|
Erdős–Tetali Theorem
In additive number theory, an area of mathematics, the Erdős–Tetali theorem is an existence theorem concerning economical additive basis, additive bases of every order. More specifically, it states that for every fixed integer h \geq 2, there exists a subset of the natural numbers \mathcal \subseteq \mathbb satisfying r_(n) \asymp \log n, where r_(n) denotes the number of ways that a natural number ''n'' can be expressed as the sum of ''h'' elements of ''B''. The theorem is named after Paul Erdős and Prasad V. Tetali, who published it in 1990. Motivation The original motivation for this result is attributed to a problem posed by S. Sidon in 1932 on ''economical bases''. An additive basis \mathcal\subseteq\mathbb is called ''economical'' (or sometimes ''thin'') when it is an additive basis of order ''h'' and :r_(n) \ll_ n^\varepsilon for every \varepsilon > 0. In other words, these are additive bases that use as few numbers as possible to represent a given ''n'', and yet repres ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Additive Number Theory
Additive number theory is the subfield of number theory concerning the study of subsets of integers and their behavior under addition. More abstractly, the field of additive number theory includes the study of abelian groups and commutative semigroups with an operation of addition. Additive number theory has close ties to combinatorial number theory and the geometry of numbers. Principal objects of study include the sumset of two subsets and of elements from an abelian group , :A + B = \, and the -fold sumset of , :hA = \underset\,. Additive number theory The field is principally devoted to consideration of ''direct problems'' over (typically) the integers, that is, determining the structure of from the structure of : for example, determining which elements can be represented as a sum from , where ' is a fixed subset.Nathanson (1996) II:1 Two classical problems of this type are the Goldbach conjecture (which is the conjecture that contains all even numbers greater than two, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-constructive
In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existence proof or ''pure existence theorem''), which proves the existence of a particular kind of object without providing an example. For avoiding confusion with the stronger concept that follows, such a constructive proof is sometimes called an effective proof. A constructive proof may also refer to the stronger concept of a proof that is valid in constructive mathematics. Constructivism is a mathematical philosophy that rejects all proof methods that involve the existence of objects that are not explicitly built. This excludes, in particular, the use of the law of the excluded middle, the axiom of infinity, and the axiom of choice. Constructivism also induces a different meaning for some terminology (for example, the term "or" has a stronger ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Van H
A van is a type of road vehicle used for transporting goods or people. There is some variation in the scope of the word across the different English-speaking countries. The smallest vans, microvans, are used for transporting either goods or people in tiny quantities. Mini MPVs, compact MPVs, and MPVs are all small vans usually used for transporting people in small quantities. Larger vans with passenger seats are used for institutional purposes, such as transporting students. Larger vans with only front seats are often used for business purposes, to carry goods and equipment. Specially equipped vans are used by television stations as mobile studios. Postal services and courier companies use large step vans to deliver packages. Word origin and usage Van meaning a type of vehicle arose as a contraction of the word caravan. The earliest records of a van as a vehicle in English are in the mid-19th century, meaning a covered wagon for transporting goods; the earliest reported r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Terence Tao
Terence Chi-Shen Tao (; born 17 July 1975) is an Australian-American mathematician, Fields medalist, and professor of mathematics at the University of California, Los Angeles (UCLA), where he holds the James and Carol Collins Chair in the College of Letters and Sciences. His research includes topics in harmonic analysis, partial differential equations, algebraic combinatorics, arithmetic combinatorics, geometric combinatorics, probability theory, compressed sensing and analytic number theory. Tao was born to Chinese immigrant parents and raised in Adelaide. Tao won the Fields Medal in 2006 and won the Royal Medal and Breakthrough Prize in Mathematics in 2014, and is a 2006 MacArthur Fellow. Tao has been the author or co-author of over three hundred research papers, and is widely regarded as one of the greatest living mathematicians. Life and career Family Tao's parents are first generation immigrants from Hong Kong to Australia.'' Wen Wei Po'', Page A4, 24 August ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joel Spencer
