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Lobb Numbers
In combinatorics, combinatorial mathematics, the Lobb number ''L''''m'',''n'' counts the number of ways that ''n'' + ''m'' open parentheses and ''n'' − ''m'' close parentheses can be arranged to form the start of a valid sequence of Dyck language, balanced parentheses. Lobb numbers form a natural generalization of the Catalan numbers, which count the number of complete strings of balanced parentheses of a given length. Thus, the ''n''th Catalan number equals the Lobb number ''L''0,''n''. They are named after Andrew Lobb, who used them to give a simple mathematical induction, inductive proof of the formula for the ''n''th Catalan number. The Lobb numbers are parameterized by two non-negative integers ''m'' and ''n'' with ''n'' ≥ ''m'' ≥ 0. The (''m'', ''n'')th Lobb number ''L''''m'',''n'' is given in terms of binomial coefficients by the formula :L_ = \frac\binom \qquad\textn \ge m \ge 0. An alternative expression for Lobb number ... [...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] |
Dyck Language
In the theory of formal languages of computer science, mathematics, and linguistics, a Dyck word is a balanced string of square brackets and The set of Dyck words forms the Dyck language. Dyck words and language are named after the mathematician Walther von Dyck. They have applications in the parsing of expressions that must have a correctly nested sequence of brackets, such as arithmetic or algebraic expressions. Formal definition Let \Sigma = \ be the alphabet consisting of the symbols and Let \Sigma^ denote its Kleene closure. The Dyck language is defined as: : \. Context-free grammar It may be helpful to define the Dyck language via a context-free grammar in some situations. The Dyck language is generated by the context-free grammar with a single non-terminal , and the production: : That is, ''S'' is either the empty string () or is " , an element of the Dyck language, the matching ", and an element of the Dyck language. An alternative context-free grammar for the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
College Mathematics Journal
The ''College Mathematics Journal'' is an expository magazine aimed at teachers of college mathematics, particular those teaching the first two years. It is published by Taylor & Francis on behalf of the Mathematical Association of America and is a continuation of ''Two-Year College Mathematics Journal''. It covers all aspects of mathematics. It publishes articles intended to enhance undergraduate instruction and classroom learning, including expository articles, short notes, problems, and "mathematical ephemera" such as fallacious proofs, quotations, cartoons, poetry, and humor. Paid circulation in 2008 was 9,000 and total circulation was 9,500. The MAA gives the George Pólya Awards annually "for articles of expository excellence" published in the ''College Mathematics Journal''. References External links *''The College Mathematics Journal''at JSTOR''The College Mathematics Journal''at Taylor & Francis Taylor & Francis Group is an international company originating i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Catalan Number
In combinatorial mathematics, the Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after the French-Belgian mathematician Eugène Charles Catalan (1814–1894). The ''n''th Catalan number can be expressed directly in terms of binomial coefficients by :C_n = \frac = \frac = \prod\limits_^\frac \qquad\textn\ge 0. The first Catalan numbers for ''n'' = 0, 1, 2, 3, ... are :1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, ... . Properties An alternative expression for ''C''''n'' is :C_n = - for n\ge 0, which is equivalent to the expression given above because \tbinom=\tfrac\tbinomn. This expression shows that ''C''''n'' is an integer, which is not immediately obvious from the first formula given. This expression forms the basis for a proof of the correctness of the formula. The Catalan numbers satisfy the recurrence relations :C_0 = 1 \quad \text \quad C_=\sum_^C_i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Induction
Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder: A proof by induction consists of two cases. The first, the base case, proves the statement for ''n'' = 0 without assuming any knowledge of other cases. The second case, the induction step, proves that ''if'' the statement holds for any given case ''n'' = ''k'', ''then'' it must also hold for the next case ''n'' = ''k'' + 1. These two steps establish that the statement holds for every natural number ''n''. The base case does not necessarily begin with ''n'' = 0, but often with ''n'' = 1, and possibly with any fixed natural number ''n'' = ''N'', establishing the truth of the statement for all natu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Gazette
