Search Results for Sequence*
Biographies
- Adleman biography
- For example, given a molecule with the sequence CATGTC, DNA polymerase will produce a new molecule with the sequence GTACAG.
- It is now possible to write a DNA sequence on a piece of paper, send it to a commercial synthesis facility and in a few days receive a test tube containing approximately molecules of DNA, all (or atleast most) of which have the described sequence.
- Adleman first assigned a random DNA sequence to each vertex and edge in the graph (the sequences are known as oligonucleotides).
- Because each DNA sequence has its Watson-Crick complement, each vertex is associated with its complement sequence.
- Once the encodings were set in place, the complementary DNA sequences for the vertices and the sequences for the edges were synthesized.
- I took a pinch (about 1014 molecules) of each of the different sequences and put them into a common test tube.
- Serre biography
- Serre's early work was on spectral sequences.
- A spectral sequence is an algebraic construction like an exact sequence, but more difficult to describe.
- Serre did not invent spectral sequences, these were invented by the French mathematician Jean Leray.
- However, in 1951, Serre applied spectral sequences to the study of the relations between the homology groups of fibre, total space and base space in a fibration.
- Serre's work led to topologists realising the importance of spectral sequences.
- The Serre spectral sequence provided a tool to work effectively with the homology of fiberings.
- For this work on spectral sequences and his work developing complex variable theory in terms of sheaves, Serre was awarded a Fields Medal at the International Congress of Mathematicians in 1954.
- Farey biography
- The Farey series (really a sequence) is defined as follows.
- Write the sequence in ascending order of magnitude beginning with the smallest.
- Then the "curious property" is that each member of the sequence is equal to the rational whose numerator is the sum of the numerators of the fractions on either side, and whose denominator is the sum of the denominators of the fractions on either side.
- Then the Farey sequence is:
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- He explains how to construct what is in fact the Farey sequence for n = 99 and Farey's "curious property" is built into his construction.
- Historical references to the Farey sequence have been examined by the authors of [The Mathematical Intelligencer 17 (2) (1995), 64-67.',3)">3].
- The standard reference for the Farey sequence is [An introduction to the theory of numbers (New York, 1945).',2)">2] in which Hardy writes:-
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- The article [The Mathematical Intelligencer 17 (2) (1995), 64-67.',3)">3] contains other interesting information on Farey's sequence, its relation to Pick's area theorem, and the inaccurate historical comments made about the sequence over many years.
- Beatty biography
- Define two sequences [nR] and [nS] where n runs through the natural numbers and, for any real number x, [x] denotes the greatest integer less than or equal to x.
- The two sequences are called 'Beatty sequences' and have the property that every natural number appears in one and only one of the two sequences.
- For example taking R = 5/π gives the sequences
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- R = 6/π gives the sequences
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- while R = 7/π gives the sequences
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- References to paper investigating Beatty sequences appear in [Canadian Math.
- Adams Frank biography
- He won a Fellowship at Trinity College, Cambridge, with his doctoral thesis on spectral sequences On Special Sequences of Self-Obstruction Invariants which he submitted in 1955.
- He returned to Cambridge in 1956 to take up the Fellowship and during this period he developed the spectral sequence which today is called the "Adams' spectral sequence".
- Babbage biography
- Babbage illustrated what his small engine was capable of doing by calculating successive terms of the sequence n2 + n + 41.
- The terms of this sequence are 41, 43, 47, 53, 61, ..
- Babbage reports that his small difference engine was capable of producing the members of the sequence n2 + n + 41 at the rate of about 60 every 5 minutes.
- The control on the sequence of operations to be carried out was by a Jacquard loom type device.
- elaborations on the points made by Menabrea, together with some complicated programs of her own, the most complex of these being one to calculate the sequence of Bernoulli numbers.
- Fibonacci biography
- A problem in the third section of Liber abaci led to the introduction of the Fibonacci numbers and the Fibonacci sequence for which Fibonacci is best remembered today:-
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- The resulting sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ..
- This sequence, in which each number is the sum of the two preceding numbers, has proved extremely fruitful and appears in many different areas of mathematics and science.
- The Fibonacci Quarterly is a modern journal devoted to studying mathematics related to this sequence.
- For unity is a square and from it is produced the first square, namely 1; adding 3 to this makes the second square, namely 4, whose root is 2; if to this sum is added a third odd number, namely 5, the third square will be produced, namely 9, whose root is 3; and so the sequence and series of square numbers always rise through the regular addition of odd numbers.
- Caccioppoli biography
- to build, in its most general case, a sequence of polyhedral approximating surfaces, whose areas tend to the area of the curved surface whether finite or infinite.
- To decide on both existence and uniqueness (and not only on existence, as Brouwer's theorem does) he provided the general concept of functional correspondence inversion, stating, in 1932, that a transformation between two Banach spaces is invertible only if it is locally invertible and if the compact sequences are the only ones to be transformed into convergent sequences.
- Tait biography
- and then the knot would be described by the sequence of crossings of length 2n where each of A, B, C, ..
- Firstly which sequences of the above type corresponded to a knot, and secondly how could it be determined when two knots described by such sequences were the same.
- Baire biography
- Around this time he discovered conditions under which a function is a limit of a sequence of continuous functions.
- Class 1 functions were those functions which were the limit of a sequence of continuous functions.
- Class 2 functions were those functions which were the limit of a sequence of Class 1 functions, while Class 3 functions were those functions which were the limit of a sequence of Class 2 functions.
- Auslander biography
- His discovery, with Idun Reiten, of almost split sequences in the early seventies is certainly one of the foundation stones of our subject.
- (1) Homological dimension and local rings, (2) Ramification theory, (3) Functors, (4) Almost split sequences and Artin algebras, (5) Some topics in representation theory, (6) Lattices over general orders, (7) Tilting theory and homologically finite subcategories, (8) Almost split sequences and commutative rings, (9) Grothendieck groups and Cohen-Macaulay approximations, and (10) Relative theory and syzygy modules ..
- Kaprekar biography
- Continuing gives the sequence
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- In fact the sequence we looked at really starts at 1
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- It is possible to have 20 consecutive Harshad numbers but one has to go to numbers greater than 1044363342786 before such a sequence is found.
- Lyndon biography
- in 1946 for a thesis on homological algebra, the work being an outstanding early step in the study of spectral sequences.
- Now we know that the difficulty lies in the additional invariants presented by the spectral sequence of that group extension.
- Turing biography
- Some of the symbols written down will form the sequences of figures which is the decimal of the real number which is being computed.
- It is impossible to decide (using another Turing machine) whether a Turing machine with a given table of instructions will output an infinite sequence of numbers.
- Lucas biography
- Lucas is best known for his results in number theory: in particular he studied the Fibonacci sequence and the associated Lucas sequence is named after him.