Joel Spencer (born April 20, 1946) is an American mathematician. He is a combinatorialist who has worked on probabilistic methods in combinatorics and on Ramsey theory. He received his doctorate from Harvard University in 1970, under the supervision of Andrew Gleason. He is currently () a professor at the Courant Institute of Mathematical Sciences of New York University. Spencer's work was heavily influenced by Paul Erdős, with whom he coauthored many papers (giving him an Erdős number of 1). In 1963, while studying at the Massachusetts Institute of Technology, Spencer became a Putnam Fellow. In 1984, Spencer received a Lester R. Ford Award. He was an Erdős Lecturer at Hebrew University of Jerusalem in 2001. In 2012, he became a fellow of the American Mathematical Society. He was elected as a fellow of the Society for Industrial and Applied Mathematics in 2017, "for contributions to discrete mathematics and theory of computing, particularly random graphs and networks, Ramsey ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noga Alon
Noga Alon (; born 1956) is an Israeli mathematician and a professor of mathematics at Princeton University noted for his contributions to combinatorics and theoretical computer science, having authored hundreds of papers. Education and career Alon was born in 1956 in Haifa, where he graduated from the Hebrew Reali School in 1974. He graduated summa cum laude from the Technion – Israel Institute of Technology in 1979, earned a master's degree in mathematics in 1980 from Tel Aviv University, and received his Ph.D. in Mathematics at the Hebrew University of Jerusalem in 1983 with the dissertation ''Extremal Problems in Combinatorics'' supervised by Micha Perles. After postdoctoral research at the Massachusetts Institute of Technology he returned to Tel Aviv University as a senior lecturer in 1985, obtained a permanent position as an associate professor there in 1986, and was promoted to full professor in 1988. He was head of the School of Mathematical Science from 1999 to 2001, ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heini Halberstam
Heini Halberstam (11 September 1926 – 25 January 2014) was a Czech-born British mathematician, working in the field of analytic number theory. He is remembered in part for the Elliott–Halberstam conjecture from 1968. Life and career Halberstam was born in Most, Czechoslovakia and died in Champaign, Illinois, US. His father died when he was very young. After Adolf Hitler's annexation of the Sudetenland, he and his mother moved to Prague. At the age of twelve, as the Nazi occupation progressed, he was one of the 669 children saved by Sir Nicholas Winton, who organized the Kindertransport, a train that allowed those children to leave Nazi-occupied territory. He was sent to England, where he lived during World War II. He obtained his PhD in 1952, from University College, London, under the supervision of Theodor Estermann. From 1962 until 1964, Halberstam was Erasmus Smith's Professor of Mathematics at Trinity College Dublin; From 1964 until 1980, Halberstam was a Professor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function Of A Real Variable
In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function (mathematics), function whose domain of a function, domain is the real numbers \mathbb, or a subset of \mathbb that contains an interval (mathematics), interval of positive length. Most real functions that are considered and studied are differentiable function, differentiable in some interval. The most widely considered such functions are the real functions, which are the real-valued functions of a real variable, that is, the functions of a real variable whose codomain is the set of real numbers. Nevertheless, the codomain of a function of a real variable may be any set. However, it is often assumed to have a structure of \mathbb-vector space over the reals. That is, the codomain may be a Euclidean space, a coordinate vector, the set of matrix (mathematics), matrices of real numbers of a given size, or an \mathbb-algebra over a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real-valued Function
In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain. Real-valued functions of a real variable (commonly called ''real functions'') and real-valued functions of several real variables are the main object of study of calculus and, more generally, real analysis. In particular, many function spaces consist of real-valued functions. Algebraic structure Let (X,) be the set of all functions from a set to real numbers \mathbb R. Because \mathbb R is a field, (X,) may be turned into a vector space and a commutative algebra over the reals with the following operations: *f+g: x \mapsto f(x) + g(x) – vector addition *\mathbf: x \mapsto 0 – additive identity *c f: x \mapsto c f(x),\quad c \in \mathbb R – scalar multiplication *f g: x \mapsto f(x)g(x) – pointwise multiplication These operations extend to partial functions from to \mathbb R, with the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
