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Klaus Roth
Klaus Friedrich Roth (29 October 1925 – 10 November 2015) was a German-born British mathematician who won the Fields Medal for proving Roth's theorem on the Diophantine approximation of algebraic numbers. He was also a winner of the De Morgan Medal and the Sylvester Medal, and a Fellow of the Royal Society. Roth moved to England as a child in 1933 to escape the Nazis, and was educated at the University of Cambridge and University College London, finishing his doctorate in 1950. He taught at University College London until 1966, when he took a chair at Imperial College London. He retired in 1988. Beyond his work on Diophantine approximation, Roth made major contributions to the theory of progression-free sets in arithmetic combinatorics and to the theory of irregularities of distribution. He was also known for his research on sums of powers, on the large sieve, on the Heilbronn triangle problem, and on square packing in a square. He was a coauthor of the book ''Sequen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wrocław
Wrocław (; german: Breslau, or . ; Silesian German: ''Brassel'') is a city in southwestern Poland and the largest city in the historical region of Silesia. It lies on the banks of the River Oder in the Silesian Lowlands of Central Europe, roughly from the Baltic Sea to the north and from the Sudeten Mountains to the south. , the official population of Wrocław is 672,929, with a total of 1.25 million residing in the metropolitan area, making it the third largest city in Poland. Wrocław is the historical capital of Silesia and Lower Silesia. Today, it is the capital of the Lower Silesian Voivodeship. The history of the city dates back over a thousand years; at various times, it has been part of the Kingdom of Poland, the Kingdom of Bohemia, the Kingdom of Hungary, the Habsburg monarchy of Austria, the Kingdom of Prussia and Germany. Wrocław became part of Poland again in 1945 as part of the Recovered Territories, the result of extensive border changes and expulsions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sylvester Medal
The Sylvester Medal is a bronze medal awarded by the Royal Society (London) for the encouragement of mathematical research, and accompanied by a £1,000 prize. It was named in honour of James Joseph Sylvester, the Savilian Professor of Geometry at the University of Oxford in the 1880s, and first awarded in 1901, having been suggested by a group of Sylvester's friends (primarily Raphael Meldola) after his death in 1897. Initially awarded every three years with a prize of around £900, the Royal Society have announced that starting in 2009 it will be awarded every two years instead, and is to be aimed at 'early to mid career stage scientist' rather than an established mathematician. The award winner is chosen by the Society's A-side awards committee, which handles physical rather than biological science awards. , 45 medals have been awarded, of which all but 10 have been awarded to citizens of the United Kingdom, two to citizens of France and United States, and one medal each has be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prussia
Prussia, , Old Prussian: ''Prūsa'' or ''Prūsija'' was a German state on the southeast coast of the Baltic Sea. It formed the German Empire under Prussian rule when it united the German states in 1871. It was ''de facto'' dissolved by an emergency decree transferring powers of the Prussian government to German Chancellor Franz von Papen in 1932 and ''de jure'' by an Allied decree in 1947. For centuries, the House of Hohenzollern ruled Prussia, expanding its size with the Prussian Army. Prussia, with its capital at Königsberg and then, when it became the Kingdom of Prussia in 1701, Berlin, decisively shaped the history of Germany. In 1871, Prussian Minister-President Otto von Bismarck united most German principalities into the German Empire under his leadership, although this was considered to be a "Lesser Germany" because Austria and Switzerland were not included. In November 1918, the monarchies were abolished and the nobility lost its political power during the Ger ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Sequence
In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers. An integer sequence may be specified ''explicitly'' by giving a formula for its ''n''th term, or ''implicitly'' by giving a relationship between its terms. For example, the sequence 0, 1, 1, 2, 3, 5, 8, 13, ... (the Fibonacci sequence) is formed by starting with 0 and 1 and then adding any two consecutive terms to obtain the next one: an implicit description. The sequence 0, 3, 8, 15, ... is formed according to the formula ''n''2 − 1 for the ''n''th term: an explicit definition. Alternatively, an integer sequence may be defined by a property which members of the sequence possess and other integers do not possess. For example, we can determine whether a given integer is a perfect number, even though we do not have a formula for the ''n''th perfect number. Examples Integer sequences that have their own name include: *Abundant numbers *Baum–Sweet sequence *Bell numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sequences (book)
''Sequences'' is a mathematical monograph on integer sequences. It was written by Heini Halberstam and Klaus Roth, published in 1966 by the Clarendon Press, and republished in 1983 with minor corrections by Springer-Verlag. Although planned to be part of a two-volume set, the second volume was never published. Topics The book has five chapters, each largely self-contained and loosely organized around different techniques used to solve problems in this area, with an appendix on the background material in number theory needed for reading the book. Rather than being concerned with specific sequences such as the prime numbers or square numbers, its topic is the mathematical theory of sequences in general. The first chapter considers the natural density of sequences, and related concepts such as the Schnirelmann density. It proves theorems on the density of sumsets of sequences, including Mann's theorem that the Schnirelmann density of a sumset is at least the sum of the Schnirelmann d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Packing In A Square
Square packing in a square is a packing problem where the objective is to determine how many squares of side one (unit squares) can be packed into a square of side . If is an integer, the answer is , but the precise, or even asymptotic, amount of wasted space for non-integer is an open question. Small numbers of squares The smallest value of a that allows the packing of n unit squares is known when n is a perfect square (in which case it is \sqrt), as well as for n=2, 3, 5, 6, 7, 8, 10, 13, 14, 15, 24, 34, 35, 46, 47, and 48. For most of these numbers (with the exceptions only of 5 and 10), the packing is the natural one with axis-aligned squares, and a is \lceil\sqrt\,\rceil, where \lceil\,\ \rceil is the ceiling (round up) function. The figure shows the optimal packings for 5 and 10 squares, the two smallest numbers of squares for which the optimal packing involves tilted squares.. The smallest unresolved case involves packing 11 unit squares into a larger square. 11 unit s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heilbronn Triangle Problem