- Define the sequence
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- Aiken biography
- whereas accounting machines handle only positive numbers, scientific machines must be able to handle negative ones as well; that scientific machines must be able to handle such functions as logarithms, sines, cosines and a whole lot of other functions; the computer would be most useful for scientists if, once it was set in motion, it would work through the problem frequently for numerous numerical values without intervention until the calculation was finished; and that the machine should compute lines instead of columns, which is more in keeping with the sequence of mathematical events.
- Working with three engineers, Aiken developed the ASCC computer (Automatic Sequence Controlled Calculator) which could carry out five operations, addition, subtraction, multiplication, division and reference to previous results.
- ASCC was controlled by a sequence of instructions on punched paper tapes.
- Lions biography
- If the quantity to be minimised has an "energy"-like term involving derivatives, then one has control on local regularity along a minimising sequence.
- Lions's clever idea was to introduce "concentration compactness" techniques which look at energy concentrations and so avoid problems which occur when examining the minimising sequences without compactness.
- Shnirelman biography
- Using these ideas of compactness of a sequence of natural numbers he was able to prove a weak form of the Goldbach conjecture showing that every number is the sum of ≤ 20 primes.
- Later significant contributions by Shnirelman include his two papers On the additive properties of numbers, and On addition of sequences published in 1940 after his death.
- Fischer biography
- In 1907 Ernst Fischer studied orthonormal sequences of functions and gave necessary and sufficient conditions for a sequence of constants to be the Fourier coefficients of a square integrable function.
- Gleason biography
- A connected locally compact group G is a projective limit of a sequence of Lie groups; and, if G has no small subgroups, then it is a Lie group.
- Chapters I to VI cover elementary logic and set theory; Chapters VII to X deal with the various "number systems" from the natural integers to the complex numbers; Chapter XI briefly returns to set theory (countable sets, cardinal numbers and the axiom of choice); finally, the last four chapters deal, respectively, with limits of complex sequences, infinite series and products, metric spaces, and the elementary theory of holomorphic functions of one variable (Cauchy integral excluded, but the logarithmic function is defined and studied).
- Stieltjes biography
- Stieltjes examined the sequence of rational functions Pn(z)/Qn(z) and the connections between the roots of the polynomials Pn(z) and Qn(z).
- This problem arose in the study of two functions arising as the limits of the sequences P2n(z)/Q2n(z) and P2n+1(z)/Q2n+1(z).
- Slutsky biography
- In 1927 he showed that subjecting a sequence of independent random variables to a sequence of moving averages generated an almost periodic sequence.
- Jones Burton biography
- And its nice to have a sequence of theorems which are useful but which can be proved by everybody.
- Elementary properties of connected point sets can be formulated into a sequence of this sort.
- Toeplitz biography
- In a joint paper with Kothe in 1934, Toeplitz introduced, in the context of linear sequence spaces, some important new concepts and theorems.
- He also relates how the isomorphism problem for sequence spaces appeared in disguise as a new problem on nuclear (F)-spaces with basis.
- Karlin biography
- In particular he concentrated on the development of mathematical and computational techniques and tools for the analysis of DNA and protein sequences.
- He worked on descriptive and statistical analysis of protein structure properties, including methods for characterizing and comparing protein structures and sequences.
- Fiorentini biography
- In 1971 Fiorentini published four papers: On relative regular sequences; Esempi di anelli di Cohen-Macaulay semifattoriali che non sono di Gorenstein; Esempi di anelli di Cohen-Macaulay che non sono di Gorenstein and (with Aldo Marruccelli) Oggetto e fondamenti della matematica.
- regular sequences and refinements of this notion, leading to special classes of rings, important in algebraic geometry (complete intersection, Gorenstein, Cohen-Macaulay, Buchsbaum, etc.);
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- Mandelbrot biography
- If the sequence z0 , z1 , z2 , z3 , ..
- If the sequence diverges from the origin, then the point is not in the set.
- Wang biography
- Wang worked on algebraic topology and discovered the 'Wang sequence', an exact sequence involving homology groups associated with fibre bundles over spheres.
- Rado Richard biography
- in 1935 under Hardy's supervision on Linear transformations of sequences.
- Some of his more minor work was in topics such as the convergence of sequences and series.
- Roth Klaus biography
- The conjecture concerned a sequence
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- If N(x) denotes the number of terms of the sequence less than x, Roth proved the conjecture that N(x)/x → 0 as x → ∞.
- Jeffery Ralph biography
- He obtained his doctorate in 1928 after submitting his dissertation The Uniform Approximation of a Sequence of Integrals and the Sequence of Functions Which Define a Definite Integral Containing a Parameter.
- Euwe biography
- In 1929 he published a mathematics paper in which he constructed an infinite sequence of 0's and 1's with no three identical consecutive subsequences of any length.
- It had always been the intention of the rules that this should not be possible, but the rule that a game is a draw if the same sequence of moves occurs three times in succession was not, as Euwe showed, sufficient.
- Markov biography
- He also studied sequences of mutually dependent variables, hoping to establish the limiting laws of probability in their most general form.
- Markov is particularly remembered for his study of Markov chains, sequences of random variables in which the future variable is determined by the present variable but is independent of the way in which the present state arose from its predecessors.
- 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 1897 he published a deep investigation into the subject of uniform convergence of sequences of real continuous functions ..
- Church biography
- He used these notions in On the concept of a random sequence (1940) where he attempted to give a logically satisfactory definition of "random sequence".
- Mises biography
- He combined the idea of a Venn limit and a random sequence of events.
- von Mises' notion of a random sequence in the context of his approach to probability theory.
- Green Sandy biography
- Finally let us mention Sandy's little book Sequences and series (1958) whose aim is stated in the Preface:-
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- Green: "Sequences and Series"
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- Behrend biography
- Two years later he again published on a new topic with his paper The uniform convergence of sequences of monotonic functions.
- In the same year of 1948 he also published Generalization of an inequality of Heilbronn and Rohrbach and Some remarks on the distribution of sequences of real numbers, with Some remarks on the construction of continuous non-differentiable functions being published in the Proceedings of the London Mathematical Society in the following year.
- Collatz biography
- Given any integer m define a sequence by putting
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- The problem asks if, for every starting value m, the sequence a(i) always reaches 1? The problem remains unsolved, but before you try a few small numbers yourself looking for a counterexample, let us say that the conjecture has been verified for all numbers m up to about 1014 .
- Lawson biography
- In a meeting held on Friday 9 March 1923 a discussion on the teaching of elementary geometry was opened by Mr A J Tressland, M.A., F.R.S.E., of the Edinburgh Academy, who advocated the adoption of the sequence in geometrical teaching contained in the schedule recently issued by a special committee of the Assistant-Masters' Association.
- De Bruijn biography
- His work on combinatorics resulted in influential notions and results of which we mention the de Bruijn-sequences of 1946 and the de Bruijn-Erdos theorem of 1948.
