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Proofs That Really Count
''Proofs That Really Count: the Art of Combinatorial Proof'' is an undergraduate-level mathematics book on combinatorial proofs of Identity (mathematics), mathematical identies. That is, it concerns equations between two integer-valued formulas, shown to be equal either by showing that both sides of the equation count the same type of mathematical objects, or by finding a bijective proof, one-to-one correspondence between the different types of object that they count. It was written by Arthur T. Benjamin and Jennifer Quinn, and published in 2003 by the Mathematical Association of America as volume 27 of their Dolciani Mathematical Expositions series. It won the Beckenbach Book Prize of the Mathematical Association of America. Topics The book provides combinatorial proofs of thirteen theorems in combinatorics and 246 numbered identities (collated in an appendix). Several additional "uncounted identities" are also included. Many proofs are based on a visual-reasoning method that the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorial Proof
In mathematics, the term ''combinatorial proof'' is often used to mean either of two types of mathematical proof: * A proof by double counting. A combinatorial identity is proven by counting the number of elements of some carefully chosen set in two different ways to obtain the different expressions in the identity. Since those expressions count the same objects, they must be equal to each other and thus the identity is established. * A bijective proof. Two sets are shown to have the same number of members by exhibiting a bijection, i.e. a one-to-one correspondence, between them. The term "combinatorial proof" may also be used more broadly to refer to any kind of elementary proof in combinatorics. However, as writes in his review of (a book about combinatorial proofs), these two simple techniques are enough to prove many theorems in combinatorics and number theory. Example An archetypal double counting proof is for the well known formula for the number \tbinom nk of ''k'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations ( Diophantine geometry). Questions in number theory are often best understood through the study of analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic object ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Proofs
A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in ''all'' possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work. Proofs employ logic expressed in mathematical symbols, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Enumerative Combinatorics
Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed. Two examples of this type of problem are counting combinations and counting permutations. More generally, given an infinite collection of finite sets ''S''''i'' indexed by the natural numbers, enumerative combinatorics seeks to describe a ''counting function'' which counts the number of objects in ''S''''n'' for each ''n''. Although counting the number of elements in a set is a rather broad mathematical problem, many of the problems that arise in applications have a relatively simple combinatorial description. The twelvefold way provides a unified framework for counting permutations, combinations and partitions. The simplest such functions are '' closed formulas'', which can be expressed as a composition of elementary functions such as factorials, powers, and so on. For instance, as shown below, the number of different possible orderings of a deck of ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Internet Archive
The Internet Archive is an American digital library with the stated mission of "universal access to all knowledge". It provides free public access to collections of digitized materials, including websites, software applications/games, music, movies/videos, moving images, and millions of books. In addition to its archiving function, the Archive is an activist organization, advocating a free and open Internet. , the Internet Archive holds over 35 million books and texts, 8.5 million movies, videos and TV shows, 894 thousand software programs, 14 million audio files, 4.4 million images, 2.4 million TV clips, 241 thousand concerts, and over 734 billion web pages in the Wayback Machine. The Internet Archive allows the public to upload and download digital material to its data cluster, but the bulk of its data is collected automatically by its web crawlers, which work to preserve as much of the public web as possible. Its web archive, the Wayback Machine, contains hundreds of b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Reviews
''Mathematical Reviews'' is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science. The AMS also publishes an associated online bibliographic database called MathSciNet which contains an electronic version of ''Mathematical Reviews'' and additionally contains citation information for over 3.5 million items as of 2018. Reviews Mathematical Reviews was founded by Otto E. Neugebauer in 1940 as an alternative to the German journal '' Zentralblatt für Mathematik'', which Neugebauer had also founded a decade earlier, but which under the Nazis had begun censoring reviews by and of Jewish mathematicians. The goal of the new journal was to give reviews of every mathematical research publication. As of November 2007, the ''Mathematical Reviews'' database contained information on over 2.2 million articles. The authors of reviews are volunte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Mathematics Teacher
