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Baker's Map
In dynamical systems theory, the baker's map is a chaotic map from the unit square into itself. It is named after a kneading operation that bakers apply to dough: the dough is cut in half, and the two halves are stacked on one another, and compressed. The baker's map can be understood as the bilateral shift operator of a bi-infinite two-state lattice model. The baker's map is topologically conjugate to the horseshoe map. In physics, a chain of coupled baker's maps can be used to model deterministic diffusion. As with many deterministic dynamical systems, the baker's map is studied by its action on the space of functions defined on the unit square. The baker's map defines an operator on the space of functions, known as the transfer operator of the map. The baker's map is an exactly solvable model of deterministic chaos, in that the eigenfunctions and eigenvalues of the transfer operator can be explicitly determined. Formal definition There are two alternative definitions of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenfunction
In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, this condition can be written as Df = \lambda f for some scalar eigenvalue \lambda. The solutions to this equation may also be subject to boundary conditions that limit the allowable eigenvalues and eigenfunctions. An eigenfunction is a type of eigenvector. Eigenfunctions In general, an eigenvector of a linear operator ''D'' defined on some vector space is a nonzero vector in the domain of ''D'' that, when ''D'' acts upon it, is simply scaled by some scalar value called an eigenvalue. In the special case where ''D'' is defined on a function space, the eigenvectors are referred to as eigenfunctions. That is, a function ''f'' is an eigenfunction of ''D'' if it satisfies the equation where λ is a scalar. The solutions to Equation may also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shift Space
In symbolic dynamics and related branches of mathematics, a shift space or subshift is a set of infinite words that represent the evolution of a discrete system. In fact, shift spaces and '' symbolic dynamical systems'' are often considered synonyms. The most widely studied shift spaces are the subshifts of finite type. Notation Let ''A'' be a finite set of states. An ''infinite'' (respectively ''bi-infinite'') ''word'' over ''A'' is a sequence \mathbf x=(x_n)_, where M=\mathbb N (respectively M=\mathbb Z) and x_n is in ''A'' for any n \in M. The shift operator \sigma acts on an infinite or bi-infinite word by shifting all symbols to the left, i.e., :\sigma(\mathbf x)_n=x_ for all ''n''. In the following we choose M=\mathbb N and thus speak of infinite words, but all definitions are naturally generalizable to the bi-infinite case. Definition A set of infinite words over ''A'' is a ''shift space'' (or ''subshift'') if it is closed with respect to the natural product topolog ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square-integrable Function
In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, square-integrability on the real line (-\infty,+\infty) is defined as follows. One may also speak of quadratic integrability over bounded intervals such as ,b/math> for a \leq b. An equivalent definition is to say that the square of the function itself (rather than of its absolute value) is Lebesgue integrable. For this to be true, the integrals of the positive and negative portions of the real part must both be finite, as well as those for the imaginary part. The vector space of square integrable functions (with respect to Lebesgue measure) forms the ''Lp'' space with p=2. Among the ''Lp'' spaces, the class of square integrable functions is unique in being compatible with an inner product, which allows notions lik ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that defines a distance function for which the space is a complete metric space. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term ''Hilbert space'' for the abstract concept that under ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitary Operator
In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Unitary operators are usually taken as operating ''on'' a Hilbert space, but the same notion serves to define the concept of isomorphism ''between'' Hilbert spaces. A unitary element is a generalization of a unitary operator. In a unital algebra, an element of the algebra is called a unitary element if , where is the identity element. Definition Definition 1. A ''unitary operator'' is a bounded linear operator on a Hilbert space that satisfies , where is the adjoint of , and is the identity operator. The weaker condition defines an ''isometry''. The other condition, , defines a ''coisometry''. Thus a unitary operator is a bounded linear operator which is both an isometry and a coisometry, or, equivalently, a surjective isometry. An equivalent definition is the following: Definition 2. A ''unitary operator'' is a bounded linear operator on a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Baker
