Nonsingular Transformation Research Articles

Nonsingular Transformations of Symmetric Lévy Processes

In this paper, we consider a group of transformations of the space of trajectories of symmetric α-stable Levy laws with stability constant α ∈ [0; 2). For α = 0, the true analog of a stable Levy process (the so-called 0-stable process) is the gamma process, whose law is quasi-invariant under the action of the group of multiplicators $ \mathcal{M}\equiv \left\{ {{M_a}:a\geq 0,\,\log a\in {L^1}} \right\} $ ; the action of Ma on a trajectory ω(∙) is given by (Maω)(t) = a(t)ω(t). For every α < 2, an appropriate conjugacy transformation sends the group $ \mathcal{M} $ to the group $ {{\mathcal{M}}_a} $ of nonlinear transformations of trajectories, and the law of the corresponding stable process is quasi-invariant under this group. We prove that for α = 2, an appropriate change of coordinates reduces the group of symmetries to the Cameron-Martin group of nonsingular translations of trajectories of the Wiener process.

Bifurcation of Periodic Solutions and Numerical Simulation for the Viscoelastic Belt

We study the bifurcation of periodic solutions for viscoelastic belt with integral constitutive law in 1: 1 internal resonance. At the beginning, by applying the nonsingular linear transformation, the system is transformed into another system whose unperturbed system is composed of two planar systems: one is a Hamiltonian system and the other has a focus. Furthermore, according to the Melnikov function, we can obtain the sufficient condition for the existence of periodic solutions and make preparations for studying the stability of the periodic solution and the invariant torus. Eventually, we need to give the phase diagrams of the solutions under different parameters to verify the analytical results and obtain which parameters the existence and the stability of the solution are based on. The conclusions not only enrich the behaviors of nonlinear dynamics about viscoelastic belt but also have important theoretical significance and application value on noise weakening and energy loss.

Evaluation of the provincial competitiveness of the Chinese high-tech industry using an improved TOPSIS method

Evaluation of the competitiveness of high-tech industry is a technical decision-making issue involving multiple criteria. It is also a practical path to promote a country’s competitiveness. However, the competitiveness indicators in high-tech industry often act and react upon one another. Moreover, different dimensions and indicator weights also affect the evaluation results. In this paper, the Mahalanobis distance is used to improve the traditional technique for order preference by similarity to ideal solution (TOPSIS). The improved TOPSIS method has the following properties: (1) an improved relative closeness which is invariant after non-singular linear transformation, and (2) the weighted Mahalanobis distance is the same as the weighted Euclidean distance when the indicators are uncorrelated. The new method is applied to evaluate the competitiveness of the Chinese high-tech industry using data from 2011. Consideration of the correlation between indicators improves the evaluation results (in terms of sorting and closeness) to a certain extent compared to the traditional TOPSIS method. The top five provinces are: Guangdong, Jiangsu, Shanghai, Beijing, and Shandong. This finding reflects the practical linkage among provinces and softens the closeness value, consistent with reality.

Method of estimating transients in induction machines

The present paper is concerned with induction motors with wound and squirrel-cage rotors. It is assumed that the magnetic field produced by the stator windings is constant in magnitude and rotates with constant angular velocity. Differential equations that describe relations between the electromagnetic torque and main electric and mechanical quantities of the induction motors under consideration are derived in detail, the geometry of the rotors of motors being fully taken into account. A special nonsingular transformation is applied to the initial systems of equations (with angular coordinates) to split them up and reduce to a lower-order (in fact, third-order) system. A local stability analysis of the resulting equations is carried out. The stable equilibrium states that correspond to the operating modes of induction motors are determined. Methods of speed control of induction motors with wound and squirrel-cage rotors are considered. The limit load problem on motors and the control problem of the speed of motors are discussed; these problems lead to the necessity to estimate the transients in induction motors. The method of estimating transients that occur due to changes of the operating parameters of motors is developed based on a modification of the nonlocal reduction method. Using this method in combination with Barbashin and Tabueva’s methods to apply to the obtained systems made it possible to find analytical estimates of the ultimate permissible loads on induction motors and the control ranges of the parameters of the system that correspond to additional external active and inductive resistances. Moreover, estimates for the region of the attraction of stable equilibrium states of systems that describe the dynamics of induction motors are obtained.

On the control of linear time-varying systems of a special form

Some classes of linear time-varying control systems of the first and second order that can be reduced to a time-invariant system by nonsingular transformation of the same order as the original system are considered. A description of the classes of systems that admit such a reduction by extending the space of states is given. By way of example, the problem of a spacecraft control in the vicinity of the collinear libration point of a planar restricted circular three-body problem is discussed.

