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Behrend's Theorem
In arithmetic combinatorics, Behrend's theorem states that the subsets of the integers from 1 to n in which no member of the set is a multiple of any other must have a logarithmic density that goes to zero as n becomes large. The theorem is named after Felix Behrend, who published it in 1935. Statement The logarithmic density of a set of integers from 1 to n can be defined by setting the weight of each integer i to be 1/i, and dividing the total weight of the set by the nth partial sum of the harmonic series (or, equivalently for the purposes of asymptotic analysis, dividing by \log n). The resulting number is 1 or close to 1 when the set includes all of the integers in that range, but smaller when many integers are missing, and particularly when the missing integers are themselves small. A subset of \ is called ''primitive'' if it has the property that no subset element is a multiple of any other element. Behrend's theorem states that the logarithmic density of any primitive s ... [...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] |
Integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of mathematics, the set of integers is often denoted by the boldface or blackboard bold \mathbb. The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the natural numbers, \mathbb is countably infinite. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and are not. The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Density
In number theory, natural density (also referred to as asymptotic density or arithmetic density) is one method to measure how "large" a subset of the set of natural numbers is. It relies chiefly on the probability of encountering members of the desired subset when combing through the interval as ''n '' grows large. Intuitively, it is thought that there are more positive integers than perfect squares, since every perfect square is already positive, and many other positive integers exist besides. However, the set of positive integers is not in fact larger than the set of perfect squares: both sets are infinite and countable and can therefore be put in one-to-one correspondence. Nevertheless if one goes through the natural numbers, the squares become increasingly scarce. The notion of natural density makes this intuition precise for many, but not all, subsets of the naturals (see Schnirelmann density, which is similar to natural density but defined for all subsets of \mathbb). If ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Felix Behrend
Felix Adalbert Behrend (23 April 1911 – 27 May 1962) was a German mathematician of Jewish descent who escaped Nazi Germany and settled in Australia. His research interests included combinatorics, number theory, and topology. Behrend's theorem and Behrend sequences are named after him. Life Behrend was born on 23 April 1911 in Charlottenburg, a suburb of Berlin. He was one of four children of Dr. Felix W. Behrend, a politically liberal mathematics and physics teacher. Although of Jewish descent, their family was Lutheran. Behrend followed his father in studying both mathematics and physics, both at Humboldt University of Berlin and the University of Hamburg, and completed a doctorate in 1933 at Humboldt University. His dissertation, ''Über numeri abundantes'' 'On abundant numbers''">abundant_number.html" ;"title="'On abundant number">'On abundant numbers''was supervised by Erhard Schmidt. With Adolf Hitler's rise to power in 1933, Behrend's father lost his job, and Behren ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Series (mathematics)
In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions: \sum_^\infty\frac = 1 + \frac + \frac + \frac + \frac + \cdots. The first n terms of the series sum to approximately \ln n + \gamma, where \ln is the natural logarithm and \gamma\approx0.577 is the Euler–Mascheroni constant. Because the logarithm has arbitrarily large values, the harmonic series does not have a finite limit: it is a divergent series. Its divergence was proven in the 14th century by Nicole Oresme using a precursor to the Cauchy condensation test for the convergence of infinite series. It can also be proven to diverge by comparing the sum to an integral, according to the integral test for convergence. Applications of the harmonic series and its partial sums include Euler's proof that there are infinitely many prime numbers, the analysis of the coupon collector's problem on how many random trials are needed to provide a complete range of responses, the co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymptotic Analysis
In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior. As an illustration, suppose that we are interested in the properties of a function as becomes very large. If , then as becomes very large, the term becomes insignificant compared to . The function is said to be "''asymptotically equivalent'' to , as ". This is often written symbolically as , which is read as " is asymptotic to ". An example of an important asymptotic result is the prime number theorem. Let denote the prime-counting function (which is not directly related to the constant pi), i.e. is the number of prime numbers that are less than or equal to . Then the theorem states that \pi(x)\sim\frac. Asymptotic analysis is commonly used in computer science as part of the analysis of algorithms and is often expressed there in terms of big O notation. Definition Formally, given functions and , we define a binary relation f(x) \sim g(x) \qu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dilworth's Theorem
