Nonsingular Transformation Research Articles

An Elementary Derivation of Kronecker Canonical Form for Linear Time-Invariant Systems

This paper demonstrates an elementary approach to derive the canonical form of a state space representation of a finite-dimensional, linear, time-invariant system under the group of transformation consisting of a state feedback, an output injection, and nonsingular coordinate transformations of state, input and output spaces. The prerequisites are under-graduate level of matrix algebra and a few facts from the linear systems theory.

Automating non-unimodular loop transformations for massive parallelism

Loop transformations have been shown to be very useful in parallelising compilation and regular array design. This paper provides a solution to the open problem of automatic rewriting loop nests for non-unimodular transformations. We present an algorithm that rewrites a loop nest under any non-singular (unimodular or non-unimodular) transformation in a mechanical manner. The algorithm works nicely with unimodular transformations being treated as a special case. The extra time complexity incurred due to non-unimodularity is polynomially bounded by the depth of the loop nest.

Graphs of nonsingular threshold transformations

After the graph structures of self-dual nonsingular (i.e. one-to-one) transformations of {0,1} n are described, a construction method of generating minimal nonsingular threshold transformations from lower-dimensional ones is presented. Theorems which concern nonsingular threshold transformations and support that procedure are also proved. Then, in addition to circular, nonsingular transformations in the author's previous paper, five classes of noncircular, nonsingular threshold transformations are given with their graph structures.

An Exact Confidence Region in Multivariate Calibration

In the multivariate calibration problem using a multivariate linear model, an exact confidence region is constructed. It is shown that the region is always nonempty and is invariant under nonsingular transformations.

An Algorithm for the Generalized Singular Value Decomposition on Massively Parallel Computers

A new algorithm for computing the generalized singular value decomposition that diagonalizes two matrices is introduced. First, two existing algorithms are studied, one due to Paige and the other to Han and Veselic. The former requires only orthogonal plane transformations, resulting in two matrices with parallel rows. The latter employs nonsingular (not necessarily orthogonal) transformations to directly diagonalize the two data matrices. For many applications, it is preferable to be given both the diagonal matrices and the transformation matrices explicitly. Our algorithm has the advantages that the nonsingular transformation is readily available, computation of transformations is simple, and the convergence test is efficient in a parallel computing environment. We present implementation results, obtained on a Connection Machine 2, to compare our new algorithm with that of Paige.

High order approximation of the Frobenius-Perron operator

We formulate a general convergence theory for the finite dimensional projection approximation of the fixed point of a class of linear operators. In particular, we explore the relation with the standard framework of stability and consistency of the approximation imply convergence. Applications include the Frobenius-Perron operator P S , corresponding to a nonsingular measurable transformation from [0, 1] to itself. Numerical experiments are also given to support the theoretical analysis.

Dynamic feedback linearization of the induction motor

Dynamic state feedback is used to achieve feedback linearization of the induction motor. It is shown that the sixth-order nonlinear dynamical model of the induction motor consisting of rotor speed, two stator currents, two rotor fluxes and an integrator in the feedback controller can be made equivalent to two third-order decoupled linear systems by nonlinear state feedback through the stator input voltages. It is further shown that a nonsingular dynamic feedback linearizing transformation exists as long as the (electromagnetic) torque put out by the motor is nonzero.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Maximal subgroups of infinite dimensional general linear groups

AbstractLet k be an infinite cardinal, F a field, and let GL(k, F) be the group of all non-singular linear transformations on a ki-dimensional vector space V over F. Various examples are given of maximal subgroups of GL(k, F). These include (i) stabilizers of families of subspaces of V which are like filters or ideals on a set, (ii) almost stabilizers of certain subspaces of V, (iii) almost stabilizers of a direct decomposition of V into two k-dimensional subspaces.It is also noted that GL(k, F) is not the union of any chain of length k of proper subgroups.

An exact solution for the one-dimensional elastic wave equation in layered media

A new approach to the direct scattering problem in one-dimensional inhomogeneous bodies is presented. A general formulation considering density, wave speed, and cross section variation is introduced and a compact form for the wave equation in the characteristic plane, with a generalized acoustical impedance profile as a parameter, is established with the aid of a nonsingular variable transformation. The wave propagation pattern in layered (or discretized continuous) media is then examined and an exact algebraic solution for the reflected wave is obtained. The resulting formula is adequate to an arbitrary input pulse, furnishing a quick computation of all multiple reflection contributions to the final echo.

Circular nonsingular threshold transformations

Circular n-dimensional Boolean transformations are those which commute with an n-cyclic permutation of variables. A minimal Boolean transformation is the one which has the minimum number of coordinates changed by it among its equivalent transformations with respect to permutations and complementations of their variables. After general results on these concepts and nonsingular threshold transformations, seven n-dimensional (and three special 6-dimensional) nonsingular threshold transformations which are circular and minimal are listed with a proof of nonsingularity. Finally, it is proved that they are all equivalent to threshold transformations with graphs composed of 2-cycles and fixed points.

