Showing papers on "Linear map published in 1997"
TL;DR: Ten examples of computed pseudospectra of thirteen highly nonnormal matrices arising in various applications are presented, each chosen to illustrate one or more mathematical or physical principles.
Abstract: If a matrix or linear operator A is far from normal, its eigenvalues or, more generally, its spectrum may have little to do with its behavior as measured by quantities such as ||An|| or ||exp(tA)||. More may be learned by examining the sets in the complex plane known as the pseudospectra of A, defined by level curves of the norm of the resolvent, ||(zI - A)-1||. Five years ago, the author published a paper that presented computed pseudospectra of thirteen highly nonnormal matrices arising in various applications. Since that time, analogous computations have been carried out for differential and integral operators. This paper, a companion to the earlier one, presents ten examples, each chosen to illustrate one or more mathematical or physical principles.
507 citations
Book•
01 Jul 1997
TL;DR: This book offers an elementary and self-contained introduction to many fundamental issues concerning approximate solutions of operator equations formulated in an abstract Banach space setting, including important topics such as solvability, computational schemes, convergence stability and error estimates.
Abstract: This book offers an elementary and self-contained introduction to many fundamental issues concerning approximate solutions of operator equations formulated in an abstract Banach space setting, including important topics such as solvability, computational schemes, convergence stability and error estimates. The operator equations under investigation include various linear and nonlinear types of ordinary and partial differential equations, integral equations and abstract evolution equations, which are frequently involved in applied mathematics and engineering applications. Chapter 1 gives an overview of a general projective approximation scheme for operator equations, which covers several well-known approximation methods as special cases, such as the Galerkin-type methods, collocation-like methods, and least-square-based methods. Chapter 2 discusses approximate solutions of compact linear operator equations, and chapter 3 studies both classical and generalized solutions, as well as the projective approximations, for general linear operator equations. Chapter 4 gives an introduction to some important concepts, such as the topological degree and the fixed point principle, with applications to projective approximations of nonlinear operator equations. Linear and nonlinear monotone operator equations and their projective approximators are investigated in chapter 5, while chapter 6 addresses basic questions in discrete and semi-discrete projective approximations for two important classes of abstract operator evolution equations. Each chapter contains well-selected examples and exercises, for the purposes of demonstrating the fundamental theories and methods developed in the text and familiarizing the reader with functional analysis techniques useful for numerical solutions of various operator equations.
301 citations
TL;DR: In this paper, the authors introduce a new potential-theoretic definition of the dimension spectrum of a probability measure for q > 1 and explain its relation to prior definitions, and apply this definition to prove that if and is a Borel probability measure with compact support in, then under almost every linear transformation from to, the q-dimension of the image of is preserved provided.
Abstract: We introduce a new potential-theoretic definition of the dimension spectrum of a probability measure for q > 1 and explain its relation to prior definitions. We apply this definition to prove that if and is a Borel probability measure with compact support in , then under almost every linear transformation from to , the q-dimension of the image of is ; in particular, the q-dimension of is preserved provided . We also present results on the preservation of information dimension and pointwise dimension. Finally, for and q > 2 we give examples for which is not preserved by any linear transformation into . All results for typical linear transformations are also proved for typical (in the sense of prevalence) continuously differentiable functions.
109 citations
TL;DR: In this paper, the converse and saturation results for Tikhonov regularization of linear ill-posed problems Tx=y, where T is a linear operator between two Hilbert spaces, were proved.
Abstract: In this paper we prove some new converse and saturation results for Tikhonov regularization of linear ill-posed problems Tx=y, where T is a linear operator between two Hilbert spaces.
93 citations
TL;DR: Some triangle conditions for the existence of a common quadratic Lyapunov function are presented in this paper, derived from a criterion for judging the semipositiveness of a linear map defined on symmetric matrices.
Abstract: Concerning a pair of linear systems, some triangle conditions for the existence of a common quadratic Lyapunov function are presented in this paper. These conditions are derived from a criterion for judging the semipositiveness of a linear map defined on symmetric matrices.
72 citations
TL;DR: A method is presented that reduces the problem of computing the radical of a matrix algebra over an arbitrary field to solving systems of semilinear equations and a simple test for isomorphism of semisimple modules is presented.
50 citations
TL;DR: In this paper, an exponential local bounded linear map is defined, i.e. such that for some, Let be exponentially localized around a site, then the response is also exponentially localized about o. This linear result is of fundamental importance to a wide variety of nonlinear problems.
