Computable simulated transformations of the Cartesian coordinates for random vectors

Рассмотрены специальные преобразования декартовых координат, позволяющие строить эффективные (экономичные) алгоритмы численного моделирования многомерных случайных величин. В качестве иллюстративных и практически значимых примеров таких преобразований рассмотрены переходы к полярным, сферическим, параболическим и цилиндрическим координатам. The purpose of the paper was to expand the range of efficient (economical) computer algorithms for simulation of multi-dimensional random variables. The authors noticed that in a number of applied problems (for example, when modelling twoor three-dimensional isotropic vectors), transitions from Cartesian to other coordinate systems (for example, to polar or spherical) are effective. In this regard, the new generalizing notation of the computable simulated transformation of Cartesian coordinates is introduced in the paper. Such transformations allow constructing effective (economical) algorithms for the numerical modelling (simulation) of multi-dimensional random variables. The problem of finding examples of constructive and practical applications for computer simulated transformations of Cartesian coordinates is formulated. Transitions to polar, spherical, parabolic and cylindrical coordinates are considered as illustrative examples of such transformations. The practical applications of computable simulated transformations of Cartesian coordinates found in the scientific literature are described in detail by the authors. These applications are associated with both computer modelling (simulation) of isotropic vectors and Gaussian distribution along with the numerical solution of boundary value problems and problems of radiation transfer. Thus, the introduced notion of the computable simulated transformations of Cartesian coordinates is quite constructive. It opens up prospects for the scientific search for new transformations of this type with the aim of using them in stochastic computer models for important processes and phenomena.

Espaces adéliques quadratiques

We study quadratic forms defined on an adelic vector space over an algebraic extension K of the rationals. Under the sole condition that a Siegel lemma holds over K, we provide height bounds for several objects naturally associated to the quadratic form, such as an isotropic subspace, a basis of isotropic vectors (when it exists) or an orthogonal basis. Our bounds involve the heights of the form and of the ambient space. In several cases, we show that the exponents of these heights are best possible. The results improve and extend previously known statements for number fields and the field of algebraic numbers.

Orthonormal Isotropic Vector Bases

Orthonormal bases of isotropic vectors for indefinite square matrices are proposed and solved. A necessary and sufficient condition is that the matrix must have zero trace. A recursive algorithm is presented for computer applications. The isotropic vectors of 3 × 3 matrices are solved explicitly. Deviatoric stresses in continuum mechanics, the existence of isotropic vectors (particularly in screw space), and stiffness synthesis by springs are shown to be related to the isotropic vector problem.