In discrete geometry and discrepancy theory, the Heilbronn triangle problem is a problem of placing points in the plane, avoiding triangles of small area. It is named after Hans Heilbronn, who conjectured that, no matter how points are placed in a given area, the smallest triangle area will be at most inversely proportional to the square of the number of points. His conjecture was proven false, but the asymptotic growth rate of the minimum triangle area remains unknown. Definition The Heilbronn triangle problem concerns the placement of n points within a shape in the plane, such as the unit square or the unit disk, for a given Each triple of points form the three vertices of a triangle, and among these triangles, the problem concerns the smallest triangle, as measured by area. Different placements of points will have different smallest triangles, and the problem asks: how should n points be placed to maximize the area of the smallest More formally, the shape may be assumed to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Large Sieve
The large sieve is a method (or family of methods and related ideas) in analytic number theory. It is a type of sieve where up to half of all residue classes of numbers are removed, as opposed to small sieves such as the Selberg sieve wherein only a few residue classes are removed. The method has been further heightened by the larger sieve which removes arbitrarily many residue classes. Name Its name comes from its original application: given a set S \subset \ such that the elements of ''S'' are forbidden to lie in a set ''Ap'' ⊂ Z/''p'' Z modulo every prime ''p'', how large can ''S'' be? Here ''A''''p'' is thought of as being large, i.e., at least as large as a constant times ''p''; if this is not the case, we speak of a ''small sieve''. History The early history of the large sieve traces back to work of Yu. B. Linnik, in 1941, working on the problem of the least quadratic non-residue. Subsequently Alfréd Rényi worked on it, using probability methods. It was only two deca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sums Of Powers
In mathematics and statistics, sums of powers occur in a number of contexts: * Sums of squares arise in many contexts. For example, in geometry, the Pythagorean theorem involves the sum of two squares; in number theory, there are Legendre's three-square theorem and Jacobi's four-square theorem; and in statistics, the analysis of variance involves summing the squares of quantities. *Faulhaber's formula expresses 1^k + 2^k + 3^k + \cdots + n^k as a polynomial in ''n'', or alternatively in term of a Bernoulli polynomial. *Fermat's right triangle theorem states that there is no solution in positive integers for a^2=b^4+c^4 and a^4=b^4+c^2. *Fermat's Last Theorem states that x^k+y^k=z^k is impossible in positive integers with ''k''>2. *The equation of a superellipse is , x/a, ^k+, y/b, ^k=1. The squircle is the case k=4, a=b. *Euler's sum of powers conjecture (disproved) concerns situations in which the sum of ''n'' integers, each a ''k''th power of an integer, equals another ''k'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrepancy Theory
In mathematics, discrepancy theory describes the deviation of a situation from the state one would like it to be in. It is also called the theory of irregularities of distribution. This refers to the theme of ''classical'' discrepancy theory, namely distributing points in some space such that they are evenly distributed with respect to some (mostly geometrically defined) subsets. The discrepancy (irregularity) measures how far a given distribution deviates from an ideal one. Discrepancy theory can be described as the study of inevitable irregularities of distributions, in measure-theoretic and combinatorial settings. Just as Ramsey theory elucidates the impossibility of total disorder, discrepancy theory studies the deviations from total uniformity. A significant event in the history of discrepancy theory was the 1916 paper of Weyl on the uniform distribution of sequences in the unit interval. __NOTOC__ Theorems Discrepancy theory is based on the following classic theorems: * T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Combinatorics
In mathematics, arithmetic combinatorics is a field in the intersection of number theory, combinatorics, ergodic theory and harmonic analysis. Scope Arithmetic combinatorics is about combinatorial estimates associated with arithmetic operations (addition, subtraction, multiplication, and division). Additive combinatorics is the special case when only the operations of addition and subtraction are involved. Ben Green explains arithmetic combinatorics in his review of "Additive Combinatorics" by Tao and Vu. Important results Szemerédi's theorem Szemerédi's theorem is a result in arithmetic combinatorics concerning arithmetic progressions in subsets of the integers. In 1936, Erdős and Turán conjectured. that every set of integers ''A'' with positive natural density contains a ''k'' term arithmetic progression for every ''k''. This conjecture, which became Szemerédi's theorem, generalizes the statement of van der Waerden's theorem. Green–Tao theorem and extensions The Gre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Salem–Spencer Set
In mathematics, and in particular in arithmetic combinatorics, a Salem-Spencer set is a set of numbers no three of which form an arithmetic progression. Salem–Spencer sets are also called 3-AP-free sequences or progression-free sets. They have also been called non-averaging sets, but this term has also been used to denote a set of integers none of which can be obtained as the average of any subset of the other numbers. Salem-Spencer sets are named after Raphaël Salem and Donald C. Spencer, who showed in 1942 that Salem–Spencer sets can have nearly-linear size. However a later theorem of Klaus Roth shows that the size is always less than linear. Examples For k=1,2,\dots the smallest values of n such that the numbers from 1 to n have a k-element Salem-Spencer set are :1, 2, 4, 5, 9, 11, 13, 14, 20, 24, 26, 30, 32, 36, ... For instance, among the numbers from 1 to 14, the eight numbers : form the unique largest Salem-Spencer set. This example is shifted by adding one to the ele ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
