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Exponential Formula
In combinatorial mathematics, the exponential formula (called the polymer expansion in physics) states that the exponential generating function for structures on finite sets is the exponential of the exponential generating function for connected structures. The exponential formula is a power-series version of a special case of Faà di Bruno's formula. Statement For any formal power series of the form f(x)=a_1 x+x^2+x^3+\cdots+x^n+\cdots\, we have \exp f(x)=e^=\sum_^\infty x^n,\, where b_n = \sum_ a_\cdots a_, and the index \pi runs through the list of all partitions \ of the set \. (When k = 0, the product is empty and by definition equals 1.) One can write the formula in the following form: b_n = B_n(a_1,a_2,\dots,a_n), and thus \exp\left(\sum_^\infty x^n \right) = \sum_^\infty x^n, where B_n(a_1,\ldots,a_n) is the nth complete Bell polynomial. Alternatively, the exponential formula can also be written using the cycle index of the symmetric group, as follows:\exp\left(\sum_^\inf ... [...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] |
Symmetric Group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \mathrm_n defined over a finite set of n symbols consists of the permutations that can be performed on the n symbols. Since there are n! (n factorial) such permutation operations, the order (number of elements) of the symmetric group \mathrm_n is n!. Although symmetric groups can be defined on infinite sets, this article focuses on the finite symmetric groups: their applications, their elements, their conjugacy classes, a finite presentation, their subgroups, their automorphism groups, and their representation theory. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set. The symmetric group is important to diverse areas of mathematics such as Galois theory, invariant theory, the representatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connected Correlation Function
In statistical mechanics, an Ursell function or connected correlation function, is a cumulant of a random variable. It can often be obtained by summing over connected Feynman diagrams (the sum over all Feynman diagrams gives the correlation functions). The Ursell function was named after Harold Ursell, who introduced it in 1927. Definition If ''X'' is a random variable, the moments ''s''''n'' and cumulants (same as the Ursell functions) ''u''''n'' are functions of ''X'' related by the exponential formula: : \operatorname(\exp(zX)) = \sum_n s_n \frac = \exp\left(\sum_n u_n \frac\right) (where \operatorname is the expectation). The Ursell functions for multivariate random variables are defined analogously to the above, and in the same way as multivariate cumulants. :u_n\left(X_1, \ldots, X_n\right) = \left.\frac \cdots \frac\log \operatorname\left(\exp\sum z_i X_i\right)\_ The Ursell functions of a single random variable ''X'' are obtained from these by setting . The first f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Feynman Diagram
In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduced the diagrams in 1948. The interaction of subatomic particles can be complex and difficult to understand; Feynman diagrams give a simple visualization of what would otherwise be an arcane and abstract formula. According to David Kaiser, "Since the middle of the 20th century, theoretical physicists have increasingly turned to this tool to help them undertake critical calculations. Feynman diagrams have revolutionized nearly every aspect of theoretical physics." While the diagrams are applied primarily to quantum field theory, they can also be used in other fields, such as solid-state theory. Frank Wilczek wrote that the calculations that won him the 2004 Nobel Prize in Physics "would have been literally unthinkable without Feynman diagra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Correlation Function
A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables. If one considers the correlation function between random variables representing the same quantity measured at two different points, then this is often referred to as an autocorrelation function, which is made up of autocorrelations. Correlation functions of different random variables are sometimes called cross-correlation functions to emphasize that different variables are being considered and because they are made up of cross-correlations. Correlation functions are a useful indicator of dependencies as a function of distance in time or space, and they can be used to assess the distance required between sample points for the values to be effectively uncorrelated. In addition, they can form the basis of rules for interpolating values at points for which there are no observations. Correlation functions used ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partition Function (mathematics)