- Johnson biography
- Johnson's proof can be generalized to include asymmetric Dirichlet priors and those finitely exchangeable sequences with linear posterior expectation of success.
- Young Laurence biography
- The book is an introduction to Lebesgue-Stieltjes integration using techniques based on monotone sequences of functions and was published by Cambridge University Press in 1927.
- Al-Uqlidisi biography
- unlike al-Samawal, al-Uqlidisi never formulates the idea of completing the sequence of powers of ten by that of their inverse after having defined the zero power.
- Rajagopal biography
- Rajagopal studied sequences, series, summability.
- Cauchy biography
- 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.
- Schmidt biography
- Schmidt defined a space H whose elements are square summable sequences of complex numbers.
- Leray biography
- Following this line he published papers which introduced sheaves, and the spectral sequence of a continuous map.
- Bachelier biography
- (namely f (xn|xs) = intR f (xn|xr) f (xr|xs) dxr where n > r > s where f are the transition densities of a Markov sequence of random variables) and the seeds of Markov Processes, weak convergence of random variables (i.e.
- Bosanquet biography
- During 1969-70 he visited the University of Western Ontario and gave another major lecture series, this time on Matrix transformations and sequence spaces with applications to summability.
- Peschl biography
- The titles of the chapters are: Algebra and geometry of complex numbers; Fundamental topological concepts, sets, sequences of complex numbers and infinite series; Functions, real and complex differentiability and holomorphy; Integral theorems and their consequences; Winding number and curves homologous to zero; Taylor development of holomorphic functions; Elementary transcendental functions; Laurent series, isolated singularities and residue calculus; Holomorphic and meromorphic functions obtained by limiting processes; Analytic continuation; and Conformal mappings.
- Pitt biography
- The second part deals with more specialised topics, such as convergence theorems and random sequences and functions.
- McDuff biography
- I had always thought of mathematics as being much more straightforward: a formula is a formula, and an algebra is an algebra, but Gelfand found hedgehogs lurking in the rows of his spectral sequences!
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- Parry biography
- The author investigates the structure of finite-state stochastic processes that are called intrinsically Markovian since they behave like Markov chains because "possible" sequences of the processes are determined by a chain rule.
- Schatten biography
- Who could forget what a sequence was after hearing Schatten describe a long corridor, stretching as far as the eye could see, with hooks regularly spaced on the wall and numbered 1, 2, 3, ..
- Bohl biography
- For example it is still unknown whether the fractional parts of (3/2)n form a uniform distribution on (0,1) or even if there is some finite subinterval of (0,1) which is avoided by the sequence.
- Moser William biography
- Moser published a fine collection of combinatorics papers jointly with Morton Abramson: A note on combinations (1966); Combinations, successions and the n-kings problem (1966); Permutations without rising or falling w-sequences (1967); Enumeration of combinations with restricted differences and cospan (1969); Generalizations of Terquem's problem (1969); The problem of the second seating and generalizations (1972); Arrays with fixed row and column sums (1973); and Linear and ring arrangements (1976).
- Lehmer Derrick N biography
- One is not apt to get a very wide view of the history of a subject by reading a hundred biographical footnotes, arranged in no sort of sequence.
- Rychlik biography
- He did excellent work on algebra and number theory, for example he generalised Hensel's ideas on g-adic numbers in 1914, later approaching them via sequences and limits unlike the 'generalised decimal expansion' approach of Hensel.
- David biography
- After she returned to University College after World War II she published A power function for tests of randomness in a sequence of alternatives and A c2'smooth' test for goodness of fit which both appeared in Biometrika in 1947.
- Koszul biography
- The main topics on which Koszul undertook research included: homology and cohomology of Lie algebras; relative cohomology; reductive subalgebras and the transgression theorem; the formalism of spectral sequences; "Koszul complexes"; proper and differentiable actions of Lie groups; slices; hermitian forms on complex homogeneous domains; bounded domains; locally flat manifolds; convex homogeneous domains; simplicial spaces; themes related to Gelfand-Fuks theory and supergeometry.
- Hardie Robert biography
- On Friday 9 March 1923 a discussion on the teaching of elementary geometry was opened by Mr A J Tressland, M.A., F.R.S.E., of the Edinburgh Academy, who advocated the adoption of the sequence in geometrical teaching contained in the schedule recently issued by a special committee of the Assistant-Masters' Association.
- Spencer biography
- In the first of these they proved that a progression-free sequence of positive integers never has positive density.
- Frohlich biography
- A careful account is given of Herbrand's description of the change in the sequence of ramification groups when passing from a Galois group to a quotient group.
- Jeans biography
- four dimensional space, a space which expands forever; a sequence of events which follows the laws of probability instead of the laws of causation; all these concepts seem to my mind to be structures of pure thought.
- Carleman biography
- If (an), n ≥ 1, is a sequence of positive numbers, then
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- Lupas biography
- The author first develops a method for constructing sequences of positive linear operators on a subspace of bounded functions on the real line.
- Betti biography
- 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.
- Moufang biography
- It is supplemented by a sequence of papers on continuum mechanics.
- Eckert Wallace biography
- In 1949 the Selective Sequence Electronic Calculator (SSEC) was built.
- Riesz biography
- Riesz introduced the idea of the 'weak convergence' of a sequence of functions ( fn(x) ).
- Kostrikin biography
- This chapter includes explanations of duality, Lie algebras, the language of category theory, and exact sequences of functors.
- Robinson biography
- The main body of the work consists of rewritten versions of the author's main contributions to the subject, which are brought into a smooth and eminently readable sequence.
- Zeckendorf biography
- Eduourd Zeckendorf was an amateur mathematician whose name is given to the property that every positive integer can be represented uniquely as the sum of non-consecutive Fibonacci numbers, the sequence defined by
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- Lagny biography
- Stewart Dugald biography
- They adduced note xvi of his "Heat" in which he had written favourably about the doctrine of the sceptical David Hume that causation was nothing more than an observed constant and invariable sequence of events.
- Robinson Raphael biography
- A typical paper on logic was Finite sequences of classes which appeared in 1945.
- Weatherburn biography
- He held this post until he retired in 1950 but his excellent sequence of research papers stopped in 1939.
- Helly biography
- First there is Helly's selection principle which says that given a sequence of functions of bounded variation which are of uniform bounded variation and uniformly bounded at a point, then there exists a subsequence which converges to a function of bounded variation.
- Wolf biography
- On a plane surface draw a sequence of parallel, equally spaced straight lines; take an absolutely cylindrical needle of length a, less than the constant interval d which separates the parallels, and drop it randomly a great number of times on the surface covered by the lines.
- Al-Umawi biography
- However, he does note that the sequence r1, r2, r3, r4, r5, ..
- Knopp biography
- Chapter III: Sets, sequences and power series.
- Hurwitz biography
- Hurwitz solved this problem completely showing that the condition held if and only if a certain sequence of determinants are all positive.