Abstract: Let S be a countable metric space with metric d, for each let , be Banach spaces, and let X,Y be the subsets of , respectively, with finite supremum norm over their factors. Let be an invertible `exponentially local' bounded linear map, i.e. such that for some , Let be exponentially localized around a site . Then the response is also exponentially localized about o. This linear result is of fundamental importance to a wide variety of nonlinear problems, including spatial localization of discrete breathers and bipolarons. For illustration, a simple application is given to equilibria of networks of bistable units. Finally, the result is generalized to maps between product spaces with arbitrary norms based on the norms on the factors.
39 citations
TL;DR: In this paper, the exact output tracking problem for non-minimum-phase systems with parameter jumps is considered and sufficient conditions are derived for the elimination of switching-introduced output transient.
Abstract: In this paper we consider the exact output tracking problem for systems with parameter jumps. Necessary and sufficient conditions are derived for the elimination of switching-introduced output transient. Previous works have studied this problem by developing a regulator that maintains exact tracking through parameter jumps (switches). Such techniques are, however, only applicable to minimum-phase systems. In contrast, our approach is applicable to nonminimum-phase systems and obtains bounded but possibly non-causal solutions. If the reference trajectories are generated by an exo-system, then we develop an exact-tracking controller in a feedback form. As in standard regulator theory, we obtain a linear map from the states of the exo-system to the desired system state which is defined via a matrix differential equation. The constant solution of this differential equation provides asymptotic tracking, and coincides with the feedback law used in standard regulator theory. The obtained results are applied to a simple flexible manipulator with jumps in the pay-load mass.
36 citations
TL;DR: In this article, the homogeneous Diophantine approximation constant L~(03B8) = lim sup q~~(q~q3B8~)-1 was shown to have bounded partial quotient.
Abstract: Let 03B8 be a real number with continued fraction expansion 03B8 = [a0, a1, a2,...], and let be a matrix with integer entries and nonzero determinant. If 03B8 has bounded partial quotients, then [a*0, a*1, a*2, ...] also has bounded partial quotients. More precisely, if aj ~ K for all sufficiently large j, then | det (M)|(K + 2) for all sufficiently large j. We also give a weaker bound valid for all with j ~ 1. The proofs use the homogeneous Diophantine approximation constant L~(03B8) = lim sup q~~(q~q03B8~)-1. We show that det(M)|L~(03B8). Manuscrit re~u le 28 avril 1997 268
21 citations
01 Jan 1997
TL;DR: In this article, the authors considered a scalar Nevanlinna-pick interpolation problem with finitely many data and assumed that the Pick matrix P is invertible and has k negative eigenvalues.
Abstract: We consider a scalar Nevanlinna-Pick interpolation problem with finitely many data and assume that the Pick matrix P is invertible and has k negative eigenvalues. We look for solutions of this problem in the class of meromorphic functions whose Nevanlinna kernel has k negative squares. The set of these solutions can be written as a fractional linear transformation of a parameter in the class of Nevanlinna functions, much as in the case K = 0. But now not the whole Nevanlinna class can be used as a parameter set. Our results are obtained through the characterization of the selfadjoint extensions of a symmetric operator in a Pontryagin space with both defect numbers equal to 1 in terms of a so called it-resolvent matrix.
21 citations
TL;DR: In this article, a completely integrable dynamical system in discrete time is studied by methods of algebraic geometry, which is associated with factorization of a linear operator acting in the direct sum of three linear spaces into a product of three operators, each acting nontrivially only in two spaces, and subsequently reversing the order of the factors.
Abstract: A completely integrable dynamical system in discrete time is studied by methods of algebraic geometry. The system is associated with factorization of a linear operator acting in the direct sum of three linear spaces into a product of three operators, each acting nontrivially only in the direct sum of two spaces, and subsequently reversing the order of the factors. There exists a reduction of the system, which can be interpreted as a classical field theory in the 2+1-dimensional space-time, whose integrals of motion coincide, in essence, with the statistical sum of an inhomogeneous 6-vertex free-fermion model on the 2-dimensional kagome lattice (here the statistical sum is a function of two parameters). This establishes a connection with the “local,” or “generalized,” quantum Yang-Baxter equation. Bibliography:10 titles.
27 Aug 1997
TL;DR: In this article, the authors studied the asymptotic behavior of attainable sets for linear periodic control systems and showed that the curve t/spl rarr/D(t), t>0 of the shape of an attainable set is not a point, but a limit cycle.
