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Bondy's Theorem
In mathematics, Bondy's theorem is a bound on the number of elements needed to distinguish the sets in a family of sets from each other. It belongs to the field of combinatorics, and is named after John Adrian Bondy, who published it in 1972. Statement The theorem is as follows: :Let ''X'' be a set with ''n'' elements and let ''A''1, ''A''2, ..., ''A''''n'' be distinct subsets of ''X''. Then there exists a subset ''S'' of ''X'' with ''n'' − 1 elements such that the sets ''A''''i'' ∩ ''S'' are all distinct. In other words, if we have a 0-1 matrix with ''n'' rows and ''n'' columns such that each row is distinct, we can remove one column such that the rows of the resulting ''n'' × (''n'' − 1) matrix are distinct. Example Consider the 4 × 4 matrix :\begin 1&1&0&1\\ 0&1&0&1\\ 0&0&1&1\\ 0&1&1&0 \end where all rows are pairwise distinct. If we delete, for example, the first column, the resulting matrix : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Family Of Sets
In set theory and related branches of mathematics, a collection F of subsets of a given set S is called a family of subsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets, set family, or a set system. The term "collection" is used here because, in some contexts, a family of sets may be allowed to contain repeated copies of any given member, and in other contexts it may form a proper class rather than a set. A finite family of subsets of a finite set S is also called a ''hypergraph''. The subject of extremal set theory concerns the largest and smallest examples of families of sets satisfying certain restrictions. Examples The set of all subsets of a given set S is called the power set of S and is denoted by \wp(S). The power set \wp(S) of a given set S is a family of sets over S. A subset of S having k elements is called a k-subset of S. The k-subsets S^ of a set S form a family of sets. Let S = \. An ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Adrian Bondy
John Adrian Bondy (born 1944 in London) is a retired English mathematician, known for his work in combinatorics and graph theory. Career Bondy received his Ph.D. in graph theory from the University of Oxford in 1969. His advisor was Dominic Welsh. Between 1969 and 1994, Bondy was ''Professor of Graph Theory'' at the University of Waterloo in Canada, and then, until his retirement, at Université Lyon 1 in France. From 1976, he was managing editor, and, between 1979 and 2004, co-editor-in-chief (together with U. S. R. Murty) of Journal of Combinatorial Theory, Series B. Throughout his career, Bondy has (co-)authored over 100 publications with 51 co-authors, including the widely influential textbook ''Graph Theory with Applications'' (with U. S. R. Murty), and supervised 12 Ph.D. students. His Erdős number is 1. Bondy was dismissed from his tenured position at the University of Waterloo in 1995, after 25 years in which he had been a major contributor to the renown of the Un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Combinatorial Theory, Series B
The ''Journal of Combinatorial Theory'', Series A and Series B, are mathematical journals specializing in combinatorics and related areas. They are published by Elsevier. ''Series A'' is concerned primarily with structures, designs, and applications of combinatorics. ''Series B'' is concerned primarily with graph and matroid theory. The two series are two of the leading journals in the field and are widely known as ''JCTA'' and ''JCTB''. The journal was founded in 1966 by Frank Harary and Gian-Carlo Rota.They are acknowledged on the journals' title pages and Web sites. SeEditorial board of JCTA Originally there was only one journal, which was split into two parts in 1971 as the field grew rapidly. An electronic, [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix (mathematics)
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, \begin1 & 9 & -13 \\20 & 5 & -6 \end is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a "-matrix", or a matrix of dimension . Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra. Therefore, the study of matrices is a large part of linear algebra, and most properties and operations of abstract linear algebra can be expressed in terms of matrices. For example, matrix multiplication represents composition of linear maps. Not all matrices are related to linear algebra. This is, in particular, the case in graph theory, of incidence matrices, and adjacency matrices. ''This article focuses on matrices related to linear algebra, and, unle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Learning Theory
In computer science, computational learning theory (or just learning theory) is a subfield of artificial intelligence devoted to studying the design and analysis of machine learning algorithms. Overview Theoretical results in machine learning mainly deal with a type of inductive learning called supervised learning. In supervised learning, an algorithm is given samples that are labeled in some useful way. For example, the samples might be descriptions of mushrooms, and the labels could be whether or not the mushrooms are edible. The algorithm takes these previously labeled samples and uses them to induce a classifier. This classifier is a function that assigns labels to samples, including samples that have not been seen previously by the algorithm. The goal of the supervised learning algorithm is to optimize some measure of performance such as minimizing the number of mistakes made on new samples. In addition to performance bounds, computational learning theory studies the t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Concept Class
In computational learning theory in mathematics, a concept over a domain ''X'' is a total Boolean function over ''X''. A concept class is a class of concepts. Concept classes are a subject of computational learning theory. Concept class terminology frequently appears in model theory associated with probably approximately correct (PAC) learning. In this setting, if one takes a set ''Y'' as a set of (classifier output) labels, and ''X'' is a set of examples, the map , i.e. from examples to classifier labels (where and where ''c'' is a subset of ''X''), ''c'' is then said to be a ''concept''. A ''concept class'' is then a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Witness Set
In computational learning theory, let ''C'' be a concept class In computational learning theory in mathematics, a concept over a domain ''X'' is a total Boolean function over ''X''. A concept class is a class of concepts. Concept classes are a subject of computational learning theory. Concept class terminolog ... over a domain ''X'' and ''c'' be a concept in ''C''. A subset ''S'' of ''X'' is a witness set for ''c'' in ''C'' if ''c''(''S'') verifies ''c'' (i.e., ''c'' is the only consistent concept with respect to ''c''(''S'')). The minimum size of a witness set for ''c'' is called the ''witness size'' or ''specification number'' and is denoted by w_C(c). The value \max\ is called the teaching dimension of ''C''. Computational learning theory {{Compu-AI-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Teaching Dimension
In computational learning theory, the teaching dimension of a concept class ''C'' is defined to be \max_\, where is the minimum size of a witness set for ''c'' in ''C''. Intuitively, this measures the number of instances that are needed to identify a concept in the class, using supervised learning with examples provided by a helpful teacher who is trying to convey the concept as succinctly as possible. This definition was formulated in 1995 by Sally Goldman and Michael Kearns, based on earlier work by Goldman, Ron Rivest, and Robert Schapire Robert Elias Schapire is an American computer scientist, former David M. Siegel '83 Professor in the computer science department at Princeton University, and has recently moved to Microsoft Research. His primary specialty is theoretical and app .... The teaching dimension of a finite concept class can be used to give a lower and an upper bound on the membership query cost of the concept class. In Stasys Jukna's book "Extremal Combinato ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Learning Theory
In computer science, computational learning theory (or just learning theory) is a subfield of artificial intelligence devoted to studying the design and analysis of machine learning algorithms. Overview Theoretical results in machine learning mainly deal with a type of inductive learning called supervised learning. In supervised learning, an algorithm is given samples that are labeled in some useful way. For example, the samples might be descriptions of mushrooms, and the labels could be whether or not the mushrooms are edible. The algorithm takes these previously labeled samples and uses them to induce a classifier. This classifier is a function that assigns labels to samples, including samples that have not been seen previously by the algorithm. The goal of the supervised learning algorithm is to optimize some measure of performance such as minimizing the number of mistakes made on new samples. In addition to performance bounds, computational learning theory studies the t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