Founded in 1920, The National Council of Teachers of Mathematics (NCTM) is a professional organization for schoolteachers of mathematics in the United States. One of its goals is to improve the standards of mathematics in education. NCTM holds annual national and regional conferences for teachers and publishes five journals. Journals NCTM publishes five official journals. All are available in print and online versions. ''Teaching Children Mathematics'' supports improvement of pre-K–6 mathematics education by serving as a resource for teachers so as to provide more and better mathematics for all students. It is a forum for the exchange of mathematics idea, activities, and pedagogical strategies, and or sharing and interpreting research. ''Mathematics Teaching in the Middle School'' supports the improvement of grade 5–9 mathematics education by serving as a resource for practicing and prospective teachers, as well as supervisors and teacher educators. It is a forum for the e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The American Statistician
''The American Statistician'' is a quarterly peer-reviewed scientific journal covering statistics published by Taylor & Francis on behalf of the American Statistical Association. It was established in 1947. The editor-in-chief is Daniel R. Jeske, a professor at the University of California, Riverside The University of California, Riverside (UCR or UC Riverside) is a public university, public Land-grant university, land-grant research university in Riverside, California. It is one of the ten campuses of the University of California system. Th .... External links * Taylor & Francis academic journals Statistics journals Publications established in 1947 English-language journals Quarterly journals 1947 establishments in the United States Academic journals associated with learned and professional societies of the United States {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SIAM Review
Society for Industrial and Applied Mathematics (SIAM) is a professional society dedicated to applied mathematics, computational science, and data science through research, publications, and community. SIAM is the world's largest scientific society devoted to applied mathematics, and roughly two-thirds of its membership resides within the United States. Founded in 1951, the organization began holding annual national meetings in 1954, and now hosts conferences, publishes books and scholarly journals, and engages in advocacy in issues of interest to its membership. Members include engineers, scientists, and mathematicians, both those employed in academia and those working in industry. The society supports educational institutions promoting applied mathematics. SIAM is one of the four member organizations of the Joint Policy Board for Mathematics. Membership Membership is open to both individuals and organizations. By the end of its first full year of operation, SIAM had 130 membe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibonacci Quarterly
The ''Fibonacci Quarterly'' is a scientific journal on mathematical topics related to the Fibonacci numbers, published four times per year. It is the primary publication of The Fibonacci Association, which has published it since 1963. Its founding editors were Verner Emil Hoggatt Jr. and Alfred Brousseau; by Clark Kimberling the present editor is Professor Curtis Cooper of the Mathematics Department of the . The ''Fibonacci Quarterly'' has an editorial board of nineteen membe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ZbMATH
zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles in pure and applied mathematics, produced by the Berlin office of FIZ Karlsruhe – Leibniz Institute for Information Infrastructure GmbH. Editors are the European Mathematical Society, FIZ Karlsruhe, and the Heidelberg Academy of Sciences. zbMATH is distributed by Springer Science+Business Media. It uses the Mathematics Subject Classification codes for organising reviews by topic. History Mathematicians Richard Courant, Otto Neugebauer, and Harald Bohr, together with the publisher Ferdinand Springer, took the initiative for a new mathematical reviewing journal. Harald Bohr worked in Copenhagen. Courant and Neugebauer were professors at the University of Göttingen. At that time, Göttingen was considered one of the central places for mathematical research, having appointed mathematicians like David Hilbert, Hermann Minkowski, Carl Runge, and Felix Klein, the great ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Library Association
The American Library Association (ALA) is a nonprofit organization based in the United States that promotes libraries and library education internationally. It is the oldest and largest library association in the world, with 49,727 members as of 2021. History During the Centennial Exposition in Philadelphia in 1876, 103 librarians, 90 men and 13 women, responded to a call for a "Convention of Librarians" to be held October 4–6 at the Historical Society of Pennsylvania. At the end of the meeting, according to Ed Holley in his essay "ALA at 100", "the register was passed around for all to sign who wished to become charter members," making October 6, 1876, the date of the ALA’s founding. Among the 103 librarians in attendance were Justin Winsor (Boston Public, Harvard), William Frederick Poole (Chicago Public, Newberry), Charles Ammi Cutter (Boston Athenaeum), Melvil Dewey, and Richard Rogers Bowker. Attendees came from as far west as Chicago and from England. The ALA wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
