential equations arising in radio research.
      
    • Cartwright and Littlewood's analysis of the van der Pol equation and its generalizations led them to explore some interesting topological methods, including the development of a fixed-point theorem for continua invariant under a homeomorphism of the plane.
      
    • in recognition of his distinguished contributions to many branches of analysis, including Tauberian theory, the Riemann zeta-function, and non-linear differential equations.
      

40. Khayyam biography
    
    • This problem in turn led Khayyam to solve the cubic equation x3 + 200x = 20x2 + 2000 and he found a positive root of this cubic by considering the intersection of a rectangular hyperbola and a circle.
      
    • Khayyam also wrote that he hoped to give a full description of the solution of cubic equations in a later work [Scripta Math.
      
    • Indeed Khayyam did produce such a work, the Treatise on Demonstration of Problems of Algebra which contained a complete classification of cubic equations with geometric solutions found by means of intersecting conic sections.
      
    • In fact Khayyam gives an interesting historical account in which he claims that the Greeks had left nothing on the theory of cubic equations.
      
    • Indeed, as Khayyam writes, the contributions by earlier writers such as al-Mahani and al-Khazin were to translate geometric problems into algebraic equations (something which was essentially impossible before the work of al-Khwarizmi).
      
    • However, Khayyam himself seems to have been the first to conceive a general theory of cubic equations.
      
    • Another achievement in the algebra text is Khayyam's realisation that a cubic equation can have more than one solution.
      
    • He demonstrated the existence of equations having two solutions, but unfortunately he does not appear to have found that a cubic can have three solutions.
      
    • Khayyam's construction for solving a cubic equation .
      

41. Arino biography
    
    • Solutions periodiques d'equations differentielles a argument retarde.
      
    • Oscillations autour d'un point stationnaire, conditions suffisantes de non-existence (1980); "Following a note by P Seguier the authors give some results on the non-existence of a nontrivial periodic solution to differential equations with delay, using mainly properties of monotonicity.
      
    • Stabilite d'un ensemble ferme pour une equation differentielle a argument retarde (1978); "Our aim is to establish a local existence result for a differential equation with delay in a reflexive Banach space, with the hypothesis of weak continuity in the second member.
      
    • Solutions oscillantes d'equations differentielles autonomes a retard (1978); "We show some results proving the existence, and specifying the behaviour, of solutions oscillating near a stationary point for some equations of the type x '(t) = L(xt) + N(xt) which have certain monotone and continuity properties.
      
    • Comportement des solutions d'equations differentielles a retard dans un espace ordonne (1980); "Using vectorial Ljapunov functionals, we give here some results related to the behaviour at infinity of solutions of a differential equation with delay in an ordered Banach space." .
      
    • Arino studied for a doctorate supervised by Maurice Gaultier and was awarded the degree in 1980 from the University of Bordeaux for his thesis Contributions a l'etude des comportements des solutions d'equations differentielles a retard par des methodes de monotonie et bifurcation.
      
    • As can be seen from this work his interest was mainly in differential equations, mainly with delay, but later he became primarily involved with applications of these ideas to biomathematics, particularly population dynamics.
      
    • His results in the field of delay differential equations stand out: oscillations, functional differential equations in infinite dimensional spaces, state-dependent delay differential equations.
      
    • In the light of this theory a cell equation involving unequal division is investigated in great detail.
      

42. Trudinger biography
    
    • in 1966 for his thesis Quasilinear Elliptical Partial Differential Equations in n Variables.
      
    • First there is the paper On the Dirichlet problem for quasilinear uniformly elliptic equations in n variables in which he extended previous work by his supervisor David Gilbarg, Olga Ladyzhenskaya and others on the solvability of the classical Dirichlet problem in bounded domains for certain second order quasilinear uniformly elliptic equations.
      
    • Secondly, in the paper The Dirichlet problem for nonuniformly elliptic equation he exploited the maximum principle to formulate general conditions for solvability of the Dirichlet problem for certain nonlinear elliptic equations.
      
    • In another 1967 paper On Harnack type inequalities and their application to quasilinear elliptic equations Trudinger examines weak solutions, subsolutions and supersolutions of certain quasilinear second order differential equations.
      
    • The book Elliptic partial differential equations of second order aimed to present (in the words of the authors):- .
      
    • the systematic development of the general theory of second order quasilinear elliptic equations and of the linear theory required in the process.
      
    • to the theory of quasilinear partial differential equations.
      
    • It had two new chapters one of which examined strong solutions of linear elliptic equations, and the other was on fully nonlinear elliptic equations.
      
    • The theory of nonlinear elliptic second order equations has continued to flourish during the last fifteen years and, in a brief epilogue to this volume, we signal some of the major advances.
      
    • In recent years, members of the programme have solved major open problems in curvature flow, affine geometry and optimal transportation, using techniques from nonlinear partial differential equations.
      

43. Sokolov biography
    
    • He also worked on functional equations and on such practical problems as the filtration of groundwater.
      
    • Other applications include On the determination of dynamic pull in shaft-lifting cables (Ukrainian) (1955) and On approximate solution of the basic equation of the dynamics of a hoisting cable (Ukrainian) (1955).
      
    • One of the topics which will always be associated with Sokolov's name is his method for finding approximate solutions to differential and integral equations.
      
    • Examples of his papers on this topic are On a method of approximate solution of linear integral and differential equations (Ukrainian) (1955), Sur la methode du moyennage des corrections fonctionnelles (Russian) (1957), Sur l'application de la methode des corrections fonctionnelles moyennes aux equations integrales non lineaires (Russian) (1957), On a method of approximate solution of systems of linear integral equations (Russian) (1961), On a method of approximate solution of systems of nonlinear integral equations with constant limits (Russian) (1963), and On sufficient tests for the convergence of the method of averaging of functional corrections (Russian) (1965).
      
    • This basic approach is developed by the author and applied to the approximate solution of Fredholm and Volterra-type integral equations of the second kind, to their nonlinear counterparts, to integral equations of mixed type, to linear and nonlinear one-dimensional boundary value problems, to initial-value problems in ordinary differential equations and to certain elliptic, hyperbolic and parabolic equations.
      
    • The first part of Sokolov's book discusses applications of his method to problems which can be modelled by linear integral equations with constant limits.
      
    • The next three parts look first at problems which can be modelled by nonlinear integral equations with constant limits and then extend the analysis to the situation where the upper limit is variable.
      
    • In the final part Sokolov examines applications of his method to integral equations of mixed type, then in a number of appendices he presents some generalisations of the method.
      

44. Cardan biography
    
    • There followed a period of intense mathematical study by Cardan who worked on solving cubic and quartic equations by radical over the next six years.
      
    • I have certainly grasped this rule, but when the cube of one-third of the coefficient of the unknown is greater in value than the square of one-half of the number, then, it appears, I cannot make it fit into the equation.
      
    • In 1540 Cardan resigned his mathematics post at the Piatti Foundation, the vacancy being filled by Cardan's assistant Ferrari who had brilliantly solved quartic equations by radicals.
      
    • In it he gave the methods of solution of the cubic and quartic equation.
      
    • In fact he had discovered in 1543 that Tartaglia was not the first to solve the cubic equation by radicals and therefore felt that he could publish despite his oath.
      
    • Solving a particular cubic equation, he writes:- .
      
    • Quadratic, cubic and quartic equations .
      
    • History Topics: Quadratic, cubic and quartic equations .
      

45. Brahmagupta biography
    
    • Brahmagupta developed some algebraic notation and presents methods to solve quardatic equations.
      
    • He presents methods to solve indeterminate equations of the form ax + c = by.
      
    • Brahmagupta perhaps used the method of continued fractions to find the integral solution of an indeterminate equation of the type ax + c = by.
      
    • Brahmagupta also solves quadratic indeterminate equations of the type ax2 + c = y2 and ax2 - c = y2.
      
    • For the equation 11x2 + 1 = y2 Brahmagupta obtained the solutions (x, y) = (3, 10), (161/5, 534/5), ..
      
    • Pell's equation .
      
    • History Topics: Quadratic, cubic and quartic equations .
      
    • History Topics: Pell's equation .
      

46. Kato biography
    
    • II in 1950, and Note on Schwinger's variational method, On the existence of solutions of the helium wave equation, Upper and lower bounds of scattering phases and Fundamental properties of Hamiltonian operators of Schrodinger type in 1951.
      
    • The course covered, thoroughly but efficiently, most of the standard material from the theory of functions through partial differential equations.
      
    • In 1962 he introduced new powerful techniques for studying the partial differential equations of incompressible fluid mechanics, the Navier-Stokes equations.
      
    • In 1983 he discovered the "Kato smoothing" effect while studying the initial-value problem associated with the Korteweg-de Vries equation, which was originally introduced to model the propagation of shallow water waves.
      
    • Another contribution to this area was On the Korteweg-de Vries equation where, in Kato's words from the paper:- .
      
    • Existence, uniqueness, and continuous dependence on the initial data are proved for the local (in time) solution of the (generalized) Korteweg-de Vries equation on the real line ..
      
    • He lectured on Abstract differential equations and nonlinear mixed problems.
      

47. Krylov Aleksei biography
    
    • There he was taught advanced mathematics by Aleksandr Nikolaevich Korkin, a student of Chebyshev, who was an expert in partial differential equations.
      
    • In 1904 he constructed a mechanical integrator to solve ordinary differential equations, being the first in Russia to make such an instrument.
      
    • He studied the acceleration of convergence of Fourier series in a paper in 1912, and studied the approximate solutions to differential equations in a paper published in 1917.
      
    • In 1931 he found a new method of solving the secular equation determining the frequency of vibrations in mechanical systems which is better than methods due to Lagrange, Laplace, Jacobi and Le Verrier.
      
    • This paper On the numerical solution of the equation by which, in technical matters, frequencies of small oscillations of material systems are determined deals with eigenvalue problems.
      
    • It is clear that, if for k = 2 and k = 3 it is easy to compose this [secular] equation, then for k = 4 the laying-out becomes cumbersome, and for values k more than 5 this is completely unrealisable in a direct way.
      
    • is to present simple methods of composition of the secular equation in the developed form, after which, its solution, i.e.
      
    • The first edition of On Some Differential Equations of Mathematical Physics Having Application to Technical Problems appeared in 1913, the second edition in 1932, and the fourth appeared in 1948 as part of Krylov's collected works.
      

48. Robinson Julia biography
    
    • Robinson was awarded a doctorate in 1948 and that same year started work on Hilbert's Tenth Problem: find an effective way to determine whether a Diophantine equation is soluble.
      
    • Hilbert in 1900 posed the problem of finding a method for solving Diophantine equations as the 10th problem on his famous list of 23 problems which he believed should be the major challenges for mathematical research this century.
      
    • Now you are going to ask how could he be sure? He couldn't check each possible method and maybe there were very involved methods that didn't seem to have anything to do with Diophantine equations but still worked.
      
    • The method of proof is based on the fact that there is a Diophantine equation say P(x,y,z,..
      
    • In 1971 at a conference in Bucharest Robinson gave a lecture Solving diophantine equations in which she set the agenda for continuing to study Diophantine equations following the negative solution to Hilbert's Tenth Problem problem.
      
    • Instead of asking whether a given Diophantine equation has a solution, ask "for what equations do known methods yield the answer?" .
      

49. Van der Pol biography
    
    • even in mathematics, his papers covered number theory, special functions, operational calculus and nonlinear differential equations.
      
    • Of course, to most mathematicians the name of van der Pol is associated with the differential equation which now bears his name.
      
    • This equation first appeared in his article On relaxation oscillation published in the Philosophical Magazine in 1926.
      
    • 35 (1960), 367-376.',4)">4], [Application of asymptotic methods in the theory of nonlinear differential equations (Russian) (Akad.
      
    • We explain the history of the development of the equation carrying his name, and also the origins of the method of finding the first approximation to the solution of this equation (the method of slowly varying coefficients).
      
    • Van der Pol did much to popularize his subject; he was an engaging lecturer, and often took the opportunity of bringing together phenomena over a wide field of science which could be elucidated by a single mathematical relation such as the equation for relaxation oscillations.
      
    • In fact van der Pol corresponded with Nikolai Mitrofanovich Krylov about the theory of nonlinear oscillations; a letter sent by van der Pol to Krylov is published in [Application of asymptotic methods in the theory of nonlinear differential equations (Russian) (Akad.
      

50. Schrodinger biography
    
    • In theoretical physics he studied analytical mechanics, applications of partial differential equations to dynamics, eigenvalue problems, Maxwell's equations and electromagnetic theory, optics, thermodynamics, and statistical mechanics.
      
    • In mathematics he was taught calculus and algebra by Franz Mertens, function theory, differential equations and mathematical statistics by Wilhelm Wirtinger (whom he found uninspiring as a lecturer).
      
    • One week later Schrodinger gave a seminar on de Broglie's work and a member of the audience, a student of Sommerfeld's, suggested that there should be a wave equation.
      
    • Within a few weeks Schrodinger had found his wave equation.
      
    • The solution of the natural boundary value problem of this differential equation in wave mechanics is completely equivalent to the solution of Heisenberg's algebraic problem.
      
    • I am simply fascinated by your [wave equation] theory and the wonderful new viewpoint it brings.
      
    • Note the wave equation!')">Schrodinger's grave at Alpbach in Austria .
      

51. Volterra biography
    
    • In 1890 Volterra showed by means of his functional calculus that the theory of Hamilton and Jacobi for the integration of the differential equations of dynamics could be extended to other problems of mathematical physics.
      
    • During the years 1892 to 1894 Volterra published papers on partial differential equations, particularly the equation of cylindrical waves.
      
    • His most famous work was done on integral equations.
      
    • He began this study in 1884 and in 1896 he published papers on what is now called 'an integral equation of Volterra type'.
      
    • He continued to study functional analysis applications to integral equations producing a large number of papers on composition and permutable functions.
      
    • He studied the Verhulst equation and the logistic curve.
      
    • He also wrote on predator-prey equations.
      

52. Galois biography
    
    • On 25 May and 1 June he submitted articles on the algebraic solution of equations to the Academie des Sciences.
      
    • Galois sent Cauchy further work on the theory of equations, but then learned from Bulletin de Ferussac of a posthumous article by Abel which overlapped with a part of his work.
      
    • Galois then took Cauchy's advice and submitted a new article On the condition that an equation be soluble by radicals in February 1830.
      
    • Galois was invited by Poisson to submit a third version of his memoir on equation to the Academy and he did so on 17 January.
      
    • .as correct as it is deep of this lovely problem: Given an irreducible equation of prime degree, decide whether or not it is soluble by radicals.
      
    • A page from Galois' Memoire sur les conditions de resolubilite des equationspar radicaus (published in his collected works in 1897) .
      

53. Ferro biography
    
    • In one sense he is well known, for his role in solving cubic equations is explained in almost every general work on the history of mathematics ever written, and yet, surprisingly, his name remains relatively unknown.
      
    • The outstanding problem which del Ferro solved was to find a formula to solve a cubic equation similar to the formula which had been known since the time of the Babylonians for solving quadratic equations.
      
    • There has been much conjecture as to whether del Ferro came to work on the solution to cubic equations as a result of a visit which Pacioli made to Bologna.
      
    • It is not known whether the two discussed the algebraic solution of cubic equations, but certainly Pacioli had included this topic in his famous treatise the Summa which he had published seven years earlier.
      
    • Four years ago when Cardano was going to Florence and I accompanied him, we saw at Bologna Hannibal della Nave, a clever and humane man who showed us a little book in the hand of Scipione del Ferro, his father-in-law, written a long time ago, in which that discovery [solution of cubic equations] was elegantly and learnedly presented.
      
    • Dal Ferro's rule for the solution of cubic equations.
      
    • The manuscript gives a method of solution which is applied to the equation 3x3 + 18x = 60.
      
    • History Topics: Quadratic, cubic and quartic equations .
      

54. Tamarkin biography
    
    • Tamarkin maintained his close friendship and academic collaboration with Friedmann and by 1908 the two were attending lectures by Steklov on partial differential equations.
      
    • I proposed to Tamarkin that he think about the asymptotic solution of differential equations (i.e.
      
    • and then submitted his thesis in 1917 on boundary value problems for linear differential equations.
      
    • It was published in English in Mathematische Zeitschrift in 1928 as Some general problems of the theory of ordinary linear differential equations and expansion of an arbitrary function in series of fundamental functions.
      
    • His research during this period continued on boundary value problems, but also included advances in mathematical physics, differential equations, and approximations.
      
    • Five papers were published in these journals in 1926 and 1927: On Laplace's integral equations; On Volterra's integro-functional equation; A new proof of Parseval's identity for trigonometric functions; On Fredholm's integral equations, whose kernels are analytic in a parameter; and The notion of the Green's function in the theory of integro-differential equations.
      
    • For example they published: On the summability of Fourier series (two papers), On a theorem of Hahn-Steinhaus, On a theorem of Paley and Wiener, On the theory of linear integral equations.
      
    • The problem of moments is the theory of an infinite system of integral equations under various hypotheses.
      

55. Walsh biography
    
    • The equation of a curve transformed as above Mr Walsh calls its 'partial equation'.
      
    • Memoir on the Invention of Partial Equations; The Theory of Partial Functions; Irish Manufactures: A New Method of Tangents; An Introduction to the Geometry of the Sphere, Pyramid and Solid Angles; General Principles of the Theory of Sound; The Normal Diameter in Curves; The Problem of Double Tangency; The Geometric Base; The Theoretic Solution of Algebraic Equations of the Higher Orders.
      
    • Thus, in a page headed Cubic Equations, he writes the name of Cardan opposite to a well-known algebraic solution, that of Walsh opposite to the same result put under another and less convenient form, and below these he gives a formula headed For a Complete Cubic by Walsh only.
      
    • Discovered the general solution of numerical equations of the fifth degree at 114 Evergreen Street, at the Cross of Evergreen, Cork, at nine o'clock in the forenoon of July 7th, 1844; exactly twenty-two years after the invention of the Geometry of Partial Equations, and the expulsion of the differential calculus from Mathematical Science.
      
    • And the falsehood of the offspring of that method, namely, the no less celebrated doctrine of fluxions, differentials, limits, etc., the boast and glory of England, France and Germany, demonstrated by the great invention of the geometry of partial equations which has superseded them, at least in my hands, and indefinitely surpassed the old system in power.
      

56. Lagrange biography
    
    • He solved the resulting system of n+1 differential equations, then let n tend to infinity to obtain the same functional solution as Euler had done.
      
    • In papers which were published in the third volume, Lagrange studied the integration of differential equations and made various applications to topics such as fluid mechanics (where he introduced the Lagrangian function).
      
    • Also contained are methods to solve systems of linear differential equations which used the characteristic value of a linear substitution for the first time.
      
    • In 1770 he also presented his important work Reflexions sur la resolution algebrique des equations which made a fundamental investigation of why equations of degrees up to 4 could be solved by radicals.
      
    • The paper is the first to consider the roots of an equation as abstract quantities rather than having numerical values.
      
    • The Mecanique analytique summarised all the work done in the field of mechanics since the time of Newton and is notable for its use of the theory of differential equations.
      
    • Extract of Lagrange's Reflexions sur la resolution algebrique des equations from his collected works (1869).
      
    • History Topics: Pell's equation .
      

57. Yau biography
    
    • Yau was awarded a Fields Medal in 1982 for his contributions to partial differential equations, to the Calabi conjecture in algebraic geometry, to the positive mass conjecture of general relativity theory, and to real and complex Monge-Ampere equations.
      
    • S-T Yau has done extremely deep and powerful work in differential geometry and partial differential equations.
      
    • The analytic problem is that of proving the existence of a solution of a highly nonlinear (complex Monge-Ampere ) differential equation.
      
    • .for his work in nonlinear partial differential equations, his contributions to the topology of differentiable manifolds, and for his work on the complex Monge-Ampere equation on compact complex manifolds.
      
    • As a result of Yau's work over the past twenty years, the role and understanding of basic partial differential equations in geometry has changed and expanded enormously within the field of mathematics.
      
    • His work has had, and will continue to have, a great impact on areas of mathematics and physics as diverse as topology, algebraic geometry, representation theory, and general relativity as well as differential geometry and partial differential equations.
      

58. Lions Jacques-Louis biography
    
    • Schwartz had made a big breakthrough in the understanding of partial differential equations which he saw should be completely recast in the context of distribution theory.
      
    • Lions was one of several students who Schwartz directed to take this new approach and his doctoral thesis developed what has become the standard variational theory of linear elliptic and evolution equations.
      
    • In one of these collaborations with Enrico Magenes, they were investigating inhomogeneous boundary problems for elliptic equations and inhomogeneous initial-boundary value problems for parabolic and hyperbolic evolution equations.
      
    • It is a work to be recommended to every serious student of partial differential equations and particularly to those who are fascinated by the manner in which modern functional analysis has aided and influenced their study.
      
    • The systems he had in mind are those described by linear and nonlinear partial differential equations.
      
    • He had already published a major work on control of systems Controle optimal de systemes gouvernes par des equations aux derivees partielles in 1968 which investigates deterministic optimisation problems involving partial differential equations.
      
    • One notable feature of this work is that Lions introduces an infinite dimensional version of the Riccati equation in it.
      
    • reports on methods of solving nonlinear boundary value problems for partial differential equations, on a theoretical and functional analysis basis.
      
    • Integral equations and numerical methods.
      

59. Brioschi biography
    
    • Francesco Brioschi was an important mathematician in the European context owing to his contributions to the theory of algebraic equations and to the applications of mathematics to hydraulics.
      
    • One of his most important results was his application of elliptical modular functions to the solution of equations of the fifth degree in 1858.
      
    • Brioschi however later went on to solve sixth degree equations using similar techniques.
      
    • In 1888, Maschke proved that a particular sixth-degree equation could be solved by using hyperelliptic functions and Brioschi then showed that any sixth-degree algebraic equation could be reduced to Maschke's equation and therefore solved using hyperelliptic functions.
      
    • In mechanics Brioschi dealt with problems of statics, proving Mobius's results by analytic means; with the integration of equations in dynamics, according to Jacobi's method; with hydrostatics; and with hydrodynamics.
      

60. Yang Hui biography
    
    • For example, if the problem reduced to the solution of a quadratic equation, then Yang would solve it numerically, then show how to solve a general quadratic equation numerically.
      
    • What Yang's method essentially reduces to is finding the determinant of the matrix of coefficients of the system of equations.
      
    • The topics covered by Yang include multiplication, division, root-extraction, quadratic and simultaneous equations, series, computations of areas of a rectangle, a trapezium, a circle, and other figures.
      
    • Then a modern solution would set up equations .
      
    • He is subtracting the second equation from the first: 300 - 100 coins, 21 - 1 Wenzhou oranges, 9 - 1 green oranges.
      
    • This is exactly what he is doing! Replace y in the above equation so that .
      

61. Whittaker biography
    
    • It also develops the theory of special functions and their related differential equations.
      
    • He studied these special functions as arising from the solution of differential equations derived from the hypergeometric equation.
      
    • His results in partial differential equations (described as 'most sensational' by Watson) included a general solution of the Laplace equation in three dimensions in a particular form and the solution of the wave equation.
      
    • He also worked on electromagnetic theory giving a general solution of Maxwell's equation, and it was through this topic that his interest in relativity arose.
      

62. Laplace biography
    
    • His next paper for the Academy followed soon afterwards, and on 18 July 1770 he read a paper on difference equations.
      
    • This paper contained equations which Laplace stated were important in mechanics and physical astronomy.
      
    • Not only had he made major contributions to difference equations and differential equations but he had examined applications to mathematical astronomy and to the theory of probability, two major topics which he would work on throughout his life.
      
    • The main mathematical approach here is the setting up of differential equations and solving them to describe the resulting motions.
      
    • In the Mecanique Celeste Laplace's equation appears but although we now name this equation after Laplace, it was in fact known before the time of Laplace.
      

63. Mathisson biography
    
    • Mathisson studied general dynamical laws governing the motion of a particle, with possibly a spin or an angular momentum, in a gravitational or electromagnetic field, and developed a powerful method for passing from field equations to particle equations.
      
    • Mathisson proved that the variational equation can be solved when it has been defined so that the equations to be imposed upon the characteristic tensor will be compatible with the variations allowed in the fields.
      
    • He obtained the equations of motion for the angular momentum and for the centre of mass with arbitrary external forces.
      
    • Finally, he calculated the linear forces for the case of no electric moment, leading to the equations for linear motion.
      
    • M Mathisson, The variational equation of relativistic dynamics, Proc.
      

64. Collatz biography
    
    • Collatz was awarded his doctorate in 1935 for his dissertation Das Differenzenverfahren mit hoherer Approximation fur lineare Differentialgleichunge (The finite difference method with higher approximation for linear differential equations).
      
    • Among his early papers are Genaherte Berechnung von Eigenwerten (1939) in which he considers various methods of approximating characteristic values, Das Hornersche Schema bei komplexen Wurzeln algebraischer Gleichungen (1940) in which he presents a more efficient way of using Horner's method to approximate the complex roots of an algebraic equation, and Schrittweise Naherungen bei Integralgleichungen und Eigenwertschranken (1940) in which inequalities between the eigenvalues of certain integral equations are studied.
      
    • Eigenwertprobleme und ihre numerische Behandlung (1945) contains three parts, the first containing a collection of practical applications which lead to boundary value problems for ordinary and partial differential equations.
      
    • This was followed by Numerische Behandlung von Differentialgleichungen (1951) which provides a comprehensive text on numerical methods for solving differential equations.
      
    • This small book gives a wealth of information on differential equations.
      
    • The book Aufgaben aus der Angewandten Mathematik (1972) (with J Albrecht) provides a collection of problems (with their solutions) on the solution of equations and systems of equations, interpolation, quadrature, approximation, and harmonic analysis.
      
    • Later texts by Collatz include Optimization problems (1975) and Differential equations (1986), the second of these being an English translation of an earlier German book.
      

65. Dahlquist biography
    
    • BESK came into operation in December 1953 and Dahlquist used the machine to solve differential equations.
      
    • During this time Dahlquist wrote a number of papers such as The Monte Carlo-method (1954), Convergence and stability for a hyperbolic difference equation with analytic initial-values (1954), and Convergence and stability in the numerical integration of ordinary differential equations (1956).
      
    • He submitted his doctoral thesis Stability and error bounds in the numerical integration of ordinary differential equations to Stockholm University in 1958, defending it in a viva in December.
      
    • Dahlquist was to use this idea throughout his research in stiff differential equations.
      
    • In the same year of 1963 he published Stability questions for some numerical methods for ordinary differential equations, an expository paper on his fundamental results concerning stability of difference approximations for ordinary differential equations.
      
    • Awarded to a young scientist (normally under 45) for original contributions to fields associated with Germund Dahlquist, especially the numerical solution of differential equations and numerical methods for scientific computing.
      
    • He has created the fundamental concepts of stability, A-stability and the nonlinear G-stability for the numerical solution of ordinary differential equations.
      

66. Goursat biography
    
    • He began teaching at the University of Paris in 1879, receiving his doctorate in 1881 from l'Ecole Normale Superieure for his thesis Sur l'equation differentialle lineaire qui admet pour integrale la serie hypergeometrique.
      
    • Goursat's papers on the theory of linear differential equations and their rational transformations, as well as his studies on hypergeometric series, Kummer's equation, and the reduction of abelian integrals form, in the words of Picard "a remarkable ensemble of works evolving naturally one from the other".
      
    • Goursat introduced the notion of orthogonal kernels and semiorthogonals in connection with Erik Fredholm's work on integral equations.
      
    • In 1891 Goursat wrote Lecons sur l'integration des equations aux derivees partielles du premier ordre.
      
    • Volume 2 explores functions of a complex variable and differential equations.
      
    • Volume 3 surveys variations of solutions and partial differential equations of the second order and integral equations and calculus of variations.
      

67. Taylor biography
    
    • He gave an account of an experiment to discover the law of magnetic attraction (1715) and an improved method for approximating the roots of an equation by giving a new method for computing logarithms (1717).
      
    • It was, wrote Taylor, due to a comment that Machin made in Child's Coffeehouse when he had commented on using "Sir Isaac Newton's series" to solve Kepler's problem, and also using "Dr Halley's method of extracting roots" of polynomial equations.
      
    • Taylor initially derived the version which occurs as Proposition 11 as a generalisation of Halley's method of approximating roots of the Kepler equation, but soon discovered that it was a consequence of the Bernoulli series.
      
    • The second version occurs as Corollary 2 to Proposition 7 and was thought of as a method of expanding solutions of fluxional equations in infinite series.
      
    • These include singular solutions to differential equations, a change of variables formula, and a way of relating the derivative of a function to the derivative of the inverse function.
      
    • Taylor, in his studies of vibrating strings was not attempting to establish equations of motion, but was considering the oscillation of a flexible string in terms of the isochrony of the pendulum.
      
    • He tried to find the shape of the vibrating string and the length of the isochronous pendulum rather than to find its equations of motion.
      

68. Gray Marion biography
    
    • in 1926 for her thesis The theory of singular ordinary differential equations of the second order having offered physics as an allied subject.
      
    • In 1925 E T Whittaker communicated the paper The equation of conduction of heat by Marion C Gray to the Royal Society of Edinburgh.
      
    • Marion C Gray and S A Schelkunoff, The approximate solution of linear differential equations.
      
    • Various papers by Gray were read to the Society: The equation of telegraphy (which appeared in volume 42 of the Proceedings and she read to the meeting of the Society in November 1923), The equation of conduction of heat (which also appeared in volume 42 of the Proceedings), and On the equation of heat (which appeared as Particular solutions of the equation of conduction of heat in one dimension in volume 43 of the Proceedings).
      

69. Tricomi biography
    
    • In this paper he studied the theory of partial differential equations of mixed type, in particular the equation .
      
    • now known as the 'Tricomi equation'.
      
    • The equation became important in describing an object moving at supersonic speed.
      
    • Of course there were no supersonic aircraft in 1923 but the equation was to play a major role in later studies of supersonic flight.
      
    • These papers cover a vast range of subjects including singular integrals, differential and integral equations, pseudodifferential operators, functional transforms, special functions, probability theory and its applications to number theory.
      
    • As well as having the 'Tricomi equation' named after him, there are also special functions called 'Tricomi functions'.
      

70. Black Fischer biography
    
    • The Black-Scholes-Merton partial differential equation for the price of a financial asset was derived in their famous paper [Journal of Political Economy, 81(3), 637-54.',15)">15], using Ito's Lemma, with the economics coming in by way of the observation, due to Merton R.
      
    • In 1973, Black published with Myron Scholes their famous paper entitled The Pricing of Options and Corporate Liabilities [Journal of Political Economy, 81(3), 637-54.',15)">15] which derived and solved the Black-Scholes-Merton differential equation thereby solving the stock-option pricing problem [Note 1] .
      
    • the model of the stock-price in continuous time was represented by a special type of differential equation (so called stochastic differential equation) which allowed for randomness in stock-price.
      
    • The differential equation for the stock-price, S(t), was:- .
      
    • When the above differential equation was integrated, it gave a stock-price distribution that was lognormal (i.e.
      
    • The famous paper The Pricing of Options and Corporate Liabilities [Journal of Political Economy, 81(3), 637-54.',15)">15] has two ways of deriving the relevant partial differential equation.
      
    • Putting a = μS(t) and b = σS(t) in Ito's Lemma and exploiting that there are no free lunches in a randomless/riskless portfolio, gave rise to the famous partial differential equation:- .
      
    • This equation is satisfied by the traded assets themselves, for example, f (S, t) = S(t) and by f (S, t) = A ert but these do not have the European call-option boundary conditions (at time T) that Black and Scholes were interested in (i.e.
      
    • By 1969, Black and Scholes had the above differential equation.
      
    • They tried to solve this partial differential equation with the European call-option boundary condition but could not solve it.
      
    • But Black and Scholes had noticed the curious absence (in the differential equation) of the investment return, μ, of the stock-price or any parameter representing the degree of preference, as to risk, on the part of option purchasers.
      
    • the return was the risk-free rate, irrespective of the purchaser's risk preferences, they found that Sprenkle's formula, with these adjustments, satisfied the partial differential equation.
      
    • It is clear that the unexpected aspect of the Black-Scholes-Merton differential equation was not at first accepted.
      
    • Inter alia, Bachelier, had shown in his thesis [The Random Character of Stock Market Prices, MIT Press, Cambridge, Massachusetts (contains the translation from French of Bachelier\'s doctoral thesis and contains Sprenkle\'s, 1961 paper).',88)">88] the close connection between random walks and the Fourier heat equation, something that was expanded on by Kac, in 1951, [Ito\'s stochastic calculus and probability theory, Tokyo, ix-xiv.
      
    • ',98)">98] and by Feynman [Review of Modern Physics, 20, 367-387.',89)">89], where it was shown that the solution of Fourier's equation could be expressed as the distribution function of a random variable arising of a large number of random walks each with n steps (and with each step size proportional to √(t/n)) and by letting n become very large (i.e.
      

71. Fox Leslie biography
    
    • We should note that Fox was undertaking numerical work solving partial differential equations which arose in engineering problems but which could not be solved by analytic techniques.
      
    • For example he published Some improvements in the use of relaxation methods for the solution of ordinary and partial differential equations (1947), A short account of relaxation methods (1948), and The solution by relaxation methods of ordinary differential equations (1949).
      
    • This book was The numerical solution of two-point boundary problems in ordinary differential equations and it is a great tribute to his expository skills that it was reprinted by Dover Publications in 1990.
      
    • The book summarises at an elementary level the methods for numerical construction of the solutions of boundary-value problems which can be expressed in terms of ordinary differential equations of orders one to four.
      
    • With a practical yet rigorous approach, methods to investigate topics such as recurrence relations, zeros of polynomials, linear equations, eigenvalues and eigenvectors, approximations, interpolation, integration, and ordinary differential equations are described and analysed.
      
    • Another collaboration between Fox and Mayers led to Numerical solution of ordinary differential equations published in 1987, four years after Fox retired.
      
    • Though there are two main types of ordinary differential equations, those of initial value type and those of boundary value type, most books until quite recently have concentrated on the former, and again until recently the boundary value problem has had little literature.
      
    • Some numerical experiments with eigenvalue problems in ordinary differential equations (1960) considers methods for solving such equations using a computer.
      
    • Partial differential equations (1963) is summarised by Fox as follows:- .
      
    • This expository paper discusses the present state of our ability to solve partial differential equations.
      
    • Outstanding problems include a determination of the error of finite-difference approximations, the automatic machine production of finite-difference formulae in complicated regions, the smoothing of physical data, and the classification of equations for computing-machine library routines.
      
    • Some of his later papers examine numerical methods for factorising polynomials, for solving elliptic partial differential equations, and methods for treating singularities in boundary value problems.
      

72. Sturm biography
    
    • One of Sturm's most famous papers Memoire sur la resolution des equations numeriques was published in 1829.
      
    • It considered the problem of determining the number of real roots of an equation on a given interval.
      
    • The 1829 paper was not the last of Sturm's work on this algebraic equations and in [Rev.
      
    • seeks to determine the mutual influence between A-L Cauchy's and Ch-F Sturm's research from 1829 to around 1840 on the roots of algebraic equations.
      
    • These were the years during which he published some important results on differential equations.
      
    • Sturm became interested in obtaining results on specific differential equations which occurred in Poisson's theory of heat.
      
    • Liouville was also working on differential equations derived from the theory of heat.
      
    • Papers of 1836-1837 by Sturm and Liouville on differential equations involved expansions of functions in series and is today well-known as the Sturm-Liouville problem, an eigenvalue problem in second order differential equations.
      

73. Hermite biography
    
    • Also like Galois he was attracted by the problem of solving algebraic equations and one of the two papers attempted to show that the quintic cannot be solved in radicals.
      
    • The letters he exchanged with Jacobi show that Hermite had discovered some differential equations satisfied by theta-functions and he was using Fourier series to study them.
      
    • He had found general solutions to the equations in terms of theta-functions.
      
