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Arnold's Cat Map
In mathematics, Arnold's cat map is a chaotic map from the torus into itself, named after Vladimir Arnold, who demonstrated its effects in the 1960s using an image of a cat, hence the name. Thinking of the torus \mathbb^2 as the quotient space \mathbb^2/\mathbb^2, Arnold's cat map is the transformation \Gamma : \mathbb^2 \to \mathbb^2 given by the formula :\Gamma \, : \, (x,y) \to (2x+y,x+y) \bmod 1. Equivalently, in matrix notation, this is :\Gamma \left( \begin x \\ y \end \right) = \begin 2 & 1 \\ 1 & 1 \end \begin x \\ y \end \bmod 1 = \begin 1 & 1 \\ 0 & 1 \end \begin 1 & 0 \\ 1 & 1 \end \begin x \\ y \end \bmod 1. That is, with a unit equal to the width of the square image, the image is sheared one unit up, then two units to the right, and all that lies outside that unit square is shifted back by the unit until it is within the square. Properties * Γ is invertible because the matrix has determinant 1 and therefore its inverse has integer entries, * Γ is area prese ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with respect to the main diagonal. So if a_ denotes the entry in the ith row and jth column then for all indices i and j. Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. Similarly in characteristic different from 2, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative. In linear algebra, a real symmetric matrix represents a self-adjoint operator represented in an orthonormal basis over a real inner product space. The corresponding object for a complex inner product space is a Hermitian matrix with complex-valued entries, which is equal to its conjugate transpose. Therefore, in linear algebra over the complex numbers, it is often assumed that a symmetric matrix refe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arnold Cat
Arnold may refer to: People * Arnold (given name), a masculine given name * Arnold (surname), a German and English surname Places Australia * Arnold, Victoria, a small town in the Australian state of Victoria Canada * Arnold, Nova Scotia United Kingdom * Arnold, East Riding of Yorkshire * Arnold, Nottinghamshire United States * Arnold, California, in Calaveras County * Arnold, Carroll County, Illinois * Arnold, Morgan County, Illinois * Arnold, Iowa * Arnold, Kansas * Arnold, Maryland * Arnold, Mendocino County, California * Arnold, Michigan * Arnold, Minnesota * Arnold, Missouri * Arnold, Nebraska * Arnold, Ohio * Arnold, Pennsylvania * Arnold, Texas * Arnold, Brooke County, West Virginia * Arnold, Lewis County, West Virginia * Arnold, Wisconsin * Arnold Arboretum of Harvard University, Massachusetts * Arnold Township, Custer County, Nebraska Other uses * Arnold (automobile), a short-lived English car * Arnold of Manchester, a former English coachbuilder * Arnold (band ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Structural Stability
In mathematics, structural stability is a fundamental property of a dynamical system which means that the qualitative behavior of the trajectories is unaffected by small perturbations (to be exact ''C''1-small perturbations). Examples of such qualitative properties are numbers of fixed points and periodic orbits (but not their periods). Unlike Lyapunov stability, which considers perturbations of initial conditions for a fixed system, structural stability deals with perturbations of the system itself. Variants of this notion apply to systems of ordinary differential equations, vector fields on smooth manifolds and flows generated by them, and diffeomorphisms. Structurally stable systems were introduced by Aleksandr Andronov and Lev Pontryagin in 1937 under the name "systèmes grossiers", or rough systems. They announced a characterization of rough systems in the plane, the Andronov–Pontryagin criterion. In this case, structurally stable systems are ''typical'', they form an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anosov Diffeomorphism
In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold ''M'' is a certain type of mapping, from ''M'' to itself, with rather clearly marked local directions of "expansion" and "contraction". Anosov systems are a special case of Axiom A systems. Anosov diffeomorphisms were introduced by Dmitri Victorovich Anosov, who proved that their behaviour was in an appropriate sense ''generic'' (when they exist at all). Dmitri V. Anosov, ''Geodesic flows on closed Riemannian manifolds with negative curvature'', (1967) Proc. Steklov Inst. Mathematics. 90. Overview Three closely related definitions must be distinguished: * If a differentiable map ''f'' on ''M'' has a hyperbolic structure on the tangent bundle, then it is called an Anosov map. Examples include the Bernoulli map,_and_Arnold's_cat_map.html" ;"title=", 1)^\infty : x \mapsto (x_0, x_1, x_2, ..., and Arnold's cat map">, 1)^\infty : x \mapsto (x_0, x_1, x_2, ... ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mixing (physics)