- Borchardt biography
- In 1881 Borchardt published an algorithm for the arithmetic-geometric mean of two elements from (two) sequences, although it was actually first proposed by Gauss in a letter to Pfaff written in 1800.
- Ribenboim biography
- But in several of them, such as powers in recurrent sequences, the author has made a number of contributions.
- Al-Qalasadi biography
- For example, the sequences ∑ n2 and ∑ n3 had been studied by al-Samawal and al-Baghdadi, and methods for computing square roots were known to the Babylonians.
- De Finetti biography
- However, his contributions to probability and statistics do not reduce to his subjective approach and in fact they include important results on finitely additive measures, processes with independent increments, sequences of exchangeable variables and associative means; see the review by M D Cifarelli and E Regazzini [Statistical Science 11 (1996), 253-282.',3)">3] for details on these.
- Yamabe biography
- A connected locally compact group G is a projective limit of a sequence of Lie groups; and, if G has no small subgroups, then it is a Lie group.
- Edmonds biography
- The product of two such sequences ∑ an and ∑ bn is defined to be ∑ cn where cn = ∑ ambn-m where the sum is again over all integers m, both positive and negative.
- Gateaux biography
- A central problem arises from the fact that generally, in infinite dimension, a subset has a volume equal to zero or infinity, and this prevents the direct extension of the Riemann integral defined through an approximating step-functions sequence.
- Loyd biography
- The mysterious feature of the puzzle is that none seem able to remember the sequence of moves whereby they feel sure they have succeeded in solving the puzzle.
- Yates biography
- The method of "partial systematic samples," based on short sections of completely enumerated sequences, is proposed for estimating the systematic sampling error.
- Magnitsky biography
- [A] student took each of the "sciences" in sequence.
- Montgomery biography
- A connected locally compact group G is a projective limit of a sequence of Lie groups; and, if G has no small subgroups, then it is a Lie group.
- Straus biography
- An earlier joint paper by Straus and Erdos was On linear independence of sequences in a Banach space (1953).
- Hemchandra biography
- Before we rush to try to change the name of the Fibonacci numbers into Hemchandra number it is worth noting that Gopala had studied these numbers in about 1135 and Indian mathematicians as early as the 7th century had looked at sequences which are produced by the familiar Fibonacci rule.
- Rudin biography
- She began publishing a sequence of four papers in 1998 aimed at characterizing the Hausdorff continuous images of compact linearly ordered spaces.
- Sonin biography
- He has a sequence of polynomials named after him - the Sonin polynomials Tnm(x) satisfy the differential equation
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- Fenyo biography
- Here, after laying the groundwork, the authors discuss sequences, Fourier transforms and the regularization of functions, and conclude with a number of applications.
- Cooper biography
- As a lecturer he could be hard to follow: sometimes the sequence of ideas came too quickly for the comfort of those in the audience with less agile minds: sometimes he overestimated the background knowledge of his audience.
- Bishop biography
- However, a good deal of Brouwer's intuitionism is rejected, notably his notions of free choice sequences, spreads and the bar theorem.
- Seidel Jaap biography
- The joint paper with Van Lint in 1966 (still cited) started a long sequence of important contributions to the theory of strongly regular graphs and design theory.
- Hirzebruch biography
- complex K-theory and its spectral sequence and various geometrical applications (with M F Atiyah),
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- Meiklejohn biography
- If there were any doubts as to his rapid promotion to the headship after only four years of teaching, these were soon dispelled, for three years later there began an almost unbroken sequence of highly-placed John Welsh Mathematical Bursars at his old University.
- Mellin biography
- In this theory, he included the possibility of high-order poles (thereby leading to the inclusion of logarithmic terms in the expansion) and to several sequences of poles yielding sums of asymptotic expansions of very general form.
- Erdos biography
- To Erdos the proof had to provide insight into why the result was true, not just provide a complicated sequence of steps which would constitute a formal proof yet somehow fail to provide any understanding.
- Montel biography
- The idea of compactness had emerged as a fundamental concept in analysis during the nineteenth century; provided a set is bounded in Rn, it is possible to define for and sequence of points, a subsequence which converges to a point of Rn (the Bolzano-Weierstrass theorem).
- Hall Marshall biography
- Hall returned to Yale where he was awarded his doctorate in 1936 for his thesis An Isomorphism Between Linear Recurring Sequences and Algebraic Rings which was supervised by Oystein Ore.
- Thomae biography
- He then went to construct the rational numbers using Weierstrass's approach, then continued with a construction of the real numbers using the Cauchy sequence type of definition already published by Cantor and Heine.
- Goldstine biography
- We wish to develop here methods that will permit us to use the coded sequence of a problem, when that problem occurs as part of a more complicated one, as a single entity, as a whole, and avoid the need for recoding it each time when it occurs as a part in a new context, i.e., in a new problem.
- Amitsur biography
- It continues with the remarkable result that every PI-algebra satisfies a power of the standard identity, the primeness property of the T-ideal of the polynomial identities of matrices, some striking properties of Capelli identities which led to the construction of new central polynomials which simplifies and sometimes extends various results in structure theory of PI-algebras and is closed with results on sequences of codimensions and cocharacters of PI-algebras.
- Aleksandrov biography
- Aleksandrov was the first to use the phrase 'kernel of a homomorphism' and around 1940-41 he discovered the ingredients of an exact sequence.
- Argand biography
- However, the fact that his name is associated with this geometrical interpretation of complex numbers is only as a result of a rather strange sequence of events.
- Shannon biography
- He gave a method of analysing a sequence of error terms in a signal to find their inherent variety, matching them to the designed variety of the control system.
- Blum biography
- An inductive inference machine produces, from any enumeration of a partial function, a certain output sequence of numbers.
- Pade biography
- It deals with the development into a continued fraction of the generating function of a sequence satisfying a difference equation.
- Banach biography
- The completeness is important as this means that Cauchy sequences in Banach spaces converge.
- Narayana biography
- The thirteenth chapter of Ganita Kaumudi was called Net of Numbers and was devoted to number sequences.
- Foster biography
- For example in 1941 he published Natural systems : the structure of abstract monotone sequences.
- Girard Albert biography
- He is also famed for being the first to formulate the (now well known) inductive definition fn+2 = fn+1 + fn for the Fibonacci sequence.
- Lemaitre biography
- The sequence of events narrated by the author shows, however, that time and again Lemaitre was accused (especially by Einstein) of using scientific reasonings "to defend a (religious) dogma of the Church".
- Alberti biography
- There is certainly a mathematical flavour to the way that Alberti has sequences of small and large chapels alternating along the sides of the main space.