    • Although an algebraic equation of the fifth degree cannot be solved in radicals, a result which was proved by Ruffini and Abel, Hermite showed in 1858 that an algebraic equation of the fifth degree could be solved using elliptic functions.
      
    • Hermite is now best known for a number of mathematical entities that bear his name: Hermite polynomials, Hermite's differential equation, Hermite's formula of interpolation and Hermitian matrices.
      

74. Richardson biography
    
    • He first developed his method of finite differences in order to solve differential equations which arose in his work for the National Peat Industries concerning the flow of water in peat.
      
    • a scheme of weather prediction which resembles the process by which the Nautical Almanac is produced in so far as it is founded upon the differential equations and not upon the partial recurrence of phenomena in their ensemble.
      
    • Making observations from weather stations would provide data which defined the initial conditions, then the equations could be solved with these initial conditions and a prediction of the weather could be made.
      
    • It was a remarkable piece of work but in a sense it was ahead of its time since the time taken for the necessary hand calculations in a pre-computer age took so long that, even with many people working to solve the equations, the solution would be found far too late to be useful to predict the weather.
      
    • However the way that Richardson modelled the causes of war was quite different, giving systems of differential equations which governed the interactions between countries caused by such things as attitudes and moods.
      
    • The equations are merely a description of what people would do if they did not stop and think.
      
    • He set up equations governing arms build-up by nations, taking into account factors such as the expense of an arms race, grievances between states, ambitions of states, etc.
      
    • Choosing different values for the various parameters in the equation he then tried to investigate when situations were stable and when they were unstable.
      

75. Ostrowski biography
    
    • One consequence of this association was his monograph Solution of equations and systems of equations which was published in 1960 and was the result of a series of lectures he had given at the National Bureau of Standards.
      
    • By 1973 the third edition of this monograph appeared, this time with a new title: Solution of equations in Euclidean and Banach spaces.
      
    • These are determinants, linear algebra, algebraic equations, multivariate algebra, formal algebra, number theory, geometry, topology, convergence, theory of real functions, differential equations, differential transformations, theory of complex functions, conformal mappings, numerical analysis and miscellany.
      
    • His work on aglebraic equations involved a study of the fundamental theorem of algebra, Galois theory, and estimating the roots of algebraic equations.
      
    • Other work of Ostrowski was on the Cauchy functional equation, the Fourier integral formula, Cauchy-Frullani integrals, and the Euler-Maclaurin formula.
      

76. Krylov Nikolai biography
    
    • He worked mainly on interpolation and numerical solutions to differential equations, where he obtained very effective formulas for the errors.
      
    • For example he published On the approximate solution of the integro-differential equations of mathematical physics (1926), and Approximation of periodic solutions of differential equations in French in 1929.
      
    • With his collaborator and former student N N Bogolyubov, he published On Rayleigh's principle in the theory of differential equations of mathematical physics and on Euler's method in calculus of variations (1927-8) and On the quasiperiodic solutions of the equations of the nonlinear mechanics.
      
    • Examples of physical systems are given which lead to the type of equation considered in the monograph.
      
    • Moreover, general statements of methods for solving equations are illustrated by the explicit solution of examples.
      
    • We present the fundamental results of the works of N M Krylov and N N Bogolyubov devoted to the establishment of effective error estimates for the Ritz method, the Bubnov-Galerkin method and the least squares method in connection with self-adjoint differential equations.
      
    • In 1939 Krylov and Bogolyubov published Sur les equations de Focker-Planck deduites dans la theorie des perturbations a l'aide d'une methode basee sur les proprietes spectrales de l'hamiltonien perturbateur (Application a la mecanique classique et a la mecanique quantique).
      

77. Piaggio biography
    
    • His most famous work, An Elementary Treatise on Differential Equations, was published by G Bell & Sons in 1920.
      
    • Here list a few articles which Piaggio published in The Mathematical Gazette: Relativity rhymes with a mathematical commentary (January 1922); Geometry and relativity (July 1922); Mathematics for evening technical students (July 1924); Mathematical physics in university and school (October 1924); Probability and its applications (July 1931); Three Sadleirian professors: A R Forsyth, E W Hobson and G H Hardy (October 1931); Mathematics and psychology (February 1933); Lagrange's equation (May 1935); Fallacies concerning averages (December 1937); and The incompleteness of "complete" primitives of differential equations (February 1939).
      
    • In the Proceedinggs of the Glasgow Mathematical Association he published Exceptional integrals of a not completely integrable total differential equation (1953).
      
    • The usual theory of a single Pfaffian equation holds if the coefficients are of class C'.
      
    • He read papers to the Society such as Note on Linear Differential Equations with constant coefficients on 10 May 1912.
      

78. Yamabe biography
    
    • This was a period when his mathematical interests began to move away from Lie groups to differential equations and differential geometry.
      
    • His next major research contribution was Kernel functions of diffusion equations.
      
    • This paper presents a new, elegant method for constructing Green's function G for the heat equation over any domain D in Euclidean space.
      
    • In the following year Yamabe published A unique continuation theorem for solutions of a parabolic differential equation written jointly with Seizo Ito.
      
    • The same theme was taken up in A unique continuation theorem of a diffusion equation which he published in 1959.
      
    • His second paper on Kernel functions of diffusion equations was published in 1959 and in the following year he published On a deformation of Riemannian structures on compact manifolds and Global stability criteria for differential systems.
      

79. Moser Jurgen biography
    
    • The difficulty that Moser had no money was overcome and he began to study the spectral theory of differential equations with Rellich as his advisor.
      
    • In 1955 several of Moser's papers were published including Singular perturbation of eigenvalue problems for linear differential equations of even order, and Nonexistence of integrals for canonical systems of differential equations.
      
    • Moser worked in ordinary differential equations, partial differential equations, spectral theory, celestial mechanics, and stability theory.
      
    • Next is Stable and random motions in dynamical systems (1973, reprinted 2001) which describes how stable behaviour and statistical behaviour take place together in analytic conservative systems of differential equations.
      
    • Here Moser examines inverse spectral theory for the one-dimensional Schrodinger equation with the aim, as he writes in the introduction, of showing that:- .
      
    • For his fundamental work on stability in Hamiltonian mechanics and his profound and influential contributions to nonlinear differential equations.
      

80. Magiros biography
    
    • This part includes papers on nonlinear differential equations, mathematical modelling of physical phenomena and linearization of nonlinear models.
      
    • Linearization by exact methods which presents various "exact" methods of linearizing problems in differential equations; On the linearization of nonlinear models of phenomena.
      
    • Linearization by approximate methods in which he points out that "approximate" linearizations may lose the whole qualitative behaviour of the original nonlinear equation; and Characteristic properties of linear and nonlinear systems in which he gives many examples, recalls the importance of identifying characteristic properties of solutions, such as the superposition property for linear systems, and the possibility of limit cycles and self-excited oscillations in nonlinear systems.
      
    • Two papers which Magiros published in 1977 are: Nonlinear differential equations with several general solutions in which he gives specific devices for finding solutions of some nonlinear ordinary differential equations; and The general solutions of nonlinear differential equations as functions of their arbitrary constants presenting some nonlinear differential equations for which, surprisingly, some superposition does occur, that is, there are families of solutions depending linearly on arbitrary constants.
      
    • covers a variety of topics from special functions and transforms to numerical methods for the solution of nonlinear differential equations and optimal control problems.
      

81. Nash biography
    
    • Meanwhile he went to Levinson to inquire about a differential equation that intervened and Levinson says it is a system of partial differential equations and if he could only [get] to the essentially simpler analog of a single ordinary differential equation it would be a damned good paper - and Nash had only the vaguest notions about the whole thing.
      
    • His research on the theory of real algebraic varieties, Riemannian geometry, parabolic and elliptic equations was, however, extremely deep and significant in the development of all these topics.
      
    • After this Nash worked on ideas that would appear in his paper Continuity of solutions of parabolic and elliptic equations which was published in the American Journal of Mathematics in 1958.
      
    • The outstanding results which Nash had obtained in the course of a few years put him into contention for a 1958 Fields' Medal but since his work on parabolic and elliptic equations was still unpublished when the Committee made their decisions he did not make it.
      

82. Copson biography
    
    • Copson studied classical analysis, asymptotic expansions, differential and integral equations, and applications to problems in theoretical physics.
      
    • by Poisson's analytical solution of the equation of wave-motions.
      
    • .The analogue of Kirchhoff's formula, due to Volterra, is derived and an interesting account is given of a method, devised by Marcel Riesz and based on the theory of fractional integration, which provides a powerful method of solving initial value problems for equations like the wave equation.
      
    • In 1975 he published Partial differential equations which covers most of the classical techniques for first and second order linear partial differential equations, giving many examples and applications to physical problems.
      

83. Fock biography
    
    • Schrodinger published his two fundamental papers on quantum theory in the spring of 1926 and Fock immediately started to develop the ideas and by the end of the year two of his own important papers on the Schrodinger equation had been published.
      
    • He became interested in the geometrization of the Dirac equation and he published an important paper in 1928 on Dirac's work on distributions.
      
    • the fundamental paper of 1935 in which the full symmetry structure of the hydrogen atom energy levels was shown to be given by the full Lorentz group; and the 1937 paper on the proper time parametrization of the Dirac equation, seminal for the later development of Schwinger's theory of field propagators and for the whole subject of parametrised field theories.
      
    • The reviewer feels that the author has made a major contribution to the understanding of gravitation theory, especially by his insistence on studying the solutions of the field equations and not merely the formal properties of the equations.
      
    • Some we have mentioned but now let us list a few: Fock space; Fock vacuum; the Fock method of quantisation; the Fock proper time method; the Hartree-Fock method; Fock symmetry; the Klein-Fock-Gordon equation; the Fock-Krylov theorem; and Dirac-Fock-Podolsky formalism.
      

84. Sobolev biography
    
    • Sobolev became interested in differential equations, a topic which would dominate his research throughout his life, and even at this stage in his career he produced new results which he published.
      
    • published a number of profound papers in which he put forward a new method for the solution of an important class of partial differential equations.
      
    • Working with Smirnov, Sobolev studied functionally invariant solutions of the wave equation.
      
    • These methods allowed them to find closed form solutions to the wave equation describing the oscillations of an elastic medium.
      
    • By 1935 Sobolev was head of the Department of the Theory of Differential Equations at the Institute.
      
    • In 1958 Sobolev was part of the Soviet delegation to the International Mathematical Union, the delegation being led by Vinogradov, and Sobolev attended the International Congress at Edinburgh that year and gave an invited address on partial differential equations.
      

85. Bring biography
    
    • This work describes Bring's contribution to the algebraic solution of equations.
      
    • Bring discovered an important transformation to simplify a quintic equation.
      
    • It enabled the general quintic equation to be reduced to one of the form .
      
    • By the time Jerrard discovered the transformation, Ruffini's work and Abel's work on the impossibility of solving the quintic and higher order equations had been published.
      
    • However, at the time of Bring's discovery, there was no hint that the quintic could not be solved by radicals and, although Bring does not claim that he discovered his transformation in an attempt to solve the quintic, it is likely that this is in fact why he was examining quintic equations.
      
    • History Topics: Quadratic, cubic and quartic equations .
      

86. Picard Emile biography
    
    • Picard made his most important contributions in the fields of analysis, function theory, differential equations, and analytic geometry.
      
    • He used methods of successive approximation to show the existence of solutions of ordinary differential equations solving the Cauchy problem for these differential equations.
      
    • Starting in 1890, he extended properties of the Laplace equation to more general elliptic equations.
      
    • Picard also discovered a group, now called the Picard group, which acts as a group of transformations on a linear differential equation.
      

87. Ito biography
    
    • Introducing the concept of regularisation, developed by Doob of the United States, I finally devised stochastic differential equations, after painstaking solitary endeavours.
      
    • He created the theory of stochastic differential equations, which describe motion due to random events.
      
    • Among them were On a stochastic integral equation (1946), On the stochastic integral (1948), Stochastic differential equations in a differentiable manifold (1950), Brownian motions in a Lie group (1950), and On stochastic differential equations (1951).
      
    • Stochastic differential equations, called "Ito Formula," are currently in wide use for describing phenomena of random fluctuations over time.
      
    • When I first set forth stochastic differential equations, however, my paper did not attract attention.
      

88. Rolle biography
    
    • He published his most important work Traite d'algebre in 1690 on the theory of equations.
      
    • He also used it to solve Diophantine linear equations.
      
    • Let us see how this idea worked: If P(x) = 0 is a given polynomial equation with real roots a and b then he constructs a polynomial P'(x), which he called the 'first cascade,' so that P'(b) = (b - a)Q(b) where Q(x) is a polynomial of lower degree.
      
    • Some basic principles of the calculus and the theory of equations can definitely be traced to their origin as incidental propositions of the method.
      
    • It amplified the concepts of limits of roots of equations, provided the fundamentals from which Maclaurin derived his formula, began modern methods of series for determining roots, and discussed the relationship of imaginary roots in equations and their derivatives.
      
    • Rolle published another important work on solutions of indeterminate equations in 1699, Methode pour resoudre les equations indeterminees de l'algebre.
      

89. Bhaskara I biography
    
    • He considers problems of indeterminate equations of the first degree and trigonometric formulae.
      
    • ',12)">12], [Ganita 23 (1) (1972), 57-79',13)">13] and [Ganita 23 (2) (1972), 41-50.',14)">14] Shukla discusses some features of Bhaskara's mathematics such as: numbers and symbolism, the classification of mathematics, the names and solution methods of equations of the first degree, quadratic equations, cubic equations and equations with more than one unknown, symbolic algebra, unusual and special terms in Bhaskara's work, weights and measures, the Euclidean algorithm method of solving linear indeterminate equations, examples given by Bhaskara I illustrating Aryabhata I's rules, certain tables for solving an equation occurring in astronomy, and reference made by Bhaskara I to the works of earlier Indian mathematicians.
      

90. Fowler biography
    
    • Early in his career, after receiving his degree, Fowler took to examining the behavior of the solutions to certain second-order differential equations.
      
    • In particular, he studied Emden's differential equation: .
      
    • Sir Arthur Eddington had originally shown that the equilibrium of gaseous stars could be found using the above equation with n = 3.
      
    • He rightly deduced Emden's equation must have other solutions.
      
    • The resulting general equation, which had considerable later influence on stellar astrophysics, was: .
      
    • These ions are closely packed leaving the free electrons to form a degenerate gas which Fowler described as "like a gigantic molecule in its lowest state." The equilibrium of the white dwarfs was later found to be described by a solution to Emden's equation as generalized by Fowler in the above equation with n = 3/2.
      

91. Smithies biography
    
    • There he took courses by G H Hardy on Fourier analysis, John Whittaker on integral equations, and Ebenezer Cunningham on mechanics.
      
    • Smithies graduated in 1933 and began research on integral equations with Hardy at Cambridge.
      
    • He won the Rayleigh Prize in 1935 for an essay on differential equations of fractional order, and was awarded his doctorate for his thesis The Theory Of Linear Integral Equation which he submitted to the University of Cambridge in 1936.
      
    • Smithies early work was on integral equations and in 1958 his text Integral equations was published by Cambridge University Press in their Cambridge Tracts in Mathematics and Mathematical Physics Series.
      
    • the present work is intended as a successor to Maxime Bocher's tract, "An introduction to the study of integral equations" (University Press, Cambridge, 1909).
      

92. Oleinik biography
    
    • Given Petrovsky's expertise in differential equations, the topology of algebraic curves and surfaces and mathematical physics, it is not difficult to see his influence on the direction that Oleinik's work would take.
      
    • In 1973 she became Head of the Department of Differential Equations at Moscow State.
      
    • The three chapters are: Basic mathematical aspects of the theory of elasticity; Homogenization of the equations of linear elasticity; Composites and perforated materials and Spectral problems in homogenization theory.
      
    • It is self-contained and the reader with background in partial differential equations and continuum mechanics can learn the homogenization techniques developed by Oleinik and her coauthors.
      
    • In 1996 Oleinik published Some asymptotic problems in the theory of partial differential equations.
      
    • Much of the book is devoted to the study of the asymptotic behaviour of solutions to nonlinear elliptic second-order equations.
      
    • Oleinik considers equations satisying Dirichlet boundary conditions and ones which satisfy Neumann boundary conditions.
      
    • Oleinik also studies the homogenization problem for linear elliptic equations in domains with the property that half is perforated and half contains no holes.
      
    • This book is a vast treasury of rigorous mathematical results about the Prandtl systems of partial differential equations in fluid dynamics.
      
    • The Prandtl equations were devised early in this century as a simpler replacement for the Navier-Stokes equations to describe viscous laminar fluid flows near boundaries to which the fluid adheres.
      
    • These replacement equations have a quite different mathematical character than the Navier-Stokes equations, and as one can easily see from this book, the theory goes along very different lines.
      

93. Schmidt biography
    
    • His doctoral dissertation was entitled Entwickelung willkurlicher Funktionen nach Systemen vorgeschriebener and was a work on integral equations.
      
    • Schmidt's main interest was in integral equations and Hilbert space.
      
    • He took various ideas of Hilbert on integral equations and combined these into the concept of a Hilbert space around 1905.
      
    • Hilbert had studied integral equations with symmetric kernel in 1904.
      
    • He showed that in this case the integral equation had real eigenvalues, Hilbert's word, and the solutions corresponding to these eigenvalues he called eigenfunctions.
      
    • Schmidt published a two part paper on integral equations in 1907 in which he reproved Hilbert's results in a simpler fashion, and also with less restrictions.
      
    • In 1908 Schmidt published an important paper on infinitely many equations in infinitely many unknowns, introducing various geometric notations and terms which are still in use for describing spaces of functions and also in inner product spaces.
      

94. Budan de Boislaurent biography
    
    • Budan is considered an amateur mathematician and he is best remembered for his discovery of a rule which gives necessary conditions for a polynomial equation to have n real roots between two given numbers.
      
    • In the early 19th century F D Budan and J B J Fourier presented two different (but equivalent) theorems which enable us to determine the maximum possible number of real roots that an equation has within a given interval.
      
    • Budan's rule was in a memoir sent to the Institute in 1803 but it was not made public until 1807 in Nouvelle methode pour la resolution des equations numerique d'un degre quelconque.
      
    • If an equation in x has n roots between zero and some positive number p, the transformed equation in (x - p) must have at least n fewer variations in sign than the original.
      
    • He quoted Lagrange to show that it would be useful to give the rules for solving numerical equations entirely by means of arithmetic, referring to algebra only if absolutely necessary.
      
    • Accordingly, the chief concern of Burdan's Nouvelle methode was to give the reader a mechanical process for calculating the coefficients of the transformed equation in (x - p).
      
    • Let us note that Charles-Francois Sturm in his famous paper Memoire sur la resolution des equations numeriques published in 1829 completely solved the problem of determining the number of real roots of an equation on a given interval.
      

95. Cherry biography
    
    • thesis Differential Equations Of Dynamics was written under guidance from Henry Baker and Ralph Fowler.
      
    • His first papers On the form of the solution of the equations of dynamics, On Poincare's theorem of 'the non-existence of uniform integrals of dynamical equations', and Note on the employment of angular variables in celestial mechanics were all published in 1924 and Some examples of trajectories defined by differential equations of a generalised dynamical type in the following year.
      
    • He undertook research on ordinary differential equations, particularly those arising from dynamics and celestial mechanics, for four years.
      
    • In 1937 he published Topological Properties of the Solutions of Ordinary Differential Equations and in 1947 he published the first part of Flow of a compressible fluid about a cylinder.
      
    • Since the author uses the solutions of Chaplygin, in the form of an infinite series of hypergeometric functions, of the linear second order partial differential equation in the hodograph variables of the potential function, this series diverges for values of the velocity whose speeds exceed the speed at infinity.
      

96. Bellman biography
    
    • His doctoral dissertation on the stability of differential equations was concerned with the behaviour of the solutions of real differential equations as the independent variable t tends to infinity.
      
    • Results from his dissertation appeared in the book Stability theory of differential equations which he published in 1953.
      
    • He went on to introduce Markovian decision problems in 1957 and in 1958 he published his first paper on stochastic control processes where he introduced what is today called the Bellman equation.
      
    • These include, in addition to those already mentioned: A Survey of the Theory of the Boundedness, Stability, and Asymptotic Behavior of Solutions of Linear and Nonlinear Differential and Difference Equations (1949); A survey of the mathematical theory of time-lag, retarded control, and hereditary processes (1954); Dynamic programming of continuous processes (1954); Dynamic programming (1957); Some aspects of the mathematical theory of control processes (1958); Introduction to matrix analysis (1960); A brief introduction to theta functions (1961); An introduction to inequalities (1961); Adaptive control processes: A guided tour (1961); Inequalities (1961); Applied dynamic programming (1962); Differential-difference equations (1963); Perturbation techniques in mathematics, physics, and engineering (1964); and Dynamic programming and modern control theory (1965).
      
    • After his death in 1984 his books continued to be published such as Partial differential equations (1985), Selective computation (1985), Methods in approximation (1986), and Wave propagation: An invariant imbedding approach (1986).
      

97. Fenyo biography
    
    • The last topic in the book is the theory of non-linear ordinary differential equations, beginning with questions of existence consequences and stability.
      
    • The third volume published in 1980, although still presenting methods for engineers, is more involved with one of Fenyo's main research topics, namely integral equations.
      
    • The second section, which is over a quarter of the book, discusses linear integral equations.
      
    • Amongst the topics covered are Volterra integral equations and their relation with ordinary differential equations, Fredholm equations, self-conjugate and non-self-conjugate integral operators, and the associated eigenvalue theory.
      
    • The third section is on applications of integral equations.
      
    • Other books by Fenyo on integral equations are Integral equations - a book of problems (Hungarian) (1957), and the four volume work (written with H-W Stolle) Theorie und Praxis der linearen Integralgleichungen (1982, 1983, 1983, 1984).
      
    • These three volumes complete the encyclopaedic work (roughly 1700 pages) by Fenyo and Stolle on the theory and application of linear integral equations.
      
    • Their thesis is that the classical theory of linear integral equations produced many ideas for the later development of the theory of linear operators, and in turn functional analysis has helped the further development of integral equations.
      

98. Bendixson biography
    
    • Bendixson also made interesting contributions to algebra when he investigated the classical problem of the algebraic solution of equations.
      
    • Abel had shown that the general equation of degree five could not be solved by radicals, while Galois had developed Galois theory which determined which equations could be solved by radicals.
      
    • Bendixson returned to Abel's original contribution and showed that Abel's methods could be extended to describe precisely which equations could be solved by radicals.
      
    • In examining periodic solutions of differential equations Bendixson used methods based on continued fractions.
      
    • The analysis problem which intrigued Bendixson more than all others was the investigation of integral curves to first order differential equations, in particular he was intrigued by the complicated behaviour of the integral curves in the neighbourhood of singular points.
      

99. Steklov biography
    
    • For his Master's thesis Steklov worked on the equations of a solid body moving in an ideal non-viscous fluid.
      
    • There were four cases to be considered in integrating the equations which arose from this problem, and two of these cases had been solved by Clebsch in 1871.
      
    • Began lecturing at the University on the integration of partial differential equations.
      
    • His lecture course on the integration of partial differential equations was to third year students and the lectures went on until April 1909.
      
    • Finished my lectures on integration of equations.
      
    • In addition to the work for his master's thesis and his doctoral thesis referred to above, he reduced problems to boundary value problems of Dirichlet type where Laplace's equation must be solved on a surface.
      

100. Bers biography
     
     • Here Bers began work on the problem of removability of singularities of non-linear elliptic equations.
       
     • The nonparametric differential equation of minimal surfaces may be considered the most accessible significant example revealing typical qualities of solutions of non-linear partial differential equations.
       
     • The author sets as his goal the development of a function theory for solutions of linear, elliptic, second order partial differential equations in two independent variables (or systems of two first-order equations).
       
     • One of the chief stumbling blocks in such a task is the fact that the notion of derivative is a hereditary property for analytic functions while this is clearly not the case for solutions of general second order elliptic equations.
       

101. Kruskal Martin biography
     
     • An important paper on astronomy was Maximal extension of Schwarzschild's metric (1960) which showed that, using what are now called Kruskal coordinates, certain solutions of the equations of general relativity which are singular at the origin are not singular away from the origin, so allowing the study of black holes.
       
     • Kruskal's later work studied soliton equations, asymptotic analysis, and surreal numbers.
       
     • He was led to asymptotic analysis in his plasma physics studies and from there to solutions of Hamiltonian equations as in Asymptotic theory of Hamiltonian and other systems with all solutions nearly periodic (1962).
       
     • Kruskal's important paper (written jointly with Clifford S Gardner, John M Greene and Robert M Miura) Korteweg-de Vries equation and generalizations.
       
     • Before it, there was no general theory for the exact solution of any important class of nonlinear differential equations.
       
     • For his influence as a leader in nonlinear science for more than two decades as the principal architect of the theory of soliton solutions of nonlinear equations of evolution.
       

102. Cholesky biography
     
     • To solve the condition equations in the method of least squares, Cholesky invented a very ingenious computational procedure which immediately proved extremely useful: it is now know as the Method of Cholesky and we describe it below.
       
     • After his death one of his fellow officers, Commandant Benoit, published Cholesky's method of computing solutions to the normal equations for some least squares data fitting problems in Note sur une methode de resolution des equation normales provenant de l'application de la methode des moindres carres a un systeme d'equations lineaires en nombre inferieure a celui des inconnues.
       
     • Application de la methode a la resolution d'un systeme defini d'equations lineaires (Procede du Commandant Cholesky), published in the Bulletin geodesique in 1924.
       
     • The beauty of the method is that it is trivial to solve equations of the type Mx = b when M is a triangular matrix.
       

103. Harriot biography
     
     • He introduced a simplified notation for algebra and his fundamental research on the theory of equations was far ahead of its time.
       
     • As an example of his abilities to solve equations, even when the roots are negative or imaginary, we reproduce his solution of an equation of degree 4.
       
     • As we have seen from the example above, Harriot did outstanding work on the solution of equations, recognising negative roots and complex roots in a way that makes his solutions look like a present day solution.
       
     • This is a major step forward in understanding which Harriot then carried forward to equations of higher degree.
       
     • History Topics: Quadratic, cubic and quartic equations .
       

104. Tikhonov biography
     
     • His first-class achievements in topology and functional analysis, in the theory of ordinary and partial differential equations, in the mathematical problems of geophysics and electrodynamics, in computational mathematics and in mathematical physics are all widely known.
       
     • He defended his habilitation thesis in 1936 on Functional equations of Volterra type and their applications to mathematical physics.
       
     • The thesis applied an extension of Emile Picard's method of approximating the solution of a differential equation and gave applications to heat conduction, in particular cooling which obeys the law given by Josef Stefan and Boltzmann.
       
     • Thus, his research on the Earth's crust lead to investigations on well-posed Cauchy problems for parabolic equations and to the construction of a method for solving general functional equations of Volterra type.
       
     • However, in 1948 he began to study a new type of problem when he considered the behaviour of the solutions of systems of equations with a small parameter in the term with the highest derivative.
       

105. Du Bois-Reymond biography
     
     • However, he continued to undertake research into applied mathematics and, as a consequence, became more and more involved with the theory of partial differential equations.
       
     • In this work he generalised Monge's idea of the characteristic of a partial differential equation from second order equations to third order equations.
       
     • Du Bois-Reymond's work is almost exclusively on calculus, in particular partial differential equations and functions of a real variable.
       
     • The standard technique to solve partial differential equations used Fourier series but Cauchy, Abel and Dirichlet had all pointed out problems associated with the convergence of the Fourier series of an arbitrary function.
       

106. Levinson biography
     
     • Norman decided to shift his field from gap and density theorems to non-linear differential equations, both ordinary and partial.
       
     • I recall our talking about this decision in 1940, and how difficult is was to move into this new field, and how hard Norman worked over a period of two or three years before he felt that he had enough mastery to obtain substantial results in this field; but this mastery he did achieve, and his outstanding contributions to non-linear differential equations were recognised officially in 1954 when the American Mathematical Society awarded Norman the Bocher Prize.
       
     • This was Theory of ordinary differential equations (written jointly with Earl Coddington) which [Norman Levinson : Selected papers of Norman Levinson (2 Vols.) (Boston, MA, 1998).',2)">2]:- .
       
     • The deep and original ideas of Norman Levinson have had a lasting impact on fields as diverse as differential and integral equations, harmonic, complex and stochastic analysis, and analytic number theory during more than half a century.
       
     • In other topics, Levinson provided the foundation for a rigorous theory of singularly perturbed differential equations.
       
     • He also made fundamental contributions to inverse scattering theory by showing the connection between scattering data and spectral data, thus relating the famous Gelfand-Levitan method to the inverse scattering problem for the Schrodinger equation.
       

107. Cramer biography
     
     • After giving the number of arbitrary constants in an equation of degree n as n2/2 + 3n/2, he deduces that an equation of degree n can be made to pass through n points.
       
     • Taking n = 5 he gives an example of finding the five constants involved in making an equation of degree 2 pass through 5 points.
       
     • This leads to 5 linear equations in 5 unknowns and he refers the reader to an appendix containing Cramer's rule for their solution.
       
     • He states a theorem by Maclaurin which says that an equation of degree n intersects an equation of degree m in nm points.
       

108. Tapia biography
     
     • in 1966 and then, in the following year submitted his thesis A Generalization of Newton's Method with an Application to the Euler-Lagrange Equation which led to the award of a Ph.D.
       
     • During this time he began to publish articles, the first one An application of a Newton-like method to the Euler-Lagrange equation in 1969 based on the work of his doctoral thesis.
       
     • In it Tapia considered the solution of the equation P(x) = 0, where P is a nonlinear mapping between Banach spaces.
       
     • He used Newton-like iterations to solve the generalized Euler-Lagrange equation of the calculus of variations.
       
     • It is also shown that this procedure can be applied to a class of two point boundary value problems containing the Euler-Lagrange equation for simple variational problems and most second order ordinary differential equations.
       

109. Gregory biography
     
     • On the latter topic he had become interested in the problem of solving quintic equations algebraically and made some interesting discoveries on Diophantine problems.
       
     • However, we now summarise these and other contributions in the hope that, despite his reluctance to publish his methods, his remarkable contributions might indeed be more widely understood: Gregory anticipated Newton in discovering both the interpolation formula and the general binomial theorem as early as 1670; he discovered Taylor expansions more than 40 years before Taylor; he solved Kepler's famous problem of how to divide a semicircle by a straight line through a given point of the diameter in a given ratio (his method was to apply Taylor series to the general cycloid); he gives one of the earliest examples of a comparison test for convergence, essentially giving Cauchy's ratio test, together with an understanding of the remainder; he gave a definition of the integral which is essentially as general as that given by Riemann; his understanding of all solutions to a differential equation, including singular solutions, is impressive; he appears to be the first to attempt to prove that π and e are not the solution of algebraic equations; he knew how to express the sum of the nth powers of the roots of an algebraic equation in terms of the coefficients; and a remark in his last letter to Collins suggests that he had begun to realise that algebraic equations of degree greater than four could not be solved by radicals.
       

110. Vandermonde biography
     
     • Vandermonde's four mathematical papers, with their dates of publication by the Academie des Sciences, were Memoire sur la resolution des equations (1771), Remarques sur des problemes de situation (1771), Memoire sur des irrationnelles de differens ordres avec une application au cercle (1772), and Memoire sur l'elimination (1772).
       
     • The first of these four papers presented a formula for the sum of the mth powers of the roots of an equation.
       
     • The paper also shows that if n is a prime less than 10 the equation xn - 1 = 0 can be solved in radicals.
       
     • Vandermonde's real and unrecognised claim to fame was lodged in his first paper, in which he approached the general problem of the solubility of algebraic equations through a study of functions invariant under permutations of the roots of the equation.
       
     • The reason for this strong claim by Muir is that, although mathematicians such as Leibniz had studied determinants earlier than Vandermonde, all earlier work had simply used the determinant as a tool to solve linear equations.
       

111. Schlafli biography
     
     • For a given system of n equations of higher degree with n unknowns, I take a linear equation with undetermined coefficients a, b, c, ..
       
     • The work concluded with an examination of the class equation of third degree curves.
       
     • Other papers which he published investigate a variety of topics such as partial differential equations, the motion of a pendulum, the general quintic equation, elliptic modular functions, orthogonal systems of surfaces, Riemannian geometry, the general cubic surface, multiply periodic functions, and the conformal mapping of a polygon on a half-plane.
       

112. Riccati biography
     
     • In the study of differential equations his methods of lowering the order of an equation and separating variables were important.
       
     • He considered many general classes of differential equations and found methods of solution which were widely adopted.
       
     • He is chiefly known for the Riccati differential equation of which he made elaborate study and gave solutions for certain special cases.
       
     • The equation had already been studied by Jacob Bernoulli, and was discussed by Riccati in a paper of 1724.
       

113. D'Ocagne biography
     
     • Nomography consists in the construction of graduated graphic tables, nomograms, or charts, representing formulas or equations to be solved, the solutions of which were provided by inspection of the tables.
       
     • These papers by d'Ocagne included Nomographie; les Calculs usuels effectues au moyen des abaques (1891); Le Calcul simplifie par les procedes mecaniques et graphiques (1894); Sur la representation monographique des equations du second degre a trois variables (1896); Theorie des equations representables par trois systemes lineaires de points cotes (1897); and Application de la methode nomographique la plus generale, resultant de la superposition de deux plans, aux equations a trois et a quatre variables (1898).
       
     • If one makes a system of geometric elements (points or lines) correspond to each of the variables connected by a certain equation, the elements of each system being numbered in terms of the values of the corresponding variable, and if the relationship between the variables established by the equation may be translated geometrically into terms of a certain relation of position, easy to set up between the corresponding geometric elements, then the set of elements constitutes a chart of the equation considered.
       
     • This is the theory of charts, that is to say the graphical representation of mathematical laws defined by equations in any number of variables, which is understood today under the name Nomography.
       

114. Heun biography
     
     • From 1886 to 1889 he lectured at the University of Munich on topics like: the theory of rational functions and their integrals, the theory of linear differential equations, introduction to the theory of linear substitutions and the general theory of differential equations.
       
     • The Heun equation is a second order linear differential equation of the Fuchsian type with four singular points.
       
     • It generalizes the hypergeometric differential equation which has three singular points, and is used today in mathematical physics, e.g.
       

115. Morawetz biography
     
     • In a series of three significant papers in the late 1950s, Cathleen Morawetz used functional analysis coupled with ingenious new estimates for an equation of mixed type, i.e.
       
     • with both elliptic and hyperbolic regions, to prove a striking new theorem for boundary value problems for partial differential equations.
       
     • During the 1970s she extended this work to examine other solutions to the wave equation.
       
     • She proved many important results relating to the non-linear wave equation.
       
     • for pioneering advances in partial differential equations and wave propagation resulting in applications to aerodynamics, acoustics and optics.
       
     • In addition to her deep contributions to partial differential equations, transonic flow, and other areas of applied mathematics, she provided guidance and inspiration to colleagues and students alike.
       
     • .for her deep and influential work in partial differential equations, most notably in the study of shock waves, transonic flow, scattering theory, and conformally invariant estimates for the wave equation.
       

116. Enskog biography
     
     • Enskog worked on the Maxwell-Boltzmann equations.
       
     • Enskog began to work on this equation for his master's degree at Uppsala and made a remarkable prediction.
       
     • Hilbert published a new approach to the Maxwell- Boltzmann equations in 1912.
       
     • How to extend the Maxwell- Boltzmann equation to include collisions of more than two bodies was not clear.
       
     • Chapman, who was still working on the Maxwell- Boltzmann equations, saw the importance of Enskog's methods and developed them further.
       

117. Riesz Marcel biography
     
     • Marcel Riesz's interests ranged from functional analysis to partial differential equations, mathematical physics, number theory and algebra.
       
     • Riesz broadened his range of interests during the 1930 when he became interested in potential theory and in partial differential equations.
       
