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Gauss–Kuzmin–Wirsing Operator
In mathematics, the Gauss–Kuzmin–Wirsing operator is the transfer operator of the Gauss map that takes a positive number to the fractional part of its reciprocal. (This is not the same as the Gauss map in differential geometry.) It is named after Carl Gauss, Rodion Kuzmin, and Eduard Wirsing. It occurs in the study of continued fractions; it is also related to the Riemann zeta function. Relationship to the maps and continued fractions The Gauss map The Gauss function (map) ''h'' is : :h(x)=1/x-\lfloor 1/x \rfloor. where \lfloor 1/x \rfloor denotes the floor function. It has an infinite number of jump discontinuities at ''x'' = 1/''n'', for positive integers ''n''. It is hard to approximate it by a single smooth polynomial. Operator on the maps The Gauss–Kuzmin–Wirsing operator G acts on functions f as : fx) = \int_0^1 \delta(x-h(y)) f(y) d y = \sum_^\infty \frac f \left(\frac \right). Eigenvalues of the operator The first eigenfunction of this operat ... [...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] |
Bernoulli Shift
In mathematics, the Bernoulli scheme or Bernoulli shift is a generalization of the Bernoulli process to more than two possible outcomes. Bernoulli schemes appear naturally in symbolic dynamics, and are thus important in the study of dynamical systems. Many important dynamical systems (such as Axiom A systems) exhibit a repellor that is the product of the Cantor set and a smooth manifold, and the dynamics on the Cantor set are isomorphic to that of the Bernoulli shift. This is essentially the Markov partition. The term ''shift'' is in reference to the shift operator, which may be used to study Bernoulli schemes. The Ornstein isomorphism theorem shows that Bernoulli shifts are isomorphic when their entropy is equal. Definition A Bernoulli scheme is a discrete-time stochastic process where each independent random variable may take on one of ''N'' distinct possible values, with the outcome ''i'' occurring with probability p_i, with ''i'' = 1, ..., ''N'', and :\sum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taylor Series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series, when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century. The partial sum formed by the first terms of a Taylor series is a polynomial of degree that is called the th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minkowski's Question Mark Function
In mathematics, the Minkowski question-mark function, denoted , is a function with unusual fractal properties, defined by Hermann Minkowski in 1904. It maps quadratic irrational numbers to rational numbers on the unit interval, via an expression relating the continued fraction expansions of the quadratics to the binary expansions of the rationals, given by Arnaud Denjoy in 1938. It also maps rational numbers to dyadic rationals, as can be seen by a recursive definition closely related to the Stern–Brocot tree. Definition and intuition One way to define the question-mark function involves the correspondence between two different ways of representing fractional numbers using finite or infinite binary sequences. Most familiarly, a string of 0's and 1's with a single point mark ".", like "11.001001000011111..." can be interpreted as the binary representation of a number. In this case this number is 2+1+\frac18+\frac1+\cdots=\pi. There is a different way of interpreting the same s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Haar Measure
In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This measure was introduced by Alfréd Haar in 1933, though its special case for Lie groups had been introduced by Adolf Hurwitz in 1897 under the name "invariant integral". Haar measures are used in many parts of analysis, number theory, group theory, representation theory, statistics, probability theory, and ergodic theory. Preliminaries Let (G, \cdot) be a locally compact Hausdorff topological group. The \sigma-algebra generated by all open subsets of G is called the Borel algebra. An element of the Borel algebra is called a Borel set. If g is an element of G and S is a subset of G, then we define the left and right translates of S by ''g'' as follows: * Left translate: g S = \. * Right translate: S g = \. Left and right translates map Borel sets onto Borel sets. A measure \mu on th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Product Topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seeming, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a categorical product of its factors, whereas the box topology is too fine; in that sense the product topology is the natural topology on the Cartesian product. Definition Throughout, I will be some non-empty index set and for every index i \in I, let X_i be a topological space. Denote the Cartesian product of the sets X_i by X := \prod X_ := \prod_ X_i and for every index i \in I, denote the i-th by \begin p_i :\;&& \prod_ X_j &&\;\to\; & X_i \\ .3ex && \left(x_j\r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cylinder Set