In mathematics, in the areas of order theory and combinatorics, Dilworth's theorem characterizes the width of any finite partially ordered set in terms of a partition of the order into a minimum number of chains. It is named for the mathematician . An antichain in a partially ordered set is a set of elements no two of which are comparable to each other, and a chain is a set of elements every two of which are comparable. A chain decomposition is a partition of the elements of the order into disjoint chains. Dilworth's theorem states that, in any finite partially ordered set, the largest antichain has the same size as the smallest chain decomposition. Here, the size of the antichain is its number of elements, and the size of the chain decomposition is its number of chains. The width of the partial order is defined as the common size of the antichain and chain decomposition. A version of the theorem for infinite partially ordered sets states that, when there exists a decomposition i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow method of checking the primality of a given number n, called trial division, tests whether n is a multiple of any integer between 2 and \sqrt. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divergence Of The Sum Of The Reciprocals Of The Primes
The sum of the reciprocals of all prime numbers diverges; that is: \sum_\frac1p = \frac12 + \frac13 + \frac15 + \frac17 + \frac1 + \frac1 + \frac1 + \cdots = \infty This was proved by Leonhard Euler in 1737, and strengthens Euclid's 3rd-century-BC result that there are infinitely many prime numbers and Nicole Oresme's 14th-century proof of the divergence of the sum of the reciprocals of the integers (harmonic series). There are a variety of proofs of Euler's result, including a lower bound for the partial sums stating that \sum_\frac1p \ge \log \log (n+1) - \log\frac6 for all natural numbers . The double natural logarithm () indicates that the divergence might be very slow, which is indeed the case. See Meissel–Mertens constant. The harmonic series First, we describe how Euler originally discovered the result. He was considering the harmonic series \sum_^\infty \frac = 1 + \frac + \frac + \frac + \cdots = \infty He had already used the following "product formula" to sho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Erdős
Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in discrete mathematics, graph theory, number theory, mathematical analysis, approximation theory, set theory, and probability theory. Much of his work centered around discrete mathematics, cracking many previously unsolved problems in the field. He championed and contributed to Ramsey theory, which studies the conditions in which order necessarily appears. Overall, his work leaned towards solving previously open problems, rather than developing or exploring new areas of mathematics. Erdős published around 1,500 mathematical papers during his lifetime, a figure that remains unsurpassed. He firmly believed mathematics to be a social activity, living an itinerant lifestyle with the sole purpose of writing mathematical papers with other mathem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subbayya Sivasankaranarayana Pillai
Subbayya Sivasankaranarayana Pillai (5 April 1901 – 31 August 1950) was an Indian mathematician specialising in number theory. His contribution to Waring's problem was described in 1950 by K. S. Chandrasekharan as "almost certainly his best piece of work and one of the very best achievements in Indian Mathematics since Ramanujan". Biography Subbayya Sivasankaranarayana Pillai was born to parents Subbayya Pillai and Gomati Ammal. His mother died a year after his birth and his father when Pillai was in his last year at school. Pillai did his intermediate course and B.Sc Mathematics in the Scott Christian College at Nagercoil and managed to earn a B.A. degree from Maharaja's college, Trivandrum. In 1927, Pillai was awarded a research fellowship at the University of Madras to work among professors K. Ananda Rau and Ramaswamy S. Vaidyanathaswamy. He was from 1929 to 1941 at Annamalai University where he worked as a lecturer. It was in Annamalai University that he did his maj ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of The London Mathematical Society
The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical Society and the Operational Research Society (ORS). History The Society was established on 16 January 1865, the first president being Augustus De Morgan. The earliest meetings were held in University College, but the Society soon moved into Burlington House, Piccadilly. The initial activities of the Society included talks and publication of a journal. The LMS was used as a model for the establishment of the American Mathematical Society in 1888. Mary Cartwright was the first woman to be President of the LMS (in 1961–62). The Society was granted a royal charter in 1965, a century after its foundation. In 1998 the Society moved from rooms in Burlington House into De Morgan House (named after the society's first president), at 57–5 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