     • He was motivated by wave propagation and in particular Dirac's relativistic equation for the electron.
       
     • In 1949, Riesz published a 223 page paper L'integrale de Riemann-Liouville et le probleme de Cauchy in which he introduced a multiple integral of Riemann-Liouville type and showed how important this idea is in the theory of the wave equation.
       
     • In Problems related to characteristic surfaces Riesz extended these ideas to obtain the solution of the wave equation for a very general class of characteristic boundaries.
       

118. Marchenko biography
     
     • Also in the 1950s he studied the asymptotic behaviour of the spectral measure and of the spectral function for the Sturm-Liouville equation.
       
     • He is well known for his original results in the spectral theory of differential equations, including the discovery of new methods for the study of the asymptotic behaviour of spectral functions and the convergence expansions in terms of eigenfunctions.
       
     • He also obtained fundamental results in the theory of inverse problems in spectral analysis for the Sturm-Liouville and more general equations.
       
     • In fact Marchenko later applied his methods to the Schrodinger equation.
       
     • The periodic case of the Korteweg-de Vries equation was solved by Marchenko in 1972.
       

119. Jordan biography
     
     • The second part entitled Sur des periodes des fonctions inverses des integrales des differentielles algebriques was on integrals of the form ∫n u dz where u is a function satisfying an algebraic equation f (u, z) = 0.
       
     • Volumes 1 and 2 contain Jordan's papers on finite groups, Volume 3 contains his papers on linear and multilinear algebra and on the theory of numbers, while Volume 4 contains papers on the topology of polyhedra, differential equations, and mechanics.
       
     • He applied his work on classical groups to determine the structure of the Galois group of equations whose roots were chosen to be associated with certain geometrical configurations.
       
     • His work on group theory done between 1860 and 1870 was written up into a major text Traite des substitutions et des equations algebraique which he published in 1870.
       
     • The publication of Traite des substitutions et des equations algebraique did not mark the end of Jordan's contribution to group theory.
       
     • Generalising a result of Fuchs on linear differential equations, Jordan was led to study the finite subgroups of the general linear group of n × n matrices over the complex numbers.
       
     • Although given Jordan's work on matrices and the fact that the Jordan normal form is named after him, the Gauss-Jordan pivoting elimination method for solving the matrix equation Ax= b is not.
       

120. Borok biography
     
     • Her undergraduate thesis on distribution theory and its applications to the theory of systems of linear partial differential equations was noted as outstanding and published in a top Russian journal.
       
     • In 1954, Valentina graduated from Kiev University and moved (following G E Shilov) to the graduate school at Moscow State University, where she received a PhD in 1957 for a thesis On Systems of Linear Partial Differential Equations with Constant Coefficients.
       
     • Her papers published in 1954-1959 contain a range of "inverse" theorems that allow partial differential equations to be characterized as parabolic or hyperbolic, by certain properties of their solutions.
       
     • In the same period she obtained formulae that made it possible to compute in simple algebraic terms the numerical parameters that determine classes of uniqueness and well-posedness of the Cauchy problem for systems of linear partial differential equations with constant coefficients.
       
     • In the early 1960s Valentina worked on fundamental solutions and stability for partial differential equations well-posed in the sense of Petrovskii.
       
     • Starting in the late 1960s, Valentina began a series of papers that lay the foundations for the theory of local and non-local boundary value problems in infinite layers for systems of partial differential equations.
       
     • In the early 1970s Valentina Borok founded a school on the general theory of partial differential equations in Kharkov.
       
     • The work of Valentina Borok and her school on boundary value problems in layers forms an important chapter in the general theory of partial differential equations.
       
     • Her other important contributions were in the area of difference, difference-differential, and functional-differential equations.
       
     • She also developed and published original lecture notes on a number of other core, as well as more specialized courses, in analysis and partial differential equations.
       

121. Leray biography
     
     • This led to a collaboration between Leray and Schauder and their joint work led to a paper Topologie et equations fonctionelles published in the Annales scientifiques de l'Ecole normale Superieure.
       
     • This 1934 paper on topology and partial differential equations is of major importance:- .
       
     • This degree is then used in an ingenious method to prove the existence of solutions to complicated partial differential equations.
       
     • He then returned to work on analysis, in particular studying differential equations arising from hydrodynamics.
       
     • He studied solutions of the initial value problem for three-dimensional Navier-Stokes equations.
       
     • He studied time dependent hyperbolic partial differential equations and also began to work on the Cauchy problem.
       
     • In particular he published a paper on the Cauchy problem for equations with variable coefficients in 1956.
       
     • He was able to generalise results in the theory of ordinary linear analytic differential equations to obtain similar results for partial differential equations.
       
     • In his hands, energy estimates for partial differential equations became combined with ideas from algebraic topology (such as fixed point theorems) in a highly original combination which cracked open the toughest problems.
       
     • Mathematician of penetration and originality, whose inventions revolutionized partial differential equations and algebraic topology.
       

122. Uhlenbeck Karen biography
     
     • Uhlenbeck is a leading expert on partial differential equations and describes her mathematical interests as follows:- .
       
     • I work on partial differential equations which were originally derived from the need to describe things like electromagnetism, but have undergone a century of change in which they are used in a much more technical fashion to look at the shapes of space.
       
     • I did some very technical work in partial differential equations, made an unsuccessful pass at shock waves, worked in scale invariant variational problems, made a poor stab at three dimensional manifold topology, learned gauge field theory and then some about applications to four dimensional manifolds, and have recently been working n equations with algebraic infinite symmetries.
       
     • She described advances in geometry that have been achieved through the study of systems of nonlinear partial differential equations.
       
     • Among other things, she sketched some aspects of Simon Donaldson's work on the geometry of four-dimensional manifolds, instantons - solutions, that is, of a certain nonlinear system of partial differential equations, the self-dual Yang-Mills equations, which were originally introduced by physicists in the context of quantum field theory.
       
     • She has also served on the editorial boards of many journals; a complete list to date is Journal of Differential Geometry (1979-81), Illinois Journal of Mathematics (1980-86), Communications in Partial Differential Equations (1983- ), Journal of the American Mathematical Society (1986-91), Ergebnisse der Mathematik (1987-90), Journal of Differential Geometry (1988-91), Journal of Mathematical Physics (1989- ), Houston Journal of Mathematics (1991- ), Journal of Knot Theory (1991- ), Calculus of Variations and Partial Differential Equations (1991- ), Communications in Analysis and Geometry (1992- ).
       
     • For her many pioneering contributions to global geometry that resulted in advances in mathematical physics and the theory of partial differential equations.
       
     • Karen Uhlenbeck is a distinguished mathematician of the highest international stature, specialising in differential geometry, non-linear partial differential equations and mathematical physics.
       

123. Hudde biography
     
     • Hudde worked on maxima and minima and the theory of equations.
       
     • He gave an ingenious method to find multiple roots of an equation which is essentially the modern method of finding the highest common factor of a polynomial and its derivative.
       
     • If in an equation two roots are equal and if it be multiplied by any arithmetical progression, i.e.
       
     • the first term by the first term of the progression, the second by the second term of the progression, and so on: I say that the equation found by the sum of these products shall have a root in common with the original equation.
       

124. Maschke biography
     
     • Maschke found working with Klein in his home in the evenings very rewarding and was fascinated with Klein's ideas on using group theory to solve algebraic equations.
       
     • Hermite, Kronecker and Brioschi had, in 1858, discovered how to solve the quintic equation by means of elliptic functions.
       
     • In 1888 Maschke proved that a particular sixth-degree equation could be solved by using hyperelliptic functions and Brioschi showed that any sixth-degree algebraic equation could be reduced to Maschke's equation and therefore solved in the same way.
       

125. Prthudakasvami biography
     
     • Prthudakasvami is best known for his work on solving equations.
       
     • The solution of a first-degree indeterminate equation by a method called kuttaka (or "pulveriser") was given by Aryabhata I.
       
     • Brahmagupta seems to have used a method involving continued fractions to find integer solutions of an indeterminate equation of the type ax + c = by.
       
     • In this commentary Prthudakasvami writes the equation 10x + 8 = x2 + 1 as: .
       
     • The whole equation is therefore .
       

126. Arbogast biography
     
     • The particular mathematical dispute which prompted the question set by the St Petersburg Academy in 1787, however, concerned the arbitrary functions which appeared when a differential equation was integrated.
       
     • d'Alembert claimed that these arbitrary functions were required to be continuous and must always be expressed in terms of algebraic or transcendental equations.
       
     • Euler argued that more general functions could be introduced when differential equations were integrated.
       
     • Do the arbitrary functions introduced when differential equations are integrated belong to any curves or surfaces either algebraic, transcendental, or mechanical, either discontinuous or produced by a simple movement of the hand? Or should they legitimately be applied only to continuous curves susceptible of being expressed by algebraic or transcendental equations? .
       

127. Konigsberger biography
     
     • Much of Konigsberger's work on differential equations was influenced by the function theory developed by his friend Fuchs.
       
     • His work on differential equations was, however, also influenced by the applications which interested Bunsen, Kirchhoff and Helmholtz, with whom he was close friends in Heidelberg.
       
     • His approach to the differential equations of analytic mechanics showed novelty [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]:- .
       
     • Konigsberger was the first to treat not merely one differential equation, but an entire system of such equations in complex variables.
       

128. Ferrari biography
     
     • Ferrari discovered the solution of the quartic equation in 1540 with a quite beautiful argument but it relied on the solution of cubic equations so could not be published before the solution of the cubic had been published.
       
     • Ferrari clearly understood the cubic and quartic equations more thoroughly than his opponent who decided that he would leave Milan that very night and thus leave the contest unresolved, so victory went to Ferrari.
       
     • Quadratic, cubic and quartic equations .
       
     • History Topics: Quadratic, cubic and quartic equations .
       

129. Spencer Tony biography
     
     • A particular strain energy function (Neo-Hookean) is chosen, and the condition for existence of an adjacent equilibrium position is obtained in the form of a transcendental equation, which is solved numerically for two loading conditions.
       
     • After introductory chapters on matrix algebra, vectors and Cartesian tensors, and an analysis of deformation and stress, the author examines the mathematical statements of the laws of conservation of mass, momentum and energy and the formulation of the mechanical constitutive equations for various classes of fluids and solids.
       
     • A procedure has been developed in previous papers for constructing exact solutions of the equations of linear elasticity in a plate (not necessarily thin) of inhomogeneous isotropic linearly elastic material in which the elastic moduli depend in any specified manner on a coordinate normal to the plane of the plate.
       
     • The essential idea is that any solution of the classical equations for a hypothetical thin plate or laminate (which are two-dimensional theories) generates, by straightforward substitutions, a solution of the three-dimensional elasticity equations for the inhomogeneous material.
       

130. Bachelier biography
     
     • ',4)">4] on Brownian Motion, in which Einstein derived the equation (the partial differential heat/diffusion equation of Fourier) governing Brownian motion and made an estimate for the size of molecules, Bachelier had worked out, for his Thesis, the distribution function for what is now known as the Wiener stochastic process (the stochastic process that underlies Brownian Motion) linking it mathematically with the diffusion equation.
       
     • In this course he may have drawn out the similarities between the diffusion of probability (the total probability of one being conserved) and the diffusion equation of Fourier (the total heat-energy being conserved).
       
     • Bachelier's work is remarkable for herein lie the theory of Brownian Motion (one of the most important mathematical discoveries of the 20th century), the connection between random walks and diffusion, diffusion of probability, curves lacking tangents (non-differentiable functions), the distribution of the Wiener process and of the maximum value attained in a given time by a Wiener process, the reflection principle, the pricing of options including barrier options, the Chapman-Kolmogorov equations in the continuous case, .
       

131. Seki biography
     
     • Ten years later Leibniz, independently, used determinants to solve simultaneous equations although Seki's version was the more general.
       
     • He studied equations treating both positive and negative roots but had no concept of complex numbers.
       
     • In 1685, he solved the cubic equation 30 + 14x - 5x2 - x3 = 0 using the same method as Horner a hundred years later.
       
     • He discovered the Newton or Newton-Raphson method for solving equations and also had a version of the Newton interpolation formula.
       
     • Among other problems studied by Seki were Diophantine equations.
       

132. Mason biography
     
     • Mason's mathematical research interests lay in differential equations, the calculus of variations and electromagnetic theory.
       
     • He developed the relation between the algebra of matrices and integral equations as well as studying boundary value problems.
       
     • He published seven papers in the Transactions of the American Mathematical Society between 1904 and 1910: Green's theorem and Green's functions for certain systems of differential equations (1904), The doubly periodic solutions of Poisson's equation in two independent variables (1905), A problem of the calculus of variations in which the integrand is discontinuous (1906), On the boundary value problems of linear ordinary differential equations of second order (1906), The expansion of a function in terms of normal functions (1907); The properties of curves in space which minimize a definite integral (1908) and Fields of extremals in space (1910).
       

133. Poisson biography
     
     • In his final year of study he wrote a paper on the theory of equations and Bezout's theorem, and this was of such quality that he was allowed to graduate in 1800 without taking the final examination.
       
     • During this period Poisson studied problems relating to ordinary differential equations and partial differential equations.
       
     • Poisson's name is attached to a wide variety of ideas, for example:- Poisson's integral, Poisson's equation in potential theory, Poisson brackets in differential equations, Poisson's ratio in elasticity, and Poisson's constant in electricity.
       

134. Pfaff biography
     
     • It investigates the use of some functional equations in order to calculate the differentials of logarithmic and trigonometrical functions as well as the binomial expansion and Taylor formula.
       
     • Pfaff did important work in analysis working on partial differential equations, special functions and the theory of series.
       
     • In the 1815 paper, which Pfaff submitted to the Berlin Academy on 11 May, he presented a transformation of a first-order partial differential equation into a differential system.
       
     • This theory of equations in total differentials is undoubtedly Pfaff's most significant contribution.
       
     • constituted the starting point of a basic theory of integration of partial differential equations which, through the work of Jacobi, Lie, and others, has developed into a modern Cartan calculus of extreme differential forms.
       

135. Silva biography
     
     • He published his first paper in Portugaliae Mathematica in 1940, this being On the numerical resolution of algebraic equations (Portuguese).
       
     • In the following year he published Problems concerning rational functions of the roots of an algebraic equation (Portuguese) in the same journal.
       
     • Determine the equation whose roots are all sums of p roots; if there is a factor in the field, then this equation has a root in the field.
       
     • Assuming that it is possible to find all roots of an equation in a field, the preceding section furnishes a method of finding the coefficients of the factor, if it exists.
       

136. Korkin biography
     
     • He submitted On Determining Arbitrary Functions in Integrals of Linear Partial Differential Equations which he defended on 11 December 1860.
       
     • On the Paris visit he was particularly interested in Bertrand's lectures on partial differential equations and in Germany Kummer's lectures on quadratic forms fascinated him.
       
     • He defended his thesis On systems of first order partial differential equations and some questions on mechanics towards the end of 1867.
       
     • One of Korkin's major contributions was to the development of partial differential equations.
       
     • Initially Korkin was unimpressed with Zolotarev's investigation of an indeterminate equation of degree three which he presented in his Master' thesis.
       

137. Dixon biography
     
     • Dixon's main area of research was in differential equations and he did early work on Fredholm integrals independently of Fredholm.
       
     • He worked both on ordinary differential equations and on partial differential equations studying abelian integrals, automorphic functions, and functional equations.
       
     • A spectacular generalisation of Dixon's beautiful identity is given by equation .31 on page 171 of [R L Graham, D E Knuth and O Patashnik, Concrete Mathematics (1989)] which must surely be the non plus ultra of the species.
       

138. Stephansen biography
     
     • During the time that she was teaching Stephansen was working on her doctoral dissertation on partial differential equations.
       
     • Euler, d'Alembert and Lagrange had studied which second order partial differential equations which could be reduced to first order and this had been generalised by the Norwegian mathematician Alf Guldberg who, in 1900, had described all those third order partial differential equations which could be reduced to second order equations.
       
     • In her thesis, Stephansen generalised Guldberg's work and succeeded in describing all those fourth order partial differential equations which could be reduced to equations of the third order.
       
     • Stephansen published another paper in 1903 on differential equations, the idea for which came out of Hilbert's course of lectures that she attended.
       
     • She continued to undertake mathematical research and wrote two further papers, this time on difference equations, which were published in 1905 and 1906.
       

139. Pell biography
     
     • Pell's equation y2 = ax2 + 1, where a is a non-square integer, was first studied by Brahmagupta and Bhaskara II.
       
     • It is often said that Euler mistakenly attributed Brouncker's work on this equation to Pell.
       
     • However the equation appears in a book by Rahn which was certainly written with Pell's help: some say entirely written by Pell.
       
     • Perhaps Euler knew what he was doing in naming the equation.
       
     • Pell's equation .
       
     • Pell's equation .
       
     • History Topics: Pell's equation .
       
     • Math Forum (Pell's equation) .
       

140. Golub biography
     
     • in 1959 for his thesis The Use of Chebyshev Matrix Polynomials in the Iterative Solution of Linear Equations Compared to the Method of Successive Overrelaxation which developed ideas in a paper by von Neumann.
       
     • In 1992 Golub, jointly with James M Ortega, published Scientific computing and differential equations.
       
     • A large part of scientific computing is concerned with the solution of differential equations, and thus differential equations are an appropriate focus for an introduction to scientific computing.
       
     • The need to solve differential equations was one of the original and primary motivations for the development of both analog and digital computers, and the numerical solution of such problems still requires a substantial fraction of all available computing time.
       
     • It is our goal in this book to introduce numerical methods for both ordinary and partial differential equations with concentration on ordinary differential equations, especially boundary value problems.
       
     • Although there are many existing packages for such problems, or at least for the main subproblems such as the solution of linear systems of equations, we believe that it is important for users of such packages to understand the underlying principles of the numerical methods.
       

141. Bocher biography
     
     • At Gottingen he also attended lecture courses by Klein on the potential function, on partial differential equations of mathematical physics and on non-euclidean geometry.
       
     • Bocher published around 100 papers on differential equations, series, and algebra.
       
     • Yet another exceptional service was rendered by his "Introduction to the Study of Integral Equations" ..
       
     • Special attention should be drawn also to his little known pamphlet on regular point of linear differential equations of the second order used for a number of years in connection with one of his courses of lectures.
       
     • When An introduction to the study of integral equations was reprinted in 1971 a reviewer wrote:- .
       
     • His final book was Lecons sur les methodes de Sturm dans la theorie des equations differentielles lineaires et leurs developpements modernes (1917) which was a record of lectures he gave in Paris in 1913-14 when he was Harvard Exchange Professor at the University of Paris.
       
     • He gave six lectures on Linear differential equations and their applications.
       
     • He was honoured with election to the National Academy of Sciences (United States) in 1909 and he served as president of the American Mathematical Society during 1909-1910 delivering his presidential address in Chicago on The published and unpublished works of Charles Sturm on algebraic and differential equations.
       
     • M Bocher: Integral equations .
       

142. Tschirnhaus biography
     
     • He showed Collins and Wallis his methods for solving equations, but these turned out to be special cases of known results.
       
     • In it he discussed several mathematical questions including the solution of higher equations.
       
     • In his letter Leibniz also criticises Tschirnhaus's solution of algebraic equations.
       
     • Tschirnhaus worked on the solution of equations and the study of curves.
       
     • He discovered a transformation which, when applied to an equation of degree n, gave an equation of degree n with no term in xn-1 and xn-2.
       
     • We have indicated above that he had already discussed his methods for solving equations with Leibniz who had pointed out difficulties.
       
     • Nevertheless Tschirnhaus published his transformation in Acta Eruditorum in 1683 and, in this article, showed how it could be used to solve the general cubic equation.
       
     • However, his belief that the method would allow an equation of any degree to be solved is false as had already been pointed out to him by Leibniz.
       
     • History Topics: Quadratic, cubic and quartic equations .
       

143. Klein biography
     
     • He owed some of his greatest successes to his development of Riemann's ideas and to the intimate alliance he forged between the later and the conception of invariant theory, of number theory and algebra, of group theory, and of multidimensional geometry and the theory of differential equations, especially in his own fields, elliptic modular functions and automorphic functions.
       
     • He showed it had equation x3y + y3z + z3x = 0 as a curve in projective space and its group of symmetries was PSL(2,7) of order 168.
       
     • Klein considered equations of degree greater than 4 and was particularly interested in using transcendental methods to solve the general equation of the fifth degree.
       

144. Lipschitz biography
     
     • He carried out many important and fruitful investigations in number theory, in the theory of Bessel functions and of Fourier series, in ordinary and partial differential equations, and in analytical mechanics and potential theory.
       
     • Lipschitz's work on the Hamilton-Jacobi method for integrating the equations of motion of a general dynamical system led to important applications in celestial mechanics.
       
     • Lipschitz is remembered for the 'Lipschitz condition', an inequality that guarantees a unique solution to the differential equation y' = f (x, y).
       
     • Peano gave an existence theorem for this differential equation, giving conditions which guarantee at least one solution.
       

145. Bernoulli Daniel biography
     
     • The third part of Mathematical exercises was on the Riccati differential equation while the final part was on a geometry question concerning figures bounded by two arcs of a circle.
       
     • He was able to give the basic laws for the theory of gases and gave, although not in full detail, the equation of state discovered by Van der Waals a century later.
       
     • Daniel worked on mechanics and again used the principle of conservation of energy which gave an integral of Newton's basic equations.
       
     • It is especially unfortunate that he could not follow the rapid growth of mathematics that began with the introduction of partial differential equations into mathematical physics.
       

146. Crank biography
     
     • His main work was on the numerical solution of partial differential equations and, in particular, the solution of heat-conduction problems.
       
     • John Crank is best known for his joint work with Phyllis Nicolson on the heat equation, where a continuous solution u(x, t) is required which satisfies the second order partial differential equation .
       
     • Crank and Nicolson's method, which is numerically stable, requires the solution of a very simple system of linear equations (a tridiagonal system) at each time level.
       

147. Kochina biography
     
     • An application of the theory of linear differential equations to some problems of ground-water motion published in 1940 is quite typical of many of her papers.
       
     • For example in 1948 she studied numerical solutions of a partial differential equation in On a nonlinear partial differential equation arising in the theory of filtration.
       
     • The papers in this book are divided into eight sections: Kinematics of atmospheric motions; Hydrodynamics; Applications of the analytical theory of linear differential equations in filtration theory; Steady flow in the presence of porous media; Unsteady motion of groundwater; Problems on oil filtration; Gas filtration through coal layers; and Filtration of liquids through porous media.
       

148. Korteweg biography
     
     • On this topic he published Sur la forme que prennent les equations du mouvement des fluids si l'on tient compte des forces capillaires causes par les variations de densite (On the form the equations of motions of fluids assume if account is taken of the capillary forces caused by density variations) in 1901.
       
     • He is remembered in particular for the Korteweg - de Vries equation on solitary waves, a courageous topic to attack since many mathematicians, including Stokes, were convinced such waves could not exist.
       
     • They found explicit, closed-form, travelling-wave solutions to the Korteweg - de Vries equation that decay rapidly.
       
     • We can mention, again showing Korteweg's versatility, his pure mathematics work on algebraic equations in papers such as Sur un theoreme remarquable, qui se rapporte a la theorie des equations algebriques a parametres reels, dont toutes les racines restent constamment reelles (1900).
       

149. Bevan-Baker biography
     
     • Huygens' geometrical construction, with its restriction that only one sheet of the envelope of the spherical wavelets is to be considered, is first justified in Chapter I by Poisson's analytical solution of the equation of wave-motions.
       
     • The analogue of Kirchhoff's formula, due to Volterra, is derived and an interesting account is given of a method, devised by Marcel Riesz and based on the theory of fractional integration, which provides a powerful method of solving initial value problems for equations like the wave equation.
       
     • A second edition of the book, which differed from the first by the addition of a new chapter on the application of the theory of integral equations to problems of diffraction theory by a plane screen, was published in 1950.
       

150. Henrici Peter biography
     
     • His next contribution Bergmans Integraloperator erster Art und Riemannsche Funktion (1952) is an elegant study of the representation of solutions of an elliptic partial differential equation in terms of analytic functions.
       
     • His first book Discrete variable methods in ordinary differential equations, published by John Wiley & Sons in 1962, quickly won international acclaim and became a classic standard text on the topic.
       
     • This book contains a comprehensive and up-to-date treatment of methods for the numerical integration of ordinary differential equations, especially those associated with initial-value problems.
       
     • There is no doubt that this book is a valuable contribution to numerical analysis, and it will certainly have an important influence on future developments in the numerical integration of ordinary differential equations.
       

151. Spence David biography
     
     • Obtaining equations under special conditions, Spence found numerical results for lift, pitching moment, and jet shape, which he compared with experimental results obtained from a wind tunnel.
       
     • By similarity considerations, the displacements are expressed in terms of the solution of a pair of nonlinear ordinary differential equations satisfying two-point boundary conditions.
       
     • We consider boundary value problems for the biharmonic equation in the open rectangle x > 0, -1 < y < 1, with homogeneous boundary conditions on the free edges y = ±1, and data on the end x = 0 of a type arising both in elasticity and in Stokes flow of a viscous fluid, in which either two stresses or two displacements are prescribed.
       
     • For such 'noncanonical' data, coefficients in the eigenfunction expansion can be found only from the solution of infinite sets of linear equations, for which a variety of methods of formulation have been proposed.
       

152. Stampacchia biography
     
     • For three years he produced outstanding examination results in a wide range of courses such as Tutorial Sessions in Analysis and in Geometry, Calculus of Variations, Theory of Functions, and Ordinary Differential Equations.
       
     • His thesis was concerned with an adaptation of an approximation procedure for Volterra integral equations due to Tonelli to boundary value problems for systems of ordinary differential equations.
       
     • From the time Stampacchia took up his appointment in Naples, his research output was impressive consisting mainly of papers on differential equations and the calculus of variations.
       
     • The years that Stampacchia spent in Pisa and Naples characterize the formation of his personality as an analyst: he was a passionate specialist in calculus of variations and in the theory of partial differential equations, a practitioner and an inspirer of research works of considerable depth and originality of thought.
       
     • His 326 page text Equations elliptiques du second ordre a coefficients discontinus was published in 1966, then in 1967 he was elected President of the Italian Mathematical Union (Unione Matematica Italiana).
       
     • On the one hand, variational inequalities have stimulated new and deep results dealing with nonlinear partial differential equations.
       

153. Hertz Heinrich biography
     
     • There were several new factors in the equation which affected the issue such as, on the negative side, his unhappiness with the working environment of engineering firms, and on the positive side, his enjoyment of the mathematics he had learnt as part of his engineering studies.
       
     • However, it may have been a wise decision to delay beginning the work as S D'Agostino [Centaurus 36 (1) (1993), 46-82.',11)">11] suggests that Hertz's derivation of Maxwell's equations in 1884 formed an important part of the structural background to his studies on the propagation of electric waves which he now carried out.
       
     • He searched for a mechanical basis for electrodynamics starting from Maxwell's equations.
       
     • Maxwell's theory is Maxwell's system of equations.
       

154. Schur biography
     
     • Third, he handled algebraic equations, sometimes proceeding to the evaluation of roots, and sometimes treating the so-called equation without affect, that is, with symmetric Galois groups.
       
     • He was also the first to give examples of equations with alternating Galois groups.
       
     • Sixth, in integral equations; .
       

155. Feller biography
     
     • He transformed the relation between Markov processes and partial differential equations.
       
     • Other papers written by Feller while still at Brown University include: On the time distribution of so-called random events (1940), On the integral equation of renewal theory (1941), On A C Aitken's method of interpolation (1943), The fundamental limit theorems in probability (1945) and Note on the law of large numbers and "fair" games (1945).
       
     • outlines some new results and open problems concerning diffusion theory where we find an intimate interplay between differential equations and measure theory in function space.
       
     • It was also the first mathematics course I took at Princeton (a course in sophomore differential equations).
       

156. Hilbert biography
     
     • Hilbert's work in integral equations in about 1909 led directly to 20th-century research in functional analysis (the branch of mathematics in which functions are studied collectively).
       
     • Making use of his results on integral equations, Hilbert contributed to the development of mathematical physics by his important memoirs on kinetic gas theory and the theory of radiations.
       
     • Many have claimed that in 1915 Hilbert discovered the correct field equations for general relativity before Einstein but never claimed priority.
       
     • In this paper the authors show convincingly that Hilbert submitted his article on 20 November 1915, five days before Einstein submitted his article containing the correct field equations.
       
     • Einstein's article appeared on 2 December 1915 but the proofs of Hilbert's paper (dated 6 December 1915) do not contain the field equations.
       
     • In the printed version of his paper, Hilbert added a reference to Einstein's conclusive paper and a concession to the latter's priority: "The differential equations of gravitation that result are, as it seems to me, in agreement with the magnificent theory of general relativity established by Einstein in his later papers".
       
     • Hilbert contributed to many branches of mathematics, including invariants, algebraic number fields, functional analysis, integral equations, mathematical physics, and the calculus of variations.
       

157. Siegel biography
     
     • Approximation of algebraic numbers by rationals and applications thereof to Diophantine equations.
       
     • These include his improvement of Thue's theorem, described above, given in his 1920 dissertation, and its application to certain polynomial Diophantine equations in two unknowns, proving an affine curve of genus at least 1 over a number field has only a finite number of integral points in 1929.
       
     • He had earlier than this in 1922, written papers on the functional equation of Dedekind's zeta functions of algebraic number fields and in 1921/23 made contributions to additive questions such as Waring type problems for algebraic number fields.
       
     • He examined Birkhoff's work on perturbation theory solutions for analytical Hamiltonian differential equations near an equilibrium point using formal power series.
       

158. Carmeli biography
     
     • Among his publications at this time are The motion of a particle of finite mass in an external gravitational field (1964), Has the geodesic postulate any significance for a finite mass? (1964), Semigenerally covariant equations of motion.
       
     • Derivation (1965), Semigenerally covariant equations of motion.
       
     • The significance of the "tail" and the relation to other equations of motion (1965), Motion of a charge in a gravitational field (1965), The equations of motion of slowly moving particles in the general theory of relativity (1965), and Equations of motion without infinite self-action terms in general relativity (1965).
       
     • During his time in this post he published papers such as Group analysis of Maxwell's equations (1969), Infinite-dimensional representations of the Lorentz group (1970), and SL(2, C) symmetry of the gravitational field dynamical variables (1970).
       
     • students, and E Leibowitz) Gauge fields : Classification and equations of motion (1989):- .
       

159. Leimanis biography
     
     • Immediately he was on his travels again, this time going to Paris where he spent a year at the Henri Poincare Institute undertaking research on differential equations and celestial mechanics.
       
     • Leimanis continued to publish and, when he was approaching 80, the paper On integration of the differential equation of central motion appeared.
       
     • Assuming that the force acting on a particle is of the form f(r)g(q), the theory of infinitesimal transformations is applied to determine the forms of f(r) and g(q) for which the differential equation of central motion is integrable by quadratures or reducible to a first-order differential equation.
       

160. Gelfond biography
     
     • Gelfond developed basic techniques in the study of transcendental numbers, that is numbers that are not the solution of an algebraic equation with rational coefficients.
       
     • He also contributed to the study of differential and integral equations and to the history of mathematics.
       
     • This book is very much in the spirit of the modern Russian school concerned with the so-called constructive theory of functions, approximative methods for the solution of differential equations, and so forth.
       
     • Also in 1952 Gelfond published the low level Solving equations in integers which was translated into English in 1960.
       

161. Peschl biography
     
     • this work lies on the common boundary between differential geometry, function theory (of one and several variables) and partial differential equations.
       
     • in the Schwarz lemma; the essence is the relation between the standard hyperbolic metric in the unit disc and the Beltrami equation, to which particular differential invariants are associated.
       
     • This situation has been generalized to different types of metrics, to equations of higher order and to more than one variable.
       
     • Partielle Differentialgleichungen erster Ordnung (1973) provides an elementary introduction to first order partial differential equations while Differential-geometrie (1973) provides a clear, elementary and concisely presented introduction to local differential geometry in Euclidean and Riemannian spaces.
       

162. Schlesinger biography
     
     • He then studied mathematics and physics at the universities of Heidelberg and Berlin between 1896 and 1887, and he received a doctorate from the University of Berlin in 1887 for a thesis on differential equations entitled: Uber lineare homogene Differentialgleichungen vierter Ordnung, zwischen deren Integralen homogene Relationen hoheren als ersten Grades bestehen.
       
     • In this paper Schlesinger formulated the problem of isomonodromy deformations for a certain matrix Fuchsian equation.
       
     • Prove the existence of linear differential equations having a prescribed monodromic group.
       
     • The paper introduces what today are known as the Schlesinger transformations and Schlesinger equations which have an important role in differential geometry.
       

163. Day biography
     
     • Within nine months he had completed his Master's degree and submitted a Master's thesis On modular equational classes.
       
     • Some of Day's early papers are: Injectives in non-distributive equational classes of lattices are trivial (1970), A note on the congruence extension property (1971), Injectivity in equational classes of algebras (1972), Splitting algebras and a weak notion of projectivity (1973), Filter monads, continuous lattices and closure systems (1975), and Splitting lattices generate all lattices (1975).
       
     • I came to work in the morning, wrote the equations down, and tried to manipulate them.
       

164. Navier biography
     
     • Navier is remembered today, not as the famous builder of bridges for which he was known in his own day, but rather for the Navier-Stokes equations of fluid dynamics.
       
     • He gave the well known Navier-Stokes equations for an incompressible fluid in 1821 while in 1822 he gave equations for viscous fluids.
       
     • We should note, however, that Navier derived the Navier-Stokes equations despite not fully understanding the physics of the situation which he was modelling.
       
     • He did not understand about shear stress in a fluid, but rather he based his work on modifying Euler's equations to take into account forces between the molecules in the fluid.
       
     • The irony is that although Navier had no conception of shear stress and did not set out to obtain equations that would describe motion involving friction, he nevertheless arrived at the proper form for such equations.
       

165. Luzin biography
     
     • Many of these mathematicians turned to other topics such as topology, differential equations, and functions of a complex variable.
       
     • In 1931 Luzin himself turned to a new area when he began to study differential equations and their application to geometry and to control theory.
       
     • Finikov had derived differential equations that determine all principal on a given surface, and Byushgens had obtained differential equations that determine surfaces which have a given linear element and admit a bending on a principal base.
       
     • However, the question of solubility of these equations, in general, remained unclear.
       
     • no example was found in which the equations ..
       
     • up to 1938, when Luzin, by means of a subtle analysis of these equations, established that the existence of a principal base is rather rare.
       

166. Saunderson biography
     
     • The chapters on algebra introduce the idea of an equation and how real life problems can be reduced to equations.
       
     • The reader is shown how to solve quadratic equations, there other topics such as magic squares are studied.
       
     • The final book presents the solution of cubic and quartic equations.
       

167. Zhang Qiujian biography
     
     • There are problems on extracting square and cube roots, problems on finding the solution to quadratic equations, problems on finding the sum of an arithmetic progression, and on solving systems of linear equations.
       
     • Zhang gives the solution by solving a quadratic equation, but his formulae are not particularly accurate.
       
     • In Chapter 3 problems which involve solving systems of equations occur.
       

168. Boruvka biography
     
     • He discussed these matters with Frantisek Vycichlo, a Prague mathematician, and their feeling was that differential equations would be a good direction to take his research team.
       
     • Boruvka had already written a paper on differential equations in 1934, but now he began to direct the research of Masaryk University towards that topic.
       
     • We discussed the matter thoroughly and arrived at the conclusion that it was essential to start pursuing the theory of differential equations which is immensely important as far as applications are concerned and which was much neglected before the war and in essence it was not at all developed.
       
     • In 1946 Boruvka became an ordinary professor at Masaryk University and in the following year he set up a Differential Equations Seminar.
       
     • The main aim of the seminar was to study global properties of linear differential equations of the nth order.
       
