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Arnold's cat map is a particularly well-known example of a hyperbolic toral automorphism, which is an automorphism of a torus given by a square unimodular matrix having no eigenvalues of absolute value 1. [3] The set of the points with a periodic orbit is dense on the torus. Actually a point is periodic if and only if its coordinates are rational.
Arnold conjectured that "the mechanism of 'transition chains' which guarantees that nonstability in our example is also applicable to the general case (for example, to the problem of three bodies)." [1] The KAM theorem and Arnold diffusion has led to a compendium of rigorous mathematical results, with insights from physics. [3] [4]
In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable.The general form of its probability density function is [2] [3] = ().
This is a list of notable theorems.Lists of theorems and similar statements include: List of algebras; List of algorithms; List of axioms; List of conjectures
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
In numerical linear algebra, the alternating-direction implicit (ADI) method is an iterative method used to solve Sylvester matrix equations.It is a popular method for solving the large matrix equations that arise in systems theory and control, [1] and can be formulated to construct solutions in a memory-efficient, factored form.
Born, however, had learned matrix algebra from Rosanes, as already noted, but Born had also learned Hilbert's theory of integral equations and quadratic forms for an infinite number of variables as was apparent from a citation by Born of Hilbert's work Grundzüge einer allgemeinen Theorie der Linearen Integralgleichungen published in 1912. [16 ...
The series includes the volumes Mechanics, Mechanics of Deformable Bodies, Electrodynamics, Optics, Thermodynamics and Statistical Mechanics, and Partial Differential Equations in Physics. Focusing on one subject each semester, the lectures formed a three-year cycle of courses that Sommerfeld repeatedly taught at the University of Munich for ...