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Appell Polynomials
In mathematics, an Appell sequence, named after Paul Émile Appell, is any polynomial sequence \_ satisfying the identity :\frac p_n(x) = np_(x), and in which p_0(x) is a non-zero constant. Among the most notable Appell sequences besides the trivial example \ are the Hermite polynomials, the Bernoulli polynomials, and the Euler polynomials. Every Appell sequence is a Sheffer sequence, but most Sheffer sequences are not Appell sequences. Appell sequences have a probabilistic interpretation as systems of moments. Equivalent characterizations of Appell sequences The following conditions on polynomial sequences can easily be seen to be equivalent: * For n = 1, 2, 3,\ldots, ::\frac p_n(x) = n p_(x) :and p_0(x) is a non-zero constant; * For some sequence \_^ of scalars with c_0 \neq 0, ::p_n(x) = \sum_^n \binom c_k x^; * For the same sequence of scalars, ::p_n(x) = \left(\sum_^\infty \frac D^k\right) x^n, :where ::D = \frac; * For n=0,1,2,\ldots, ::p_n(x+y) = \sum_^n \bi ... [...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] |
Pincherle Derivative
In mathematics, the Pincherle derivative T' of a linear operator T: \mathbb \to \mathbb /math> on the vector space of polynomials in the variable ''x'' over a field \mathbb is the commutator of T with the multiplication by ''x'' in the algebra of endomorphisms \operatorname(\mathbb . That is, T' is another linear operator T': \mathbb \to \mathbb /math> :T' := ,x= Tx-xT = -\operatorname(x)T,\, (for the origin of the \operatorname notation, see the article on the adjoint representation) so that :T'\=T\-xT\\qquad\forall p(x)\in \mathbb This concept is named after the Italian mathematician Salvatore Pincherle (1853–1936). Properties The Pincherle derivative, like any commutator, is a derivation, meaning it satisfies the sum and products rules: given two linear operators S and T belonging to \operatorname\left( \mathbb \right), #(T + S)^\prime = T^\prime + S^\prime; #(TS)^\prime = T^\prime\!S + TS^\prime where TS = T \circ S is the composition of operators. One also ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dover Publications
Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward and Blanche Cirker. It primarily reissues books that are out of print from their original publishers. These are often, but not always, books in the public domain. The original published editions may be scarce or historically significant. Dover republishes these books, making them available at a significantly reduced cost. Classic reprints Dover reprints classic works of literature, classical sheet music, and public-domain images from the 18th and 19th centuries. Dover also publishes an extensive collection of mathematical, scientific, and engineering texts. It often targets its reprints at a niche market, such as woodworking. Starting in 2015, the company branched out into graphic novel reprints, overseen by Dover acquisitions editor and former comics writer and editor Drew Ford. Most Dover reprints are photo facsimiles of the originals, retaining the original pagination and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Advances In Mathematics
''Advances in Mathematics'' is a peer-reviewed scientific journal covering research on pure mathematics. It was established in 1961 by Gian-Carlo Rota. The journal publishes 18 issues each year, in three volumes. At the origin, the journal aimed at publishing articles addressed to a broader "mathematical community", and not only to mathematicians in the author's field. Herbert Busemann writes, in the preface of the first issue, "The need for expository articles addressing either all mathematicians or only those in somewhat related fields has long been felt, but little has been done outside of the USSR. The serial publication ''Advances in Mathematics'' was created in response to this demand." Abstracting and indexing The journal is abstracted and indexed in: * |
Wick Product
In probability theory, the Wick product is a particular way of defining an adjusted product of a set of random variables. In the lowest order product the adjustment corresponds to subtracting off the mean value, to leave a result whose mean is zero. For the higher order products the adjustment involves subtracting off lower order (ordinary) products of the random variables, in a symmetric way, again leaving a result whose mean is zero. The Wick product is a polynomial function of the random variables, their expected values, and expected values of their products. The definition of the Wick product immediately leads to the Wick power of a single random variable and this allows analogues of other functions of random variables to be defined on the basis of replacing the ordinary powers in a power-series expansions by the Wick powers. The Wick powers of commonly-seen random variables can be expressed in terms of special functions such as Bernoulli polynomials or Hermite polynomials. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Appell Polynomials
In mathematics, a polynomial sequence \ has a generalized Appell representation if the generating function for the polynomials takes on a certain form: :K(z,w) = A(w)\Psi(zg(w)) = \sum_^\infty p_n(z) w^n where the generating function or kernel K(z,w) is composed of the series :A(w)= \sum_^\infty a_n w^n \quad with a_0 \ne 0 and :\Psi(t)= \sum_^\infty \Psi_n t^n \quad and all \Psi_n \ne 0 and :g(w)= \sum_^\infty g_n w^n \quad with g_1 \ne 0. Given the above, it is not hard to show that p_n(z) is a polynomial of degree n. Boas–Buck polynomials are a slightly more general class of polynomials. Special cases * The choice of g(w)=w gives the class of Brenke polynomials. * The choice of \Psi(t)=e^t results in the Sheffer sequence of polynomials, which include the general difference polynomials, such as the Newton polynomials. * The combined choice of g(w)=w and \Psi(t)=e^t gives the Appell sequence of polynomials. Explicit representation The generalized Appell polynomials ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Umbral Calculus