The partition function or configuration integral, as used in probability theory, information theory and dynamical systems, is a generalization of the definition of a partition function in statistical mechanics. It is a special case of a normalizing constant in probability theory, for the Boltzmann distribution. The partition function occurs in many problems of probability theory because, in situations where there is a natural symmetry, its associated probability measure, the Gibbs measure, has the Markov property. This means that the partition function occurs not only in physical systems with translation symmetry, but also in such varied settings as neural networks (the Hopfield network), and applications such as genomics, corpus linguistics and artificial intelligence, which employ Markov networks, and Markov logic networks. The Gibbs measure is also the unique measure that has the property of maximizing the entropy for a fixed expectation value of the energy; this underlies the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bell Polynomials
In combinatorial mathematics, the Bell polynomials, named in honor of Eric Temple Bell, are used in the study of set partitions. They are related to Stirling and Bell numbers. They also occur in many applications, such as in the Faà di Bruno's formula. Definitions Exponential Bell polynomials The ''partial'' or ''incomplete'' exponential Bell polynomials are a triangular array of polynomials given by :B_(x_1,x_2,\dots,x_) = \sum \left(\right)^\left(\right)^\cdots\left(\right)^, where the sum is taken over all sequences ''j''1, ''j''2, ''j''3, ..., ''j''''n''−''k''+1 of non-negative integers such that these two conditions are satisfied: :j_1 + j_2 + \cdots + j_ = k, :j_1 + 2 j_2 + 3 j_3 + \cdots + (n-k+1)j_ = n. The sum :B_n(x_1,\dots,x_n)=\sum_^n B_(x_1,x_2,\dots,x_) is called the ''n''th ''complete exponential Bell polynomial''. Ordinary Bell polynomials Likewise, the partial ''ordinary'' Bell polynomial is defined by :\hat_(x_1,x_2,\ldots,x_) = \sum \frac x_1^ x_2^ \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cycle Index
Cycle, cycles, or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in social sciences ** Business cycle, the downward and upward movement of gross domestic product (GDP) around its ostensible, long-term growth trend Arts, entertainment, and media Films * ''Cycle'' (2008 film), a Malayalam film * ''Cycle'' (2017 film), a Marathi film Literature * ''Cycle'' (magazine), an American motorcycling enthusiast magazine * Literary cycle, a group of stories focused on common figures Music Musical terminology * Cycle (music), a set of musical pieces that belong together **Cyclic form, a technique of construction involving multiple sections or movements **Interval cycle, a collection of pitch classes generated from a sequence of the same interval class **Song cycle, individually complete songs designed to be performe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bell Polynomial
In combinatorial mathematics, the Bell polynomials, named in honor of Eric Temple Bell, are used in the study of set partitions. They are related to Stirling and Bell numbers. They also occur in many applications, such as in the Faà di Bruno's formula. Definitions Exponential Bell polynomials The ''partial'' or ''incomplete'' exponential Bell polynomials are a triangular array of polynomials given by :B_(x_1,x_2,\dots,x_) = \sum \left(\right)^\left(\right)^\cdots\left(\right)^, where the sum is taken over all sequences ''j''1, ''j''2, ''j''3, ..., ''j''''n''−''k''+1 of non-negative integers such that these two conditions are satisfied: :j_1 + j_2 + \cdots + j_ = k, :j_1 + 2 j_2 + 3 j_3 + \cdots + (n-k+1)j_ = n. The sum :B_n(x_1,\dots,x_n)=\sum_^n B_(x_1,x_2,\dots,x_) is called the ''n''th ''complete exponential Bell polynomial''. Ordinary Bell polynomials Likewise, the partial ''ordinary'' Bell polynomial is defined by :\hat_(x_1,x_2,\ldots,x_) = \sum \frac x_1^ x_2^ \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Empty Product
In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operation in question), just as the empty sum—the result of adding no numbers—is by convention zero, or the additive identity. When numbers are implied, the empty product becomes one. The term ''empty product'' is most often used in the above sense when discussing arithmetic operations. However, the term is sometimes employed when discussing set-theoretic intersections, categorical products, and products in computer programming; these are discussed below. Nullary arithmetic product Definition Let ''a''1, ''a''2, ''a''3, ... be a sequence of numbers, and let :P_m = \prod_^m a_i = a_1 \cdots a_m be the product of the first ''m'' elements of the sequence. Then :P_m = P_ a_m for all ''m'' = 1, 2, ... provided that we use the convention P_0 = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