     • Boruvka's publications on this topic include Sur les integrales oscilatoires des equations differentielles lineaires du second ordre (1953), Remark on the use of Weyr's theory of matrices for the integration of systems of linear differential equations with constant coefficients (Czech) (1954), Uber eine Verallgemeinerung der Eindeutigkeitssatze fur Integrale der Differentialgleichung y' = f (x, y) (1956), and Sur la transformation des integrales des equations differentielles lineaires ordinaires du second ordre (1956).
       

169. Brouncker biography
     
     • Brouncker gave a method of solving the diophantine equation .
       
     • See our article Pell's equation for more details.
       
     • It is believed that Euler made an error in naming the equation 'Pell's equation', and that he was intending to acknowledge the outstanding contribution made by Brouncker.
       
     • It is interesting to think that if Euler had not made this error then Brouncker, instead of being relatively unknown as a mathematician, would be universally known through 'Brouncker's equation'.
       
     • Pell's equation .
       
     • History Topics: Pell's equation .
       

170. Petersen biography
     
     • The interest he had shown in ruler and compass constructions when he was at school had continued to influence his research topic and his doctoral thesis was entitled On equations which can be solved by square roots, with application to the solution of problems by ruler and compass.
       
     • If the equation of degree 2n can be solved by square roots, one of the roots can be expressed by n such different square roots, where each can appear several times.
       
     • His research was on a wide variety of topics from algebra and number theory to geometry, analysis, differential equations and mechanics.
       
     • He published The theory of algebraic equations in 1877 which was written in a concise style, treating as many topics as possible without using Galois theory.
       

171. Guo Shoujing biography
     
     • We should now look at the rather remarkable work which Guo did on spherical trigonometry and solving equations.
       
     • To solve this equation Guo used a numerical method similar to Horner's method.
       
     • The equation has two real roots, the smaller being the solution to the problem while the other, being numerically larger than the length of the arc, was rightly discarded by Guo.
       
     • Two of the coefficients of the equation, namely the constant term and the coefficient of x2, involve the length a of the arc, so require a value to be chosen for π.
       

172. Levi-Civita biography
     
     • the main mathematical and physical questions discussed by Einstein and Levi-Civita in their 1915 - 1917 correspondence: the variational formulation of the gravitational field equations and their covariance properties, and the definition of the gravitational energy and the existence of gravitational waves.
       
     • Its major achievements are two: a derivation of the equations of motion of n point masses, free from the subtle errors besetting most of the standard treatments; and a careful discussion of the possible contributions, in the Einsteinian approximation, of the finite size and internal constitution of the bodies involved.
       
     • He also wrote on the theory of systems of ordinary and partial differential equations.
       
     • In [Italian mathematics between the two world wars (Pitagora, Bologna, 1987), 125-141.',18)">18] the authors argue that Levi-Civita was interested in the theory of stability and qualitative analysis of ordinary differential equations for three reasons: his interest in geometry and geometric models; his interest in classical mechanics and celestial mechanics, in particular, the three-body problem; and his interest in stability of movement in the domain of analytic mechanics.
       
     • Their results include the conception of the localized induction approximation for the induced velocity of thin vortex filaments, the derivation of the intrinsic equations of motion, the asymptotic potential theory applied to vortex tubes, the derivation of stationary solutions in the shape of helical vortices and loop-generated vortex configurations, and the stability analysis of circular vortex filaments.
       
     • In 1933 Levi-Civita contributed to Dirac's equations of quantum theory.
       

173. Lie biography
     
     • Although not on the permanent staff, Sylow taught a course, substituting for Broch, in which he explained Abel's and Galois' work on algebraic equations.
       
     • Lie had started examining partial differential equations, hoping that he could find a theory which was analogous to the Galois theory of equations.
       
     • the theory of differential equations is the most important discipline in modern mathematics.
       
     • He examined his contact transformations considering how they affected a process due to Jacobi of generating further solutions of differential equations from a given one.
       
     • It was during the winter of 1873-74 that Lie began to develop systematically what became his theory of continuous transformation groups, later called Lie groups leaving behind his original intention of examining partial differential equations.
       

174. Infeld biography
     
     • During this time he wrote six joint papers with Max Born - examples of their papers in the Proceedings of the Royal Society are Foundations of the new Field Theory (1934) and On the Quantization of the New Field Equations (1935).
       
     • For example he published (jointly with A Einstein and B Hoffmann) The gravitational equations and the problem of motion (1938) and a second part, jointly with Einstein, two years later.
       
     • the gravitational field equations, satisfied in regions free of matter, imply the vanishing of certain surface integrals taken over 2-dimensional surfaces enclosing spatial regions containing the particles responsible for the field.
       
     • In obtaining this result the authors made use of a normalizing condition restricting the choice of coordinates; with it they were able to show that the vanishing of the surface integrals led to equations of motion for the particles.
       
     • Other papers he published around this time include (with P R Wallace, one of his doctoral students) The equations of motion in electrodynamics (1940), On the Theory of Brownian Motion (1940), On a new treatment of some eigenvalue problems (1941), A generalization of the factorization method for solving eigenvalue problems (1942), and Clocks, rigid rods and relativity theory (1943).
       
     • The problem is a substantial one, because the field equations are non-linear, and because they are connected by differential identities.
       

175. Arnold biography
     
     • Arnold has also made innumerable and fundamental contributions to the theory of differential equations, symplectic geometry, real algebraic geometry, the calculus of variations, hydrodynamics, and magneto- hydrodynamics.
       
     • The areas are Dynamical Systems, Differential Equations, Hydrodynamics, Magnetohydrodynamics, Classical and Celestial Mechanics, Geometry, Topology, Algebraic Geometry, Symplectic Geometry, and Singularity Theory.
       
     • He published Problemes ergodiques de la mecanique classique (with A Avez) (1967), Ordinary differential equations (Russian) (1971), Mathematical methods of classical mechanics (Russian) (1974), Supplementary chapters to the theory of ordinary differential equations (Russian) (1978), Singularity theory (1981), Singularities of differentiable mappings (Russian) (with A N Varchenko and S M Gusein-Zade) (1982), Catastrophe theory (1984), Huygens and Barrow, Newton and Hooke (Russian) (1989), Contact geometry and wave propagation (1989), Singularities of caustics and wave fronts (1990), The theory of singularities and its applications (1991), Topological invariants of plane curves and caustics (1994), Lectures on partial differential equations (Russian) (1997), Topological methods in hydrodynamics (with B A Khesin) (1998), and Arnold problems (Russian) (2000).
       
     • for his deep and influential work in a multitude of areas of mathematics, including dynamical systems, differential equations, and singularity theory.
       
     • In classical hydrodynamics the basic equations of an ideal fluid were derived by Euler in 1757 and major steps towards understanding them were taken by Helmholtz in 1858, and Kelvin in 1869.
       

176. Hopf Eberhard biography
     
     • Another important contribution from this period was the Wiener-Hopf equations, which he developed in collaboration with Norbert Wiener from the Massachusetts Institute of Technology.
       
     • By 1960, a discrete version of these equations was being extensively used in electrical engineering and geophysics, their use continuing until the present day.
       
     • Other work which he undertook during this period was on stellar atmospheres and on elliptic partial differential equations.
       
     • An example of this was the dropping of Hopf's name from the discrete version of the so called Wiener-Hopf equations, which are currently referred to as "Wiener filter".
       
     • His interests and principal achievements were in the fields of partial and ordinarydifferential equations, calculus of variations, ergodic theory, topological dynamics, integral equations, differential geometry, complex function theory and functional analysis.
       

177. Atkinson biography
     
     • A new phase of his work began when he began to study eigenfunction expansions both for difference equations and differential equations.
       
     • We shall present the theory of certain recurrence relations in the spirit of the theory of boundary problems for differential equations.
       
     • Second, we shall present the theory of boundary problems for certain ordinary differential equations, emphasizing cases in which the coefficients may be discontinuous, or may have singularities of delta function type.
       
     • Finally, we give some account of theories which unify the topics of differential and difference equations, relying mainly on the method of replacement by integral equations.
       

178. Lemaitre biography
     
     • Einstein was at the conference and he spoke to Lemaitre in Brussels telling him that the ideas in his 1927 paper had been presented by Friedmann in 1922, but he also said that although he thought Lemaitre's solutions of the equations of general relativity were mathematically correct, they presented a solution which was not feasible physically.
       
     • In 1942 he published L'iteration rationnelle in which he discussed Gauss's method of successive approximations applied to a system of two equations in two unknowns to determine the orbit of a planet from three observations.
       
     • Lemaitre then applied these ideas to accelerate the orthodox process of iteration, taking the Picard iterative solution of first order differential equations as an example.
       
     • He applied the same techniques in another paper published in the same year, namely Integration d'une equation differentielle par iteration rationnelle.
       
     • In Sur un cas limite du probleme de Stormer (1945) he studied trajectories of an electron in the neighborhood of lines of force of a magnetic dipole field, then returned to his study of numerical solutions to first order differential equations in Interpolation dans la methode de Runge-Kutta (1947).
       
     • Is it possible to account for the existence of more or less permanent concentrations of galaxies in which no single galaxy remains long in the same place? The two-fold purpose of the paper is to delineate the underlying mechanical model and to write down the fundamental equations of the problem.
       
     • It is shown how these equations can be applied toward the solution of the well-known problem of uniform distribution in a homogeneous, expanding universe.
       

179. Kolchin biography
     
     • Other papers around this time were Algebraic matric groups (1946) and The Picard-Vessiot theory of homogeneous linear ordinary differential equations (1946).
       
     • His deep and abiding interest has always been in the application of the powerful and clarifying techniques of algebra to problems in the theory of differential equations.
       
     • Following the tradition set by Joseph Fels Ritt (1893 - 1951), the founding father of differential algebra, his desire has been to remove the algebraic aspects of differential equations from analysis.
       
     • It is intended that such a theory bear to algebraic groups the same relation that the theory of differential equations bears to the theory of algebraic equations.
       
     • Algebraic groups can be viewed as groups in the category of algebraic varieties, where the latter are taken to be locally given as sets of simultaneous solutions of algebraic equations.
       

180. Halphen biography
     
     • He was led to extend results due to Max Noether which, in turn, had him examine projective transformations which fix certain differential equations.
       
     • A characterisation of such invariant differential equations appeared in Halphen's doctoral dissertation On differential invariants which he presented in 1878.
       
     • Halphen made major contributions to linear differential equations and algebraic space curves.
       
     • He examined problems in the areas of systems of lines, classification of space curves, enumerative geometry of plane conics, singular points of plane curves, projective geometry and differential equations, elliptic functions, and assorted questions in analysis.
       
     • For example, in 1880 he won the Grand Prix of the Academie des Sciences for his work on linear differential equations.
       
     • Other work such as that on linear differential equations was overtaken by Lie group methods.
       

181. Kneser biography
     
     • After writing this thesis on algebraic functions and equations, he then worked on space curves.
       
     • Adolf Kneser's early work was on algebraic functions and equations.
       
     • One of these areas is that of linear differential equations; in particular he worked on the Sturm-Liouville problem and integral equations in general.
       
     • He wrote an important text on integral equations.
       
     • the first to introduce Hilbert's new methods into analysis in his textbook on integral equations.
       

182. Ford biography
     
     • Ford read the paper On the Roots of a Derivative of a Rational Function to the meeting of the Society on Friday 14 May 1915, the paper On the Oscillation Functions derived from a Discontinuous Function to the meeting on 11 June 1915, and the paper A method of solving algebraic equations to the meeting on 12 January 1917.
       
     • Following his contributions to the war effort, Ford joined the faculty at the Rice Institution, Houston, Texas and while there he published papers such as On the closeness of approach of complex rational fractions to a complex irrational number (1925), The Solution of Equations by the Method of Successive Approximations (1925), On motions which satisfy Kepler's first and second laws (1927/28), and The limit points of a group (1929).
       
     • Two significant books published by Ford are Automorphic Functions (1929) and Differential Equations (1933, second edition 1955).
       
     • See reviews at THIS LINK for Differential Equations and THIS LINK for Automorphic Functions.
       
     • Some of the papers are related to the fields of Ford's major interests: complex functions, interpolation, differential equations, and numerical analysis.
       
     • L R Ford - Differential Equations .
       

183. Caccioppoli biography
     
     • After 1930 Caccioppoli devoted himself to the study of differential equations and he provided existence theorems for both linear and non-linear problems.
       
     • His idea was to use a topological- functional approach to the study of differential equations.
       
     • Carrying on in this way Caccioppoli, in 1931, extended in some cases Brouwer's fixed point theorem, and applied his results to existence problems of both partial differential equations and ordinary differential equations.
       
     • In the period between 1933 and 1938 Caccioppoli applied his method to elliptic equations, providing the a priori upper bound for their solutions, in a more general way than Bernstein did for the two-dimensional case.
       
     • In 1935 he dealt with the question introduced in 1900 by Hilbert during the International Congress of Mathematicians, namely whether or not the solutions of analytical elliptic equations are analytic.
       

184. John biography
     
     • He applied this in his study of general properties of linear partial differential equations, convex geometry and the mathematical theory of water waves.
       
     • It was in this period that John introduced the space of functions of bounded mean oscillations which plays a fundamental role in harmonic analysis and nonlinear elliptic equations.
       
     • He retired in 1981 but at this time his work was concentrating on the theory of nonlinear wave equations.
       
     • For anyone interested in the analysis of partial differential equations, the work of Fritz John is especially rewarding.
       
     • He wrote by now classical papers in convexity, ill-posed problems, the numerical treatment of partial differential equations, quasi-isometry and blow-up in nonlinear wave propagation.
       

185. Krasnosel'skii biography
     
     • For example Positive solutions of operator equations (1962) which studied the existence, uniqueness, and properties of positive solutions of linear and non-linear equations in a partially ordered Banach space, Vector fields in the plane (1963) which the angular variation of a plane vector field relative to a curve, and Displacement operators along trajectories of differential equations (1966) which is described by C Olech as follows:- .
       
     • For example Approximate solution of operator equations (1969):- .
       
     • is devoted to the investigation of approximate methods of solving operator equations.
       

186. Wantzel biography
     
     • Wantzel is famed for his work on solving equations by radicals.
       
     • In 1845 Wantzel, continuing his researches into equations, gave a new proof of the impossibility of solving all algebraic equations by radicals.
       
     • In meditating on the researches of these two mathematicians, and with the aid of principles we established in an earlier paper, we have arrived at a form of proof which appears so strict as to remove all doubt on this important part of the theory of equations.
       
     • It was he who first gave the integration of differential equations of the elastic curve.
       

187. Clairaut biography
     
     • The following year Clairaut studied the differential equations now known as 'Clairaut's differential equations' and gave a singular solution in addition to the general integral of the equations.
       
     • In 1739 and 1740 he published further work on the integral calculus, proving the existence of integrating factors for solving first order differential equations (a topic which also interested Johann Bernoulli, Reyneau and Euler).
       
     • The algebra book was an even more scholarly work and took the subject up to the solution of equations of degree four.
       

188. Schauder biography
     
     • While Schauder was in Paris he collaborated with J Leray and their joint work led to a paper Topologie et equations fonctionelles published in the Annales scientifiques de l'Ecole normale Superieure.
       
     • This 1934 paper on topology and partial differential equations is of major importance [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]:- .
       
     • This degree is then used in an ingenious method to prove the existence of solutions to complicated partial differential equations.
       
     • His last work was to generalise results of Courant, Friedrichs and Lewy on hyperbolic partial differential equations.
       
     • In particular, Schauder's formulation of a fixed point theorem originated a new, extremely fruitful method in the theory of differential equations, known as Schauder's method ..
       
     • Schauder's fixed point theorem and his skillful use of function space techniques to analyse elliptic and hyperbolic partial differential equations are contributions of lasting quality.
       

189. Ramanathan biography
     
     • Another fascinating paper is Ramanujan's modular equations (1990).
       
     • The present author commences with a very informative historical survey of modular equations.
       
     • Of course, in a paper of only 18 pages in length, the author can only discuss a small portion of Ramanujan's modular equations and he concentrates therefore on equations of composite degree.
       
     • He gives some proofs, shows connections to previous work, and offers insights into how Ramanujan may have discovered some of his equations.
       

190. Hille biography
     
     • Kirsti Hille wrote the article [Integral Equations Operator Theory 4 (3) (1981), 304-306.',5)">5] after the death of her husband.
       
     • Hille's main work was on integral equations, differential equations, special functions, Dirichlet series and Fourier series.
       
     • Among Hille's other texts were Analytic function theory Vol 1 (1959), Vol 2 (1964); Analysis Vol 1 (1964), Vol 2 (1966); Lectures on ordinary differential equations (1969); Methods in classical and functional analysis (1972); and Ordinary differential equations in the complex domain (1976).
       

191. Wazewski biography
     
     • Wazewski made important contributions to the theory of ordinary differential equations, partial differential equations, control theory and the theory of analytic spaces.
       
     • was to bring him fame and lead to the development of a new school of differential equations.
       
     • he succeeded in applying with amazing effect the topological notion of retract (introduced by K Borsuk) to the study of the solutions of differential equations.
       
     • Lefschetz considered his method of retracts one of the most important achievements in the theory of differential equations since the war.
       

192. Calderon biography
     
     • Calderon, on the other hand, with his background as an engineer, saw that such operators held an important key to understanding the theory of partial differential equations.
       
     • In 1958 Calderon published one of his most important results on uniqueness in the Cauchy problem for partial differential equations.
       
     • for his ground-breaking work on singular integral operators leading to their application to important problems in partial differential equations, including his proof of uniqueness in the Cauchy problem, the Atiyah-Singer index theorem, and the propagation of singularities in nonlinear equations..
       
     • Calderon's techniques have been absorbed as standard tools of harmonic analysis and are now propagating into nonlinear analysis, partial differential equations, complex analysis, and even signal processing and numerical analysis.
       

193. Al-Samawal biography
     
     • In Book 2 of al-Bahir al-Samawal describes the theory of quadratic equations but, rather surprisingly, he gave geometric solutions to these equations despite algebraic methods having been fully described by al-Khwarizmi, al-Karaji, and others.
       
     • Al-Samawal also described the solution of indeterminate equations such as finding x so that a xn is a square, and finding x so that axn + bxn-1 is a square.
       
     • The final book of al-Bahir contains an interesting example of a problem in combinatorics, namely to find ten unknowns given the 210 equations which give their sums taken 6 at a time.
       
     • Of course such a system of 210 equations need not be consistent and al-Samawal gave the 504 conditions which are necessary for the system to be consistent.
       

194. Church biography
     
     • Early contributions included the papers On irredundant sets of postulates (1925), On the form of differential equations of a system of paths (1926), and Alternatives to Zermelo's assumption (1927).
       
     • For example he published Remarks on the elementary theory of differential equations as area of research in 1965 and A generalization of Laplace's transformation in 1966.
       
     • The first examines ideas and results in the elementary theory of ordinary and partial differential equations which Church feels may encourage further investigation of the topic.
       
     • The paper includes a discussion of a generalization the Laplace transform which he extends to non-linear partial differential equations.
       
     • This generalization of the Laplace transform is the topic of study of the second paper, again using the method to obtain solutions of second-order partial differential equations.
       

195. Mitchell biography
     
     • He worked on an idea of Ron's of incorporating higher derivatives into methods for Ordinary Differential Equations, apparently one of the few times Ron strayed away from PDE's to ODE's.
       
     • He was mainly interested at that time in finite difference methods for both ordinary and partial differential equations.
       
     • He was joined in March 1964 by Donald Kershaw, whose main interests were differential and integral equations, and some students, including Alistair Watson.
       
     • In this talk Olec Zienkiewicz described instabilities they had experienced in converting their successful finite element codes for structural problems into codes for solving the Navier-Stokes and related equations in fluid dynamics.
       
     • Some of the problems arose from Mathematical Biology, on which "Mano" Manoranjan did much of his PhD work, but Ron was also interested in solitons, particularly those arising from the Korteweg-de Vries and Schrodinger equations.
       

196. Deligne biography
     
     • He also worked closely with Jean-Pierre Serre, leading to important results on the l-adic representations attached to modular forms, and the conjectural functional equations of L-functions.
       
     • Andre Weil gave for the first time a theory of varieties defined by equations with coefficients in an arbitrary field, in his Foundations of Algebraic Geometry (1946).
       
     • Weil's work on polynomial equations led to questions on what properties of a geometric object can be determined purely algebraically.
       
     • Weil's work related questions about integer solutions to polynomial equations to questions in algebraic geometry.
       
     • He conjectured results about the number of solutions to polynomial equations over the integers using intuition on how algebraic topology should apply in this novel situation.
       

197. Remez biography
     
     • He gave courses at these institutions on analysis, differential equations and differential geometry while undertaking research for his doctorate.
       
     • from Kiev State University in 1929 with his thesis Methods of Numerical Integration of Differential Equations with an Estimate of Exact Limits of Allowable Errors.
       
     • Remez generalised Chebyshev-Markov characterisation theory and used it to obtain approximate solutions of differential equations.
       
     • He also worked on approximate solutions of differential equations and the history of mathematics.
       
     • The book has two parts: Part I - Properties of the solution of the general Chebyshev problem; Part II - Finite systems of inconsistent equations and the method of nets in Chebyshev approximation.
       

198. Baker Alan biography
     
     • This was awarded for his work on Diophantine equations.
       
     • [Diophantine equations], carrying a history of more than one thousand years, was, until the early years of this century, little more than a collection of isolated problems subjected to ingenious ad hoc methods.
       
     • It was A Thue who made the breakthrough to general results by proving in 1909 that all Diophantine equations of the form .
       
     • Turan goes on to say that Carl Siegel and Klaus Roth generalised the classes of Diophantine equations for which these conclusions would hold and even bounded the number of solutions.
       
     • He proved that for equations of the type f (x, y) = m described above there was a bound B which depended only on m and the integer coefficients of f with .
       

199. Van Vleck biography
     
     • Almost all Van Vleck's research papers were in the fields of function theory and differential equations.
       
     • For example he published On the determination of a series of Sturm's functions by the calculation of a single determinant (1899), On linear criteria for the determination of the radius of convergence of a power series (1900), On the convergence of continued fractions with complex elements (1901), A determination of the number of real and imaginary roots of the hypergeometric series (1902), On an extension of the 1894 memoir of Stieltjes (1903), and On the extension of a theorem of Poincare for difference-equations (1912).
       
     • Of the American Mathematical Society sometime president, and editor of the Transactions; always wise counsellor and leader; creative mathematician and successful investigator in the theory of functions, and in the theories of differential and difference equations and of functional equations; for these eminent services in mathematics, and especially for your important researches concerning functional equations and analytic continued fractions.
       

200. Gordan biography
     
     • Moving to Konigsberg, Gordan studied under Jacobi, then he moved to Berlin where he began to become interested in problems concerning algebraic equations.
       
     • In the year 1874-75 when Gordan and Klein were together at Erlangen they undertook a joint research project examining groups of substitutions of algebraic equations.
       
     • They investigated the relationship between PSL(2,5) and equations of degree five.
       
     • Later Gordan went on to examine the relation between the group PSL(2,7) and equations of degree seven, then he studied the relation of the group A6 to equations of degree six.
       

201. McClintock biography
     
     • One paper treats difference equations as differential equations of infinite order and others look at quintic equations which are soluble algebraically.
       
     • This led he to restate difference equations as differential equations of infinite order.
       

202. Saint-Venant biography
     
     • Perhaps his most remarkable work was that which he published in 1843 in which he gave the correct derivation of the Navier-Stokes equations.
       
     • Seven years after Navier's death, Saint-Venant re-derived Navier's equations for a viscous flow, considering the internal viscous stresses, and eschewing completely Navier's molecular approach.
       
     • Why his name never became associated with those equations is a mystery.
       
     • We should remark that Stokes, like Saint-Venant, correctly derived the Navier-Stokes equations but he published the results two years after Saint-Venant.
       
     • In 1871 he derived the equations for non-steady flow in open channels.
       

203. Murnaghan biography
     
     • Harry Bateman had been appointed there in 1912 and his interests in partial differential equations fitted perfectly with Murnaghan's interests at the time.
       
     • Arriving at Johns Hopkins University in Baltimore, Murnaghan began doctoral studies working on differential equations which arose in the study of radio-active decay.
       
     • Of course this meant that he was deeply involved in solving differential equations, and indeed he also wrote papers on this topic.
       
     • It covers topics such as: vectors and matrices; Fourier series; boundary value problems; Legendre and Bessel functions; integral equations; the calculus of variations and dynamics; and the operational calculus.
       
     • The first of these is a short book of less that 100 pages written for engineers and scientists, while the second consists of 19 lectures on such topics as: the Fourier integral; the Laplace integral transformation; the differential equations of Laguerre and Bessel; properties of special functions; asymptotic series for an error function, and for certain Bessel functions.
       

204. Reizins biography
     
     • He graduated in 1948 with distinction and became a member of the Department of Mathematical Analysis at the University while he undertook research on differential equations under Arvids Lusis.
       
     • However, he continued to undertake research in mathematics and in 1951 his first paper The behaviour of the integral curves of a system of three differential equations in the neighbourhood of a singular point was published by the Latvian Academy of Sciences.
       
     • thesis he studied the qualitative behaviour of homogeneous differential equations and obtained results that were highly regarded by specialists.
       
     • Of the many other important contributions made by Reizins we should mention in particular his work on Pfaff's equations and his contributions to the history of mathematics.
       
     • Other important historical papers include Mathematics in University of Latvia 1919-1969 (1975, joint with E Riekstins) and From the History of the General Theory of Ordinary Differential Equations (1977).
       

205. Jeffery biography
     
     • He did one years teacher training in 1911 but he was already undertaking research and his first paper On a form of the solution of Laplace's equation suitable for problems relating to two spheres was read to the Royal Society in 1912.
       
     • He made effective use of Whittaker's general solution to Laplace's equation which Whittaker found in 1903.
       
     • Jeffery also worked on general relativity and produced exact solutions to Einstein's field equations in certain special cases.
       

206. Osgood biography
     
     • Osgood's main work was on the convergence of sequences of continuous functions, solutions of differential equations, the calculus of variations and space filling curves.
       
     • In 1898 Osgood published an important paper on the solutions of the differential equation dy/dx = f(x, y) satisfying the prescribed initial conditions y(a) = b.
       
     • Some papers over the next few years included: Sufficient conditions in the calculus of variations (1900), On a fundamental property of a minimum in the calculus of variations and the proof of a theorem of Weierstrass's (1901), A Jordan curve of positive area (1903), On Cantor's theorem concerning the coefficients of a convergent trigonometric series, with generalizations (1909), On the gyroscope (1922), and On normal forms of differential equations (1925).
       

207. Peano biography
     
     • In 1886 Peano proved that if f (x, y) is continuous then the first order differential equation dy/dx = f (x, y) has a solution.
       
     • Four years later Peano showed that the solutions were not unique, giving as an example the differential equation dy/dx = 3y2/3 , with y(0) = 0.
       
     • The following year he discovered, and published, a method for solving systems of linear differential equations using successive approximations.
       

208. Friedmann biography
     
     • In his last year at the University he was working on an essay on the subject I assigned: 'Find all orthogonal substitutions such that the Laplace equation, transformed for the new variables, admits particular solutions in the form of a product of two functions, one of which depends only on one, and the other on the other two variables'.
       
     • Also in this letter he asked Steklov's advice on integrating equations he had obtained from theoretically modelling bombs dropping.
       
     • In reality it turns out that the solution given in it does not satisfy the field equations.
       

209. Fibonacci biography
     
     • Indeed, although mainly a book about the use of Arab numerals, which became known as algorism, simultaneous linear equations are also studied in this work.
       
     • Fibonacci proves that the root of the equation is neither an integer nor a fraction, nor the square root of a fraction.
       
     • And because it was not possible to solve this equation in any other of the above ways, I worked to reduce the solution to an approximation.
       

210. Wallis biography
     
     • He also discovered methods of solving equations of degree four which were similar to those which Harriot had found but Wallis claimed that he made the discoveries himself, not being aware of Harriot's contributions until later.
       
     • He also criticises Descartes' Rule of Signs stating, quite correctly, that the rule which determines the number of positive and the number of negative roots by inspection, is only valid if all the roots of the equation are real.
       
     • History Topics: Pell's equation .
       

211. Goldstine biography
     
     • Most of the paper is taken up with the more difficult problem of determining such conditions when the class of admissible points is (1940) required to satisfy an equation of an abstract functional character.
       
     • There followed A generalized Pell equation.
       
     • In particular the School used a Bush analyser, designed by Vannevar Bush, specifically to integrate systems of ordinary differential equations.
       

212. Bouquet biography
     
     • Bouquet and Briot developed Cauchy's work on the existence of integrals of a differential equation.
       
     • For example Etude des fonctions d'une variable imaginaire (Study of functions with one imaginary variable); Recherches sur les proprietes des fonctions definies par des equations differentielles (Research on the properties of functions defined by differential equations); and Memoire sur l'integration des equations differentielles au moyen des fonctions elliptiques (Memoir on the integration of differential equations by means of elliptic functions).
       

213. Jeffrey biography
     
     • He did one years teacher training in 1911 but he was already undertaking research and his first paper On a form of the solution of Laplace's equation suitable for problems relating to two spheres was read to the Royal Society in 1912.
       
     • He made effective use of Whittaker's general solution to Laplace's equation which Whittaker found in 1903.
       
     • Jeffrey also worked on general relativity and produced exact solutions to Einstein's field equations in certain special cases.
       

214. Dyson biography
     
     • The first, written in 1941 (published in 1944) is A proof that every equation has a root.
       
     • there are so many proofs of the theorem that every equation has a root that it seems almost criminal to produce another.
       
     • The historical account of the breakdown in communications between mathematicians and physicists and of the lack of interest in Maxwell's equations constitutes an indictment of the mathematical community.
       

215. Sridhara biography
     
     • We give details below of Sridhara's rule for solving quadratic equations as given by Bhaskara II.
       
     • Sridhara was one of the first mathematicians to give a rule to solve a quadratic equation.
       
     • Multiply both sides of the equation by a known quantity equal to four times the coefficient of the square of the unknown; add to both sides a known quantity equal to the square of the coefficient of the unknown; then take the square root.
       

216. Stampioen biography
     
     • In 1633 he challenged Descartes to a public competition by giving him a geometric problem whose solution involved the solution of a quartic equation.
       
     • In fact Stampioen's criticism was fair for although Descartes had taken the geometric problem and derived the correct quartic equation, he left the problem there without solving the quartic.
       
     • The problem which Stampioen was interested in came as a consequence of using the Cardan-Tartaglia formula to solve cubic equations.
       

217. Spitzer biography
     
     • A rapid treatment of the Boltzmann equation, in an appendix, brings us in Chapter 2 to the transport equation for a fluid.
       
     • This is joined with Maxwell's equations, and the simple limits of high and low magnetic fields are briefly considered.
       

218. Koch biography
     
     • Von Koch's first results were on infinitely many linear equations in infinitely many unknowns.
       
     • In 1891 he wrote the first of two papers on applications of infinite determinants to solving systems of differential equations with analytic coefficients.
       
     • Yet this work can be said to be the first step on the long road which eventually led to functional analysis, since it provided Fredholm with the key for the solution of his integral equation.
       

219. Betti biography
     
     • His early work is in the area of equations and algebra.
       
     • In 1854 Betti showed that the quintic equation could be solved in terms of integrals resulting in elliptic functions.
       
     • Although Jordan, in his Traite des substitutions et des equations algebriques (1870) credits Betti with having filled the gaps in Galois' arguments and with having been the first to establish the sequence of Galois' theorems rigorously, the fact is that Betti's work contains substantial obscurities and errors.
       

220. Mohr Ernst biography
     
     • From this time on his work was on applied mathematics, mainly fluid dynamics and differential equations, but he also published the occasional paper on polynomials.
       
     • One of these 1951 papers looks at the numerical solution of the differential equation dy/dx= f (x, y).
       
     • Hermann Weyl, in two pioneering papers, described a class of differential equations in the limit point case where complete determination of the continuous spectrum is possible.
       

221. Evans biography
     
     • His doctoral dissertation Volterra's integral equation of the second kind with discontinuous kernel was published in the Transactions of the American Mathematical Society in two parts in 1910 and 1911.
       
     • His work dealt with potential theory, functional analysis, integral equations and the problem of minimal surfaces, the Plateau Problem.
       
     • Among the important texts he wrote were Functional equations and their applications (1918), The logarithmic potential (1927), and Mathematical Introduction to economics (1930).
       

222. Nicolson biography
     
     • Phyllis Nicolson is best known for her joint work with John Crank on the heat equation, where a continuous solution u(x, t) is required which satisfies the second order partial differential equation .
       
     • Crank and Nicolson's method, which is numerically stable, requires the solution of a very simple system of linear equations (a tridiagonal system) at each time level.
       

223. Kurschak biography
     
     • Another topic which Kurschak investigated was the differential equations of the calculus of variations.
       
     • He proved invariance of the differential equations he was considering under contact transformations.
       
     • a second-order differential expressions to provide the equation belonging to the variation of a multiple integral.
       

224. Arf biography
     
     • At that time, I was thinking about making a list of the algebraic equations or Galois algebraic equations that could be solved.
       
     • Arf presented a paper On a generalization of Green's formula and its application to the Cauchy problem for a hyperbolic equation to the volume Studies in mathematics and mechanics presented to Richard von Mises in 1954.
       

225. Chang biography
     
     • Chang's research interests include the study of certain geometric types of nonlinear partial differential equations.
       
     • The Ruth Lyttle Satter Prize is awarded to Sun-Yung Alice Chang for her deep contributions to the study of partial differential equations on Riemannian manifolds and in particular for her work on extremal problems in spectral geometry and the compactness of isospectral metrics within a fixed conformal class on a compact 3-manifold.
       
     • Yang and I have solved the partial differential equation of Gaussian/scalar curvatures on the sphere by studying the extremal functions for certain variation functionals.
       
     • The course was entitled "Geometric PDE" and described using analytic tools like that of partial differential equations to solve problems in geometry.
       
     • model differential equations like that of the Gaussian curvature equations on compact surfaces, the prescribing curvature equations and the evolution equations related to the curvature flows.
       

226. Schubert Hans biography
     
     • Uber eine lineare Integrodifferentialgleichung mit Zusatzkern (1950) looked at certain aerodynamical problems which lead to integrodifferential equations.
       
     • At Halle Schubert taught a variety of different courses such as differential and integral calculus, partial differential equation, and integral equations.
       

227. Coble biography
     
     • His interests in research relate to finite geometries and the group theory related to them, and to Cremona transformations related to the Galois theory of equations.
       
     • His early papers, written while he was at Johns Hopkins University, include: On the relation between the three-parameter groups of a cubic space curve and a quadric surface (1906); An application of the form-problems associated with certain Cremona groups to the solution of equations of higher degree (1908); An application of Moore's cross-ratio group to the solution of the sextic equation (1911); An application of finite geometry to the characteristic theory of the odd and even theta functions (1913); and Point sets and allied Cremona groups (1915).
       

228. Bernoulli Nicolaus(I) biography
     
     • There he worked on geometry and differential equations.
       
     • Other problems he worked on involved differential equations.
       
     • He also made significant contributions in studying the Riccati equation.
       

229. Sripati biography
     
     • His work on equations in this chapter contains the rule for solving a quadratic equation and, more impressively, he gives the identity: .
       
     • Other mathematics included in Sripati's work includes, in particular, rules for the solution of simultaneous indeterminate equations of the first degree that are similar to those given by Brahmagupta .
       

230. Vessiot biography
     
     • In 1892 he submitted his doctoral dissertation on groups of linear transformations, in particular studying the action of these groups on the independent solutions of a differential equation.
       
     • Vessiot applied continuous groups to the study of differential equations.
       
     • He extended results of Drach (1902) and Cartan (1907) and also extended Fredholm integrals to partial differential equations.
       