IP-rigidity and eigenvalue groups

AbstractWe examine the class of increasing sequences of natural numbers which are IP-rigidity sequences for some weakly mixing probability-preserving transformation. This property is closely related to the uncountability of the eigenvalue group of a corresponding non-singular transformation. We give examples, including a super-lacunary sequence which is not IP-rigid.

On Constructing Ergodic Hyperfinite Equivalence Relations of Non-Product Type

AbstractProduct type equivalence relations are hyperfinitemeasured equivalence relations, which, up to orbit equivalence, are generated by product type odometer actions. We give a concrete example of a hyperfinite equivalence relation of non-product type, which is the tail equivalence on a Bratteli diagram. In order to show that the equivalence relation constructed is not of product type we will use a criterion called property A. This property, introduced by Krieger for non-singular transformations, is defined directly for hyperfinite equivalence relations in this paper.

Random perturbations of non-singular transformations on [0,1

We consider random perturbations of non-singular measurable transformations S on [0,1]. By using the spectral decomposition theorem of Komornik and Lasota, we prove that the existence of the invariant densities for random perturbations of S. Moreover the densities for random perturbations with small noise strongly converges to the deinsity for Perron-Frobenius operator corresponding to S with respect to L1([0,1])-norm.

Research on the Sufficient Condition for the Existence of Periodic Solution of FGM Subjected to Aero-Thermal Load

In this paper, the average equations are given through using the multi-scale approach method. By using the Melinkov function, the nonsingular linear transformation and the Poincaré map, the sufficient condition for existence of periodic solution of the nonlinear dynamical system about the FGM subjected to aero-thermal load is derived.

Measurable time-restricted sensitivity

We develop two notions of sensitivity to initial conditions for measurable dynamical systems, where the time before divergence of a pair of paths is at most an asymptotically logarithmic function of a measure of their initial distance. In the context of probability measure-preserving transformations on a compact space, we relate these notions to the metric entropy of the system. We examine one of these notions for classes of non-measure-preserving, nonsingular transformations.

A piecewise constant method for Frobenius–Perron operators via delta function approximations

Let S:[0,1]→[0,1] be a nonsingular transformation such that the corresponding Frobenius–Perron operator PS:L1(0,1)→L1(0,1) has a stationary density f∗. We develop a piecewise constant method for the numerical computation of f∗, based on the approximation of Dirac’s delta function via pulse functions. We show that the numerical scheme out of this new approach is exactly the classic Ulam’s method. Numerical results are given for several one dimensional test mappings.

A Null-Space-based Technique for the Estimation of Linear-Time Invariant Structured State-Space Representations

Estimating the order as well as the matrices of a linear state-space model is now an easy problem to solve. However, it is well-known that the state-space matrices are unique modulo a non-singular similarity transformation matrix. This could have serious consequences if the system being identified is a real physical system. Indeed, if the true model contains physical parameters, then the identified system could no longer have the physical parameters in a form that can be extracted easily. The question addressed in this paper then is, how to recover the physical parameters once the system has been identified in a fully-parameterized form. The novelty of our approach is on transforming the bilinear equations arising from the similarity transformation equations as a null-space problem. We show that the null-space of a certain matrix contains the physical parameters. Extracting the physical parameters then requires the solution of a non-convex optimization problem in a reduced dimensional space. By assuming that the physical state-space form is identifiable and the initial fully-parameterized model is consistent, the solution of this optimization problem is unique. The proposed algorithm is presented, along with an example of a physical system.

Birkhoff's ergodic theorem and the piecewise-constant maximum entropy method for Frobenius–Perron operators

Let (X, Σ, σ) be a finite measure space and S: X→X be a nonsingular transformation such that the corresponding Frobenius–Perron operator P S : L 1(X)→L 1(X) has a stationary density f*. We propose a piecewise-constant maximum entropy method for the numerical recovery of f* and give its relation to the classic Birkhoff's individual ergodic theorem. An advantage of the piecewise-constant method over the current maximum entropy method based on polynomial basis functions is that a nonlinear system of equations is not needed for solving the related moment problem. Numerical results are given for several one dimensional test mappings.

Distribution of an arbitrary linear transformation of internally Studentized residuals of multivariate regression with elliptical errors

This paper derives the matrix-variate distribution of an arbitrary non-singular linear transformation of Studentized multivariate observations. The joint distributions of the major commonly utilized Studentized versions of (multivariate) regression residuals are obtained as special cases of the matrix-variate distribution introduced in the paper. As applications, the paper discusses testing for outliers, omitted variables, and general linear hypothesis in terms of Studentized linear combinations of residuals.