In mathematics before the 1970s, the term umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain "shadowy" techniques used to "prove" them. These techniques were introduced by John Blissard and are sometimes called Blissard's symbolic method. They are often attributed to Édouard Lucas (or James Joseph Sylvester), who used the technique extensively. Short history In the 1930s and 1940s, Eric Temple Bell attempted to set the umbral calculus on a rigorous footing. In the 1970s, Steven Roman, Gian-Carlo Rota, and others developed the umbral calculus by means of linear functionals on spaces of polynomials. Currently, ''umbral calculus'' refers to the study of Sheffer sequences, including polynomial sequences of binomial type and Appell sequences, but may encompass systematic correspondence techniques of the calculus of finite differences. The 19th-century umbral calculus The method is a notational procedure used for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup of ''G'' if the restriction of ∗ to is a group operation on ''H''. This is often denoted , read as "''H'' is a subgroup of ''G''". The trivial subgroup of any group is the subgroup consisting of just the identity element. A proper subgroup of a group ''G'' is a subgroup ''H'' which is a proper subset of ''G'' (that is, ). This is often represented notationally by , read as "''H'' is a proper subgroup of ''G''". Some authors also exclude the trivial group from being proper (that is, ). If ''H'' is a subgroup of ''G'', then ''G'' is sometimes called an overgroup of ''H''. The same definitions apply more generally when ''G'' is an arbitrary semigroup, but this article will only deal with subgroups of groups. Subgroup tests Suppose th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel. The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood and fully classified. Definition An abelian group is a set A, together with an operation \cdot that combines any two elements a and b of A to form another element of A, denoted a \cdot b. The symbo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-abelian Group
In mathematics, and specifically in group theory, a non-abelian group, sometimes called a non-commutative group, is a group (''G'', ∗) in which there exists at least one pair of elements ''a'' and ''b'' of ''G'', such that ''a'' ∗ ''b'' ≠ ''b'' ∗ ''a''. This class of groups contrasts with the abelian groups. (In an abelian group, all pairs of group elements commute). Non-abelian groups are pervasive in mathematics and physics. One of the simplest examples of a non-abelian group is the dihedral group of order 6. It is the smallest finite non-abelian group. A common example from physics is the rotation group SO(3) in three dimensions (for example, rotating something 90 degrees along one axis and then 90 degrees along a different axis is not the same as doing them in reverse order). Both discrete groups and continuous groups may be non-abelian. Most of the interesting Lie groups are non-abelian, and these play an important role in gauge theory. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Umbral Calculus
In mathematics before the 1970s, the term umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain "shadowy" techniques used to "prove" them. These techniques were introduced by John Blissard and are sometimes called Blissard's symbolic method. They are often attributed to Édouard Lucas (or James Joseph Sylvester), who used the technique extensively. Short history In the 1930s and 1940s, Eric Temple Bell attempted to set the umbral calculus on a rigorous footing. In the 1970s, Steven Roman, Gian-Carlo Rota, and others developed the umbral calculus by means of linear functionals on spaces of polynomials. Currently, ''umbral calculus'' refers to the study of Sheffer sequences, including polynomial sequences of binomial type and Appell sequences, but may encompass systematic correspondence techniques of the calculus of finite differences. The 19th-century umbral calculus The method is a notational procedure used for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Émile Appell
:''M. P. Appell is the same person: it stands for Monsieur Paul Appell''. Paul Émile Appell (27 September 1855, in Strasbourg – 24 October 1930, in Paris) was a French mathematician and Rector of the University of Paris. Appell polynomials and Appell's equations of motion are named after him, as is rue Paul Appell in the 14th arrondissement of Paris and the minor planet 988 Appella. Life Paul Appell entered the École Normale Supérieure in 1873. He was elected to the French Academy of Sciences in 1892. In 1895, he became a Professor at the École Centrale Paris. Between 1903 and 1920 he was Dean of the Faculty of Science of the University of Paris, then Rector of the University of Paris from 1920 to 1925. Appell was the President of the Société astronomique de France (SAF), the French astronomical society, from 1919 to 1921. His daughter Marguerite Appell (1883–1969), who married the mathematician Émile Borel, is known as a novelist under her pen-name Camille Marbo. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