In physics, a dynamical system is said to be mixing if the phase space of the system becomes strongly intertwined, according to at least one of several mathematical definitions. For example, a measure-preserving transformation ''T'' is said to be strong mixing if : \lim_ \, \mu(T^A \cap B) = \mu(A) \cdot \mu(B) whenever ''A'' and ''B'' are any measurable sets and μ is the associated measure. Other definitions are possible, including weak mixing and topological mixing. The mathematical definition of mixing is meant to capture the notion of physical mixing. A canonical example is the Cuba libre: suppose one is adding rum (the set ''A'') to a glass of cola. After stirring the glass, the bottom half of the glass (the set ''B'') will contain rum, and it will be in equal proportion as it is elsewhere in the glass. The mixing is uniform: no matter which region ''B'' one looks at, some of ''A'' will be in that region. A far more detailed, but still informal description of mixin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ergodic
In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies that the average behavior of the system can be deduced from the trajectory of a "typical" point. Equivalently, a sufficiently large collection of random samples from a process can represent the average statistical properties of the entire process. Ergodicity is a property of the system; it is a statement that the system cannot be reduced or factored into smaller components. Ergodic theory is the study of systems possessing ergodicity. Ergodic systems occur in a broad range of systems in physics and in geometry. This can be roughly understood to be due to a common phenomenon: the motion of particles, that is, geodesics on a hyperbolic manifold are divergent; when that manifold is compact, that is, of finite size, those orbits return to the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Transitivity
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, drinks, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by boldface , or blackboard bold \mathbb. A rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: ), or eventually begins to repeat the same finite sequence of digits over and over (example: ). This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see ). A real number that is not rational is called irrational. Irrational numbers include , , , and . Since the set of rational numbers is countable, and the set of real numbers is uncountable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Periodic Orbit
In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time. Iterated functions Given a mapping ''f'' from a set ''X'' into itself, :f: X \to X, a point ''x'' in ''X'' is called periodic point if there exists an ''n'' so that :\ f_n(x) = x where f_n is the ''n''th iterate of ''f''. The smallest positive integer ''n'' satisfying the above is called the ''prime period'' or ''least period'' of the point ''x''. If every point in ''X'' is a periodic point with the same period ''n'', then ''f'' is called ''periodic'' with period ''n'' (this is not to be confused with the notion of a periodic function). If there exist distinct ''n'' and ''m'' such that :f_n(x) = f_m(x) then ''x'' is called a preperiodic point. All periodic points are preperiodic. If ''f'' is a diffeomorphism of a differentiable manifold, so that the derivative f_n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenvalues
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by \lambda, is the factor by which the eigenvector is scaled. Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated. Formal definition If is a linear transformation from a vector space over a field into itself and is a nonzero vector in , then is an eigenvector of if is a scalar multiple of . This can be written as T(\mathbf) = \lambda \mathbf, where is a scalar in , known as the eigenvalue, characteristic value, or characteristic root ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unimodular Matrix
In mathematics, a unimodular matrix ''M'' is a square integer matrix having determinant +1 or −1. Equivalently, it is an integer matrix that is invertible over the integers: there is an integer matrix ''N'' that is its inverse (these are equivalent under Cramer's rule). Thus every equation , where ''M'' and ''b'' both have integer components and ''M'' is unimodular, has an integer solution. The ''n'' × ''n'' unimodular matrices form a group called the ''n'' × ''n'' general linear group over \mathbb, which is denoted \operatorname_n(\mathbb). Examples of unimodular matrices Unimodular matrices form a subgroup of the general linear group under matrix multiplication, i.e. the following matrices are unimodular: * Identity matrix * The inverse of a unimodular matrix * The product of two unimodular matrices Other examples include: * Pascal matrices * Permutation matrices * the three transformation matrices in the ternary tree of primitive Pythagorean ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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