- Mendelsohn biography
- He wrote papers on a wide variety of combinatorial problems, for example: Symbolic solution of card matching problems (1946), Applications of combinatorial formulae to generalizations of Wilson's theorem (1949), Representations of positive real numbers by infinite sequences of integers (1952), A problem in combinatorial analysis (1953), The asymptotic series for a certain class of permutation problems (1956), and Some elementary properties of ill conditioned matrices and linear equations (1956).
- 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).
- Feller biography
- Some of the first papers reviewed by Mathematical Reviews were written by Feller himself such as Completely monotone functions and sequences (Duke Journal, 1939) and Die Grundlagen der Volterraschen Theorie des Kampfes ums Dasein in wahrscheinlichkeitstheoretischer Behandlung (1939).
- Freitag biography
- Most appropriately, the Fibonacci Quarterly chose to honour her not on her 90th birthday, but on the threshold of her 89th year, since 89 is a number in the Fibonacci sequence.
- Rennie biography
- This paper was published in the Journal of the Australian Mathematical Society, as were On sequences of integrable functions (1962) and On a class of inequalities (1963).
- Antonelli biography
- Each of these had to be routed to the proper bank of electronics and performed in sequence - not simply a linear progression but a parallel one, for the ENIAC, amazingly, could conduct many operations simultaneously.
- Hurewicz biography
- Hurewicz is best remembered for two remarkable contributions to mathematics, his discovery of the higher homotopy groups in 1935-36, and his discovery of exact sequences in 1941.
- Cantor biography
- Cantor published a paper on trigonometric series in 1872 in which he defined irrational numbers in terms of convergent sequences of rational numbers.
- Poincare biography
- Being given a sequence of propositions, he finds that all follow logically from the first.
- Remez biography
- He proved results about bounded polynomials and created general operator methods of sequence approximation.
- Young Alfred biography
- In 1934 Young realised the significance of the sequence in which the standard tableau can be written and the following year he again related his work to that of Frobenius and Schur.
- Brouwer biography
- His constructive theories were not easy to set up since the notion of a set could not be taken as a basic concept but had to be built up using more basic notions which, in Brouwer's case, were choice sequences.
- Leslie biography
- They adduced note xvi of his Heat in which he had written favourably about the doctrine of the sceptical David Hume that causation was nothing more than an observed constant and invariable sequence of events.
- Titeica biography
- It describes the work he did on the lattice of mutually conjugated lines on a surface and the Laplace sequence of such lattices [Balkan J.
- Nash-Williams biography
- In the paper necessary and sufficient conditions are given so that a knight can visit each square exactly once in a single infinite sequence of moves.
- Laplace biography
- the small probability of collision of the Earth and a comet can become very great in adding over a long sequence of centuries.
- Clarke Joan biography
- Joan Murray's greatest achievement was to establish the sequence of gold unicorns and heavy groats of James III and James IV, an extremely complex series which caused great difficulty for previous students.
- Bolzano biography
- The paper gives a proof of the intermediate value theorem with Bolzano's new approach and in the work he defined what is now called a Cauchy sequence.
- Levy Paul biography
- This involved extending the calculus of functions of a real variable to spaces where the points are curves, surfaces, sequences or functions.
- Xu Yue biography
- The lower system is based on the sequence of powers of 10
.
- Hahn biography
- In 1923 he introduced what today is known as the Hahn sequence space.
- Hajek biography
- Hajek developed the property of sequences of pairs of probability measures from ideas due to de la Vallee Poussin.
- Post biography
- for full generality a complete analysis would have to be given of all possible ways in which the human mind could set up finite processes for generating sequences.
History Topics
- Real numbers 2
- Among the forms of the completeness property he implicitly assumed are that a bounded monotone sequence converges to a limit and that the Cauchy criterion is a sufficient condition for the convergence of a series.
- He does say that a real number is the limit of a sequence of rational numbers but he is assuming here that the real numbers are known.
- He says nothing about the need for the sequence to be what we call today a Cauchy sequence and this is necessary if one is to define convergence of a sequence without assuming the existence of its limit.
- be a sequence of rationals approaching B closer and closer.
- Then the product AB will be the limit of the sequence of rational numbers Ab, Ab', Ab'', ..
- Bolzano, on the other hand, showed that bounded Cauchy sequence of real numbers had a least upper bound in 1817.
- His definition of a real number was made in terms of convergent sequences of rational numbers and is explained in [Casopis Pest.
- Cauchy himself does not seem to have understood the significance of his own "Cauchy sequence" criterion for defining the real numbers.
- Two years after the publication of Hankel's monograph, Meray published Remarques sur la nature des quantites in which he considered Cauchy sequences of rational numbers which, if they did not converge to a rational limit, had what he called a "fictitious limit".
- Essentially Heine looks at Cauchy sequences of rational numbers.
- He defines an equivalence relation on such sequences by defining
.
- to be equivalent if the sequence of rational numbers a1 - b1, a2 - b2 , a3 - b3 , a4 - b4 , ..
- Heine then introduced arithmetic operations on his sequences and an order relation.
- Particular care is needed to handle division since sequences with a non-zero limit might still have terms equal to 0.
- His numbers were Cauchy sequences of rational numbers and he used the term "determinate limit".
- If this distance has a rational relation to the unit of measure, then it is expressed by a rational quantity in the domain of rational numbers; otherwise, if the point is one known through a construction, it is always possible to give a sequence of rationals a1 , a2 , a3 , ..
- Knots and physics
- would then be described by the sequence of crossings of length 2n where each of A, B, C, ..
- Tait called the sequence the "scheme of the knot".
- Firstly which sequences of the above type correspond to a knot, and secondly how could it be determined when two knots described by such sequences were the same.
- However there were some other problems, for example although a sequence of length 10, say, might represent a knot it might be one with less than 5 crossings.
- However this is not good enough for there might be a sequence of moves which first increase the number of crossings, then further moves reduce to a fewer number of crossings than were there originally.
- If we interpret Tait in a form that he seems to have used the conjecture, namely that two alternating diagrams without nugatory crossings representing the same prime knot are related by a sequence of twists, then we get what has been called Tait's second conjecture.
- Pi history
- The effect of this procedure is to define an increasing sequence
.
- and a decreasing sequence
.
- such that both sequences have limit π.
- In this and all subsequent computer expansions the number of 7's does not differ significantly from its expectation, and indeed the sequence of digits has so far passed all statistical tests for randomness.
Go directly to this paragraph
- Calculus history
- Archimedes constructed an infinite sequence of triangles starting with one of area A and continually adding further triangles between the existing ones and the parabola to get areas
.
Go directly to this paragraph
- Leibniz thought of variables x, y as ranging over sequences of infinitely close values.
- He introduced dx and dy as differences between successive values of these sequences.
- Mathematical games
- Fibonacci, already mentioned above, is famed for his invention of the sequence 1, 1, 2, 3, 5, 8, 13, ..
- In fact a wealth of mathematics has arisen from this sequence and today a Journal is devoted to topics related to the sequence.