''The Mathematical Gazette'' is an academic journal of mathematics education, published three times yearly, that publishes "articles about the teaching and learning of mathematics with a focus on the 15–20 age range and expositions of attractive areas of mathematics." It was established in 1894 by Edward Mann Langley as the successor to the Reports of the Association for the Improvement of Geometrical Teaching. Its publisher is the Mathematical Association. William John Greenstreet was its editor for more than thirty years (1897–1930). Since 2000, the editor is Gerry Leversha. Editors * Edward Mann Langley: 1894-1896 * Francis Sowerby Macaulay: 1896-1897 * William John Greenstreet: 1897-1930 * Alan Broadbent: 1930-1955 * Reuben Goodstein: 1956-1962 * Edwin A. Maxwell: 1962-1971 * Douglas Quadling Douglas Arthur Quadling (1926–2015) was an English mathematician, school master and educationalist who was one of the four drivers behind the School Mathematics Project (SMP) i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integers
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 Countable set, countably infinite. An integer may be regarded as a real number that can be written without a fraction, fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and are not. The integers form the smallest Group (mathematics), group and the smallest ring (mathematics), ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the term in the polynomial expansion of the binomial power ; this coefficient can be computed by the multiplicative formula :\binom nk = \frac, which using factorial notation can be compactly expressed as :\binom = \frac. For example, the fourth power of is :\begin (1 + x)^4 &= \tbinom x^0 + \tbinom x^1 + \tbinom x^2 + \tbinom x^3 + \tbinom x^4 \\ &= 1 + 4x + 6 x^2 + 4x^3 + x^4, \end and the binomial coefficient \tbinom =\tfrac = \tfrac = 6 is the coefficient of the term. Arranging the numbers \tbinom, \tbinom, \ldots, \tbinom in successive rows for n=0,1,2,\ldots gives a triangular array called Pascal's triangle, satisfying the recurrence relation :\binom = \binom + \binom. The binomial coefficients occur in many areas of mathematics, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Sum
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics) through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance. For a long time, the idea that such a potentially infinite summation could produce a finite result was considered paradoxical. This paradox was resolved using the concept of a limit during the 17th century. Zeno's paradox of Achilles and the tortoise illustrates this counterintuitive property of infinite sums: Achilles runs after a tortoise, but when he reaches the position of the tortoise at the beginning of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bertrand's Ballot Theorem
In combinatorics, Bertrand's ballot problem is the question: "In an election where candidate A receives ''p'' votes and candidate B receives ''q'' votes with ''p'' > ''q'', what is the probability that A will be strictly ahead of B throughout the count?" The answer is :\frac. The result was first published by W. A. Whitworth in 1878, but is named after Joseph Louis François Bertrand who rediscovered it in 1887. In Bertrand's original paper, he sketches a proof based on a general formula for the number of favourable sequences using a recursion relation. He remarks that it seems probable that such a simple result could be proved by a more direct method. Such a proof was given by Désiré André, based on the observation that the unfavourable sequences can be divided into two equally probable cases, one of which (the case where B receives the first vote) is easily computed; he proves the equality by an explicit bijection. A variation of his method is popularly known as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Allen Whitworth
William Allen Whitworth (1 February 1840 – 12 March 1905) was an English mathematician and a priest in the Church of England.. Education and mathematical career Whitworth was born in Runcorn; his father, William Whitworth, was a school headmaster, and he was the oldest of six siblings. He was schooled at the Sandicroft School in Northwich and then at St John's College, Cambridge, earning a B.A. in 1862 as 16th Wrangler. He taught mathematics at the Portarlington School and the Rossall School, and was a professor of mathematics at Queen's College in Liverpool from 1862 to 1864. He returned to Cambridge to earn a master's degree in 1865, and was a fellow there from 1867 to 1882. Mathematical contributions As an undergraduate, Whitworth became the founding editor in chief of the ''Messenger of Mathematics'', and he continued as its editor until 1880. He published works about the logarithmic spiral and about trilinear coordinates, but his most famous mathematical publication is the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty."Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory", Alan Stuart and Keith Ord, 6th Ed, (2009), .William Feller, ''An Introduction to Probability Theory and Its Applications'', (Vol 1), 3rd Ed, (1968), Wiley, . The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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