Error estimates of the Markov finite approximation of the Frobenius-Perron operator

IN THE practical application of the modern ergodic theory, one of the main problems is related to the computation of the absolutely continuous invariant measures for nonsingular transformations on measure spaces (see Lasota and Mackey [l]). A simple example is constructed in Li [2] to show that the round-off error can completely dominate the calculation if one is to compute the invariant measure directly using a sequence of Cesaro sums. An alternative is to compute the invariant density function for the Frobenius-Perron operator P,: L’(0, 1) + L’(0, 1) associated with the nonsingular measurable transformations S: [0, 11 + [0, 11. The Frobenius-Perron operator P, is defined by

Singularly perturbed linear boundary value problems

In this paper an alternative approach to the method of asymptotic expansions for the study of a singularly perturbed linear system with multiparameters and multiple time scales is developed. The method consists of developing a non-singular linear transformation that transforms an arbitrary n-time scale system into diagonal form. This fast and slow mode decomposition provides a modern technique to find an approximate solution of the original system in terms of the solution of an auxiliary system corresponding to the decoupled system. Furthermore, the decoupled system provides a useful mechanism to relate the asymptotic behavior of the solution of the original system and the solution of the degenerate system relative to the original system.

A new measure of overall potential influence in linear regression

In this article we introduce a new influence measure and a graphical display for the detection of influential observations in linear regression. We demonstrate the need for such a new influence measure by giving examples where existing influence measures, numerous as they may be, may fail to detect influential observations. The existing influence measures are procedures each of which is designed to detect the influence of observations on a specific regression result. The proposed diagnostic is a measure of overall potential influence. Fitting a regression model in practice can have many purposes and the user may prefer to begin with an overall measure. This diagnostic possesses several desirable properties that many of the frequently used diagnostics do not generally possess: (a) invariance to location and scale in the response variable. (b) invariance to nonsingular transformations of the explanatory variables, (c) it is an additive function of measures of leverage and of residual error, and (d) it is monotonically increasing in the leverage values and in the squared residuals. The new influence measure is also supplemented by a graphical display which shows the source of influence. This graph is suitable for both single- and multiple-case influence. Analytical and empirical comparisons between the proposed measure and the existing ones indicate that our measure is superior to existing influence measures.

Linear sections of the general linear group: A geometric approach

In this paper we investigate the subset ϑ( n, q) of the general linear group Gl( n+1) consisting of all elements which have no nontrivial fixed points. In particular we show that there is a bijection between this set and the set of n-dimensional subspaces of the projective space PG(2n+1,q) which satisfy the intersection property of being skew to three given spaces which are themselves pairwise disjoint. We obtain this bijection by associating to each A∈ ϑ( n, q) first the row space of the row echelon matrix [ I A] and then the projectivization of this space obtained by identifying scalar multiples and discarding the vector 0. Classical results for PG(3, q) then provide the basis for a recurrence relation which expresses ϑ( n, q) in terms of ϑ( n−1, q) and ϑ( n−2, q). Because the development of this recurrence is constructive it provides, as an application, an efficient method for not only enumerating but for exhibiting all the nonsingular linear transformations A∈ Gl( n+1) such that A + id is nonsingular as well.

Absorbing cantor sets and trapping structures

AbstractIt it shown that a minimal attractor for a continuous, lebesgue non-singular transformation on an interval with no wandering intervals is either a periodic orbit, a finite collection of intervals, a simply attracting cantor set, or an absorbing cantor set.

The Converse of the Dominated Ergodic Theorem in Hurewicz Setting

AbstractThe converse of the dominated ergodic theorem in infinite measure spaces is extended to non-singular transformations, i.e. transformations that only preserve the measure of null sets. An inverse weak maximal inequality is given and then applied to obtain related results in Orlicz spaces.

Projection solutions of Frobenius‐Perron operator equations

We construct in this paper the first order and second order piecewise polynomial finite approximation schemes for the computation of invariant measures of a class of nonsingular measurable transformations on the unit interval of the real axis. These schemes are based on the Galerkin's projection method forL1-spaces and are proved to be convergent for the class of Frobenius-Perron operators.

The classification of non-singular actions, revisited

AbstractA simpler treatment is given of the theorems of Dye and Krieger concerning the classification of non-singular ergodic transformations up to orbit equivalence.

Linear cyclic codes of wordlength v over GF(q s ) which are also linear cyclic codes of wordlength sv over GF(q)

Any nonsingular linear transformation φ: GF(qs) → GF(qs) can be used to treat a linear cyclic code ρ of wordlength v over GF(qs) as a linear code φ(ρ) of Wordlength sv over GF(q). This paper determines those linear cyclic codes ρ and transformations φ for which the resulting linear code φ(ρ) is also cyclic.

A synthesis of multidimensional judgements consistent with linear transformations

We give a characterization of the weighted arithmetic mean onRn by solving a functional equation motivated by a problem of synthesizing multidimensional judgements consistent with nonsingular linear transformations.