Abstract: We study in a more general setup a phenomenon concerning the asymptotic behavior of attainable sets for linear autonomous control systems Oseevich's main result (1991) consists in discovering a simple behavior in a long run of shapes of attainable sets, unlike that of attainable sets itself Here, shape stands for the entity of all images of a set under nonsingular linear transformations More precisely, the shapes of attainable sets for linear autonomous control systems always possess a limit as t/spl rarr//spl infin/ in a natural metric of the infinite-dimensional space of forms At present the range of this phenomenon is not clear-cut In a search for its limits we consider here the asymptotic behavior of attainable sets for linear periodic control systems Our main result establishes both a similarity and a distinction between the case under consideration and the autonomous case We show that the curve t/spl rarr/D(t), t>0 of forms of attainable sets approaches, in general, not a point, but a limit cycle
01 Jan 1997
TL;DR: In this paper, numerical methods for solving problems involving total variation (TV) regularization for semidefinite quadratic minimization problems arising from illposed inverse problems have been considered.
Abstract: We consider numerical methods for solving problems involving total variation (TV) regularization for semidefinite quadratic minimization problems minu ‖Ku−z‖2 arising from illposed inverse problems. HereK is a compact linear operator, and z is data containing inexact or partial information about the “true” u. TV regularization entails adding to the objective function a penalty term which is a scalar multiple of the total variation of u; this term formally appears as (a scalar times) the L1 norm of the gradient of u. The advantage of this regularization is that it improves the conditioning of the optimization problem while not penalizing discontinuities in the reconstructed image. This approach has enjoyed significant success in image denoising and deblurring, laser interferometry, electrical tomography, and estimation of permeabilities in porus media flow models. The Euler equation for the regularized objective functional is a quasilinear elliptic equation of the form [ K∗K+ A(u) ] u = −K∗z. Here, A(u) is a standard self-adjoint second order elliptic operator in which the coefficient κ depends on u, by [κ(u)](x) = 1/|∇u(x)|. Following the literature, we approach the Euler equation by means of fixed point iterations, resulting in a sequence of linear subproblems. In this paper we present results from numerical experiments in which we use the preconditioned conjugate gradient method on the linear subproblems, with various multilevel iterative methods used as preconditioners.
TL;DR: In this article, the decay rate of singular values of A tending to zero is defined as an interval of ill-posedness and motivated its meaning by considering lower and upper bounds for the rates of the condition numbers occurring in the numerical solution process of the discretized problem.
Abstract: The degree of ill-posedness of a linear inverse problem is an important knowledge base to select appropriate regularization methods for the stable approximate solution of such a problem. In this paper, we consider ill-posedness measures for a linear ill-posed operator equation Ax = y, where the compact linear operator A X -* Y maps between infinite dimensional Hilbert spaces. Using the decay rate of singular values of A tending to zero we define an interval of ill-posedness and motivate its meaning by considering lower and upper bounds for the rates of the condition numbers occurring in the numerical solution process of the discretized problem. An equivalent interval information is obtained when compactness measures as E-entropy or C-capacity are exploited alternatively. For the specific case X L2(0, 1), the space change problem of shifting the space X along a Sobolev scale is treated. In detail, we study the change of the interval of ill-posedness if the solutions are restricted to the Sobolev space W2 (0, 11 . The results of these considerations are a warning to characterize the ill-posedness of a problem superficial. Moreover, the interdependences between ill-posedess measures, embedding operators, Hilbert and Sobolev scales are discussed.
Book•
25 Sep 1997
TL;DR: This book discusses linear vector spaces, linear Transformations, Finite Dimensional Euclidean Spaces and Cartesian Tensors, and four-Tensors.
Abstract: Chapter 1: Linear Vector Spaces Chapter 2: Linear Transformations Chapter 3: Finite Dimensional Euclidean Spaces and Cartesian Tensors Chapter 4: Four-Tensors Chapter 5: Applications Appendix 1: Background Appendix 2: Solutions for Selected Problems
TL;DR: Modeling of linear features, particularly straight lines and circles, in two-dimensional and three-dimensional spaces, is developed, with specific consideration of independent descriptors, followed by the derivation of the four-parameter 2D transformations in terms of the line and circle parameters.
Abstract: Modeling of linear features, particularly straight lines and circles, in two-dimensional (2D) and three-dimensional (3D) spaces, is developed, with specific consideration of independent descriptors. This is followed by the derivation of the four-, six-, and eight-parameter 2D transformations in terms of the line and circle parameters, as well as 3D linear transformation. Minimum configuration, redundant cases, and restrictions regarding special cases are carefully noted and are compared to the cases of using points only. Many useful geometric constraints between linear features, which provide significant information, are identified and corresponding equations are developed. Constraints involving relative geometric information relating features to each other are considered in addition to those constraints relating features to the reference coordinate system. Finally, the standard point-based photogrammetric collinearity equations are replaced by those based on correspondence between image and object linear features. Results are presented describing experiments where the coordinates of points are perturbed using Gaussian random noise to emulate real data. Several thoughts are listed regarding the future applications of this research in practice.