A baker is a tradesperson who bakes and sometimes sells breads and other products made of flour by using an oven or other concentrated heat source. The place where a baker works is called a bakery. History Ancient history Since grains have been a staple food for millennia, the activity of baking is a very old one. Control of yeast, however, is relatively recent.Wayne Gisslen, ''Professional Baking'' (4th ed.: John Wiley & Sons, 2005), p. 4. By the fifth and sixth centuries BCE, the ancient Greeks used enclosed ovens heated by wood fires; communities usually baked bread in a large communal oven. Greeks baked dozens and possibly hundreds of types of bread; Athenaeus described seventy-two varieties. In ancient Rome several centuries later, the first mass production of breads occurred, and "the baking profession can be said to have started at that time." Ancient Roman bakers used honey and oil in their products, creating pastries rather than breads. In ancient Rome, bakers (L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function (mathematics)
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the function and the set is called the codomain of the function.Codomain ''Encyclopedia of Mathematics'Codomain. ''Encyclopedia of Mathematics''/ref> The earliest known approach to the notion of function can be traced back to works of Persian mathematicians Al-Biruni and Sharaf al-Din al-Tusi. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topologically Mixing
In mathematics, mixing is an abstract concept originating from physics: the attempt to describe the irreversible thermodynamic process of mixing in the everyday world: mixing paint, mixing drinks, industrial mixing, ''etc''. The concept appears in ergodic theory—the study of stochastic processes and measure-preserving dynamical systems. Several different definitions for mixing exist, including ''strong mixing'', ''weak mixing'' and ''topological mixing'', with the last not requiring a measure to be defined. Some of the different definitions of mixing can be arranged in a hierarchical order; thus, strong mixing implies weak mixing. Furthermore, weak mixing (and thus also strong mixing) implies ergodicity: that is, every system that is weakly mixing is also ergodic (and so one says that mixing is a "stronger" notion than ergodicity). Informal explanation The mathematical definition of mixing aims to capture the ordinary every-day process of mixing, such as mixing paints, dr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strong Mixing
In mathematics, mixing is an abstract concept originating from physics: the attempt to describe the irreversible thermodynamic process of mixing in the everyday world: mixing paint, mixing drinks, industrial mixing, ''etc''. The concept appears in ergodic theory—the study of stochastic processes and measure-preserving dynamical systems. Several different definitions for mixing exist, including ''strong mixing'', ''weak mixing'' and ''topological mixing'', with the last not requiring a measure to be defined. Some of the different definitions of mixing can be arranged in a hierarchical order; thus, strong mixing implies weak mixing. Furthermore, weak mixing (and thus also strong mixing) implies ergodicity: that is, every system that is weakly mixing is also ergodic (and so one says that mixing is a "stronger" notion than ergodicity). Informal explanation The mathematical definition of mixing aims to capture the ordinary every-day process of mixing, such as mixing paints, dri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Baker's Map Mixing
A baker is someone who primarily bakes and sells bread. Baker and Bakers may also refer to: Brands and companies * Baker Skateboards * Baker's Chocolate * Baker's Drive-Thru, chain of fast-food restaurants in Southern California * Baker Hughes, an oilfield services company * Baker McKenzie, a law firm headquartered in Chicago * Baker's Supermarkets, U.S. supermarket chain * Baker's bourbon, a Jim Beam product * Baker Publishing Group * ''Baker's Biographical Dictionary of Musicians'' Law * ''Baker v. Carr'', a 1962 US Supreme Court case that holds that the federal government can review state election districts * '' Baker v. Nelson'', a 1972 Minnesota Supreme Court case holding that gay people do not have a right to marriage * '' Baker v. Vermont'', a 1999 Vermont Supreme Court case holding that gay people have the right to marry in Vermont * Baker Act, common alternate name for the Florida Mental Health Act Military * Baker (military code-name), a series of training exercis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lebesgue Measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called ''n''-dimensional volume, ''n''-volume, or simply volume. It is used throughout real analysis, in particular to define Lebesgue integration. Sets that can be assigned a Lebesgue measure are called Lebesgue-measurable; the measure of the Lebesgue-measurable set ''A'' is here denoted by ''λ''(''A''). Henri Lebesgue described this measure in the year 1901, followed the next year by his description of the Lebesgue integral. Both were published as part of his dissertation in 1902. Definition For any interval I = ,b/math>, or I = (a, b), in the set \mathbb of real numbers, let \ell(I)= b - a denote its length. For any subset E\subseteq\mathbb, the Lebesgue oute ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