In mathematics, the cylinder sets form a basis of the product topology on a product of sets; they are also a generating family of the cylinder σ-algebra. General definition Given a collection S of sets, consider the Cartesian product X = \prod_ Y of all sets in the collection. The canonical projection corresponding to some Y\in S is the function p_ : X \to Y that maps every element of the product to its Y component. A cylinder set is a preimage of a canonical projection or finite intersection of such preimages. Explicitly, it is a set of the form, \bigcap_^n p_^ \left(A_i\right) = \left\ for any choice of n, finite sequence of sets Y_1,...Y_n\in S and subsets A_ \subseteq Y_i for 1 \leq i \leq n. Here x_Y\in Y denotes the Y component of x\in X. Then, when all sets in S are topological spaces, the product topology is generated by cylinder sets corresponding to the components' open sets. That is cylinders of the form \bigcap_^n p_^ \left(U_i\right) where for each i, U_i is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indicator Function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\in A, and \mathbf_(x)=0 otherwise, where \mathbf_A is a common notation for the indicator function. Other common notations are I_A, and \chi_A. The indicator function of is the Iverson bracket of the property of belonging to ; that is, :\mathbf_(x)= \in A For example, the Dirichlet function is the indicator function of the rational numbers as a subset of the real numbers. Definition The indicator function of a subset of a set is a function \mathbf_A \colon X \to \ defined as \mathbf_A(x) := \begin 1 ~&\text~ x \in A~, \\ 0 ~&\text~ x \notin A~. \end The Iverson bracket provides the equivalent notation, \in A/math> or to be used instead of \mathbf_(x)\,. The function \mathbf_A is sometimes denoted , , , or even just . Nota ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term "Fréchet space." Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces. Definition A Banach space is a complete norme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Baire Space (set Theory)
In set theory, the Baire space is the set of all infinite sequences of natural numbers with a certain topology. This space is commonly used in descriptive set theory, to the extent that its elements are often called "reals". It is denoted NN, ωω, by the symbol \mathcal or also ωω, not to be confused with the countable ordinal obtained by ordinal exponentiation. The Baire space is defined to be the Cartesian product of countably infinitely many copies of the set of natural numbers, and is given the product topology (where each copy of the set of natural numbers is given the discrete topology). The Baire space is often represented using the tree of finite sequences of natural numbers. The Baire space can be contrasted with Cantor space, the set of infinite sequences of binary digits. Topology and trees The product topology used to define the Baire space can be described more concretely in terms of trees. The basic open sets of the product topology are cylinder sets, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brigitte Vallée
Brigitte Vallée (née Salesse) (born 6 June 1950, in Courbevoie, Hauts-de-Seine, France) is a French mathematician and computer scientist. She entered the École Normale Supérieure de Jeunes Filles in 1970, and received her PhD in 1986 at the University of Caen (Lattice reduction algorithms in small dimensions). Her doctoral advisor was Jacques Stern. Vallée has been Director of Research at the French CNRS at Université de Caen, since 2001 and specialized in computational number theory and analysis of algorithms. Amongst the algorithms she studied are the celebrated Lenstra–Lenstra–Lovász lattice basis reduction algorithm, LLL algorithm used for basis reductions in Lattice (group), Euclidean lattice and the different Euclidean algorithms to determine Greatest common divisor, GCD. The main tool used to achieve her results is the so-called ''dynamical analysis''. Loosely speaking, it is a mix between analysis of algorithms and dynamical systems. Brigitte Vallée greatly cont ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philippe Flajolet
Philippe Flajolet (; 1 December 1948 – 22 March 2011) was a French computer scientist. Biography A former student of École Polytechnique, Philippe Flajolet received his PhD in computer science from University Paris Diderot in 1973 and state doctorate from Paris-Sud 11 University in 1979. Most of Philippe Flajolet's research work was dedicated towards general methods for analyzing the computational complexity of algorithms, including the theory of average-case complexity. He introduced the theory of analytic combinatorics. With Robert Sedgewick of Princeton University, he wrote the first book-length treatment of the topic, the 2009 book entitled ''Analytic Combinatorics''. In 1993, together with Rainer Kemp, Helmut Prodinger and Robert Sedgewick, Flajolet initiated the successful series of workshops and conferences which was key to the development of a research community around the analysis of algorithms, and which evolved into the AofA—International Meeting on Combinatori ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