TL;DR: In this article, it was shown that if k is even and F is algebraically closed of characteristic ≠ 2, then any such linear operator T on Sn (F) must be a congruence map, possibly followed by negation.
Abstract: LetSn (F) denote the set of all n × n symmetric matrices over the field F. Let k be a positive integer such that k ≤ n. Alinear operator T on Sn (F) is said to be a rank-k preserver provided that it maps the set of all rank k matrices into itself. We show here that if k is even and F is algebraically closed of characteristic ≠ 2, then any such T must be a congruence map. The corresponding result for k odd has already been established. Now suppose that r is a positive integer such that 2r≤ n. We also consider a linear operator T on the space of n × n hermitian matrices, which maps the set of matrices whose inertia is (r,r,n – 2r) into itself. In the real symmetric case we show that if n ≥ 4r any such T must be a congruence map, possibly followed by negation. This proves the Johnson-Pierce conjecture for this inertia class, improving known results. An analogous result is obtained in the hermitian case.
TL;DR: In this article, the problem of separation of variables in a dyon-dyon system is discussed, and a linear transformation is obtained between fundamental bases of this system and a comparison with a 4D isotropic oscillator is carried out.
Abstract: The problem of separation of variables in a dyon-dyon system is discussed. A linear transformation is obtained between fundamental bases of this system. Comparison of the dyon-dyon system with a 4D isotropic oscillator is carried out.
Book•
15 Jul 1997
TL;DR: In this article, the authors propose a method for computing the Eigenvalues of Operators Acting on Euclidean Spaces (OOS) and Eigenvectors of operators acting on OOS.
Abstract: PRE-CALCULUS PREREQUISITES. CALCULUS PREREQUISITES. Vector Algebra. Applications of Vector Algebra to Analytic Geometry. Linear Spaces. Linear Transformations and Matrices. Determinants. Eigenvalues and Eigenvectors. Eigenvalues of Operators Acting on Euclidean Spaces. Applications to Linear Differential Equations. Applications to Systems of Differential Equations. The Method of Successive Approximations. Answers to Exercises. Index.
TL;DR: In this article, it was shown that semi-hyperbolic mappings in Banach spaces are locally ψ -contracting, where ψ is the Hausdorff measure of noncompactness.
TL;DR: In this paper, the Poincar\'e superalgebra in ten and eleven dimensions was shown to admit a linear realization, and the generators of the nonlinear and linear group transformations were shown to fall into equivalent representations of the super algebra.
Abstract: The new extensions of the Poincar\'e superalgebra recently found in ten and eleven dimensions are shown to admit a linear realization. The generators of the nonlinear and linear group transformations are shown to fall into equivalent representations of the superalgebra. The parametrization of the coset space $G/H$, with $G$ a given extended supergroup and $H$ the Lorentz subgroup, that corresponds to the linear transformations is presented.
TL;DR: In this paper, it was shown that the topological dimension of the set x e X: Φ (x) = Ψ (x), where X is a continuous, linear, surjective operator with relatively compact range is not less than the dimension of Φ−1 (0).
Abstract: The aim of this Note is to prove the following result: Let X, Y be two Banach spaces, Φ: X → Y a continuous, linear, surjective operator, and Ψ: X → Y be a continuous operator with relatively compact range. Then, the topological dimension of the set x e X: Φ (x) = Ψ (x) is not less than the dimension of Φ−1 (0).
Patent•
08 Aug 1997TL;DR: A linear transformation of the image signal on a segment with an arbitrary geometrical shape includes the following steps: the sub-division of the segment into sub-segments of regular shapes (rectangular, square or linear shapes), the application of an orthogonal linear transformation to each subsegment, the combining of the coefficients coming from the first transformation into classes of coefficients according to a predetermined criterion, and the iteration of the transformation on the class of coefficients, and which can be applied to the processing of images for encoding.
Abstract: A linear transformation of the image signal on a segment with an arbitrary geometrical shape includes the following steps: the sub-division of the segment into sub-segments of regular shapes (rectangular, square or linear shapes), the application of an orthogonal linear transformation to each sub-segment, the combining of the coefficients coming from the first transformation into classes of coefficients according to a predetermined criterion, and the iteration of the transformation on the classes of coefficients, and which can be applied to the processing of images for encoding.