231. Clausius biography
     
     • The basic equation set up by Clausius was therefore dQ = dU + dW where dQ was the increment in the heat, dU was the change in energy of the body, and dW was the change in external work done.
       
     • The Clausius-Clapeyron equation appears which expresses the relation between the pressure and temperature at which two phases of a substance are in equilibrium.
       
     • Clausius deliberately made choices in setting up the equations so that they were:- .
       

232. Mathieu Emile biography
     
     • From his late twenties his main efforts were devoted to the then unfashionable continuation of the great French tradition of mathematical physics, and he extended in sophistication the formation and solution of partial differential equations for a wide range of physical problems.
       
     • He discovered these functions, which are special cases of hypergeometric functions, while solving the wave equation for an elliptical membrane moving through a fluid.
       
     • The Mathieu functions are solutions of the Mathieu equation which is .
       

233. Feigenbaum biography
     
     • In 1973 it had been conjectured that the behaviour of the logistic equation was the same in a qualitative sense for all g(x) which have a maximum value and decrease monotonically on either side of this maximum.
       
     • The remarkable result obtained by Feigenbaum was to show that not only was the behaviour qualitatively similar but there was a very precise mathematical result which held for all such logistic equations.
       
     • Feigenbaum did not actually work with the precise logistic equation which May studied and in fact his work was independent of that by May.
       

234. Recorde biography
     
     • The book was the Second Part of Arithmetic, The Grounde of Artes being the first, covering the extraction of roots, the theory of equations and arithmetic with surds.
       
     • In his study of quadratic equations, Recorde does not allow solutions which are negative, but he does allow negative coefficients.
       
     • He makes good use of the sum and product of the roots stressing that for the equation .
       

235. Plucker biography
     
     • The characteristic features of Plucker's analytic geometry were already present in this work, namely, the elegant operations with algebraic symbols occurring in the equations of conic sections and their pencils.
       
     • This work also contains the celebrated 'Plucker equations' relating the order and class of a curve.
       
     • In this way of specifying coordinates, a point has a linear equation, namely that of all lines through the point while a line has a pair of numbers namely the x and y coordinates of where it cuts the axes.
       

236. Sylvester biography
     
     • He was a very active researcher and by the time he resigned the chair of natural philosophy in 1841 he had published fifteen papers on fluid dynamics and algebraic equations.
       
     • In 1851 he discovered the discriminant of a cubic equation and first used the name 'discriminant' for such expressions of quadratic equations and those of higher order.
       

237. Descartes biography
     
     • Harriot's work on equations, however, may indeed have influenced Descartes who always claimed, clearly falsely, that nothing in his work was influenced by the work of others.
       
     • Descartes' geometric solution of a quadratic equation .
       
     • History Topics: Quadratic etc equations .
       

238. Legendre biography
     
     • In "Elements" Legendre gave a simple proof that π is irrational, as well as the first proof that π2 is irrational, and conjectured that π is not the root of any algebraic equation of finite degree with rational coefficients.
       
     • I have thought that what there was better to do in the problem of comets was to start out from the immediate data of observation, and to use all means to simplify as much as possible the formulas and the equations which serve to determine the elements of the orbit.
       
     • His method involved three observations taken at equal intervals and he assumed that the comet followed a parabolic path so that he ended up with more equations than there were unknowns.
       

239. Hill biography
     
     • Examples of papers he published in the Annals of Mathematics include: On the lunar inequalities produced by the motion of the ecliptic (1884), Coplanar motion of two planets, one having a zero mass (1887), On differential equations with periodic integrals (1887) (these differential equations are now called Hill's differential equation), On the interior constitution of the earth as respects density (1888), The secular perturbations of two planets moving in the same plane; with application to Jupiter and Saturn (1890), On intermediate orbits (1893), Literal expression for the motion of the Moon's perigee (1894) and Application of Chebyshev's principle in the projection of maps (1908).
       

240. Stokes biography
     
     • After he had deduced the correct equations of motion Stokes discovered that again he was not the first to obtain the equations since Navier, Poisson and Saint-Venant had already considered the problem.
       
     • The work also discussed the equilibrium and motion of elastic solids and Stokes used a continuity argument to justify the same equation of motion for elastic solids as for viscous fluids.
       

241. Al-Quhi biography
     
     • The geometric problems that al-Quhi studied usually led to quadratic or cubic equations.
       
     • One, which requires the solution of a quadratic equation, had been found by Abu Kamil in the ninth century.
       
     • The other, which requires the solution of a quartic equation, is the one presented by al-Quhi.
       

242. Redei biography
     
     • In 1953 L Redei published his famous article "Die 2-Ringklassen-gruppe des quadratischen Zahlkorpers und die Theorie des Pell-schen Gleichung", after many years of investigation of Pell's equation.
       
     • He gave a unified theory for the structure of class groups of real quadratic number fields and conditions for solvability of Pell's equation and other indeterminate equations.
       

243. Voronoy biography
     
     • He was awarded a Master's Degree in 1894 for a dissertation on the algebraic integers associated with the roots of an irreducible cubic equation.
       
     • In the essay I am now presenting, results from the general theory of algebraic integers are applied to the particular case of numbers depending on the root of an irreducible equation x3 = rx + s.
       
     • In our exposition the resolution of these questions is based on a detailed study of the solutions of third-degree equations relative to a prime and a composite modulus.
       

244. Bombelli biography
     
     • It is unclear exactly how Bombelli learnt of the leading mathematical works of the day, but of course he lived in the right part of Italy to be involved in the major events surrounding the solution of cubic and quartic equations.
       
     • Scipione del Ferro, the first to solve the cubic equation was the professor at Bologna, Bombelli's home town, but del Ferro died the year that Bombelli was born.
       
     • History Topics: Quadratic, cubic and quartic equations .
       

245. Cowling biography
     
     • The equations of Boltzmann and Maxwell are then developed, Enskog's generalization of Maxwell's equation of transfer being given.
       
     • In the important chapter on the non-uniform state for a simple gas, use is made of Enskog's method of solving the integral equation and of Burnett's calculation of certain quantities A and B with the aid of Sonine's polynomials.
       

246. Wilkins Ernest biography
     
     • In 1944 four of his papers appeared: On the growth of solutions of linear differential equations; Definitely self-conjugate adjoint integral equations; Multiple integral problems in parametric form in the calculus of variations; and A note on skewness and kurtosis.
       
     • In the following year he published The differential difference equation for epidemics in the Bulletin of Mathematical Biophysics.
       

247. Fine Henry biography
     
     • Two further paper On the functions defined by differential equations with an extension of the Puiseux polygon construction to these equations, and Singular solutions of ordinary differential equations appeared in 1889 and 1890 respectively.
       
     • He gave his retiring address as president on An unpublished theorem of Kronecker respecting numerical equations.
       

248. Boltzmann biography
     
     • The equations of Newtonian mechanics are reversible in time and Poincare proved that if a mechanical system is in a given state it will return infinitely often to a state arbitrarily close to the given one.
       
     • The actual irreversibility of natural phenomena thus proves the existence of processes that cannot be described by mechanical equations, and with this the verdict on scientific materialism is settled.
       
     • Boltzmann continued to defend his belief in atomic structure and in a 1905 publication Populare Schriften he tried to explain how the physical world could be described by differential equations which represented the macroscopic view without representing the underlying atomic structure.
       
     • May I be excused for saying with banality that the forest hides the trees for those who think that they disengage themselves from atomistics by the consideration of differential equations.
       

249. Von Neumann biography
     
     • He was notorious for dashing out equations on a small portion of the available blackboard and erasing expressions before students could copy them.
       
     • It was then that he became aware of the mysteries underlying the subject of non-linear partial differential equations.
       
     • His work, from the beginnings of the Second World War, concerns a study of the equations of hydrodynamics and the theory of shocks.
       
     • The phenomena described by these non-linear equations are baffling analytically and defy even qualitative insight by present methods.
       

250. De Vries biography
     
     • Gustav de Vries's name is well known to mathematicians because of the work of his doctoral dissertation which contained the Korteweg-de Vries equation.
       
     • On 1 December 1894 de Vries had an oral examination on his thesis Bijdrage tot de kennis der lange golven which contained the famous Korteweg-de Vries equation.
       
     • They found explicit, closed-form, travelling-wave solutions to the Korteweg - de Vries equation that decay rapidly.
       
     • to commemorate the centennial of the equation by and named after Korteweg and de Vries.
       

251. Liouville biography
     
     • This he did in October of 1830 but even at this stage he had written a number of papers which he had submitted to the Paris Academy on electrodynamics, partial differential equations and the theory of heat.
       
     • His work on boundary value problems on differential equations is remembered because of what is called today Sturm-Liouville theory which is used in solving integral equations.
       
     • Sturm and Liouville examined general linear second order differential equations and examined properties of their eigenvalues, the behaviour of the eigenfunctions and the series expansion of arbitrary functions in terms of these eigenfunctions.
       

252. Fourier biography
     
     • Having left St Benoit in 1789, he visited Paris and read a paper on algebraic equations at the Academie Royale des Sciences.
       
     • The second objection was made by Biot against Fourier's derivation of the equations of transfer of heat.
       
     • the manner in which the author arrives at these equations is not exempt of difficulties and that his analysis to integrate them still leaves something to be desired on the score of generality and even rigour.
       
     • If they had illuminated this branch of physics by important and general views and had greatly perfected the analysis of partial differential equations, if they had established a principal element of the theory of heat by fine experiments ..
       

253. Wheeler biography
     
     • After winning a scholarship to study for her master's degrees at the University of Iowa, she was awarded the degree for a thesis The extension of Galois theory to linear differential equations in 1904.
       
     • After returning to the United States, where her husband was by now Dean of Engineering, she taught courses in the theory of functions and differential equations.
       
     • in 1909, her thesis Biorthogonal Systems of Functions with Applications to the Theory of Integral Equations being the one written originally at Gottingen.
       
     • Under his guidance she worked on integral equations studying infinite dimensional linear spaces.
       

254. Simon biography
     
     • in 1971 for his thesis Interior Gradient Bounds for Non-Uniformly Elliptic Equations.
       
     • supervisor James H Michael) Sobolev and mean-value inequalities on generalized submanifolds of Rn (1973); Global estimates of Holder continuity for a class of divergence-form elliptic equations (1974); (with Richard M Schoen and Shing-Tung Yau) Curvature estimates for minimal hypersurfaces (1975); Interior gradient bounds for non-uniformly elliptic equations (1976); and Remarks on curvature estimates for minimal hypersurfaces (1976).
       
     • This development began with his 1983 paper "Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems." The first stage of his work on general singular sets is principally described in "Cylindrical tangent cones and the singular set of minimal submanifolds" (1993), and the remaining work appears in his paper "Rectifiability of the singular set of energy minimizing maps" (1995).
       

255. Kronecker biography
     
     • students to hear that Kronecker was questioned at his oral on a wide range of topics including the theory of probability as applied to astronomical observations, the theory of definite integrals, series and differential equations, as well as on Greek, and the history of philosophy.
       
     • The topics on which he lectured were very much related to his research: number theory, the theory of equations, the theory of determinants, and the theory of integrals.
       
     • We have already indicated that Kronecker's primary contributions were in the theory of equations and higher algebra, with his major contributions in elliptic functions, the theory of algebraic equations, and the theory of algebraic numbers.
       

256. Hesse biography
     
     • the special forms of linear equation and of planar equation that Hesse used in these books are called Hesse's normal form of the linear equation and of the planar equation in all modern textbooks on the discipline.
       

257. Birkhoff biography
     
     • Birkhoff read Poincare's works on differential equations and celestial mechanics and he learnt more, and was more strongly influenced in the direction his research was taking, by Poincare than from his supervisor.
       
     • The doctoral thesis which Birkhoff submitted was entitled Asymptotic Properties of Certain Ordinary Differential Equations with Applications to Boundary Value and Expansion Problems and it led to the award of his Ph.D.
       
     • Birkhoff's work on linear differential equations, difference equations and the generalised Riemann problem mostly all arose from the basis he laid in his thesis.
       

258. Abu Kamil biography
     
     • The Book on algebra by Abu Kamil is in three parts: (i) On the solution of quadratic equations, (ii) On applications of algebra to the regular pentagon and decagon, and (iii) On Diophantine equations and problems of recreational mathematics.
       
     • The Book of rare things in the art of calculation is concerned with solutions to indeterminate equations.
       
     • Sesiano in [Centaurus 21 (2) (1977), 89-105.',11)">11] discusses Abu Kamil's work on indeterminate equations and he argues that his methods are very interesting for three reasons.
       

259. Zaanen biography
     
     • his study of the theory of linear integral equations.
       
     • Measure and integral, Banach and Hilbert space, linear integral equations (1953) which contained much of his own research as well as material from a lecture course by N G de Bruijn.
       
     • Its main feature is the emphasis laid on integral equations and especially on those with symmetrizable kernel, a domain of research in which we owe to the author many personal results.
       
     • Measure and integral, Banach and Hilbert space, linear integral equations (1953), we have already mentioned above.
       

260. Hadamard biography
     
     • The topic proposed for the prize had been one on geodesics and Hadamard's work in studying the trajectories of point masses on a surface led to certain non-linear differential equations whose solution also gave properties of geodesics.
       
     • Matrices whose determinants satisfied equality in the relation are today called Hadamard matrices and are important in the theory of integral equations, coding theory and other areas.
       
     • In particular he worked on the partial differential equations of mathematical physics producing results of outstanding importance.
       
     • He continued to produce books and papers of the highest quality, publishing perhaps his most famous text Lectures on Cauchy's problem in linear partial differential equations in 1922.
       

261. Drach biography
     
     • Drach viewed Emile Picard's application, in 1887, of Galois theory to linear differential equations as a model of perfection and he tried to extend Galois theory to differential equations in general, building on the work of Lie and Vessiot in addition to that of Emile Picard.
       
     • Other papers by Drach include three published in 1908: Sur les systemes completement orthogonaux de l'espace euclidien a n dimensions; Recherches sur certaines deformations remarquables a reseau conjugue persistant; and Sur le probleme logique de l'integration des equations differentielles.
       
     • After the war ended he published his geometric approach to such problems in L'equation differentielle de la balistique exterieure et son integration par quadratures (1920).
       
     • Drach's results can be compared with the modern treatment of the same class of equations.
       
     • Another example of his work is Sur la theorie des corps plastiques et l'equation d'Airy-Tresca which he published in 1946.
       
     • Around the same time Drach published two papers on partial differential equations: Sur les equations aux derivees partielles du premier ordre dont les caracteristiques sont lignes asymptotiques des surfaces integrales (1947); and Sur des equations aux derivees partielles du premier et du second ordre dont les caracteristiques sont lignes asymptotiques des surfaces integrales (1948).
       

262. Kumano-Go biography
     
     • His supervisor was M Nagumo who supervised his work on the singular perturbation of second order partial differential equations.
       
     • During these years Kumano-Go published a series of papers which studied the local and global uniqueness of the solutions of the Cauchy problem for partial differential equations.
       
     • This is Partial differential equations which was again written in Japanese and was published in 1978.
       
     • This is a textbook which in addition to studying partial differential equations provides an introduction to pseudo-differential operators.
       

263. Levy Hyman biography
     
     • Levy's main work was in numerical methods, numerical solution of differential equations, finite difference equations and statistics.
       
     • Among other mathematical works he published were Numerical Studies in Differential Equations (1934), Elements of the Theory of Probability (1936), and Finite Difference Equations (1958).
       

264. Samoilenko biography
     
     • In 1963 he defended his candidate-degree thesis Application of Asymptotic Methods to the Investigation of Nonlinear Differential Equation with Irregular Right-Hand Side.
       
     • In 1974 Samoilenko became a professor and headed the Integral and Differential Equations section within the Department of Mechanics and Mathematics at the Kiev State University.
       
     • In 1987 Samoilenko was appointed head of the Department of Ordinary Differential Equations at the Institute of Mathematics of the Ukrainian Academy of Sciences in Kiev.
       
     • Samoilenko worked on both linear and nonlinear ordinary differential equations.
       
     • In the 1960s he studied nonlinear ordinary differential equations with impulsive action publishing papers such as Systems with pulses at given times (1967).
       
     • His work on boundary-value problems led to papers Numerical-analytic method for the investigation of systems of ordinary differential equations (2 parts both published in 1966) and many other innovative works.
       
     • His most original contribution was the numeric-analytic method for the study of periodic solutions of differential equations with periodic right hand side.
       
     • The latter is known as the method of successive changes of variables and its aim is to ensure the convergence of the iteration process in solving systems of nonlinear differential equations.
       
     • Their work continued over a long period and was written up in the important joint monograph Impulsive Differential Equations (Russian) in 1987.
       
     • In addition to the work mentioned above they worked jointly on the theory of multifrequency oscillation, then later on a system of evolutionary equations with periodic and conditional periodic coefficients.
       
     • This last work was done in collaboration with D Martyniuk and the three of them published, in 1984, the monograph Systems of Evolution Equations with Periodic and Quasiperiodic Coefficients (Russian) giving an excellent account of their results.
       
     • For example, with Mytropolsky and V L Kulik, he wrote Investigation of Dichotomy of Linear Systems of Differential Equations Using Lyapunov Functions (Russian) published in 1990.
       
     • In 1992 they published Numerical-analytic methods in the theory of boundary value problems for ordinary differential equations.
       
     • The book is devoted to the theory of generalized inverses of operators in a Banach space and its applications to linear and weakly nonlinear boundary-value problems for various classes of functional-differential equations, including systems of ordinary differential and difference equations, systems of differential equations with delay, systems with impulse action, and integro-differential systems.
       
     • A recent book by Samoilenko, written with Yu V Teplinskii, is Elements of the mathematical theory of evolution equations in Banach spaces (Ukranian) (2008).
       
     • The book Differential equations : Examples and problems (Russian) (1984) written with S A Krivosheya and N A Perestyuk contains the following authors' summary:- .
       
     • We give the solutions of typical problems in a course on ordinary differential equations.
       
     • The text is structured so as to develop practical skills in students for solving and investigating differential equations describing evolutionary processes in different fields of natural science.
       
     • He is on the Editorial Board of: Nonlinear Oscillations; the Ukrainian Mathematical Journal; Reports of the Ukrainian Academy of Sciences; the Bulletin of the Ukrainian Academy of Sciences; the Ukrainian Mathematical Bulletin; In the World of Mathematics; the Memoirs on Differential Equations and Mathematical Physics; the Miskolc Mathematical Notes; the Georgian Mathematical Journal; and the International Journal of Dynamical Systems and Differential Equations.
       

265. Shtokalo biography
     
     • Shtokalo worked mainly in the areas of differential equations, operational calculus and the history of mathematics.
       
     • After 1945 he became particularly interested in the qualitative and stability theory of solutions of systems of linear ordinary differential equations in the Lyapunov sense and in the 1940s and 1950 published a series of articles and three monographs in these areas.
       
     • Shtokalo's work had a particular impact on linear ordinary differential equations with almost periodic and quasi-periodic solutions.
       
     • He extended the applications of the operational method to linear ordinary differential equations with variable coefficients.
       

266. Monge biography
     
     • The four memoirs that Monge submitted to the Academie were on a generalisation of the calculus of variations, infinitesimal geometry, the theory of partial differential equations, and combinatorics.
       
     • Over the next few years he submitted a series of important papers to the Academie on partial differential equations which he studied from a geometrical point of view.
       
     • .the composition of nitrous acid, the generation of curved surfaces, finite difference equations, partial differential equations (1785); double refraction and the structure of Iceland spar, the composition of iron, steel, and cast iron, and the action of electricity sparks on carbon dioxide gas (1786); capillary phenomena (1787); and the causes of certain meteorological phenomena (1788); and a study in physiological optics (1789).
       

267. Lexell biography
     
     • Lexell made a detailed investigation of exact equations differential equations.
       
     • In addition Lexell developed a theory of integrating factors for differential equations at the same time as Euler but, although it has often been thought that he learnt of the technique from Euler, the author of [Istor.-Mat.
       
     • Lexell did work in analysis on topics other than differential equations, for example he suggested a classification of elliptic integrals and he worked on the Lagrange series.
       

268. Floquet biography
     
     • Floquet submitted his doctoral thesis Sur la theorie des equations differentielles lineaires (On the theory of linear differential equations) to the Faculty of Science in Paris on 8 April 1879.
       
     • For example he published three papers with the title Sur les equations differentielles lineaires a coefficients periodiques (On linear differential equations with periodic coefficients), two in 1881 and the third in 1883.
       
     • In 1884 he published Addition a un memoire sur les equations differentielles lineaires (Addition to a memoir on linear differential equations) and Sur les equations differentielles lineaires a coefficients doublement periodiques (On linear differential equations with double periodic coefficients).
       
     • For example he published Sur une classe d'equations differentielles lineaires non homogenes (1887), Sur une propriete de la surface xyz = l3 (1888), Sur le mouvement d'un fil dans un plan fixe (1889), Sur l'equation de Lame (1895), Sur le mouvement d'un point ou d'un fil glissant sur un plan horizontal fixe lorsqu'on tient compte de la rotation de la terre et du frottement (1898), Sur les equations intrinseques du mouvement d'un fil et sur le calcul de sa tension (1901), and L'astronome Messier (1902).
       

269. Jacobi biography
     
     • By the time Jacobi left school he had read advanced mathematics texts such as Euler's Introductio in analysin infinitorum and had been undertaking research on his own attempting to solve quintic equations by radicals.
       
     • Kummer had made advances beyond what Jacobi had achieved on third-order differential equations and Jacobi wrote to his brother Moritz in 1836 describing how Kummer had managed to solve problems which had defeated him.
       
     • Jacobi carried out important research in partial differential equations of the first order and applied them to the differential equations of dynamics.
       

270. Bernoulli Jacob biography
     
     • In May 1690 in a paper published in Acta Eruditorum, Jacob Bernoulli showed that the problem of determining the isochrone is equivalent to solving a first-order nonlinear differential equation.
       
     • After finding the differential equation, Bernoulli then solved it by what we now call separation of variables.
       
     • In 1696 Bernoulli solved the equation, now called "the Bernoulli equation", .
       

271. Ritt biography
     
     • Ritt resigned his position at the Naval Observatory and began working for his doctorate which was awarded in 1917 for his thesis On a general class of linear homogeneous differential equations of infinite order with constant coefficients.
       
     • Ritt had begun a new major research topic in the 1930s when he began to create a theory of ordinary and partial differential equations.
       
     • The first book was Differential equations from an algebraic standpoint (1932) and the second, a very major revision and extension of the first, was Differential Algebra (1950).
       
     • In the last three years of his life Ritt began a deep study of the applications of Lie theory to homogeneous differential equations.
       

272. Peterson biography
     
     • The dissertation contains a derivation of two equations equivalent to those of Mainardi and Codazzi, and in it Peterson outlined a proof of the fundamental theorem of surface theory.
       
     • His main work is in differential geometry but he obtained an honorary doctorate for his work on partial differential equations.
       
     • This was in 1879 from the Novorossiiskii University of Odessa in recognition for his outstanding contributions to the theory of characteristics of partial differential equations [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]:- .
       
     • by means of a uniform general method, he deduced nearly all the devices known at that time for finding general solutions of different classes of equations.
       
     • These included: Sur l'integration des equations aux derivees partielles; Sur la deformation des surfaces du second ordre; Sur les courbes tracees sur les surfaces; and Sur les relations et les affinites entre les surfaces courbes.
       

273. Malfatti biography
     
     • Malfatti wrote an important work on equations of the fifth degree.
       
     • His papers dealt with many subjects from probability to mechanics and he participated in the debate around Ruffini's attempt to prove the impossibility of solving (in the meaning of that period) equations of higher degree than four.
       
     • These include: Problems and methods of mathematical analysis in the work of Gianfrancesco Malfatti, Contributions of Gianfrancesco Malfatti to combinatorial analysis and to the theory of finite difference equations, The work of Malfatti in the realm of mechanics, The geometrical research of Gianfrancesco Malfatti, Gianfrancesco Malfatti and the theory of algebraic equations, and Gianfrancesco Malfatti and the support problem.
       

274. Ricci-Curbastro biography
     
     • In 1875 Ricci-Curbastro was awarded a doctorate for his thesis On Fuchs's research concerning linear differential equations.
       
     • Three of these articles appeared in Nuovo Cimento in 1877 and, in the same year, an article appeared in Giornale di matematiche di Battaglini which Dini had asked him to write on Lagrange's problem on a system of linear differential equations.
       
     • Ricci-Curbastro's early work was in mathematical physics, particularly on the laws of electric circuits and differential equations.
       
     • In the paper, applications are given by Ricci-Curbastro and Levi-Civita to the classification of the quadratic forms of differentials and there are other analytic applications; they give applications to geometry including the theory of surfaces and groups of motions; and mechanical applications including dynamics and solutions to Lagrange's equations.
       

275. Edge biography
     
     • The equation of the scroll of tangents of the common curve of two quadrics is due to Cayley in 1850.
       
     • Salmon, in his famous text, gave an equation in covariant form.
       
     • Edge gave a procedure for finding this equation in 1979.
       
     • Bring's curve was first studied in Klein's 1884 book in connection with the transformation to reduce the general quintic equation to the form x5 + ex + f = 0.
       

276. Cockle biography
     
     • Most of his work, however, was in pure mathematics where he studied algebra, the theory of equations, and differential equations.
       
     • Describing Cockle's contributions to differential equations Harley explained that his [Proc.
       
     • mode of dealing with the theory of differential equations was marked by originality and independence of mind.
       

277. Horner biography
     
     • Horner is largely remembered only for the method, Horner's method, of solving algebraic equations ascribed to him by Augustus De Morgan and others.
       
     • This discussion is somewhat moot because the method was anticipated in 19th century Europe by Paolo Ruffini (it won him the gold medal offered by the Italian Mathematical Society for Science who sought improved methods for numerical solutions to equations), but had, in any case, been considered by Zhu Shijie in China in the thirteenth century.
       
     • Horner made other mathematical contributions, however, publishing a series of papers on transforming and solving algebraic equations, and he also applied similar techniques to functional equations.
       

278. Brodetsky biography
     
     • The 5th International Congress for Applied Mechanics was held at Cambridge, Massachusetts, in 1938 and Brodetsky delivered a paper on the equations of motion of an airplane.
       
     • Sections: Equations of motion, coefficients of statical and dynamical stability k, t; first approximation.
       
     • Sections: Equations of motion, first approximation.
       
     • Sections: Symmetrical aeroplane: equations of motion, first approximation.
       

279. Curry biography
     
     • He was given a topic in the theory of differential equations by George Birkhoff but he began reading books on logic which seemed to him far more interesting that his research topic.
       
     • Curry now made his final change in direction and decided to give up his doctoral studies on differential equations and to write a doctoral dissertation on logic.
       
     • this advantage, of course, implies a restriction on the scope of the treatment, because it is limited to the rational aspects such as arise from ordinary linear differential equations with constant coefficients.
       
     • For the more general cases of partial differential equations, fractional operators, etc., the theory of integral transforms is doubtless unavoidable.
       

280. Szekeres biography
     
     • The reference to chaos theory here refers in particular to his interest in Feigenbaum's functional equation.
       
     • Another unusual quality was George's interest in computational and "experimental" mathematics, which he maintained until his last paper on Abel's equation.
       
     • In the mid-90s, I worked with him on Feigenbaum's functional equation.
       
     • We wrote programs to solve several cases of this equation, and I was very impressed by this 80 year-old who knew more about how to actually get computers to do real mathematics than many of my younger colleagues.
       

281. Wu Wen-Tsun biography
     
     • He based his method on the idea of a characteristic set which had been introduced by Joseph Ritt in his algebraic and algorithmic approach to differential equations.
       
     • It is in Chapter 4 that Wu explains how to translate geometrical problems into polynomial equations.
       
     • In 2000 Wu published Mathematics mechanization : Mechanical geometry theorem-proving, mechanical geometry problem-solving and polynomial equations-solving.
       

282. Raphson biography
     
     • His election to that Society was on the strength of his book Analysis aequationum universalis which was published in 1690 contained the Newton method for approximating the roots of an equation.
       
     • Newton-Raphson method of solving equations .
       

283. Dionysodorus biography
     
     • The Dionysodorus we are interested in here is the mathematician Dionysodorus who Eutocius states solved the problem of the cubic equation using the intersection of a parabola and a hyperbola.
       
     • Strabo distinguishes this Dionysodorus from Dionysodorus of Amisene and it is now thought that the Dionysodorus referred to by Pliny is not the mathematician who solved the problem of the cubic equation.
       
     • Shortly after Cronert published details of the fragments of papyri relating to Dionysodorus which had been found at Herculaneum, Schmidt published a commentary on the material in which he argued convincingly that the Dionysodorus who solved the cubic equation using the intersection of a parabola and a hyperbola was the Dionysodorus of Caunus referred to in the Herculaneum papyrus.
       

284. Ferrers biography
     
     • They range over such subjects as quadriplanar co-ordinates, Lagrange's equations and hydrodynamics.
       
     • In 1853 Sylvester published On Mr Cayley's impromptu demonstration of the rule for determining at sight the degree of any symmetrical function of the roots of an equation expressed in terms of the coefficients in the Philosophical Magazine.
       

285. Euclid biography
     
     • Euclid's geometric solution of a quadratic equation .
       
     • History Topics: Quadratic, cubic and quartic equations .
       

286. Bellavitis biography
     
     • enables us to express by means of formulae the results of geometric constructions, to represent geometric propositions by means of equations, and to replace a logical argument by the transformation of equations.
       
     • In algebra he continued Ruffini's work on the numerical solution of algebraic equations and he also worked on number theory.
       

287. Sommerfeld biography
     
     • In this thesis he studied the representation of arbitrary functions by the eigenfunctions of partial differential equations and other given sets of functions.
       
     • His work on this topic contains important theory of partial differential equations.
       
     • He lectured on a wide range of topics, giving lectures on probability and also on the partial differential equations of physics.
       

288. De Giorgi biography
     
     • In 1955 De Giorgi gave an important example which showed nonuniqueness for solutions of the Cauchy problem for partial differential equations of parabolic type whose coefficents satisfy certain regularity conditions.
       
     • In the following year he proved what has become known as "De Giorgi's Theorem" concerning the Holder continuity of solutions of elliptic partial differential equations of second order.
       
     • The authors of this paper are all students of De Giorgi and they describe his contributions to geometric measure theory, the solution of Hilbert's XIXth problem in any dimension, the solution of the n-dimensional Plateau problem, the solution of the n-dimensional Bernstein problem, some results on partial differential equations in Gevrey spaces, convergence problems for functionals and operators, free boundary problems, semicontinuity and relaxation problems, minimum problems with free discontinuity set, and motion by mean curvature.
       

289. Haselgrove biography
     
     • The problem of stellar evolution is expressed, mathematically, by a set of non-linear partial differential equations describing the variation of density and temperature as a function of time and of distance from the star centre.
       
     • In The solution of non-linear equations and of differential equations with two-point boundary conditions (1961) Haselgrove suggests general iterative techniques, based on an n-dimensional extension of the Newton-Raphson process.
       

290. Markov biography
     
     • He wrote his first mathematics paper while at the Gymnasium but his results on integration of linear differential equations which were presented in the paper were not new.
       
     • Markov graduated in 1878 having won the gold medal for submitting the best essay for the prize topic set by the faculty in that year - On the integration of differential equations by means of continued fractions.
       
     • During his lectures he did not bother about the order of equations on the blackboard, nor about his personal appearance.
       

291. Thymaridas biography
     
     • Thymaridas also gave methods for solving simultaneous linear equations which became known as the 'flower of Thymaridas'.
       
     • For the n equations in n unknowns .
       
     • He also shows how certain other types of equations can be put into this form.
       

292. Lehmer Derrick biography
     
     • The chapter headings are: Lucas' functions; Tests for primality; Continued fractions; Bernoulli numbers and polynomials; Diophantine equations; Numerical functions; Matrices; Power residues; Analytic number theory; Partitions; Modular forms; Cyclotomy; Combinatorics; Sieves; Equation solving; Computing techniques; and Miscellaneous.
       

293. Pfeiffer biography
     
     • Pfeiffer did important work on partial differential equations following on from the methods developed by Lie and Lagrange.
       
     • He showed how to find integrals of a general system of partial differential equations by using sequential complete systems instead of passing to Jacobian systems.
       
     • Pfeiffer also constructed all the infinitesimal operators of a system of equations.
       

294. Todd John biography
     
     • Solution of differential equations by recurrence relations (1950); Experiments on the inversion of a 16 × 16 matrix (1953); Experiments in the solution of differential equations by Monte Carlo methods (1954); The condition of the finite segments of the Hilbert matrix (1954); Motivation for working in numerical analysis (1954); and A direct approach to the problem of stability in the numerical solution of partial differential equations (1956).
       

295. Titchmarsh biography
     
     • Other topics to which he made major contributions included entire functions of a complex variable and, working with Hardy, integral equations.
       
     • From 1939 Titchmarsh concentrated on the theory of series expansions of eigenfunctions of differential equations, work which helped to resolve problems in quantum mechanics.
       
     • His work on this topic occupied him for the last 25 years of his life and he published much of it in Eigenfunction Expansions Associated with Second-Order Differential Equations (1946, 1958).
       

296. Koopmans biography
     
     • He showed that the desired result is obtainable by the straightforward solution of a system of equations involving the costs of the materials at their sources and the costs of shipping them by alternative routes.
       
     • He also devised a general mathematical model of the problem that led to the necessary equations.
       
     • In Identification problems in economic model construction (1949) he used matrix methods to study structural equations within a linear economic model.
       

297. Painleve biography
     
     • He worked on differential equations, particularly studying their singular points, and on mechanics.
       
     • His interest in mechanics was a natural one since this subject provided a natural setting for applications of the results which he had proved for differential equations.
       
     • He solved, using Painleve functions, differential equations which Poincare and Emile Picard had failed to solve, showing, as Hadamard wrote, that:- .
       

298. Boersma biography
     
     • The discussion centres around the use of integral representation theory to reduce such problems to Fredholm integral equations which are suitable for the study of low frequency oscillations.
       
     • The results of the airfoil analysis are infinite systems of linear equations, from which numerical results can be obtained by truncation.
       
     • Complex Function Theory, Applied Analysis and Partial Differential Equations which provided the interesting combination of mathematical theory applied to physics problems.
       

299. Friedrichs biography
     
     • He collaborated with Lewy on linear hyperbolic partial differential equations and they wrote a joint paper in 1927, and another joint paper, with Courant and Lewy, considered the stability of difference schemes for partial differential equations.
       
     • He was now interested in operator on Hilbert spaces and applied these tools to initial value problem for hyperbolic equations.
       

300. Sonin biography
     
     • He continued working on his doctorate, essentially equivalent to the German habilitation, and after submitting a thesis on partial differential equations of the second order to the Moscow University he was awarded the degree in 1874.
       
     • He has a sequence of polynomials named after him - the Sonin polynomials Tnm(x) satisfy the differential equation .
       

301. Kirchhoff biography
     
     • Kirchhoff considered an electrical network consisting of circuits joined at nodes of the network and gave laws which reduce the calculation of the currents in each loop to the solution of algebraic equations.
       
     • An early form of the theory had been developed by Germain and Poisson but it was Navier who gave the correct differential equation a few years later.
       

302. Chebyshev biography
     
     • Chebyshev submitted a paper on The calculation of roots of equations in which he solved the equation y = f (x) by using a series expansion for the inverse function of f.
       

303. Narayana biography
     
     • He did this by using an indeterminate equation of the second order, Nx2 + 1 = y2, where N is the number whose square root is to be calculated.
       
     • If x and y are a pair of roots of this equation with x < y then √N is approximately equal to y/x.
       
     • History Topics: Pell's equation .
       

304. Christoffel biography
     
     • Christoffel published papers on function theory including conformal mappings, geometry and tensor analysis, Riemann's o-function, the theory of invariants, orthogonal polynomials and continued fractions, differential equations and potential theory, light, and shock waves.
       