Marginal Distributions of Random Vectors Generated by Affine Transformations of Independent Two-Piece Normal Variables

Marginal probability density and cumulative distribution functions are presented for multidimensional variables defined by nonsingular affine transformations of vectors of independent two-piece normal variables, the most important subclass of Ferreira and Steel's general multivariate skewed distributions. The marginal functions are obtained by first expressing the joint density as a mixture of Arellano-Valle and Azzalini's unified skew-normal densities and then using the property of closure under marginalization of the latter class.

Succinct Representation of Codes with Applications to Testing

Motivated by questions in property testing, we search for linear error-correcting codes that have the “single local orbit” property, i.e., they are specified by a single local constraint and its translations under the symmetry group of the code. We show that the dual of every “sparse” binary code whose coordinates are indexed by elements of $\mathbb{F}_{2^n}$ for prime $n$ and whose symmetry group includes the group of nonsingular affine transformations of $\mathbb{F}_{2^n}$ has the single local orbit property. (A code is said to be sparse if it contains polynomially many codewords in its block length.) In particular this class includes the dual-BCH codes for whose duals (i.e., for BCH codes) simple bases were not known. Our result gives the first short ($O(n)$-bit, as opposed to the natural $\exp(n)$-bit) description of a low-weight basis for BCH codes. The interest in the single local orbit property comes from the recent result of Kaufman and Sudan (STOC 2008) that shows that the duals of codes that have the single local orbit property under the affine symmetry group are locally testable. When combined with our main result, this shows that all sparse affine-invariant codes over the coordinates $\mathbb{F}_{2^n}$ for prime $n$ are locally testable. If, in addition to $n$ being prime, $2^n-1$ does not have large divisors, then we get that every sparse cyclic-invariant code also has the single local orbit. In particular this implies that BCH codes of such length are generated by a single low-weight codeword and its cyclic shifts.

A non-product type non-singular transformation which satisfies Krieger’s Property A

In this paper we give an example of an ergodic non-singular transformation of type III, which satisfies Krieger’s Property A, but which is not orbit equivalent to any product type odometer.

A Maximum Entropy Method Based on Piecewise Linear Functions for the Recovery of a Stationary Density of Interval Mappings

Let S:[0,1]→[0,1] be a nonsingular transformation such that the corresponding Frobenius-Perron operator PS:L1(0,1)→L1(0,1) has a stationary density f∗. We propose a maximum entropy method based on piecewise linear functions for the numerical recovery of f∗. An advantage of this new approximation approach over the maximum entropy method based on polynomial basis functions is that the system of nonlinear equations can be solved efficiently because when we apply Newton’s method, the Jacobian matrices are positive-definite and tri-diagonal. The numerical experiments show that the new maximum entropy method is more accurate than the Markov finite approximation method, which also uses piecewise linear functions, provided that the involved moments are known. This is supported by the convergence rate analysis of the method.

Group-Theoretic Dimension of Stationary Symmetric α-Stable Random Fields

The growth rate of the partial maximum of a stationary stable process was first studied in the works of Samorodnitsky (2004a,b), where it was established, based on the seminal works of Rosi\'nski (1995,2000), that the growth rate is connected to the ergodic theoretic properties of the flow that generates the process. The results were generalized to the case of stable random fields indexed by Z^d in Roy and Samorodnitsky (2008), where properties of the group of nonsingular transformations generating the stable process were studied as an attempt to understand the growth rate of the partial maximum process. This work generalizes this connection between stable random fields and group theory to the continuous parameter case, that is, to the fields indexed by R^d.

Affine equivalence of cubic homogeneous rotation symmetric functions

Homogeneous rotation symmetric Boolean functions have been extensively studied in recent years because of their applications in cryptography. Little is known about the basic question of when two such functions are affine equivalent. The simplest case of quadratic rotation symmetric functions which are generated by cyclic permutations of the variables in a single monomial was only settled in 2009. This paper studies the much more complicated cubic case for such functions. A new concept of patterns is introduced, by means of which the structure of the smallest group G n , whose action on the set of all such cubic functions in n variables gives the affine equivalence classes for these functions under permutation of the variables, is determined. We conjecture that the equivalence classes are the same if all nonsingular affine transformations, not just permutations, are allowed. Our method gives much more information about the equivalence classes; for example, in this paper we give a complete description of the equivalence classes when n is a prime or a power of 3.