TL;DR: In this paper, the creation of dark solitons from an arbitrary initial pulse in the system, described by the nonlinear Schrodinger equation, is considered by applying the variational method to the corresponding linear spectral problem, and the procedure for calculation of approximate values of the lowest and higher-order discrete eigenvalues from spectral data of known (trial) potential is proposed.
Abstract: The creation of dark solitons from an arbitrary initial pulse in the system, described by the nonlinear Schr\"odinger equation, is considered by applying the variational method to the corresponding linear spectral problem. The initial pulse is a potential in the linear operator of the Zakharov-Shabat eigenvalue problem and the discrete spectrum of the problem determines the number and parameters of emerged solitons. The procedure for calculation of approximate values of the lowest- and higher-order discrete eigenvalues from spectral data of known (trial) potential is proposed. The application of this procedure to some examples shows qualitative agreement between variational and exact results.
TL;DR: In this paper, a linear algebra course to engineering freshmen was presented, and one of the highlights of the course was eigenvector theory, and in particular the diagonalization of linear operators on finite-dimensional vector spaces (i.e., of square real or complex matrices).
Abstract: INTRODUCTION. As it often happens, everything began with a mistake. I was teaching for the third year in a row a linear algebra course to engineering freshmen. One of the highlights of the course was eigenvector theory, and in particular the diagonalization of linear operators on finite-dimensional vector spaces (i.e., of square real or complex matrices). Toward the end of the course I assigned a standard homework: prove that the matrix
08 Oct 1997
TL;DR: A neural network for the analysis of local independent components of an input signal is introduced, a modification of Kohonen's adaptive-subspace self-organizing map which consists of weight matrices adapted to represent linear transformations which locally minimize statistical dependence among pattern vector components.
Abstract: We introduce a neural network for the analysis of local independent components of an input signal. The network is a modification of Kohonen's adaptive-subspace self-organizing map. The map units consist of weight matrices adapted to represent linear transformations which locally minimize statistical dependence among pattern vector components. Training of the map is carried out in episodes comprising pattern vectors sampled from adjacent time instants or spatial locations. The use of episodes produces independent directions which are preserved in translations of the input signal. The independent components modeled by each map unit are estimated with a nonlinear Hebbian-like learning rule, which searches for weight vectors maximizing a measure of non-Gaussianity of the scalar product of weight and pattern vectors. For demonstration, the method was applied to the segmentation of a composition image of four periodic texture fields. The spatial convolution masks, created by the map for the extraction of independent components, represent distinct frequences of particular directions.
TL;DR: In this article, a linear transformation of the Poincare superalgebra in 10 and 11 dimensions was shown to admit a linear realization, and the generators of the nonlinear and linear group transformations were shown to fall into equivalent representations of the super algebra.
Abstract: The new extensions of the Poincare superalgebra found recently in 10 and 11 dimensions are shown to admit a linear realization. The generators of the nonlinear and linear group transformations are shown to fall into equivalent representations of the superalgebra. The parametrization of the coset space G/H, with G a given extended supergroup and H the Lorentz subgroup, that corresponds to the linear transformations is presented.
TL;DR: In this paper, the authors examined the behavior of a 3-branched linear map which resulted from computations of the periodically perturbed Oregonator and showed that the linear map describes quite satisfactorily the period composition predicted by the chemical model and by the more elaborated nonlinear hyperbolic maps that were described in a previous publication.
Abstract: In the present paper we examine the behavior of a 3-branched linear map which resulted from computations of the periodically perturbed Oregonator. It turns out that the linear map describes quite satisfactorily the period composition predicted by the chemical model and by the more elaborated nonlinear hyperbolic maps that were described in a previous publication. In particular, we have shown that in any interval of the control parameter, two periodic orbits, differing by one point, coexist with one being stable and the other unstable. This phenomenon is, to the best of our knowledge, described for the first time in the literature.
TL;DR: This paper shows that the geometrical method not only is effective in the analysis of the associative memory models, but also gives a good insight into ways of improving associatives memory models and facilitates the understanding of the concept formation.
Abstract: This paper proposes a geometrical method of analyzing associative memory models. The geometrical method has the following two features: first, the state transitions of the associative memory model are subdivided into a linear transformation part based on the connection weight matrix and a nonlinear transformation part based on the sign function ; second, a flow defined on the spherical surface is introduced. The flow consists of the dynamics on the spherical surface generated by the connection weight matrix and represents the dynamics of the model. The properties of the flow change suddenly when the memory exceeds a certain limit. This paper analyzes the relation of this change of flow and of the stability of the memorized vector itself. This paper shows that the geometrical method not only is effective in the analysis of the associative memory models, but also gives a good insight into ways of improving associative memory models and facilitates the understanding of the concept formation.