     • How does one compare someone who worked solely in one area with another who contributed to many areas? Again how does one compare someone who worked on differential equations with a geometer? Despite the obvious difficulties, and minor differences of opinion, it is still surprising how much agreement there is on such a ranking.
       
     • It is difficult to compare a differential geometer with a function theorist, or those working on ordinary and partial differential equations with numerical analysts.
       

305. Straus biography
     
     • An approximate solution of the field equations for empty space is obtained and the gravitational potentials thus determined are required to piece together continuously with the known gravitational potentials for a pressure free, spatially constant density of matter.
       
     • This presented a new derivation of the field equations which was necessary since the derivation in Einstein's single authored paper published in the previous year was based on an error.
       
     • Algebraic equations satisfied by roots of natural numbers.
       

306. Atiyah biography
     
     • Subsequently (in collaboration with I M Singer) he established an important theorem dealing with the number of solutions of elliptic differential equations.
       
     • Beyond these linear problems, gauge theories involved deep and interesting nonlinear differential equations.
       
     • In particular, the Yang-Mills equations have turned out to be particularly fruitful for mathematicians.
       

307. Ayyangar biography
     
     • His papers include: Ancient Hindu Mathematics (1921); The Hindu sine Table (1923-24); The mathematics of Aryabhata (1926); The Hindu Arabic numerals (2 parts) (1928,1929); Bhaskara and samclishta kuttaka (1929-30); New light on Bhaskara's chakravala or cyclic method of solving indeterminate equations of the second degree in two variables (1929-30); New proofs of old theorems - Apollonius and Brahmagupta (1920-30); Astronomy - past and present (1930); Some glimpses of ancient Hindu mathematics (1933); Fourteen calendars (1937); A new continued fraction (1937-38); The Bhakshali manuscript (1939); Theory of the nearest square continued fraction (2 parts) (1940, 1941); Peeps into India's mathematical past (1945); and Remarks on Bhaskara's approximation to the sine of an angle (1950).
       
     • He read the paper On the Sexi-Sectional Equation at a meeting of the Society on Friday 7 November 1924.
       

308. Ostrogradski biography
     
     • He made important contributions to partial differential equations, elasticity and to algebra publishing over 80 reports and giving lectures.
       
     • His important work on ordinary differential equations considered methods of solution of non-linear equations which involved power series expansions in a parameter alpha.
       

309. FitzGerald biography
     
     • This was Electricity and Magnetism by Maxwell which, for the first time, contained the four partial differential equations, now known as Maxwell's equations.
       
     • His first work On the equations of equilibrium of an elastic surface filled in cases of a problem studied by Lagrange.
       

310. Wedderburn biography
     
     • He began mathematical research while still an undergraduate and his first paper, On the isoclinal lines of a differential equation of the first order was published in the Proceedings of The Royal Society of Edinburgh in 1903.
       
     • Two other papers which he published in the same year in publications of the Royal Society of Edinburgh were on the scalar functions of a vector and on an application of quaternions to differential equations.
       

311. Bolza biography
     
     • While at Clark, Bolza published the important paper On the theory of substitution groups and its application to algebraic equations in the American Journal of Mathematics.
       
     • Immediately after his return to Germany Bolza continued teaching and research, in particular on function theory, integral equations and the calculus of variations.
       
     • Bolza returned to Chicago for part of 1913 giving lecturers during the summer on function theory and integral equations.
       

312. Cole biography
     
     • Cole returned to Harvard and wrote a thesis A Contribution to the Theory of the General Equation of the Sixth Degree which, as the title indicates, studied equations of degree 6.
       

313. Moisil biography
     
     • While working there he wrote the paper On a class of systems of equations with partial derivatives from mathematical physics.
       
     • Before reading this work Moisil had worked on differential equations, the theory of functions and mechanics.
       
     • Among Moisil's other books we mention: Associated matrices of systems of partial differential equations.
       

314. Killing biography
     
     • Lie algebras were introduced by Lie in about 1870 in his work on differential equations.
       
     • Finally, before we leave our discussion of Killing's work, it is worth noting that he introduced the term 'characteristic equation' of a matrix.
       

315. Simpson biography
     
     • By way of compensation, however, the Newton-Raphson method for solving the equation f (x) = 0 is, in its present form, due to Simpson.
       
     • Newton described an algebraic process for solving polynomial equations which Raphson later improved.
       

316. Maxwell biography
     
     • Maxwell showed that a few relatively simple mathematical equations could express the behaviour of electric and magnetic fields and their interrelation.
       
     • The four partial differential equations, now known as Maxwell's equations, first appeared in fully developed form in Electricity and Magnetism (1873).
       

317. Abraham biography
     

318. Apollonius biography
     
     • He gives propositions determining the centre of curvature which lead immediately to the Cartesian equation of the evolute.
       
     • Included in it are a series of propositions which, though worked out by the purest geometrical methods, actually lead immediately to the determination of the evolute of each of the three conics; that is to say, the Cartesian equations of the evolutes can be easily deduced from the results obtained by Apollonius.
       

319. Zygmund biography
     
     • Among other topics, he worked on summability of numerical series, summability of general orthogonal series, trigonometric integrals, sets of uniqueness, summability of Fourier series, differentiability of functions, smooth functions, approximation theory, absolutely convergent Fourier series, multipliers and translation invariant operators, conjugate series and Taylor series, lacunary trigonometric series, series of independent random variables, random trigonometric series, the Littlewood-Paley, Luzin and Marcinkiewicz functions, boundary values of analytic and harmonic functions, singular integrals, partial differential equations and interpolation operators.
       
     • Their famous joint papers over the next few years on singular integrals and partial differential equations, the most significant of which appeared in 1952, have had a major impact on modern analysis.
       
     • For outstanding contributions to Fourier analysis and its applications to partial differential equations and other branches of analysis, and for his creation and leadership of the strongest school of analytical research in the contemporary mathematical world.
       

320. Zaremba biography
     
     • Much of Zaremba's research work was in partial differential equations and potential theory.
       
     • He studied elliptic equations and in particular contributed to the Dirichlet principle.
       
     • And as for my speciality, why, how could I forget the splendid results in the domain of mixed boundary problems and of harmonic functions, as well as of hyperbolic equations, research by means of which he opened a new path along which contemporary knowledge will proceed in the near future.
       

321. Vallee Poussin biography
     
     • Vallee Poussin's first mathematical research was on analysis, in particular concentrating on integrals and solutions of differential equations.
       
     • One of his first papers in 1892 on differential equations was awarded a prize by the Belgium Academy.
       
     • Volume 2 covered multiple integrals, differential equations, and differential geometry.
       

322. Penrose biography
     
     • In this paper Penrose defined a generalized inverse X of a complex rectangular (or possibly square and singular) matrix A to be the unique solution to the equations AXA = A, XAX = X, (AX)T = AX, (XA)T = XA.
       
     • He used this generalized inverse for problems such as solving systems of matrix equations, and finding a new type of spectral decomposition.
       
     • In the following year Penrose published On best approximation solutions of linear matrix equations which used the generalized inverse of a matrix to find the best approximate solution X to AX = B where A is rectangular and non-square or square and singular.
       
     • His development of Twistor Theory has produced a beautiful and productive approach to the classical equations of mathematical physics.
       

323. Montroll biography
     
     • He used both his expertise in chemistry and mathematics in his thesis Applications of the characteristic value theory of integral equations in which he applied integral equations to the study of imperfect gases.
       
     • In the proceeding of the conference Nonlinear equations in abstract spaces held in 1977 at the University of Texas he published On some mathematical models of social phenomena in which he examined models for population growth and statistical models of other social phenomena.
       

324. Andrews biography
     
     • Andrews had published three papers by the time he had completed his thesis work: An asymptotic expression for the number of solutions of a general class of Diophantine equations (1961); A lower bound for the volume of strictly convex bodies with many boundary lattice points (1963); and On estimates in number theory (1963).
       
     • This last paper, in the American Mathematical Monthly, gave a method for finding an upper bound for the number of solutions of a Diophantine equation of the form y = f (x).
       

325. Rouche biography
     
     • In 1875 Rouche published a two page paper Sur la discussion des equations du premier degre in volume 81 of Comptes Rendus of the Academie des Sciences.
       
     • This short paper contained his result on solving systems of linear equations.
       
     • This is the well-known criterion which says that a system of linear equations has a solution if and only if the rank of the matrix of the associated homogeneous system is equal to the rank of the augumented matrix of the system.
       

326. Riemann biography
     
     • In October he set to work on his lectures on partial differential equations.
       
     • Riemann studied the convergence of the series representation of the zeta function and found a functional equation for the zeta function.
       

327. Petrovsky biography
     
     • Petrovsky's main mathematical work was on the theory of partial differential equations, the topology of algebraic curves and surfaces, and probability.
       
     • Petrovsky also worked on the boundary value problem for the heat equation and this was applied to both probability theory and work of Kolmogorov.
       

328. Laurent Hermann biography
     
     • He wrote 30 books and a fair number of papers on infinite series, equations, differential equations and geometry.
       
     • The last three volumes are devoted entirely to the solution and application of ordinary and partial differential equations.
       

329. Wright Sewall biography
     
     • Another paper by Wright which shows his mathematical approach to the subject is The differential equation of the distribution of gene frequencies which he published in 1945.
       
     • He derives differential equations which are satisfied by the probability density function of the distribution of gene frequencies under certain conditions.
       

330. Milne Archibald biography
     
     • He read papers at meetings of the Society such as Notes on the equation of the parabolic cylinder on Friday 9 January 1914, The Conformal Representation of the Quotient of two Bessel Functions on 24 January 1916, and Note on the Peano-Baker method of solving linear differential equations on 11 February 1916.
       

331. Runge biography
     
     • Runge then worked on a procedure for the numerical solution of algebraic equations in which the roots were expressed as infinite series of rational functions of the coefficients.
       
     • There were three standard methods for the numerical solution of such equations, namely by Newton, Bernoulli and Graffe, and the method found by Runge had all three of the standard methods as special cases.
       
     • He worked out many numerical and graphical methods, gave numerical solutions of differential equations, etc.
       

332. Codazzi biography
     
     • The formulas give two relations between the first and second quadratic forms over a surface together with an equation, already found by Gauss, which gives necessary and sufficient conditions for the existence of a surface which admits two given quadratic forms.
       
     • 6 (2) (1979), 137-163.',3)">3] shows that, in 1853, Karl M Peterson, then a student of Minding at the University of Dorpat (now named Tartu), submitted a dissertation containing a derivation of two equations equivalent to those of Mainardi and Codazzi and outlining a proof of the fundamental theorem of surface theory.
       

333. Brocard biography
     
     • The text consists of brief descriptive paragraphs, with diagrams and equations of these curves.
       
     • In 1876, Brocard asked if the only solutions to the equation n! + 1 = m2, in positive integers (n, m), are (4, 5), (5, 11), (7, 71).
       

334. Borchardt biography
     
     • Borchardt's doctoral work, on non-linear differential equations, was supervised by Jacobi and submitted in 1843.
       
     • Borchardt also generalised results of Kummer on equations determining the secular disturbances of the planets.
       
     • In several further papers Borchardt applied the theory of determinants to algebraic equations, mostly in connection with symmetric functions, the theory of elimination, and interpolation.
       

335. Baudhayana biography
     
     • The Sulbasutra of Baudhayana contains geometric solutions (but not algebraic ones) of a linear equation in a single unknown.
       
     • Quadratic equations of the forms ax2 = c and ax2 + bx = c appear.
       

336. Gutzmer biography
     
     • He submitted his theses Uber gewisse partielle Differentialgleichungen hoherer Ordnung (On certain partial differential equations of higher order) to the University of Halle-Wittenberg and was awarded his doctorate on 13 January 1893.
       
     • He submitted his thesis Zur Theorie der adjungierten Differentialgleichungen (On the theory of adjoint differential equations) to the University of Halle-Wittenberg on 23 April 1896 and worked there as a privatdozent until March 1899.
       
     • Among the advanced courses he taught we list: ordinary differential equations, analytic mechanics, calculus of variations, number theory, higher algebra, function theory and the theory of algebraic curves.
       

337. Julia biography
     
     • Volume 3 contains four parts: (i) Functional equations and conformal mapping; (ii) Conformal mapping; (iii) General lectures; and (iv) Isolated works in analysis on Implicit function defined by the vanishing of an active function, and on certain series.
       
     • Volume 4 is again in four parts: (i) Functional calculus and integral equations; (ii) Quasianalyticity; (iii) Various techniques of analysis; and (iv) Works concerning Hilbert space.
       
     • The applications to the theory of matrices and equations, which are largely implicit, in certain of the more abstract treatments, are elaborated here with a wealth of detail which renders them unusually accessible to the student.
       

338. Dudeney biography
     
     • Other puzzles simply reduced to systems of linear equations if a mathematical solution was sought.
       
     • Problem 11 from the same book reduces to a quadratic equation:- .
       

339. Golab biography
     
     • Professor Golab dealt with different fields of mathematics such as geometry, topology, algebra, analysis, logic, functional and differential equations, the theory of numerical methods and various applications of mathematics.
       
     • may be considered as the father figure of the Polish school of functional equations.
       
     • All Polish mathematicians working in the theory of functional equations are - directly or indirectly - pupils of Professor Golab.
       

340. Barnes biography
     
     • His early work was concerned with various aspects of the gamma function, including generalisations of this function given by the so-called Barnes G-function, which satisfies the equation .
       
     • He also considered second-order linear difference equations connected with the .
       

341. Praeger biography
     
     • Bernhard Neumann suggested a problem to her which she solved and so published her first paper Note on a functional equation while still an undergraduate.
       
     • Praeger had studied the functional equation x(n+1) - x(n) = x2(n), where x2(n) = x(x(n)) and x is an integer-valued function of the integer variable n, and found a three-parameter family of solutions.
       
     • In fact Praeger wrote one joint paper with her husband, Note on primitive permutation groups and a Diophantine equation, which was published by the journal Discrete Mathematics in 1980.
       

342. Antonelli biography
     
     • However the war had ended before the machine came into service but it was still used for the numerical solution of differential equations as intended.
       
     • Petzinger, in [Wall Street Journal (November 1996).',3)">3], describes the way that McNulty used ENIAC to solve differential equations after the construction of the machine was complete in February 1946:- .
       
     • The first task was breaking down complex differential equations into the smallest possible steps.
       

343. Mytropolsky biography
     
     • Mytropolsky has made major contributions to the theory of oscillations and nonlinear mechanics as well as the qualitative theory of differential equations.
       
     • Using a method of successive substitutes, he constructed a general solution for a system of nonlinear equations and studied its behaviour in the neighbourhood of the quasi-periodic solution.
       
     • This work was to lead to further advances by the Kiev school, in particular they applied asymptotic methods to partial and functional differential equations.
       

344. Herglotz biography
     
     • In this last paper Herglotz solved Abel's integral equation which results from the inversion of measured seismic travel times into a velocity-depth function.
       
     • There are two sections, one of five chapters on classical theory of the mechanics of continua based on Hamilton's principle and another of four chapters on partial differential equations.
       

345. Ulugh Beg biography
     
     • This excellent book records the main achievements which include the following: methods for giving accurate approximate solutions of cubic equations; work with the binomial theorem; Ulugh Beg's accurate tables of sines and tangents correct to eight decimal places; formulae of spherical trigonometry; and of particular importance, Ulugh Beg's Catalogue of the stars, the first comprehensive stellar catalogue since that of Ptolemy.
       
     • The calculation is built on an accurate determination of sin 1° which Ulugh Beg solved by showing it to be the solution of a cubic equation which he then solved by numerical methods.
       

346. Gauss biography
     
     • In the first volume he discussed differential equations, conic sections and elliptic orbits, while in the second volume, the main part of the work, he showed how to estimate and then to refine the estimation of a planet's orbit.
       
     • Gauss used the Laplace equation to aid him with his calculations, and ended up specifying a location for the magnetic South pole.
       

347. Lame biography
     
     • He used them to transform Laplace's equation into ellipsoidal coordinates and so separate the variables and solve the resulting equation.
       
     • This happened with curvilinear coordinates for he was led to study the equation .
       

348. Cantor biography
     
     • the numbers which are roots of polynomial equations with integer coefficients, were countable.
       
     • A transcendental number is an irrational number that is not a root of any polynomial equation with integer coefficients.
       

349. Poincare biography
     
     • His thesis was on differential equations and the examiners were somewhat critical of the work.
       
     • The idea was to come in an indirect way from the work of his doctoral thesis on differential equations.
       
     • He can be said to have been the originator of algebraic topology and, in 1901, he claimed that his researches in many different areas such as differential equations and multiple integrals had all led him to topology.
       

350. Denjoy biography
     
     • These great mathematicians gave Denjoy a strong background in complex function theory, continued fractions and differential equations and set him on the road to his great discoveries.
       
     • In 1934 he wrote that his greatest achievements had been the integration of derivatives, the computation of the coefficients of a converging trigonometric series, a theorem on quasi-analytic functions, and differential equations on a torus.
       
     • Choquet, very fairly, suggests that Denjoy's work on differential equations on a torus, not nearly so highly rated by Denjoy himself, is one of his most influential pieces of work and has [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]:- .
       

351. Ortega biography
     
     • In the second part of the book, devoted mostly to geometry, Ortega gives a method of extracting square roots very accurately using Pell's equation, which is surprising since a general solution to Pell's equation does not appear to have been found before Fermat over 100 years later.
       
     • Pell's equation .
       

352. Lobachevsky biography
     
     • Despite this heavy administrative load, Lobachevsky continued to teach a variety of different topics such as mechanics, hydrodynamics, integration, differential equations, the calculus of variations, and mathematical physics.
       
     • In 1834 Lobachevsky found a method for the approximation of the roots of algebraic equations.
       
     • This method of numerical solution of algebraic equations, developed independently by Graffe to answer a prize question of the Berlin Academy, is today a particularly suitable method for using computers to solve such problems.
       

353. Bell John biography
     
     • We may say that when the state-vector is α+ or α- respectively, sz is equal to planck /2 and - planck /2 respectively, but, if one restricts oneself to the Schrodinger equation, sx and sy just do not have values.
       
     • It is clear that we could try to recover realism and determinism if we allowed the view that the Schrodinger equation, and the wave-function or state-vector, might not contain all the information that is available about the system.
       
     • Nevertheless it would appear natural that the possibility of supplementing the Schrodinger equation with hidden variables would have been taken seriously.
       

354. Ramanujan biography
     
     • Ramanujan was shown how to solve cubic equations in 1902 and he went on to find his own method to solve the quartic.
       
     • He devoloped relations between elliptic modular equations in 1910.
       
     • Ramanujan worked out the Riemann series, the elliptic integrals, hypergeometric series and functional equations of the zeta function.
       

355. Darboux biography
     
     • Darboux generalised results of Kummer giving a system defined by a single equation with many interesting properties.
       
     • This integral was introduced in a paper on differential equations of the second order which he wrote in 1870.
       

356. Fermi biography
     
     • He showed great talents, especially in mathematics and by the time he left elementary school at the age of ten he was puzzling out how the equation x2 + y2 = r2 represented a circle.
       
     • In his essay Fermi derived the system of partial differential equations for a vibrating rod, then used Fourier analysis to solve them.
       

357. Rahn biography
     
     • The book, written in German, contains an example of Pell's equation.
       
     • Pell's equation .
       
     • History Topics: Pell's equation .
       

358. Sokhotsky biography
     
     • His doctoral dissertation On definite integrals and functions with applications to expansion of series was an early investigation of the theory of singular integral equations.
       
     • One of the first to approach problems of the theory of singular integral equations, Sokhotsky in this work considered important boundary properties of integrals of the type of Cauchy and, essentially, arrived at the so-called formulas of I Plemelj (1908).
       
     • His work is important in the development of the theory of functions, in particular having applications in the theory of hypergeometric series and differential equations.
       

359. Sylow biography
     
     • Finding Abel's papers on the solvability of algebraic equations by radicals more interesting, Sylow was led from them (by the professor in applied mathematics, Carl Bjerknes) to Galois.
       
     • made myself acquainted with newer works, particularly in the theory of equations.
       
     • In his lectures Sylow explained Abel's and Galois's work on algebraic equations.
       

360. Germain biography
     
     • Lagrange, who was one of the judges in the contest, corrected the errors in Germain's calculations and came up with an equation that he believed might describe Chladni's patterns.
       
     • She demonstrated that Lagrange's equation did yield Chladni's patterns in several cases, but could not give a satisfactory derivation of Lagrange's equation from physical principles.
       

361. Al-Kashi biography
     
     • He also considered the equation associated with the problem of trisecting an angle, namely a cubic equation.
       
     • He was not the first to look at approximate solutions to this equation since al-Biruni had worked on it earlier.
       

362. Mahavira biography
     
     • He also discussed integer solutions of first degree indeterminate equation by a method called kuttaka.
       
     • An example of a problem given in the Ganita Sara Samgraha which leads to indeterminate linear equations is the following: .
       

363. Tauber biography
     
     • Of lesser importance is Tauber's work on differential equations and the gamma function, but let us give the title of one of his papers on this latter topic, namely uber die unvollstandigen Gammafunktionen (1906).
       
     • In particular his papers Uber die Hypothekenversicherung (1897) and Gutachten fur die sechste internationale Tagung der Versicherungswissenschaften (1909) contain his formulation of the Tisiko equation.
       

364. Frobenius biography
     
     • On the algebraic solution of equations, whose coefficients are rational functions of one variable.
       
     • The theory of linear differential equations.
       
     • In his work in group theory, Frobenius combined results from the theory of algebraic equations, geometry, and number theory, which led him to the study of abstract groups.
       

365. Milne-Thomson biography
     
     • In 1948 he published Applications of elliptic functions to wind tunnel interference while in 1957 he wrote a review paper A general solution of the equations of hydrodynamics which M G Scherberg reviews as follows:- .
       
     • For example he wrote Consistency equations for the stresses in isotropic elastic and plastic materials (1942), and Stress in an infinite half-plane (1947).
       
     • He gave two lectures in Madrid in 1951 on the elements of finite elasticity theory, the first lecture covering the topics of deformation tensors, stress, equations of motion, and energy.
       

366. Eckert Wallace biography
     
     • The first is the development of the theory or the solution of the differential equations of motion expressing the coordinates of the moon as explicit functions of time.
       
     • In order to bring the Tables within even their present length, various parts of the basic equations were curtailed whenever permissible in the light of observational requirements (as then visualised).
       
     • Eckert therefore decided not to recompute new tables but to compute the ephemerides directly from Brown's equations.
       

367. Goldbach biography
     
     • He also studied equations and worked out in his correspondence with Euler how to provide a quick test for whether an algebraic equation has a rational root.
       

368. Meissel biography
     
     • Meissel's mathematical interests covered the following fields: number theory (in particular, properties of prime numbers), theta functions, elliptic functions, spherical trigonometry, hydrodynamics, ordinary differential equations, asymptotic expansions, and Bessel functions.
       
     • a forerunner (in the theory of Bessel functions, in connection with Emden's equation etc.).
       

369. Boole biography
     
     • Boole had begun to correspond with De Morgan in 1842 and when in the following year he wrote a paper On a general method of analysis applying algebraic methods to the solution of differential equations he sent it to De Morgan for comments.
       
     • Boole also worked on differential equations, the influential Treatise on Differential Equations appeared in 1859, the calculus of finite differences, Treatise on the Calculus of Finite Differences (1860), and general methods in probability.
       

370. Wilczynski biography
     
     • By that time he had published over a dozen papers in astronomy, but his interests moved towards differential equations which arose in his study of the dynamics of astronomical objects.
       
     • From there his interests became pure mathematical interests in differential equations.
       
     • But Wilczynski was the first ever to appreciate, demonstrate and exploit the utility of completely integrable systems of linear homogeneous differential equations for projective differential geometry.
       

371. D'Alembert biography
     
     • He was a pioneer in the study of partial differential equations and he pioneered their use in physics.
       
     • The article contains the first appearance of the wave equation in print but again suffers from the defect that he used mathematically pleasing simplifications of certain boundary conditions which led to results which were at odds with observation.
       

372. Cauchy biography
     
     • He did important work on differential equations and applications to mathematical physics.
       
     • Numerous terms in mathematics bear Cauchy's name:- the Cauchy integral theorem, in the theory of complex functions, the Cauchy-Kovalevskaya existence theorem for the solution of partial differential equations, the Cauchy-Riemann equations and Cauchy sequences.
       

373. Tarski biography
     
     • In 1968 Tarski wrote another famous paper Equational logic and equational theories of algebras in which he presented a survey of the metamathematics of equational logic as it then existed as well as giving some new results and some open problems.
       

374. Sundman biography
     
     • To regularize the singularity of the differential equations of motion, in the 1912 paper mentioned above, Sundman introduced a new independent variable which regularizes the motion within a band of finite breadth.
       
     • It is a matter of construction a machine for solving systems of second order differential equations.
       
     • Although designed to calculate astronomical perturbations, the machine essentially functions as an integrator for differential equations and could be used for a large number of other problems.
       

375. Einstein biography
     
     • This seemed to contradict classical electromagnetic theory, based on Maxwell's equations and the laws of thermodynamics which assumed that electromagnetic energy consisted of waves which could contain any small amount of energy.
       
     • In fact Hilbert submitted for publication, a week before Einstein completed his work, a paper which contains the correct field equations of general relativity.
       

376. Duhamel biography
     
     • He published articles such as Sur les equations generales de la propagation de la chaleur dans les corps solides dont la conductibilite n'est pas la meme dans tous les sens (1832) and Sur la methode generale relative au mouvement de la chaleur dans les corps solides plonges dans des milieux dont la temperature varie avec le temps (1833) in the Journal of the Ecole Polytechnique.
       
     • Duhamel worked on partial differential equations and applied his methods to the theory of heat, to rational mechanics, and to acoustics.
       
     • 'Duhamel's principle' in partial differential equations arose from his contributions to the distribution of heat in a solid with a variable boundary temperature.
       

377. Coulson biography
     
     • In almost every case the fundamental problem is the same, since it consists in solving the standard equation of wave motion; the various applications differ chiefly in the conditions imposed upon these solutions.
       
     • Let us indicate the contents of the 156 page book: The equation of wave motion; Waves on strings; Waves in membranes; Longitudinal waves in bars and springs; Waves in liquids; Sound waves; Electric waves; General considerations.
       

378. Aryabhata I biography
     
     • It also contains continued fractions, quadratic equations, sums of power series and a table of sines.
       
     • This work is the first we are aware of which examines integer solutions to equations of the form by = ax + c and by = ax - c, where a, b, c are integers.
       

379. Leibniz biography
     
     • Another major mathematical work by Leibniz was his work on determinants which arose from his developing methods to solve systems of linear equations.
       
     • History Topics: Quadratic, cubic and quartic equations .
       

380. Bjerknes Vilhelm biography
     
     • Vilhelm Bjerknes and his associates at Bergen succeeded in devising equations relating the measurable components of weather, but their complexity precluded the rapid solutions needed for forecasting.
       
     • The next step forward in the mathematical approach was due to Richardson in 1922 when he reduced the complicated equations produced by Bjerknes's Bergen School to long series of simple arithmetic operations.
       

381. Patodi biography
     
     • His doctoral thesis, Heat equation and the index of elliptic operators, was supervised by M S Narasimhan and S Ramanan and the degree was awarded by the University of Bombay in 1971.
       
     • An analytic approach, via the heat equation yields easily a formula for the index of an elliptic operator on a compact manifold: but, the formula involves an integrand containing too many derivatives of the symbol, while from the Atiyah-Singer index theorem one would expect only two derivatives to figure.
       

382. Taussky-Todd biography
     
     • While in Gottingen Taussky also edited Artin's lectures in class field theory (1932), assisted Emmy Noether in her class field theory and Courant with his differential equations course.
       
     • For the first time I realised the beauty of research on differential equations - something that my former boss, Professor Courant, had not been able to instil in me.
       

383. Toeplitz biography
     
     • When he arrived there Hilbert was completing his theory of integral equations.
       
     • A major joint project with Hellinger to write a major encyclopaedia article on integral equations, which they worked on for many years, was completed during this time and appeared in print in 1927.
       

384. Scheffe biography
     
     • He wrote a doctoral thesis on differential equations and was awarded his PhD in 1935.
       
     • Scheffe's doctoral dissertation The Asymptotic Solutions of Certain Linear Differential Equations in Which the Coefficient of the Parameter May Have a Zero was supervised by Rudolph E Langer.
       

385. Plancherel biography
     
     • He applied his results in the theory of hyperbolic and parabolic partial differential equations.
       
     • In algebra Plancherel obtained results on quadratic forms and their applications, to the solvability of systems of equations with infinitely many variables and to the theory of commutative Hilbert algebras (theorem of Plancherel-Godement).
       

386. Shannon biography
     
     • At the Massachusetts Institute of Technology he also worked on the differential analyser, an early type of mechanical computer developed by Vannevar Bush for obtaining numerical solutions to ordinary differential equations.
       
     • The most important results [mostly given in the form of theorems with proofs] deal with conditions under which functions of one or more variables can be generated, and conditions under which ordinary differential equations can be solved.
       

387. Banach biography
     
     • However, an exception was made to allow him to submit On Operations on Abstract Sets and their Application to Integral Equations.
       
     • The theory generalised the contributions made by Volterra, Fredholm and Hilbert on integral equations.
       

388. Orlicz biography
     
     • In recent decades those spaces have been used in analysis, constructive theory of functions, differential equations, integral equations, probability, mathematical statistics, etc.
       

389. Minding biography
     
     • Minding also worked on differential equations, algebraic functions, continued fractions and analytic mechanics.
       
     • In differential equations he used integrating factor methods.
       

390. Engel biography
     
     • He also wrote on continuous groups and partial differential equations, translated works of Lobachevsky from Russian to German, wrote on discrete groups, Pfaffian equations and other topics.
       

391. Bourgain biography
     
     • Bourgain's work touches on several central topics of mathematical analysis: the geometry of Banach spaces, convexity in high dimensions, harmonic analysis, ergodic theory, and finally, nonlinear partial differential equations from mathematical physics.
       
     • ',2)">2] contains a survey relating to Bourgain's work on nonlinear partial differential equations from mathematical physics, including later results than was covered in the articles describing his work up to the award of the Fields Medal.
       

392. Bortolotti biography
     
     • In the 1940 paper on Babylonian mathematics, Bortolotti gives a summary of problems published by Neugebauer but argues that the fact that large series of examples for quadratic equations are made up from the same roots demonstrates that this pair of roots has an 'arcane mystic property'.
       
     • It is also wrong to deny the existence of approximations to irrational square roots, to assume a geometrical basis of the quadratic equations or to deny the existence of texts of this type in the Hellenistic period.
       

393. Woods biography
     
     • Woods own description of his 1953 paper The relaxation treatment of singular points in Poisson's equation states:- .
       
     • If F is harmonic or is a solution to Poisson's equation, it may have singular points in the field or on the boundary at which it (a) has finite values, but has infinite derivatives, (b) has logarithmic infinities, or (c) has simple discontinuities.
       

394. Loewy biography
     
     • Loewy worked on linear groups, the algebraic theory of differential equations and actuarial mathematics.
       
     • He also published papers (in German) in the Transactions of the American Mathematical Society such as: On the reducibility of real groups of linear homogeneous substitutions (1903); On group theory, with applications to the theory of linear homogeneous differential equations (1904); and On completely reducible groups that belong to a group of linear homogeneous substitutions (1905).
       

395. Remak biography
     
     • these equations are very awkward to handle mathematically.
       
     • There is, however, work in progress concerning the numerical solution of linear equations with several unknowns using electrical circuits.
       

396. Libri biography
     
     • A little-known consequence of these disputes is that Liouville made his famous announcement of Evariste Galois's important work on the theory of equations in response to an attack by Libri in 1843.
       
     • However he made many contributions to number theory and to the theory of equations.
       

397. Al-Karaji biography
     
     • He does not treat equations above the second degree except for ones which can easily be reduced to at most second degree equations followed by the extraction of roots.
       

398. Francais Francois biography
     
     • Francois worked on partial differential equations and his memoir of 1795 on this topic was developed further and presented to the Academie des Sciences in 1797.
       
     • Lacroix praised Francais' work and described it as making a major contribution to the study of partial differential equations; however, it was not published.
       

399. McCowan biography
     
     • A regular attendee at meetings of the Edinburgh Mathematical Society, he presented the papers: On a representation of elliptic integrals by curvilinear arcs (12 June 1891); On the solution of non-linear partial differential equations of the second order (13 May 1892); and Note on the solution of partial differential equations by the method of reciprocation (11 November 1892).
       

400. Ehrenfest biography
     
     • In 1917 and 1920 Ehrenfest published papers investigating the problem of the extent to which the three-dimensional nature of physical space is determined by the structure of basic physical equations or is reflected in these basic equations.
       

401. Gelfand biography
     
     • Another important area of his work is that on differential equations where he worked on the inverse Sturm-Liouville problem.
       
     • He worked on computational mathematics, developing general methods for solving the equations of mathematical physics by numerical means.
       

402. Hoyle biography
     
     • In 1945 he published On the integration of the equations determining the structure of a star which discussed the most advantageous way of integrating the equations of stellar equilibrium.
       

403. Sharkovsky biography
     
     • He also works in the theory of functional and functional differential equations, and the study of difference equations and their application.
       

404. Mauchly biography
     
     • In particular the School used a Bush analyser, designed by Vannevar Bush specifically to integrate systems of ordinary differential equations.
       
     • Von Neumann was working on this project and became involved with the ENIAC computer and used it to solve systems of partial differential equations which were crucial in the work on atomic weapons at Los Almos.
       

405. Kutta biography
     
     • It contains the now famous Runge-Kutta method for solving ordinary differential equations.
       
     • The former contains the Runge-Kutta method for solving ordinary differential equations while the latter contains the Zhukovsky- Kutta (Joukowski -Kutta) theorem giving the lift on an aerofoil.
       

406. La Hire biography
     
     • He began with their focal definitions and applied Cartesian analytic geometry t the study of equations and the solution of indeterminate problems; he also displayed the Cartesian method for solving certain types of equations by intersections of curves.
       

407. Stevin biography
     
     • In the latter Stevin presented a unified treatment for solving quadratic equations and a method for finding approximate solutions to algebraic equations of all degrees.
       

408. Duarte biography
     
     • He published papers on the general solution of a diophantine equation of the third degree x3 + y3 + z3 - 3xyz = v3, simplified Kummer's criterion and gave a simple proof of the impossibility of solving the Fermat equation x3 + y3 + z3 = 0 in nonzero integers.
       

409. Laguerre biography
     
     • Laguerre studied approximation methods and is best remembered for the special functions the Laguerre polynomials which are solutions of the Laguerre differential equations.
       
     • This memoir of Laguerre is significant not only because of the discovery of the Laguerre equations and polynomials and their properties, but also because it contains one of the earliest infinite continued fractions which was known to be convergent.
       

410. Rado Ferenc biography
     
     • Treated are: Nomograms for equations with two variables, with three variables (6 types), order and class of nomograms, nomograms with several variables, projective and homographic transformation of nomograms, classification of nomograms.
       
     • He published papers such as the following written in Romanian: Two theorems concerning the separation of variables in nomography (1955); On rhomboidal nomograms (1956); The best projective transformation of the scales of alignment nomograms (1957); and Functional equations in connection with nomography (1958).
       

411. Hurewicz biography
     
     • Lectures on ordinary differential equations is a beautiful introduction to ordinary differential equations which again reflects the clarity of his thinking and the quality of his writing.
       

412. Vashchenko biography
     
     • After his studies in Kazan, he returned to Kiev and he taught at Kiev Cadet School from 1855 to 1862, receiving his Master's Degree in 1862 for a dissertation on the operational method and its application to solving linear differential equations.
       