- Fibonacci writes out the first 13 terms of the sequence but does not give the recurrence relation which generates it.
- Topology history
- This process really began in 1817 when Bolzano removed the association of convergence with a sequence of numbers and associated convergence with any bounded infinite subset of the real numbers.
Go directly to this paragraph
- p satisfies the property that given any ε > 0 there is an infinite sequence (pn) of points of S with |p - pn | < ε.
- Schmidt in 1907 examined the notion of convergence in sequence spaces, extending methods which Hilbert had used in his work on integral equations to generalise the idea of a Fourier series.
Go directly to this paragraph
- Schmidt's work on sequence spaces has analogues in the theory of square summable functions, this work being done also in 1907 by Schmidt himself and independently by Frechet.
Go directly to this paragraph
- Pell's equation
- We can now generate a sequence of solutions (x,y):
.
- Moreover, the pattern in most of the recurring sequence is "palindromic".
- up to the last element, the second half of the periodic sequence is the first half in reverse.
- The last number in the repeating sequence is double the integer part of the square root.
- Golden ratio
- Someone has written a note which clearly shows that they knew that the ratio of adjacent terms in the Fibonacci sequence tend to the golden number.
- He, like the annotator of Pacioli's Euclid, knows that the ratio of adjacent terms of the Fibonacci sequence tends to the golden ratio and he states this explicitly in a letter he wrote in 1609.
- The result that the quotients of adjacent terms of the Fibonacci sequence tend to the golden ratio is usually attributed to Simson who gave the result in 1753.
- Perfect numbers
- Here 'double proportion' means that each number of the sequence is twice the preceding number.
- This Catalan sequence is 2p - 1 where
.
- However checking whether the fourth term of this sequence, namely 2p - 1 for p = 170141183460469231731687303715884105727, is prime is well beyond what is possible.
- Fractal Geometry
- The orbit for a starting point, x0 , is the sequence [Introduction to Fractals and Chaos (London, 1995).',2)">2]
.
- If this sequence goes off to infinity, then the set is disconnected.
- Abstract groups
- What we have given here is part of a sequence of development which we might call the English school.
- There were other bits in the sequence between these major contributions which we have omitted.
- Orbits
- He also won the Academie des Sciences of 1766 for work on the orbits of the moons of Jupiter where he gave a mathematical analysis to explain an observed inequality in the sequence of eclipses of the moons.
- The discovery of Uranus at distance 19.2 was close to the next term of the sequence 19.6.
- Squaring the circle
- James Gregory developed a deep understanding of infinite sequences and convergence.
- He applied these ideas to the sequences of areas of the inscribed and circumscribed polygons of a circle and tried to use the method to prove that there was no plane construction for squaring the circle.
- Physical world
- Bourbaki 1
- However, if references were to be only to texts which came earlier in the sequence, it would be necessary to know what the first books would contain in some detail to allow work to go forward on the following books.
- Wave versus matrix
- For example we try to understand the idea of convergence of a sequence by plotting points.
- Fair book
- There is now a sequence of problems involving artillery
.
- function concept
- This line was taken further in 1885 when Weierstrass showed that any continuous function is the limit of a uniformly convergent sequence of polynomials.
- Real numbers 1
- He then goes on to consider the circle as the limit of a sequence of polygons of more and more sides.
- Classical time
- Mathematics almost certainly began through the study of time, particularly the need to record sequences of events.
- Mathematics and Architecture
- He established the ratios of the sequence of notes in a scale still used in Western music.
- Infinity
- Three years later Fermat identified an important property of the positive integers, namely that it did not contain an infinite descending sequence.
- Real numbers 3
- Provided we have some finite way of specifying the n-th term in a Cauchy sequence of rationals we have a finite description of the resulting real number.
- Abstract linear spaces
- The parallel development in analysis was to move from spaces of concrete objects such as sequence spaces towards abstract linear spaces.
Go directly to this paragraph
- Prime numbers
- Mental arithmetic
- When Hunter interviewed Aitken in 1961 he had before him a record of the 1930's test and he asked Aitken if he remembered being asked to recite a random sequence of words.
Famous Curves
No matches from this section
Societies etc
- BMC 1994
- Dunwoody, M JFolding sequences for inaccessible groups
.
- Dunwoody, M JFolding sequences for inaccessible groups
.
- Academy of Scientists Leopoldina
- As a consequence the following is the resulting sequence cities in which it was based: Schweinfurt, Nuremberg, Augsburg, Altdorf, Erfurt, Halle, Nuremberg, Erlangen, Bonn, Breslau, Jena, Dresden, and Halle.
- BMC 1991
- Ray, N Sequences of polynomials in topology, combinatorics and formal calculus
.
- BMC 1969
- Adams, J FGeneralisations of the so-called Adams spectral sequence
.
- BMC 1993
- Chu, C-H Arithmetic means of sequences of functions and operators
.
- BMC 1997
- Mitchell, C J De Bruijn sequences, discrete logarithms and cryptography
.
- BMC 1967
- Garling, D J HTopological sequence spaces
.
- BMC 1958
- Kreisel, GThe meaning of Brouwer's free choice sequences
.
References
- References for Wallis
- I A Golovinskii, Interpolation of sequences in the work of Wallis and Euler (Russian), in History and methodology of the natural sciences XX (Russian) (Moscow, 1978), 62-68.
- References for Huygens
- C J Scriba, Gregory's converging double sequence : a new look at the controversy between Huygens and Gregory over the 'analytical' quadrature of the circle, Historia Math.
- References for Beatty
- K B Stolarsky Beatty sequences, continued fractions, and certain shift operators, Canadian Math.
- References for Mises
- M van Lambalgen, Randomness and foundations of probability : von Mises' axiomatisation of random sequences, in Statistics, probability and game theory (Hayward, CA, 1996), 347-367.
- References for Leray
- H Miller, Leray in Oflag XVIIA : the origins of sheaf theory, sheaf cohomology, and spectral sequences, Jean Leray (1906-1998), Gaz.
- References for Gregory
- C J Scriba, Gregory's converging double sequence: a new look at the controversy between Huygens and Gregory over the 'analytical' quadrature of the circle, Historia Math.
- References for Catalan
- P J Larcombe and P D C Wilson, On the trail of the Catalan sequence, Math.
- References for Toeplitz
- G Kothe, Toeplitz and the theory of sequence spaces, in Toeplitz centennial, Tel Aviv, 1981 (Basel-Boston, Mass., 1982), 575-584.
Additional material
- Aitken: 'Statistical Mathematics
- The reader is recommended to experiment with simple repeated trials of this kind, and for future reference to record the results in sequence, in the order in which they occur.
- Here the invariability of the configurative part of S, whether symmetrical or unsymmetrical, is tacitly assumed, and attention is concentrated upon the sequence of trials, and the incidence of E in these.
- A succession of n trials then gives a sequence
.