     • In particular he worked on the theory of linear differential equations, the theory of probability (see [A N Bogolyubov (ed.), On the history of the mathematical sciences 167 \'Naukova Dumka\' (Kiev, 1984), 36-39.',3)">3]) and non-euclidean geometry.
       
     • He published The Symbolic Calculus and its Application to the Integration of Linear Differential Equations in 1862.
       

413. Gronwall biography
     
     • In 1898, at the age 21, he was the author of ten mathematical papers and received his doctor's degree at Uppsala University for the thesis On system of linear total differential equations particularly with 2n-periodic coefficients.
       
     • Gronwall's work contains classical analysis (Fourier series, Gibbs phenomenon, summability theory, Laplace and Legendre series), differential and integral equations, analytic number theory (transcendental numbers, divisor function, L-function of Dirichlet), complex function theory (Dirichlet L-series, conformal mappings, univalent functions), differential geometry, mathematical physics (problems of elasticity, ballistics, induction, potential theory, kinetic theory of gases, optics), nomography, atomic physics (wave mechanics of hydrogen and helium atom, lattice theory of crystals) and physical chemistry where he is especially known as a very important contributor.
       

414. Hedrick biography
     
     • He was awarded a doctorate by Gottingen in February 1901 for a dissertation, supervised by Hilbert, Uber den analytischen Charakter der Losungen von Differentialgleichungen (On the analytic character of solutions of differential equations).
       
     • This strengthen his interests in differential equations, the calculus of variations, and functions of a real variable which he would work on for the rest of his life.
       

415. Bombieri biography
     
     • The award was made for his major contributions to the study of the prime numbers, to the study of univalent functions and the local Bieberbach conjecture, to the theory of functions of several complex variables, and to the theory of partial differential equations and minimal surfaces.
       
     • He has significantly influenced number theory, algebraic geometry, partial differential equations, several complex variables, and the theory of finite groups.
       

416. Bernoulli Johann biography
     
     • Integration to Bernoulli was simply viewed as the inverse operation to differentiation and with this approach he had great success in integrating differential equations.
       
     • He summed series, and discovered addition theorems for trigonometric and hyperbolic functions using the differential equations they satisfy.
       

417. Morishima biography
     
     • This paper On the Diophantine equation xp + yp = czp published in the Proceedings of the American Mathematical Society was, as all of Morishima's work, subject to the criticism that he did not give full enough explanations.
       
     • The authors obtain elegant criteria generalising the classical Wieferich and Mirimanoff criteria for the first case of Fermat's equation.
       

418. Maddison biography
     
     • When she first reached Bryn Mawr College, Maddison continued to work on this topic but later, advised by Scott, she began to work on singular solutions of differential equations.
       
     • in 1896 for her thesis On Singular Solutions of Differential Equations of the First Order in Two Variables and the Geometrical Properties of Certain Invariants and Covariants of Their Complete Primitives and in the same year appointed as Reader in Mathematics at Bryn Mawr.
       

419. Chuquet biography
     
     • The sections on equations cover quadratic equations where he discusses two solutions.
       

420. Thiele biography
     
     • One of his most important contributions to actuarial science was a differential equation for the net premium reserve Vt at time t for a life insurance, namely .
       
     • Although, as we have said, this differential equation is Thiele's most significant contribution to actuarial science, he never published the result.
       

421. Fontaine des Bertins biography
     
     • His papers are rather confused, and ignorant of the work of others, but do contain some very original ideas in the calculus of variations, differential equations and the theory of equations.
       

422. Levy Paul biography
     
     • Not only did Levy contribute to probability and functional analysis but he also worked on partial differential equations and series.
       
     • He undertook a large-scale work on generalised differential equations in functional derivatives.
       

423. Mises biography
     
     • His Institute rapidly became a centre for research into areas such as probability, statistics, numerical solutions of differential equations, elasticity and aerodynamics.
       
     • He classified his own work, not long before his death, into eight areas: practical analysis, integral and differential equations, mechanics, hydrodynamics and aerodynamics, constructive geometry, probability calculus, statistics and philosophy.
       

424. Kovalevskaya biography
     
     • The three papers were on Partial differential equations, Abelian integrals and Saturn's Rings.
       
     • The first of these three articles was still a valuable paper however, because it contained an exposition of Weierstrass's theory for integrating certain partial differential equations.
       

425. Beltrami biography
     
     • Some of Beltrami's last work was on a mechanical interpretation of Maxwell's equations.
       
     • (dated December, 1888) is devoted to the mechanical interpretation of Maxwell's equations.
       

426. Hormander biography
     
     • After Marcel Riesz retired in 1952 Hormander began working on the theory of partial differential equations.
       
     • In 1962 the International Congress was held in Stockholm and Hormander, as well as being heavily involved in the organisation, received a Fields Medal for his work on partial differential equations.
       

427. Lukacs biography
     
     • In 1942 Lukacs had made an important contribution to mathematical statistics by introducing, for the first time, the method of differential equations in characteristic function theory.
       
     • Other topics to which Lukacs made major contributions include characterisations of distributions, stability of characterisation results and functional equations.
       

428. Zorawski biography
     
     • He spent time in Leipzig where he studied continuous groups of transformations now called Lie groups, and Gottingen where he studied differential equations.
       
     • After returning to Krakow, Zorawski continued to teach courses on analytical and synthetic geometry, differential geometry, the formal theory of the differential equations, the theory of the forms, and the theory of the Lie groups.
       

429. Stepanov biography
     
     • He returned to Moscow in 1915 and, much influenced by Egorov and Luzin, he worked on periodic functions and differential equations.
       
     • In the qualitative theory of differential equations he worked on the general theory of dynamical systems studied by G D Birkhoff.
       
     • Besides writing articles on the study of almost periodic trajectories and on a generalisation of Birkhoff's ergodic theorem (which found an important application in statistical physics), Stepanov organised a seminar on the qualitative methods of the theory of differential equations (1932) that proved of great importance for the creation of the Soviet scientific school in this field.
       
     • A graduate-level text Qualitative Theory of Differential Equations by Stepanov and his student Viktor V Nemytskii became a classic, the 1960 edition being reprinted in 1989.
       
     • It considers existence and continuity theorems, integral curves of a system of two differential equations, systems of n-differential equations, general theory of dynamical systems, systems with an integral invariant, and many related topics.
       

430. Pontryagin biography
     
     • He began to study applied mathematics problems, in particular studying differential equations and control theory.
       
     • Another book by Pontryagin Ordinary differential equations appeared in English translation, also in 1962.
       

431. Boutroux biography
     
     • There he lectured at the College de France on functions which are the solutions of first order differential equations.
       
     • He worked on multiform functions and also continued Painleve's work on singularities of differential equations.
       

432. Livsic biography
     
     • He was particularly interested in the courses in complex variable, integral equations and differential equations.
       

433. Wald biography
     
     • seasonal corrections to time series, approximate formulas for economic index numbers, indifference surfaces, the existence and uniqueness of solutions of extended forms of the Walrasian system of equations of production, the Cournot duopoly problem, and finally, in his much used work written with Mann (1943), stochastic difference equations.
       

434. Bohl biography
     
     • He graduated in 1887 with a degree in mathematics having won a Gold Medal for an essay he wrote on The Theory of Invariants of Linear Differential Equations in 1886.
       
     • Bohl's doctoral dissertation applied topological methods to systems of differential equations.
       

435. Fubini biography
     
     • In this area he worked on differential equations, analytic functions and functions of several complex variables.
       
     • Another analysis topic he studied was non-linear integral equations.
       

436. Warschawski biography
     
     • The first was a single author paper On the solution of the Lichtenstein-Gershgorin integral equation in conformal mapping.
       
     • Theory while the second, On the solution of the Lichtenstein-Gershgorin integral equation in conformal mapping.
       

437. Pacioli biography
     
     • During this time Pacioli worked with Scipione del Ferro and there has been much conjecture as to whether the two discussed the algebraic solution of cubic equations.
       
     • History Topics: Quadratic, cubic and quartic equations .
       

438. Cartwright biography
     
     • The Radio Research Board of the Department of Scientific and Industrial Research produced a memorandum regarding certain differential equations which came out of modelling radio and radar work.
       
     • They began to collaborate studying the equations.
       

439. Cramer Harald biography
     
     • One interesting paper by Cramer over this period which we should note is one he published in 1920 discussing prime number solutions x, y to the equation ax + by = c, where a, b, c are fixed integers.
       
     • Note that if a = b = 1 then the question of whether this equation has a solution for all c is Goldbach's conjecture, while if a = 1, b = -1, c = 2, then the question about prime solutions to x = y + 2 is the twin prime conjecture.
       

440. Cartan biography
     
     • Cartan worked on continuous groups, Lie algebras, differential equations and geometry.
       
     • By 1904 Cartan was writing papers on differential equations and in many ways this work is his most impressive.
       

441. Laurent Pierre biography
     
     • Find the limiting equations that must be joined to the indefinite equations in order to determine completely the maxima and minima of multiple integrals.
       

442. Smeal biography
     
     • Among Smeals' publications are (with Ernest Frederick John Love) The psychrometric formula (1911), (with S Brodetsky) On Graeffe's method for complex roots of algebraic equations (1924) and The equations of the gravitational field in orthogonal coordinates (1926).
       

443. Quillen biography
     
     • for a thesis on partial differential equations in 1964 entitled Formal Properties of Over-Determined Systems of Linear Partial Differential Equations.
       

444. Fermat biography
     
     • Fermat posed further problems, namely that the sum of two cubes cannot be a cube (a special case of Fermat's Last Theorem which may indicate that by this time Fermat realised that his proof of the general result was incorrect), that there are exactly two integer solutions of x2 + 4 = y3 and that the equation x2 + 2 = y3 has only one integer solution.
       
     • History Topics: Pell's equation .
       

445. Sitter biography
     
     • He found solutions to Einstein's field equations in the absence of matter.
       
     • This is a particularly simple solution of the field equations of general relativity for an expanding universe.
       
     • He is not a cold, dispassionate juggler of Greek letters, a balancer of equations, but rather an artist in whom wild flights of the imagination are restrained by the formalism of a symbolic language and the evidence of observation.
       

446. Aepinus biography
     
     • During this period he undertook research in several different areas of mathematics including algebraic equations, solving partial differential equations, and on negative numbers.
       

447. Al-Mahani biography
     
     • However, he was led to an equation involving cubes, squares and numbers which he failed to solve after giving it lengthy meditation.
       
     • Therefore, this solution was declared impossible until the appearance of Ja'far al-Khazin who solved the equation with the help of conic sections.
       

448. Watson biography
     
     • Watson worked on a wide variety of topics, all within the area of complex variable theory, such as difference equations, differential equations, number theory and special functions.
       

449. Stoilow biography
     
     • Stoilow's thesis advisor was Emile Picard, and in 1914 he submitted his doctoral thesis Sur une classe de fonctions de deux variables definies par les equations lineaires aux derivees partielles.
       
     • He did, however, publish his first paper in 1914, namely Sur les integrales des equations lineaires aux derivees partielles a deux variables independantes.
       
     • He published two further, namely Sur les fonctions quadruplement periodiques (1915) and Sur l'integration des equations lineaires aux derivees partielles et la methode des approximations successives (1916), before publishing his doctoral thesis in 1916.
       
     • He published three further papers in 1919 including Sur les singularites mobiles des integrales des equations lineaires aux derivees partielles et sur leur integrale generale, and two further papers in 1920.
       
     • Before he took up his first university appointment in 1919, Stoilow concentrated on the theory of partial differential equations in the complex domain.
       

450. Haar biography
     
     • He examined the standard systems of orthonormal trigonometric functions and also orthonormal systems related to Sturm-Liouville differential equations.
       
     • After the work of his thesis, which we gave some details of above, he went on to study partial differential equations with applications to elasticity theory.
       

451. Grosswald biography
     
     • For the second edition of the text published in 1984, Grosswald had added material on L-functions and primes in arithmetic progressions, the arithmetic of number fields, and Diophantine equations.
       
     • In Bessel polynomials Grosswald studies: the relationship between Bessel functions and Bessel polynomials, differential equations and differential recurrence relations satisfied by the generalized Bessel polynomials, recurrence relations satisfied by the generalized Bessel polynomials, orthogonality properties of the generalized Bessel polynomials and the corresponding moment problem, relations of the generalized Bessel polynomials and the classical orthogonal polynomials, generating functions, Rodrigues-type formulas, the generalized Bessel polynomials and continued fractions, the zeros of the Bessel polynomials, algebraic properties of the Bessel polynomials, and the Galois group of the Bessel polynomials.
       

452. Morera biography
     
     • Morera studied the fundamental problems which arise in dynamics with particular regard to the use of the Pfaff method applied to Jacobian systems of partial differential equations and to the problem of Lie transformations of the canonical equations of motion.
       

453. Lagny biography
     
     • In about 1690 he developed a method of giving approximate solutions of algebraic equations and, in 1694, Halley published a twelve page paper in the Philosophical Transactions of the Royal Society giving his method of solving polynomial equations by successive approximation which is essentially the same as that given by Lagny a few years earlier.
       

454. Householder biography
     
     • He started publishing on this new topic with Some numerical methods for solving systems of linear equations which appeared in 1950.
       
     • In a remarkable series of papers he effectively classified the algorithms for solving linear equations and computing eigensystems, showing that in many cases essentially the same algorithm had been presented in a large variety of superficially quite different algorithms.
       

455. Sintsov biography
     
     • During this period he was being advised on research topics by Vasil'ev and, following his advice, he wrote his Master's Thesis The Theory of Connexes in Space in Connection with the Theory of First Order Partial Differential Equations.
       
     • Clebsch constructed the geometry of a ternary connex and applied it to the theory of ordinary differential equations.
       
     • Of course through his many years of research his interests varied but the main areas on which he worked were the theory of conics and applications of this geometrical theory to the solution of differential equations and, perhaps most important of all, the theory of nonholonomic differential geometry.
       
     • These were first published during the years 1927-1940 and include: A generalization of the Enneper-Beltrami formula to systems of integral curves of the Pfaffian equation Pdx + Qdy + Rdz = 0 (1927); Properties of a system of integral curves of Pfaff's equation, Extension of Gauss's theorem to the system of integral curves of the Pfaffian equation Pdx + Qdy + Rdz = 0 (1927); Gaussian curvature, and lines of curvature of the second kind (1928); The geometry of Mongian equations (1929); Curvature of the asymptotic lines (curves with principal tangents) for surfaces that are systems of integral curves of Pfaffian and Mongian equations and complexes (1929); On a property of the geodesic lines of the system of integral curves of Pfaff's equation (1936); Studies in the theory of Pfaffian manifolds (special manifolds of the first and second kind) (1940) and Studies in the theory of Pfaffian manifolds (1940).
       
     • There he studied the geometry of Monge equations and he introduced the important ideas of asymptotic line curvature of the first and second kind.
       
     • In 1903 he published two papers on the functional equation f (x, y) + f (y, z) = f (x, z), now called the 'Sintsov equation,' which are discussed by Detlef Gronau in [Notices of the South African Mathematical Society 31 (1) (2000), 1-8.',4)">4].
       
     • But before, it was Moritz Cantor who proposed these equations (there are two equations).
       
     • Cantor quotes these equations as examples of equations in three variables which can be solved by the method of differential calculus due to Niels Henrik Abel.
       

456. Wilkinson biography
     
     • He began to put his greatest efforts into the numerical solution of hyperbolic partial differential equations, using finite difference methods and the method of characteristics.
       
     • He worked on numerical methods for solving systems of linear equations and eigenvalue problems.
       

457. Nekrasov biography
     
     • He also investigated mathematical questions which were related to these applications, in particular writing important works on non-linear integral equations.
       
     • In the same year another important work on the applications of integral equations to aerodynamics was published.
       

458. Milne biography
     
     • Milne combined the two approaches and came up with an integral equation of great mathematical interest which is now known as Milne's integral equation.
       

459. Schubert biography
     
     • Algebraically, the solution of the problems of enumerative geometry amounts to finding the number of solutions for certain systems of algebraic equations with finitely many solutions.
       
     • Since the direct algebraic solution of the problems is possible only in the simplest cases, mathematicians sought to transform the system of equations, by continuous variation of the constants involved, into a system for which the number of solutions could be determined more easily.
       

460. Drinfeld biography
     
     • He discussed the concepts of quantum groups and quantization, and also talked about Poisson groups, Lie bi-algebras and the classical Yang-Baxter equation.
       
     • The interactions between mathematics and mathematical physics studied by Atiyah led to the introduction of instantons - solutions, that is, of a certain nonlinear system of partial differential equations, the self-dual Yang-Mills equations, which were originally introduced by physicists in the context of quantum field theory.
       

461. Hamming biography
     
     • His doctoral dissertation Some Problems in the Boundary Value Theory of Linear Differential Equations was supervised by Waldemar Trjitzinsky.
       
     • Hamming also worked on numerical analysis, integrating differential equations, and the Hamming spectral window which is much used in computation for smoothing data before Fourier analysing it.
       

462. Bottasso biography
     
     • Torino, 1912) Bottasso underlined the analogy between vector homography and integral equations, and used vector homography to solve integral equations.
       

463. Bernstein Sergi biography
     
     • This problem, posed at the 1900 Congress, was on analytic solutions of elliptic differential equations.
       
     • He studied for his Master's degree at Kharkov, continuing his way through Hilbert's Problems by solving the Twentieth on the analytic solution of Dirichlet's problem for a wide class of non-linear elliptic equations.
       

464. Crighton biography
     
     • He gave a mathematical model in which the problem reduce to solving two singular integral equations with Cauchy-type kernels, and with variable coefficients.
       
     • Solving the equations he showed that he boundary converts the energy stored in the turbulent boundary layer into the sound waves which generate noise.
       

465. Catalan biography
     
     • Two consecutive whole numbers, other than 8 and 9, cannot be consecutive powers; otherwise said, the equation xm - yn = 1 in which the unknowns are positive integers only admits a single solution.
       
     • Tijdeman proved in 1976 that Catalan's equation had only finitely many solutions, still quite far from showing that there is precisely one.
       

466. Knopp biography
     
     • Volume 1 covers numbers, functions, limits, analytic geometry, algebra, set theory; volume 2 covers differential calculus, infinite series, elements of differential geometry and of function theory; and volume 3 covers integral calculus and its applications, function theory, differential equations.
       
     • Friedrich Losch added a fourth volume in 1980 to cover more modern material: set theory, Lebesgue measure and integral, topological spaces, vector spaces, functional analysis, integral equations.
       

467. Hurwitz biography
     
     • He worked on how to derive class number relations from modular equations.
       
     • Further topics studied by Hurwitz include complex function theory, the roots of Bessel functions, and difference equations.
       

468. Durell biography
     
     • Contents include the properties of the triangle and the quadrilateral; equations, sub-multiple angles, and inverse functions; hyperbolic, logarithmic, and exponential functions; and expansions in power-series.
       
     • Further topics encompass the special hyperbolic functions; projection and finite series; complex numbers; de Moivre's theorem and its applications; one- and many-valued functions of a complex variable; and roots of equations.
       

469. Schwartz biography
     
     • This has led to extensive studies of topological vector spaces beyond the familiar categories of Hilbert and Banach spaces, studies that, in turn, have provided useful new insights in some areas of analysis proper, such as partial differential equations or functions of several complex variables.
       
     • I think every reader of his cited paper, like myself, will have left a considerable amount of pleasant excitement, on seeing the wonderful harmony of the whole structure of the calculus to which the theory leads and on understanding how essential an advance its application may mean to many parts of higher analysis, such as spectral theory, potential theory, and indeed the whole theory of linear partial differential equations ..
       

470. Radon biography
     
     • In mathematics he took lecture courses by Hans Hahn (one on Theoretical arithmetic and one on the Foundations of geometry), Wilhelm Wirtinger (Ordinary differential equations) and Franz Mertens (one on Algebra and one on Number theory) among others.
       
     • While creating a theory of absolutely additive set functions which, heretofore, has barely been investigated, the author succeeds with the development of a theory that contains the theory of integral equations, linear and bilinear forms in infinitely many variables, as a special case.
       

471. Hutton biography
     
     • The first volume looks at topics such as: arithmetic including discussion of square and cube roots, arithmetical and geometrical progressions, compound interest, double position and permutations and combinations; logarithms; algebra including the study of quadratic equations and the Cardan-Tartaglia method for cubic equations; geometry which follows the approach in Euclid's Elements; surveying; and conic sections.
       

472. MacRobert biography
     
     • Its special features are an emphasis on geometrical methods, extensive discussion of special functions and second-order differential equations, and a profusion of illustrative examples.
       
     • It is a very useful text-book on special functions, and an introduction to their application to partial differential equations of mathematical physics.
       

473. Wangerin biography
     
     • He taught many courses at the University of Halle including: linear partial differential equations; calculus of variations; theory of elliptical functions; synthetic geometry; hydrostatics and capillarity theory; theory of space curves and surfaces; analytic mechanics; potential theory and spherical harmonics; celestial mechanics; the theory of the map projections; hydrodynamics; and the partial differential equations of mathematical physics.
       

474. Lefschetz biography
     
     • He tackled problems related to dissipative nonlinear ordinary differential equations but did not take the usual approach of using linear theory to tackle nonlinear differential equations.
       

475. Forsyth biography
     
     • Famous texts which Forsyth published before his 1893 work Theory of Functions of a complex variable , are A treatise on differential equations (1885), and Theory of differential equations published in six volumes between 1890 and 1906.
       

476. Wronski biography
     
     • A piece of work which he had undertaken during this period resulted in a publication Resolution generale des equations de tous degres in 1812 claiming to show that every equation had an algebraic solution.
       
     • For good measure, it contains a summary of the "general solution of the fifth degree equation".
       

477. Girard Pierre biography
     
     • one may learn to find the equation for some solid as one finds the equation for a curved plane.
       

478. Kloosterman biography
     
     • Kloosterman was examining the number of solutions in integers xn, to the equation .
       
     • He had managed to find, provided s ≥ 5 and the an satisfy suitable congruence conditions, an asymptotic formula for the number of solution to the equation (*).
       

479. Subbotin biography
     
     • He had already published two papers prior to submitting his Master's thesis, one was On the determination of singular points of analytic functions while the second was published in France and was on singular points of certain differential equations.
       
     • Later he worked in celestial mechanics producing new methods of calculating orbits from three observations based on solving the Euler-Lambert equations.
       

480. Stiefel biography
     
     • They feel that from the point of view of the applications to stability and vibrational questions in mechanics the variational approach is the most suitable one (as compared with the approach by differential or integral equations).
       
     • Whenever possible, we derive the basic differential equations or at least we interpret them.
       

481. Mercer biography
     
     • Mercer was a mathematical analyst of originality and skill; he made noteworthy advances in the theory of integral equations, and especially in the theory of the expansion of arbitrary functions in series of orthogonal functions.
       
     • Mercer's theorem about the uniform convergence of eigenfunction expansions for kernels of operators appears in his 1909 paper Functions of positive and negative types and their connection with the theory of integral equations published in the Philosophical Transactions.
       

482. Mayer Adolph biography
     
     • Mayer worked on differential equations, the calculus of variations and mechanics.
       
     • His work on the integration of partial differential equations and a search to determine maxima and minima using variational methods brought him close to the investigations which Lie was carrying out around the same time.
       

483. Gateaux biography
     
     • He recalled that Volterra introduced this notion to study problems including an hereditary phenomenon, but also that it was used by others (Jacques Hadamard and Paul Levy) to study some problems of mathematical physics - such as the equilibrium problem of fitted elastic plates - finding a solution to equations with functional derivatives, or, in other words, by calculating a relation between this functional and its derivative.
       
     • Though I am still mobilized, I work on lectures I should read at the College de France on the functions of lines and the equations with functional derivatives and at this occasion I would like to develop several chapters of the theory.
       

484. Bott biography
     
     • We mentioned Smale above, and the second was Daniel Quillen who wrote his thesis Formal Properties of Over-Determined Systems Of Linear Partial Differential Equations at Harvard.
       
     • The main themes of the papers included in [Volume 4] are the geometry and topology of the Yang-Mills equations and the rigidity phenomena of vector bundles.
       

485. Brisson biography
     
     • His favourite topic was partial differential equations and two important papers he submitted to the Paris Academy applied functional calculus using a symbolic scheme to solve linear differential equations.
       

486. Lerch biography
     
     • He also studied elliptic functions and integral equations.
       
     • He is remembered today for his solution of integral equations in operator calculus and for the 'Lerch formula' for the derivative of Kummer's trigonometric expansion for log G(v).
       

487. Franklin biography
     
     • He worked on the four colour problem and also published books on calculus, differential equations, complex variable and Fourier series.
       
     • In particular he wrote Differential equations for electrical engineers (1933), Treatise on advanced calculus (1940), The four colour problem (1941), Methods of advanced calculus (1944), Fourier methods (1949), Differential and integral calculus (1953), Functions of a complex variable (1958) and Compact calculus (1963).
       

488. Caratheodory biography
     
     • He added important results to the relationship between first order partial differential equations and the calculus of variations.
       
     • Caratheodory wrote many fine books including Lectures on Real Functions (1918), Conformal representation (1932), Calculus of Variations and Partial Differential Equations (1935), Geometric Optics (1937), Real functions Vol.
       

489. Lavrentev biography
     
     • In the 1940s he developed the theory of quasi-conformal mappings which gave a new geometrical approach to partial differential equations.
       
     • Other topics where he made substantial contributions where the theory of sets, the general theory of functions, and the theory of differential equations.
       

490. Nikodym biography
     
     • the Radon-Nikodym theorem and derivative, the Nikodym convergence theorem, the Nikodym-Grothendieck boundedness theorem), in functional analysis (the Radon-Nikodym property of a Banach space, the Frechet-Nikodym metric space, a Nikodym set), projections onto convex sets with applications to Dirichlet problem, generalized solutions of differential equations, descriptive set theory and the foundations of quantum mechanics.
       
     • Some of his other books were: Introduction to differential calculus, (Warsaw, 1936) (jointly with his wife), Theory of tensors with applications to geometry and mathematical physics, I, (Warsaw, 1938), Differential Equations, (Poznan, 1949).
       

491. Jonquieres biography
     
     • He also worked on algebra, in particular the theory of equations, and, in the latter part of his life, on the theory of numbers where he examined Diophantine equations and the distribution of primes.
       

492. Wintner biography
     
     • Wintner published on analysis, number theory, differential equations and probability (with several joint papers with Norbert Wiener).
       
     • A study of certain astronomical equations led Wintner to consider almost periodic functions.
       

493. Thue biography
     
     • His contributions to the theory of Diophantine equations are discussed in [Normat 38 (4) (1990), 153-159; 192.
       
     • In fact Thue wrote 35 papers on number theory, mostly on the theory of Diophantine equations, and these are reproduced in [Selected mathematical papers of Axel Thue, Introduction by Carl Ludwig Siegel (Oslo, 1977).',2)">2].
       

494. Eells biography
     
     • In it the whole geometry or topology of the spaces involved play a role, rather than just the equations describing the behaviour or motion in small areas.
       
     • This he did with "Global Analysis" in 1971-72, "Geometry of the Laplace Operator" in 1976-77, and "Partial Differential Equations in Differential Geometry", in 1989-90.
       

495. Dirichlet biography
     
     • In 1852 he studied the problem of a sphere placed in an incompressible fluid, in the course of this investigation becoming the first person to integrate the hydrodynamic equations exactly.
       
     • These series had been used previously by Fourier in solving differential equations.
       

496. Smoluchowski biography
     
     • He taught a variety of courses: potential theory, mechanics, electricity, optics, thermodynamics, kinetic theory of gases, differential equations, and mathematical physics.
       
     • Smoluchowski made many contributions to physics and mathematics, particularly to the theory of Brownian motion, stochastic processes and related problems, of which the most important are the 'Smoluchowski equations' bearing his name.
       

497. Skolem biography
     
     • It was entitled Einige Satze uber ganzzahlige Losungen gewisser Gleichungen und Ungleichungen, and was on integral solutions of certain algebraic equations and inequalities.
       
     • Skolem was remarkably productive publishing around 180 papers on topics such as Diophantine equations, mathematical logic, group theory, lattice theory and set theory.
       

498. Somov biography
     
     • During this time he wrote his first mathematical work on algebraic equations Theory of determinate algebraic equations of higher degree which was published in 1838 [Dictionary of Scientific Biography (New York 1970-1990).',1)">1].
       

499. Eutocius biography
     
     • 4, of the auxiliary problem amounting to the solution by means of conics of the cubic equation (a - x) x2 = b c2.
       
     • the solutions (a) by Diocles of the original problem of II.4 without bringing in the cubic, (b) by Dionysodorus of the auxiliary cubic equation.
       

500. Gohberg biography
     
     • In addition to Gohberg's outstanding work in analysis and in particular in operator theory and matrix methods, he founded the major international journal Integral equations and operator theory in the late 1980s.
       
     • is devoted to the publication of current research in integral equations, operator theory and related topics with emphasis on the linear aspects of the theory.
       

501. Beurling biography
     
     • Beurling worked on the theory of generalized functions, differential equations, harmonic analysis, Dirichlet series and potential theory.
       

502. Apastamba biography
     
     • The general linear equation was solved in the Apastamba's Sulbasutra.
       

503. Stirling biography
     
     • In the minutes of a meeting of the Royal Society of London on 4 April 1717, when Brook Taylor lectured on extracting roots of equations and on logarithms, it is recorded:- .
       

504. Spence biography
     
     • He published on algebraic and differential equations (1814) and other manuscripts were edited and published in 1820 by John Herschel.
       

505. Ferrel biography
     
     • However this assumption is not realistic, but the realistic assumption that the friction is proportional to the square of the velocity produced non-linear equations which were much more difficult to treat.
       

506. Wigner biography
     
     • epoch-making work on how symmetry is implemented in quantum mechanics, the determination of all the irreducible unitary representations of the Poincare group, and his work with Bargmann on realizing those irreducible unitary representations as the Hilbert spaces of solutions of relativistic wave equations, ..
       

507. Hua biography
     
     • Hua wrote several papers with H S Vandiver on the solution of equations in finite fields and with I Reiner on automorphisms of classical groups.
       

508. Segre Corrado biography
     
     • Among other important work which Segre produced was an extension of ideas of Darboux on surfaces defined by certain differential equations.
       

509. Pairman biography
     
     • Her thesis advisor was George Birkhoff and after submitting her thesis Expansion Theorems for Solution of a Fredholm's Linear Homogeneous Integral Equation of the Second Kind with Kernel of Special Non-Symmetric Type she was awarded a Ph.D.
       
     • Pairman joined the Edinburgh Mathematical Society in January 1917, and read the paper On a difference equation due to Stirling to the meeting of the Society on 11 January 1918, and the paper A new form of the remainder in Newton's interpolation formula to the next meeting of the Society on 8 February.
       
     • In 1927 she published a joint mathematics paper On a class of integral equations with discontinuous kernels with Rudolph E Langer, who was a friend who graduated from Harvard in a June 1922 ceremony as Eleanor Pairman and Bancroft Brown and, like Eleanor, had George Birkhoff as his thesis advisor.
       

510. Cox Elbert biography
     
     • In 1925 Cox was awarded his doctorate for his thesis Polynomial solutions of difference equations.
       

511. Burgess biography
     
     • And Concurrency of lines joining vertices of a triangle to opposite vertices of triangles on its sides; determinants connected with the periodic solutions of Mathieu's equation.
       

512. Gram biography
     
     • Gram later published this work in the Journal fur Mathematik and it proved to be of fundamental importance in the development of the theory of integral equations.
       

513. Servois biography
     
     • Servois worked in projective geometry, functional equations and complex numbers.
       

514. Hecht biography
     
     • His later texts covered topics such as quadratic and cubic equations, differential and integral calculus, and arithmetic and geometry.
       

515. Jordanus biography
     
     • In De numeris datis Jordanus gives results on solving quadratic equations similar to those given by al-Khwarizmi except general forms are given rather than the numerical examples of the earlier text.
       

516. Linnik biography
     
     • The systems of diophantine equations studied by these methods and the flows of lattice points introduced by these methods are closely related to the behaviour of the ideal classes of the corresponding algebraic fields.
       

517. Khinchin biography
     
     • It was first published in 1943 and the eight lectures it contains are: Continuum; Limits; Functions; Series; Derivative; Integral; Series expansions of functions; and Differential equations.
       

518. Tinseau biography
     
     • Tinseau wrote on the theory of surfaces, working out the equation of a tangent plane at a point on a surface, and he generalised Pythagoras's theorem proving that the square of a plane area is equal to the sum of the squares of the projections of the area onto mutually perpendicular planes.
       

519. Chowla biography
     
     • He wrote on additive number theory (lattice points, partitions, Waring's problem), analysis, Bernoulli numbers, class invariants, definite integrals, elliptic integrals, infinite series, the Weierstrass approximation theorem), analytic number theory (Dirichlet L-functions, primes, Riemann and Epstein zeta functions), binary quadratic forms and class numbers, combinatorial problems (block designs, difference sets, Latin squares), Diophantine equations and Diophantine approximation, elementary number theory (arithmetic functions, continued fractions, and Ramanujan's tau function), and exponential and character sums (Gauss sums, Kloosterman sums, trigonometric sums).
       

520. Cassels biography
     
     • After further papers on Diophantine equations and Diophantine approximation he wrote a series of five papers on Some metrical theorems in Diophantine approximation.
       

521. De Bruijn biography
     
     • He began publishing papers on combinatorics relevant to his work during this period such as The problem of optimum antenna current distribution (1946), A combinatorial problem (1946), On the zeros of a polynomial and of its derivative (1946), and A note on van der Pol's equation (1946) [Applied Logic Series 28 (Kluwer Academic Publishers, Dordrecht, 2003).',1)">1]:- .
       

522. Feldman biography
     
     • The last part of the book describes Alan Baker's work on linear forms in the logarithms of algebraic numbers and its applications to Diophantine equations and to the determination of imaginary quadratic fields with class number 1 or 2.
       

523. Atwood biography
     
     • Atwood also published on equations for the use of Hadley's quadrant.
       

524. Riesz biography
     
     • He built on ideas introduced by Frechet in his dissertation, using Frechet's ideas of distance to provide a link between Lebesgue's work on real functions and the area of integral equations developed by Hilbert and his student Schmidt.
       

525. Novikov Sergi biography
     
     • We should mention especially Sergei's uncle Mstislav Keldysh who made major contributions to complex function theory, differential equations and applications to aerodynamics.
       
     • These include a systematic study of finite-gap solutions of two-dimensional integrable systems, formulation of the equivalence of the classification of algebraic-geometric solutions of the KP equation with the conformal classification of Riemann surfaces, and work (with Krichever) on "almost commuting" operators that appear in string theory and matrix models ("Krichever-Novikov algebras", now widely used in physics).
       

526. Stewartson biography
     
     • Keith Stewartson's abiding passion in mathematical research lay in the solution of the equations governing the motion of liquids and gases, and in the comparison of his theoretical predictions with experiment and observation.
       

527. Blades biography
     
     • For example, he communicated On Spheroidal Harmonics and Allied Functions, by Mr G B Jeffery to the meeting on Friday 11 June 1915 and Transformations of Axes for Whittaker's Solution of Laplace's Equation, by Dr G B Jeffery to the meeting on Friday 9 March 1917.
       

528. Al-Farisi biography
     
     • He noted the impossibility of giving an integer solution to the equation .
       

529. Rosanes biography
     
     • he scribbled equations which his students never quite saw because as he wrote he hid them with his body and as he moved along he rubbed them out with his sponge.
       

530. Kostrikin biography
     
     • The meaning of an algebraic concept can be of a number-theoretic or geometric nature, and frequently its roots lie in computational aspects of mathematics and in the solution of equations.
       

531. Karsten biography
     
     • He wrote an important article in 1768 Von den Logarithmen vermeinter Grossen in which he discussed logarithms of negative and imaginary numbers, giving a geometric interpretation of logarithms of complex numbers as hyperbolic sectors, based on the similarity of the equations of the circle and of the equilateral hyperbola.
       

532. Wang Yuan biography
     
     • However he did write a number of books such as: (with Hua Loo Keng) Applications of number theory to numerical analysis (1978); Goldbach Conjecture (1984); (with Hua Loo Keng) Popularising mathematical methods in the People's Republic of China (1989); Diophantine equations and inequalities in algebraic number fields (1991); (with Fang Kai-Tai) Number theoretic methods in statistics (1994); Hua Loo Keng (1995); and (with Fong Yuen) Calculus (1997).
       

533. Jia Xian biography
     
     • He generalised a method of finding square roots and cube roots to finding nth roots, for n > 3, and then extended the method to solving polynomial equations of arbitrary degree.
       

534. Bartholin biography
     
     • The problem is the first example of an inverse tangent problem which in modern notation results in requiring the solution to the differential equation .
       

535. Eckmann biography
     
     • Peter Hilton, who had been a personal friend of Eckmann's for many years spoke in detail of Eckmann's research in topology: continuous solutions of systems of linear equations, a group-theoretical proof of the Hurwitz-Radon theorem, complexes with operators, spaces with means, simple homotopy type.
       

536. Peirce Benjamin biography
     
     • For example An Elementary Treatise on Plane Trigonometry (1835), First Part of an Elementary Treatise on Spherical Trigonometry (1936), An Elementary Treatise on Sound (1936), An Elementary Treatise on Algebra : To which are added Exponential Equations and Logarithms (1937), An Elementary Treatise on Plane and Solid Geometry (1937), An Elementary Treatise on Plane and Spherical Trigonometry (1940), and An Elementary Treatise on Curves, Functions, and Forces Vol 1 (1841), Vol 2 (1846).
       

537. Hahn biography
     
     • These include a report on integral equation he wrote in 1911, his modification of Hellinger's theory of invariants of quadratic forms, in which he dispensed with the use of the Hellinger integral, and his work on duality in Banach spaces, culminating with his proof of the Hahn-Banach theorem in 1927.
       

538. Pars biography
     
     • He based his treatment on the theorem of Lagrange that he called the fundamental equation, which he proceeded to translate into six different forms, each exploited in appropriate contexts.
       

539. Holder biography
     
     • He began to study the Galois theory of equations and from there he was led to study compostion series of groups.
       

540. Young Andrew biography
     
     • He read the paper On the quasi-periodic solutions of Mathieu's differential equation to the Society at its meeting on Friday 13 February 1914.
       

541. Meders biography
     
     • Adolf Kneser, who had been taught by Kronecker and written a thesis on algebraic functions and equations, was the professor at Dorpat.
       

542. Jyesthadeva biography
     
     • Other mathematical results presented by Jyesthadeva include topics studied by earlier Indian mathematicians such as integer solutions of systems of first degree equation solved by the kuttaka method, and rules of finding the sines and the cosines of the sum and difference of two angles.
       

543. Spanier biography
     
     • Interestingly, one of Spanier's theories, now called Alexander-Spanier homology, is currently being applied to analyse differential equations - a return to Poincare's original use of algebraic topology.
       

544. Bondi biography
     
     • The exact relativistic form of the equation of hydrostatic support by an isotropic pressure is found in an especially convenient form.
       

545. Heine biography
     
     • Before arriving at Halle, Heine published on partial differential equations and during his first few years teaching at Halle he wrote papers on the theory of heat, summation of series, continued fractions and elliptic functions.
       

546. Newton biography
     
     • [Newton] brought me the other day some papers, wherein he set down methods of calculating the dimensions of magnitudes like that of Mr Mercator concerning the hyperbola, but very general; as also of resolving equations; which I suppose will please you; and I shall send you them by the next.
       

547. Zolotarev biography
     
     • He then continued his studies at the Faculty of Physics and Mathematics investigating an indeterminate equation of degree three.
       

548. Chaplygin biography
     
     • Chaplygin also developed methods of approximation for solving differential equations.
       

549. Post biography
     
     • thesis was on mathematical logic, and we shall discuss it further in a moment, but first let us note that Post wrote a second paper as a postgraduate, which was published before his first paper, and this was a short work on the functional equation of the gamma function.
       

550. Machin biography
     
     • Machin had explained to Taylor in Child's Coffeehouse how to use Newton's series to solve Kepler's problem and also how Halley's method finds roots of polynomial equations.
       

551. Vranceanu biography
     
     • Other topics he studied include the absolute differential calculus of congruences, analytical mechanics, partial differential equations of the second order, non-holonomic unitary theory, conformal connection spaces, metrics in spherical and projective spaces, Lie groups, global differential geometry, discrete groups of affine connection spaces, locally Euclidean connection spaces, Riemannian spaces of constant connection, differentiable varieties, embedding of lens spaces into Euclidean space, tangent vectors of spheres and exotic spheres, the equivalence method, non-linear connection spaces, and the geometry of mechanical systems.
       

552. Delsarte biography
     
     • He worked during that year at the private mansion of the Foundation, undertaking research for his doctoral thesis and also working on his first two papers Sur les rotations dans l'espace fonctionnel and E de certaines equations integrales qui generalisent celles de Fredholm which were published by the Academy of Science.
       
     • At Nancy he developed a new course on differential equations in the academic year 1933-34 and in the following year, also at Nancy, he gave a course on Riemann spaces and relativity.
       
     • He published a series of papers on this topic in 1934-35: Les fonctions moyenne-periodiques (1934); Application de la theorie des fonctions moyenne-periodiques a la resolution de certaines equations integrales (1934); Application de la theorie des fonctions moyenne-periodiques a la resolution des equations de Fredholm-Norlund (1935); and Les fonctions moyenne-periodiques (1935).
       

553. Zeeman biography
     
     • I suppose I am particularly fond of having unknotted spheres in 5-dimensions, of spinning lovely examples of knots in 4-dimensions, of proving Poincare's Conjecture in 5-dimensions, of showing that special relativity can be based solely on the notion of causality, and of classifying dynamical systems by using the Focke-Plank equation.
       

554. Kneser Hellmuth biography
     
     • For example he produced a beautiful solution to the functional equation f ( f (x) ) = ex which he published in 1950, and the deep understanding he achieved of the strange properties of manifolds without a countable basis of neighbourhoods between 1958 and 1964.
       

555. Blackwell biography
     
     • The most interesting thing I remember from calculus was Newton's method for solving equations.
       

556. Lanczos biography
     
     • He worked on relativity theory and after writing his dissertation Relation of Maxwell's Aether Equations to Functional Theory he sent a copy to Einstein.
       

557. Fatou biography
     
     • Using existance theorems for the solutions to differential equations, Fatou was able to prove rigorously certian results on planetary orbits which Gauss had suggested by only verified with an intuitive argument.
       

558. Poinsot biography
     
     • In addition Poinsot worked on number theory and on this topic he studied Diophantine equations, how to express numbers as the difference of two squares and primitive roots.
       

559. Segre Beniamino biography
     
     • By 1931 when he was appointed to the chair at Bologna he already had 40 publications in algebraic geometry, differential geometry, topology and differential equations.
       

560. Goldstein biography
     
     • He studied numerical solutions to steady-flow laminar boundary-layer equations in 1930.
       

561. Kalicki biography
     
     • Kalicki worked on logical matrices and equational logic and published 13 papers on these topics from 1948 until his death five years later.
       

562. Berwick biography
     
     • Berwick also gave, in 1915, necessary and sufficient conditions for a quintic equation to be soluble by radicals.
       

563. Graffe biography
     
     • Graffe is best remembered for his method of numerical solution of algebraic equations, developed to answer a prize question of the Berlin Academy of Sciences.
       

564. Cosserat biography
     
     • He studied functional equations of the sphere and ellipsoid before Fredholm.
       

565. Kempe biography
     
     • Kempe was taught mathematics by Cayley and graduated in 1872 with distinction in mathematics and in the same year he published his first mathematical paper A general method of solving equations of the nth degree by mechanical means.
       

566. Weingarten biography
     
     • In this work he reduced the problem of finding all surfaces isometric to a given surface to the problem of determining all solutions to a partial differential equation of the Monge-Ampere type.
       

567. Moulton biography
     
     • His books include An Introduction to Celestial Mechanics (1902), An introduction to astronomy (1906), Descriptive astronomy (1912), Periodic orbits (1920) The Nature of the World and Man (1926), Differential equations (1930), Astronomy (1931), and Consider the Heavens (1935).
       

568. Coulomb biography
     
     • From examination of many physical parameters, he developed a series of two-term equations, the first term a constant and the second term varying with time, normal force, velocity, or other parameters.
       

569. Weierstrass biography
     
     • ., from the differential equation defining this function, was the first mathematical task I set myself; and its fortunate solution made me determined to devote myself wholly to mathematics; I made this decision in my seventh semester ..
       

570. Dini biography
     
     • In this last work he devoted a chapter to integral equations in which he presented many of his own innovative ideas.
       

571. Griffiths Brian biography
     
     • We give the titles of a few of his mathematical education article which give an overview of his interests in that topic: Pure mathematicians as teachers of applied mathematicians (1968); Mathematics Education today (1975); Successes and failures of mathematical curricula in the past two decades (1980); Simplification and complexity in mathematics education (1983); The implicit function theorem: technique versus understanding (1984); A critical analysis of university examinations in mathematics (1984); Cubic equations, or where did the examination question come from? (1994); The British Experience of Teaching Geometry since 1900 (1998); and The Divine Proportion, matrices and Fibonacci numbers (2008).
       

572. Plessner biography
     
     • Then in session 1921/22 he studied in Berlin where von Mises lectured on differential and integral equations, Bieberbach on differential geometry and Schur on algebra.
       

573. Appell biography
     
     • He then wrote on algebraic functions, differential equations and complex analysis.
       

574. Roy biography
     
     • This work led to partial differential equations which could only be solved numerically, but at this time the Applied Mathematics Department had no computing facilities.
       

575. Peres biography
     
     • Peres' work on analysis and mechanics was always influenced by Volterra, extending results of Volterra's on integral equations.
       

576. Young Lai-Sang biography
     
     • It is generally regarded as a study of the iteration of maps, of time evolution of differential equations, and of group actions on manifolds.
       

577. Wiener Norbert biography
     
     • In 1914 he went to Gottingen to study differential equations under Hilbert, and also attended a group theory course by Edmund Landau.
       

578. Blichfeldt biography
     
     • Some of the many topics that he covered were diophantine approximations, orders of linear homogeneous groups, theory of geometry of numbers, approximate solutions of the integers of a set of linear equations, low-velocity fire angle, finite collineation groups, and characteristic roots.
       

579. Archimedes biography
     
     • History Topics: Pell's equation .
       

580. Moser Leo biography
     
     • These include On the sum of digits of powers (1947), Some equations involving Euler's totient (1949), Linked rods and continued fractions (1949), On the danger of induction (1949) and A theorem on the distribution of primes (1949).
       

581. Dantzig George biography
     
     • It was a system with nine equations in seventy-seven unknowns.
       

582. Polkinghorne biography
     
     • Also in 1955 he published Temporally ordered graphs and bound state equations and On the classification of fundamental particles.
       

583. Blades Edward biography
     
     • For example, he communicated On Spheroidal Harmonics and Allied Functions, by Mr G B Jeffery to the meeting on Friday 11 June 1915 and Transformations of Axes for Whittaker's Solution of Laplace's Equation, by Dr G B Jeffery to the meeting on Friday 9 March 1917.
       

584. Noether Max biography
     
     • This result showed that given two algebraic curves f (x, y) = 0, g(x, y) = 0 which intersect in a finite number of isolated points, then the equation of an algebraic curve which passes through all those points of intersection can be expressed in the form af + bg = 0, where a and b are polynomials in x and y, is and only if certain conditions are satisfied.
       

585. Lang biography
     
     • Your famous theorem in Diophantine equations earned you the distinguished Cole Prize of the American Mathematical Society.
       

586. Cunningham biography
     
     • He wrote on linear differential equations, prompted by Pearson's work and other work related to statistics.
       

587. Dehn biography
     
     • I attended his course in Non-linear Partial Differential Equations.
       

588. Uhlenbeck biography
     
     • He extended Boltzmann's equation to dense gasses and wrote two important papers on Brownian motion.
       

589. Gegenbauer biography
     
     • The Gegenbauer polynomials are solutions to the Gegenbauer differential equation and are generalizations of the associated Legendre polynomials.
       

590. Xu Guangqi biography
     
     • The brilliant "tian yuan" or "coefficient array method" or "method of the celestial unknown" for solving equations which had been expounded with such skill by Li Zhi in the 13th century was no longer understood in China.
       

591. Egorov biography
     
     • Egorov also worked on integral equations and a theorem in the theory of functions of a real variable is named after him.
       

592. Helly biography
     
     • His thesis was on Fredholm equations.
       

593. Bayes biography
     
     • This notebook contains a considerable amount of mathematical work, including discussions of probability, trigonometry, geometry, solution of equations, series, and differential calculus.
       

594. Sankara biography
     
     • It is a text which covers the standard mathematical methods of Aryabhata I such as the solution of the indeterminate equation by = ax ± c (a, b, c integers) in integers which is then applied to astronomical problems.
       

595. Rado biography
     
     • His first paper On the roots of algebraic equations was published in 1921 and in the following year he published his first paper on conformal mappings.
       

596. Ivory biography
     
     • Ivory wrote several articles for encyclopaedias, including the influential Equations in Encyclopaedia Britannica.
       

597. Airy biography
     
     • This text was one of eleven books which Airy published, some of the others being Trigonometry (1825), Gravitation (1834), and Partial differential equations (1866).
       

598. Widman biography
     
     • Widman used Cossist notation, as was usual at that time, discussing 24 different types of equations.
       

599. Kelland biography
     
     • He wrote analytical papers on General Differentiation (1839), and Differential Equations (1853), and gave a geometrical Theory of Parallels outlining a version of non-Euclidean geometry.
       

600. Fagnano Giulio biography
     
     • Fagnano suggested new methods of solving equations of degree 2, 3 and 4.
       

601. Borel biography
     
     • In [Enseignement mathematique 11 (1965), 1-95.',8)">8] Borel's mathematical work is divided into the following topics: Arithmetic; Numerical series; Set theory; Measure of sets; Rarefaction of a set of measure zero; Real functions of real variables; Complex functions of complex variables; Differential equations; Geometry; Calculus of probabilities; and Mathematical physics.
       

602. Osipovsky biography
     
     • His most famous work was the three-volume handbook A Course of Mathematics (1801-1823) which covered function theory, differential equations, and the calculus of variations.
       

603. Bliss biography
     
     • They were An existence theorem for a differential equation of the second order, with an application to the calculus of variations and Sufficient condition for a minimum with respect to one-sided variations.
       

604. Newman biography
     
     • The first was an early inroad on Hilbert's Fifth Problem, in which he proved that abelian continuous groups do not have arbitrarily small subgroups, the second was a simplified proof of a difficult fixed point theorem of Cartwright and Littlewood arising in the study of differential equations.
       

605. Bernoulli Johann(III) biography
     
     • In the field of mathematics he worked on probability, recurring decimals and the theory of equations.
       

606. Kellogg biography
     
     • In 1908 he published three papers, namely Potential functions on the boundary of their regions of definition and Double distributions and the Dirichlet problem, both in the Transactions of the American Mathematical Society, and A necessary condition that all the roots of an algebraic equation be real in the Annals of Mathematics.
       

607. Elliott biography
     
     • All ageing mathematicians should be particularly pleased to learn that a second piece of work by Elliott, which was again of major importance, was his contribution to the theory of integral equations which he made after he retired.
       

608. Bruns biography
     
     • He worked on the three-body problem showing that the series solutions of the Lagrange equations can change between convergent to divergent for small perturbations of the constants on which the coefficients of the time depend.
       

609. Walsh Joseph biography
     
     • The topics he taught, rotating them from year to year, included calculus, algebra, mechanics, differential equations, complex variable, probability, number theory, potential theory, approximation theory, and function theory.
       

610. Francais Jacques biography
     
     • It was a work on the integration of first order partial differential equations, but the memoir had been lost so there are few details as to its precise contents.
       

611. Rees biography
     
     • In 1931 Rees graduated with her doctorate for a thesis entitled Division algebras associated with an equation whose group has four generators.
       

612. Wiltheiss biography
     
     • His doctoral studies on systems of hyperelliptic differential equations were supervised by Weierstrass and he submitted his thesis Die Umkehrung einer Gruppe von Systemen allgemeiner hyperelliptischer Differentialgleichungen to the University of Berlin.
       

613. Aronhold biography
     
     • Certain linear partial differential equations which he came across in his work are characteristic of invariant theory and are named after him.
       

614. Janovskaja biography
     
     • An analysis is given for the problem of finding geometric solutions for algebraic equations of degree higher than two by locating points of intersection of conic sections with other curves.
       

615. Menabrea biography
     
     • The principle of Menabrea states that the elastic energy of a body in perfect elastic equilibrium is a minimum with respect to any possible system of stress-variation compatible with the equations of the statics of continua in addition to the boundary conditions.
       

616. Crelle biography
     
     • Crelle realised the importance of Abel's work and published several articles by him in this first volume, including his proof of the insolubility of the quintic equation by radicals.
       

617. Smirnov biography
     
     • Smirnov was a very active member of this circle, for example lecturing on the theory of algebraic equations, particularly the work of Goursat and Appell.
       

618. Birkhoff Garrett biography
     
     • He attended a course on potential theory given by Oliver Kellogg which gave him a good understanding of differential equations.
       

619. Urysohn biography
     
     • At this stage Urysohn was interested in analysis, in particular integral equations, and this was the topic of his habilitation.
       

620. Mayer Tobias biography
     
     • with respect to the inequalities of motions, from that famous theory of the great Newton, which that eminent mathematician Eulerus first elegantly reduced to general analytic equations.
       

621. Fuss biography
     
     • Most of Fuss's papers are solutions to problems posed by Euler on spherical geometry, trigonometry, series, differential geometry and differential equations.
       

622. Picken biography
     
     • He read papers to the Society such as A Proof of the Addition Theorem in Trigonometry to the meeting on Friday 9 December 1904, On a Direct Method of Obtaining the Foci and Directrices from the General Equation of the Second Degree to the meeting on 9 June 1905, On Simson Line and Related Theory: and An Exercise in Geometric Generality (communicated by A W Young) on 8 May 1914.
       

623. Lyapunov biography
     
     • (4) (1993), 3-47.',8)">8] include: stability, particularly the stability of critical points; the construction and the application of the Lyapunov function; stability of functional- differential equations; the second Lyapunov method; and the method of the Lyapunov vector function in stability theory and nonlinear analysis.
       

624. Cohen biography
     
     • In addition to his work on set theory, Cohen has worked on differential equations and harmonic analysis.
       

625. Kodaira biography
     
     • These include applications of Hilbert space methods to differential equations which was an important topic in his early work and was largely the result of influence by Weyl.
       

626. Bochner biography
     
     • He also published papers on the gamma function, the zeta function and partial differential equations.
       

627. Aleksandrov Aleksandr biography
     
     • These first three works were all as a result of his mathematical work with Delone but also in 1934 he published two physics papers on quantum mechanics On the calculation of the energy of a bivalent atom by Fok's method and Remark on the commutation rule in Schrodinger's equation.
       

628. Lindemann biography
     
     • Ferdinand von Lindemann was the first to prove that π is transcendental, that is, π is not the root of any algebraic equation with rational coefficients.
       

629. Luke biography
     
     • His work on these topics led him to require much information on special functions and he was led to develop tables of special functions and to use numerical techniques to solve equations.
       

630. Fano biography
     
     • Early studies deal with line geometry and linear differential equations with algebraic coefficients ..
       

631. Lindelof biography
     
     • Lindelof's first work in 1890 was on the existence of solutions for differential equations.
       

632. Dougall biography
     
     • Examples of papers he read at meeting of the Society are Elementary Proof of the Collinearity of the Mid Points of the Diagonals of a Complete Quadrilateral on Friday 12 February 1897; Methods of Solution of the Equations of Elasticity on 10 December 1897; and Notes on Spherical Harmonics on 12 December 1913.
       

633. Frenicle de Bessy biography
     
     • History Topics: Pell's equation .
       

634. Mertens biography
     
     • Mertens is perhaps best known for his determination of the sign of Gauss sums, his work on the irreducibility of the cyclotomic equation, and the hypothesis which bears his name.
       

635. Baker biography
     
     • Its contents are as follows: Euclid's theory of parallel lines; Propositions of incidence; The symbolic representation and Pappus' theorem; Theorems proved from the propositions of incidence; The fundamental hypothesis; The symbols of the real points of a line; Involution and harmonic ranges; Related ranges and pencils; Conics; Assignment of two absolute points, properties of circles; The parabola; The rectangular hyperbola; Theorems on conics; Length and distance; Equation of conic and line.
       

636. McShane biography
     
     • McShane is famous for his work in the calculus of variations, Moore-Smith theory of limits, the theory of the integral, stochastic differential equations, and ballistics.
       

637. Hippocrates biography
     
     • Hippocrates' book also included geometrical solutions to quadratic equations and included early methods of integration.
       

638. Casorati biography
     
     • Differential equations were of great interest to him and his research in this area was undertaken with the aim of making the existing theories more accurate and more complete.
       

639. Chisini biography
     
     • Indeed, Enriques immediately recognised his talent, led him to obtain a degree in mathematics in 1912, and engaged him as assistant and coauthor in the writing of the treatise Lezioni sulla teoria geometrica delle equazioni e delle funzioni algebriche (Lessons on the geometric theory of equations and algebraic functions).
       

640. Grandi biography
     
     • Grandi also applied the term "clelies" to the curves determined by certain trigonometric equations involving the sine function .
       

641. Krawtchouk biography
     
     • He wrote papers on differential and integral equations, studying both their theory and applications.
       

642. Schmetterer biography
     
     • was mostly concerned with differential equations in the field of aerodynamics.
       

643. Aryabhata II biography
     
     • In Mahasiddhanta Aryabhata II gives in about twenty verses detailed rules to solve the indeterminate equation: by = ax + c.
       

644. Routh biography
     
     • In fact the impact of this prize winning work was very significant since Thomson and Tait rewrote for the second edition of their text Natural philosophy treatise the part dealing with equations of motion using Routh's developments.
       

645. Trail biography
     
     • His widely used Elements of Algebra, which he published for his students in 1770, ranged from first principles to equations of all orders and included applications to problem solving, physics and geometry.
       

646. Dirac biography
     
     • Also in 1928 he found a connection between relativity and quantum mechanics, his famous spin-1/2 Dirac equation.
       

647. Kaluznin biography
     
     • The school provided a solid background in mathematics, including topics in the foundations of analysis, differential equations and complex variables.
       

648. Bouvard biography
     
     • Using all the data at his disposal, Bouvard produced a system of 77 equations but was unable to find a possible orbit for the planet from them.
       

649. Weil biography
     
     • At this time he was particularly fascinated by solving Diophantine equations.
       

650. Aida biography
     
     • Aida explained the use of algebraic expressions and the construction of equations.
       

651. Petzval biography
     
     • He was influenced by the work of Liouville and wrote both a long paper and a two volume treatise on the Laplace transform and its application to ordinary linear differential equations.
       

652. Herstein biography
     
     • Among the methods and problems discussed in some detail are a derivation of the Slutsky equation via the calculus, a problem in Welfare Economics treated by the theory of convex sets, matrix theory as applied to international trade, and a game-theoretical approach to the personnel assignment problem.
       

653. Zhukovsky biography
     
     • Today it is known as the Kutta-Joukowski theorem, since Kutta pointed out that the equation also appears in his 1902 dissertation.
       

654. Campbell biography
     
     • In a paper published two years later "On the Theory of Simultaneous Partial Differential Equations" he develops a system of formulas by which it may be determined whether such a system is or is not integrable.
       

655. Freundlich biography
     
     • His occasional inability to comprehend these ideas had the salutary effect of making Einstein seek to simplify their mathematical formulation, for if one of Felix Klein's pupils could not make sense of his equations who could? Through his intimate contact with Einstein, Freundlich was the first to become thoroughly acquainted with the fundamental principles of the new gravitational theory and, as Einstein himself remarks in the foreword of Freundlich's book, he was particularly well qualified as its exponent because he had been the first to attempt to put it to the test.
       

656. Stieltjes biography
     
     • Stieltjes also contributed to ordinary and partial differential equations, the gamma function, interpolation, and elliptic functions.
       

657. Pythagoras biography
     
     • For example they solved equations such as a (a - x) = x2 by geometrical means.
       

658. Fefferman biography
     
     • Fefferman's work on partial differential equations, Fourier analysis, in particular convergence, multipliers, divergence, singular integrals and Hardy spaces earned him a Fields Medal at the International Congress of Mathematicians at Helsinki in 1978.
       
     • Professor Charles Fefferman's contributions and ideas have had an impact on the development of modern analysis, differential equations, mathematical physics and geometry, with his most recent work including his sharp (computable) solution of the Whitney extension problem.
       

659. Aiken biography
     
     • These plans were made for a very specific purpose, for Aiken's research had led to a system of differential equations which had no exact solution and which could only be solved using numerical techniques.
       

660. Marcinkiewicz biography
     
     • For in the field of real variable Marcinkiewicz had exceptionally strong intuition and technique, and the results he obtained in the theory of conjugate functions, had they been extended to functions of several variables might have given (as we see clearly now) a strong push to the theory of partial differential equations.
       

661. Davenport biography
     
     • At the most advanced level he wrote a monograph Analytic methods for Diophantine equations and Diophantine inequalities (1962) which includes many of his contributions extending the Hardy-Littlewood method.
       

662. Cremona biography
     
     • The geometric method is principally a use of terms or descriptive relations instead of equations.
       

663. Faber biography
     
     • Only in the 1980s was Faber's idea seen to be an important ingredient for the efficient solution of partial differential equations.
       

664. Galerkin biography
     
     • His visits around European construction sites ended around 1914 but his academic work then turned to the area for which he is today best known, namely the method of approximate integration of differential equations known as the Galerkin method.
       

665. Macaulay biography
     
     • What ideas were there then in this work? The main theme underlying the book is the problem of solving equations of systems of polynomials in several variables.
       

666. Seifert biography
     
     • Seifert, still able to do mathematical research, worked on differential equations and wrote a series of papers on the topic through the war years.
       

667. Adams biography
     
     • He began this work in 1851 when elected as President of the Royal Astronomical Society and he presented a paper to the Royal Society in 1853 in which he showed that Laplace had omitted terms from his equations which were not negligible.
       

668. Mineur biography
     
     • He was awarded his doctorate in 1924 for a thesis on functional equations in which he established an addition theorem for Fuchsian functions.
       

669. Burkhardt biography
     
     • Other topics on which Burkhardt published papers included groups, differential equations, differential geometry and mathematical physics.
       

670. Mellin biography
     
     • He also extended his transform to several variables and applied it to the solution of partial differential equations.
       

671. Polya biography
     
     • He also worked on conformal mappings and potential theory, and he was led to study boundary value problems for partial differential equations and the theory of various functionals connected with them.
       

672. Davidov biography
     
     • As well as his work on the equilibrium of a floating body, Davidov also worked on partial differential equations, elliptic functions and the application of probability to statistics.
       

673. Schwarzschild biography
     
     • Schwarzschild's relativity papers give the first exact solution of Einstein's general gravitational equations, giving an understanding of the geometry of space near a point mass.
       

674. Weinstein biography
     
     • In examining singular partial differential equations he introduced a new branch of potential theory and applied the results to many different situations including flow about a wedge, flow around lenses and flow around spindles.
       

675. Morse biography
     
     • Morse theory is important in the field of global analysis which is the study of ordinary and partial differential equations from a global or topological point of view.
       

676. Rudio biography
     
     • He reduced this problem to the problem of solving a differential equation.
       

677. Butters biography
     
     • He also contributed to the mathematical work of the Society, For example at the meeting of the Society on Friday 11 January 1889, J Watt Butters discussed the solution of an algebraic equation.
       

678. Wolf Frantisek biography
     
     • Consideration of the Schrodinger equation leads to perturbation problems for partial differential operators, where the change may occur in the coefficients of the operator or in boundary conditions.
       

679. Lowenheim biography
     
     • Lowenheim analysed and improved upon the customary methods of solving equations in the calculus of classes or domains (that is, set theory in its Peirce-Schroder [Charles Peirce and Ernst Schroder] setting) and proved what is now known as Lowenheim's general development theorem for functions of functions.
       

680. Robertson biography
     
     • Around this time he built on de Sitter's solution of the equations of general relativity in an empty universe and developed what are now called Robertson-Walker spaces [Biographical Memoirs National Academy of Sciences 51 (1980), 343-361.',2)">2]:- .
       

681. Grave biography
     
     • In particular he worked on Galois theory, ideals and equations of the fifth degree.
       

682. Erdelyi biography
     
     • He also worked on asymptotic analysis, fractional integration and singular partial differential equations.
       

683. Bromwich biography
     
     • T J I'A Bromwich's method for solving the source-free Maxwell equations for electromagnetic waves.
       

684. Loewner biography
     
     • We should also note that Loewner's proof uses the Loewner differential equation which has been studied extensively since he introduced it, and was used by de Branges in his celebrated proof of the Bieberbach conjecture.
       

685. Russell Scott biography
     
     • This is now recognised as a fundamental ingredient in the theory of 'solitons', applicable to a wide class of nonlinear partial differential equations.
       

686. Todhunter biography
     
     • He also wrote some more elementary texts, for example Algebra (1858), Trigonometry (1859), Theory of Equations (1861), Euclid (1862), Mechanics (1867) and Mensuration (1869).
       

687. Bergman biography
     
     • This was Erhard Schmidt who had been awarded his doctorate by the University of Gottingen for a thesis on integral equations written under Hilbert's supervision.
       
     • Bergman used the theory of integral equations as developed by Erhard Schmidt and David Hilbert [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]:- .
       
     • This led him further to a general theory of integral operators that map arbitrary analytic functions into solutions of various partial differential equations.
       
     • While at Brown University he participated in the Summer School in 1941 Advanced instruction and research in mechanics which resulted in the publications Partial Differential Equations and Fluid dynamics.
       
     • Several years ago Stefan Bergman discovered that essentially the same is true for a vast class of partial differential equations which includes the potential equation as the simplest case.
       
     • Bergman gave explicit formulae which allow a solution of a given differential equation to derive from an arbitrarily chosen analytic function (in some instances from a pair of real functions) and proved that all solutions can be derived in this way.
       
     • They consider a special type of differential equation, yet more general than the potential equation, and build up a system of solutions in close analogy to the procedure followed in the theory of analytic functions.
       
     • In 1953 Bergman and Schiffer published Kernel functions and elliptic differential equations in mathematical physics.
       
     • In this book the authors collect their researches of the last few years on elliptic partial differential equations.
       
     • The second part lays more stress on rigor, and treats fundamental solutions, reduction of boundary value problems to integral equations, orthonormal systems and kernel functions, eigenvalue problems associated with the kernels, variational theory of domain functions, comparison domains, basic existence theorems, and dependence of solutions on the boundary data or on the coefficients of the differential equation.
       
     • The presentation is in an easy flowing style, and the material should prove to be a most useful guide to those interested in the more advanced theory of linear elliptic partial differential equations.
       
     • Bergman published Integral operators in the theory of linear partial differential equations in 1961.
       
     • This treatise gives a summary of the author's numerous contributions from 1926 to 1961 to the theory of solutions of linear partial differential equations in two and three real variables by means of integral operators which usually involve analytic functions of one, or sometimes two, complex variables.
       
     • Awards are made every year or two in: 1) the theory of the kernel function and its applications in real and complex analysis; or 2) function-theoretic methods in the theory of partial differential equations of elliptic type with attention to Bergman's operator method.
       

688. Peirce B O biography
     
     • master of the methods dealing with the partial differential equations of mathematical physics.
       

689. Flato biography
     
     • The second is the cohomological study of nonlinear representations of covariance groups of nonlinear partial differential equations which leads to important mathematical developments with nontrivial physical consequences ..
       

690. Gershgorin biography
     
     • Semyon Aranovich Gershgorin studied at Petrograd Technological Institute from 1923, becoming Professor in 1930, and from 1930 he worked in the Leningrad Mechanical Engineering Institute on algebra, theory of functions of complex variables, numerical methods and differential equations.
       

691. Watson Henry biography
     
     • In addition to these books he wrote on Lagrange's method and Monge's method for solving partial differential equations and, jointly with Galton, he wrote On the probability of extinction of families.
       

692. Brink biography
     
     • Finally we give some examples of Brink's papers: A new integral test for the convergence and divergence of infinite series (1918); A new sequence of integral tests for the convergence and divergence of infinite series (1919); The May Meeting of the Minnesota Section (1927); Recent Publications: Reviews: Studies in the History of Statistical Method - With Special Reference to Certain Education Problems (1929); The May Meeting of the Minnesota Section (1930); A Simplified Integral Test for the Convergence of Infinite Series (1931); Recent Publications: Reviews: Differential Equations (1932); The Annual Meeting of the Minnesota Section (1937); and College Mathematics During Reconstruction (1944).
       

693. Mackey biography
     
     • The new material in the present book is concentrated in the last 50 pages and it centres around lattice models in statistical mechanics, PDEs in hydrodynamics, Kac-Moody Lie algebras, and the Korteweg-de Vries equation.
       

694. Dionis biography
     
     • Dionis du Sejour also worked on the theory of equations, not attaining the depth of results of Bezout or Lagrange.
       

695. Valyi biography
     
     • His doctoral dissertation was on the theory of the propeller which led to his developing a theory of partial differential equations of the second order.
       

696. Williams biography
     
     • In 1925 Cox was awarded his doctorate by Cornell University for his thesis Polynomial solutions of difference equations.
       

697. Velez-Rodriguez biography
     
     • Her doctoral work consisted of studying differential equations which arose in the study of astronomical orbits.
       

698. Hensel biography
     
     • He showed, at least for quadratic forms, that an equation has a rational solution if and only if it has a solution in the p-adic numbers for each prime p and a solution in the reals.
       

699. Huygens biography
     
     • History Topics: Quadratic, cubic and quartic equations .
       

700. Bethe biography
     
     • Oddly, though, he left the neutrino out of the proton-proton reaction equation.
       

701. Stueckelberg biography
     
     • Occasionally all three of us would gather around the table while Kramers wrote out a few equations, illustrating how they fitted together to explain some atomic property or other.
       

702. Chen biography
     
     • An outstanding and original mathematician, Chen's work falls naturally into three periods: his early work on group theory and links in the three sphere; his subsequent work on formal differential equations, which gradually developed into his most powerful and important work; and his work on iterated integrals and homotopy theory, which occupied him for the last twenty years of his life.
       

703. Fox biography
     
     • He submitted two further papers in June 1925, The Expression of Hypergeometric Series in Terms of Similar Series, and Some Further Contributions to the Theory of Null Series and Their Connexion with Null Integrals to the same Proceedings; both were published in 1927 as was his next paper A Generalization of an Integral Equation Due to Bateman which he submitted in 1926.
       
     • Fox's main contributions were on hypergeometric functions, integral transforms, integral equations, the theory of statistical distributions, and the mathematics of navigation.
       

704. Babbage biography
     
     • He wrote two major papers on functional equations in 1815 and 1816.
       

705. Schramm biography
     
     • Schramm was recognized for his development of stochastic Loewner equations and for his contributions to the geometry of Brownian curves in the plane.
       

706. Schwarz biography
     
     • An idea from this work, in which he constructed a function using successive approximations, led Emile Picard to his existence proof for solutions of differential equations.
       

707. Krull biography
     
     • his earlier studies, but also dealt with other fields of mathematics: group theory, calculus of variations, differential equations, Hilbert spaces.
       

708. Turan biography
     
     • Their importance first of all is that they lead to interesting deep problems of a completely new type; they have quite unexpectedly surprising consequences in many branches of mathematics - differential equations, numerical algebra, and various branches of function theory.