- Let m be the number of 1's in this sequence.
- Granted the postulate of this limit p for one sequence of trials upon S, can we accept the more stringent postulate that the same limiting value p is obtained for any other infinite sequence of trials on S? Not without further assumptions, for one might imagine a mechanism sufficiently delicate to throw heads with a coin, or an ace with a die, on almost all occasions.
- Another difficulty is that the tendency of relative frequency m/n towards a limit p is different in nature from the corresponding tendency to a limit which mathematicians have discerned and used in the infinite sequences of mathematical analysis.
- To take a classical example, in the sequence defining a certain simple geometric series,
.
- , each being numerically half its predecessor, so that, given a small number ε, such as 1/1000000, we can always find some term sufficiently far along the sequence, after and including which all terms deviate from 2/3 by less than ε.
- Thus 2/3 is the limit of this sequence.
- These writers admit only certain sequences A of suitable postulated properties, including that of limiting ratio; but some logical difficulties remain, and the modified formulations lose the primitive simplicity in which they originated.
- The approximately constant element in our sequences A, namely the almost stable frequency ratio of E, must reflect - at least so our intuition suggests - the constant element of S, such as the rigid configuration of a coin or die; the irregularity which we name randomness doubtless reflects the variable part of S, such as the initial position, velocity and angular velocity of projection.
- Green: 'Sequences and Series
- Green: Sequences and Series
.
- One of the early texts in the series was Sequences and Series by J A Green.
- Sequences and Series
.
- It is true that a certain sophisticated skill is necessary for the construction of proofs of even quite elementary theorems involving, for example, the definition of the limit of a sequence, and that the acquisition of such skill would take more time than the non-specialist mathematician can spare.
- http://www-history.mcs.st-andrews.ac.uk/Extras/Green_Sequences.html
.
- Konrad Knopp: Texts
- The first is Infinite Sequences and Series which contains the following publisher's information:-
.
- He develops the theory of infinite sequences and series from its beginnings to a point where the reader will be in a position to investigate more advanced stages on his own.
- In the treatment of sequences and series that follows, he covers arbitrary and null sequences; sequences and sets of numbers; convergence and divergence; Cauchy's limit theorem; main tests for sequences; and infinite series.
- 1.1 Preliminary remarks concerning sequences and series
.
- Sequences and Series
.
- 2.1 Arbitrary sequences.
- Null sequences
.
- 2.2 Sequences and sets of numbers
.
- 2.5 The main tests for sequences
.
- Volume I contains more than 300 elementary problems dealing with fundamental concepts, infinite sequences and series, functions of a complex variable, conformal mapping, and more.
- Infinite Sequences and Series
.
- Limits of Sequences.
- Edmund Whittaker: 'Physics and Philosophy
- His approach is mainly historical and is a natural sequence to his History of the Theories of Aether and Electricity.
- We cannot have an infinite sequence of movers and moved, because we would then have no first mover and therefore no other mover.
- We cannot have an infinite sequence of causes because, if we had no first cause, there would be no subsequent cause or effect.
- The sequence of causes cannot go back to infinity but must end at the "Creation".
- that they are subject to change, that there cannot be an infinite sequence of causes, that there is order in the world.
- Ferrar: 'Textbook of Convergence
- The book develops the theory of convergence on the basis of two fundamental assumptions (one about upper bounds, one about irrational number as the limit of a sequence of rational numbers).
- A First Course in the Theory of Sequences and Series
.
- Bounds: Monotonic Sequences
.
- Gibson History 5 - James Gregory
- form an increasing sequence, v1 , v2 , ..
- a decreasing sequence and these are connected by the relations
.
- The two sequences (S) are called a "converging series," the corresponding pairs un , vn are called "converging terms," and the common limit is called "the termination of the series." It is from this beginning that the term "convergence" comes into use in connection with series.
- Edmund Landau: 'Foundations of Analysis' Prefaces
- I will refrain from speaking at length about the fact that often even Dedekind's fundamental theorem (or the equivalent theorem in the development of the real numbers by means of fundamental sequences) is not included in the basic material; so that such matters as the mean-value theorem of the differential calculus, the corollary of the mean-value theorem to the effect that a function having a zero derivative in some interval is constant in that interval, or, say, the theorem that a monotonically decreasing bounded sequence of numbers converges to a limit, are given without any proof or, worse yet, with a supposed proof which in reality is no proof at all.
- Kuratowski: 'Introduction to Topology
- This generality has not only a methodological significance; in modern mathematics there is a characteristic tendency to confer upon the set of objects considered in a given investigation (be these functions, sequences or curves).
- Their generality is sufficient for the majority of important applications; in particular, subsets of n-dimensional Euclidean space, sequence, spaces (of Hilbert.
- Ernest Hobson addresses the British Association in 1910, Part 3
- In the process of discovery the chains in a sequence of logical deduction do not at first arise in their final order in the mind of the mathematical discoverer.
- As an abstract formulation of the idea of determination in its most general sense, the notion of functionality includes and transcends the more special notion of causation as a one-sided determination of future phenomena by means of present conditions; it can be used to express the fact of the subsumption under a general law of past, present, and future alike, in a sequence of phenomena.
- It is found that the lack of a regular order in the sequence of propositions increases the difficulty of the examiner in appraising the performance of the candidates, and in standardising the results of examinations.
- Centenary of John Leslie
- Leslie had dared to speak of Hume with approval, stating that he was the first to treat of causation (cause and effect) in a truly philosophical manner, and had remarked that "the unsophisticated notions of mankind are in perfect unison with the deductions of logic, and imply nothing more at bottom in the relation of cause and effect than a constant and invariable sequence." On this the Edinburgh Presbytery charged him with "having laid a foundation for rejecting all the argument that is derived from the works of God, to prove either His Being, or His Attributes." The protest which was tendered by the ministers to the patrons of the chair, then the Provost and Town Council, stated that they were obliged by charter to act with the advice of the ministers.
- The present branch of the family is descended from his eldest brother Alexander, in five generations to the present day, and with a perfect classical sequence of the names Alexander and James.
- A CONTRIBUTION TO THE MATHEMATICAL THEORY OF BIG GAME HUNTING
- enumerable dense set of points, from which can be extracted a sequence
.
- this sequence, bearing with us suitable equipment.
- EMS 1934 Colloquium
- Secondly, it lays down a cosmological principle, viz.: that the sequence of an events occurring in the universe and observed by one observer A must be identical with the sequence observed by any other observer B.
- EMS 1934 Colloquium 2.html
- Secondly, it lays down a cosmological principle, viz.: that the sequence of an events occurring in the universe and observed by one observer A must be identical with the sequence observed by any other observer B.
- Library of Mathematics
- Sequences and Series J A Green
.
- Thomas Muir: 'History of determinants
- In this way any reader who will take the trouble to look up the sequence xi., xi.
- R L Wilder: 'Cultural Basis of Mathematics III
- 360), these people have no conception of one event leading up to another, and chronological sequence is unimportant.
- Semple and Kneebone: 'Algebraic Projective Geometry
- Some of the geometries that can be obtained in this way, such as euclidean geometry, affine geometry, and projective geometry, are very well known; others, such as inversive geometry (which arises from the group of all transformations that can be resolved into finite sequences of inversions with respect to circles) are known but not usually studied in much detail; and yet others are presumably ignored altogether.
- Kuratowski: 'Introduction to Set Theory
- We can also, thanks to this, extend the sequence of natural numbers, introducing numbers which characterize the power of infinite sets (called the cardinal numbers); in particular, to sets having the same power as the set of all natural numbers (or the countably infinite sets) we assign the cardinal number a to the set of all real numbers we assign the number c (the power of the continuum).
- Zariski and Samuel: 'Commutative Algebra
- We could not include, without completely disrupting the balance of this volume, the results which require the use of truly homological methods (e.g., torsion and extension functors, complexes, spectral sequences).
- Thomson EMS Tests.html
- Ernest Hobson addresses the British Association in 1910, Part 2
- The result to be obtained appears in the form of a limit, corresponding to an interminable sequence of arithmetical operations.
- A D Aleksandrov's view of Mathematics
- Second, they occur in a sequence of increasing degrees of abstraction, going very much further in this direction than the abstractions of other sciences.
- James Jeans: 'Physics and Philosophy' I
- The sequence of events has now passed beyond human control.
- Skolem: 'Abstract Set Theory
- The simple infinite sequence.
- EMS obituary
- If there were any doubts as to his rapid promotion to the headship after only four years of teaching, these were soon dispelled, for three years later there began an almost unbroken sequence of highly-placed John Welsh Mathematical Bursars at his old University.
- Wave versus matrix mechanics
- For example we try to understand the idea of convergence of a sequence by plotting points.
- The Tercentenary of the birth of James Gregory
- His daughter Janet married John Gregory of Aberdeen, who had studied at St Mary's College in this University; and thereafter for two hundred years their descendants occupied Scottish Chairs of Mathematics, Medicine, Chemistry, History or Philosophy in an almost unbroken sequence.
- Eddington: 'Mathematical Theory of Relativity' Preface
- The matter has been rewritten, the sequence of the argument rearranged in many places, and numerous additions made throughout; so that the work is now expanded to three times its former size.
- D'Arcy Thompson on Plato and Planets
- But we are further told, and here the difficulty begins, that the whorls differ from one another in respect of 'breadth of rim': 'The first and outermost whorl is that which has its circular rim the broadest, and the sixth whorl comes next to it in regard to breadth of rim; and, proceeding in order of breadth, the fourth whorl comes third, and the eighth fourth, and the seventh fifth, and the fifth sixth, and the third seventh, and the second eighth.' Thus we have now a new classification of the heavenly bodies, in the following sequence:
.
- Turnbull and Aitken: 'Canonical Matrices
- While we have tried to include all the principal features of the theory and have sought to make the sequence of argument reasonably fluent, even allowing ourselves moderate latitude in digression and explanation, we have, at the same time, aimed at a certain compactness in the formulae and demonstrations.
- Coulson: 'Electricity
- In olden days, following the historical sequence that we have already indicated, it was usual to develop the subject of magnetism quite separately from electrostatics, starting from the existence of permanent magnets.
- Alfred Tarski: 'Cardinal Algebras
- Each of them is constituted by a set of arbitrary elements and by two operations, that of binary addition and that of addition of infinite sequences.
- Heinrich Tietze on Numbers, Part 2
- This systematic representation of numbers is intimately linked with the perception - already developed by Archimedes in his grains of sand theory - that the sequence of numbers 1, 2, 3, - - - is infinite.
- Ledermann: 'Complex Numbers
- I should like to thank my friend and colleague Dr J A Green for a number of valuable suggestions, especially in connection with the chapter on convergence, which is a sequel to his volume Sequences and Series in this Library.
- A I Khinchin on Information Theory
- Accepting completely McMillan's mathematical apparatus, he avoids following Shannon's original path and constructs a proof, using the completely new and apparently very fruitful idea of a "distinguishable set of sequences", the principal features of which will be explained below.
Quotations
- A quotation by McDuff
- I had always thought of mathematics as being much more straightforward: a formula is a formula, and an algebra is an algebra, but Gel'fand found hedgehogs lurking in the rows of his spectral sequences!
.
- Quotations by Euler
- Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.
- Quotations by Descartes
- These long chains of perfectly simple and easy reasonings by means of which geometers are accustomed to carry out their most difficult demonstrations had led me to fancy that everything that can fall under human knowledge forms a similar sequence; and that so long as we avoid accepting as true what is not so, and always preserve the right order of deduction of one thing from another, there can be nothing too remote to be reached in the end, or to well hidden to be discovered.
- Quotations by Adams Frank
- the so-called Adams Spectral Sequence ..
Chronology
- Mathematical Chronology
- It also introduces the famous sequence of numbers now called the "Fibonacci sequence".
- He also defines what is now called Pascal's triangle and shows how to sum certain sequences.
- Simson notes that in the Fibonacci sequence the ratio between adjacent numbers approaches the golden ratio.
- Aleksandrov introduces exact sequences.
- Serre uses spectral sequences to the study of the relations between the homology groups of fibre, total space and base space in a fibration.
- Serre is awarded a Fields Medal for his work on spectral sequences and his work developing complex variable theory in terms of sheaves.
- Menasco and Thistlethwaite prove the knot theory conjecture known as "Tait's Second Conjecture", namely that any two reduced alternating diagrams of the same prime knot are related by a sequence of twists.
- Chronology for 1100 to 1300
- It also introduces the famous sequence of numbers now called the "Fibonacci sequence".
- Chronology for 1950 to 1960
- Serre uses spectral sequences to the study of the relations between the homology groups of fibre, total space and base space in a fibration.
- Serre is awarded a Fields Medal for his work on spectral sequences and his work developing complex variable theory in terms of sheaves.
- Chronology for 1990 to 2000
- Menasco and Thistlethwaite prove the knot theory conjecture known as "Tait's Second Conjecture", namely that any two reduced alternating diagrams of the same prime knot are related by a sequence of twists.
- Chronology for 1940 to 1950
- Aleksandrov introduces exact sequences.
- Chronology for 1930 to 1940
- Aleksandrov introduces exact sequences.
- Chronology for 1300 to 1500
- He also defines what is now called Pascal's triangle and shows how to sum certain sequences.
- Chronology for 1740 to 1760
- Simson notes that in the Fibonacci sequence the ratio between adjacent numbers approaches the golden ratio.
This search was performed by Kevin Hughes' SWISH and Ben Soares' HistorySearch Perl script
JOC/BS August 2001