线性系统的纯点谱和弱Lyapunov相似

本文引入了线性系统的弱Ｌｙａｐｕｎｏｖ相似的概念，并且得到：具有纯点谱的有界线性系统弱Ｌｙａｐｕｎｏｖ相似于常系数对角线线性系统．     

Some remarks on the concept of proportionality

Proportionality plays a role in many principles of fairness or justice. In particular, it is often invoked in electoral systems aiming at a similarity of opinion distributions in the parliament and in the electorate at large. The proportional systems of representation (PR systems, for short) strive for maximal similarity between these two distributions. Unfortunately, the concept of proportionality is imprecise in two senses: it is vague and ambiguous. In other words, once a clear criterion of what we mean by proportionality is given, different PR systems may differ in the degree of proportionality achieved in any given election. This is what will be referred to as the vagueness of proportionality (of election outcomes). Proportionality is, however, also ambiguous in that it refers to different things depending on how the voters are expected to signal their opinions and on what is it that one wishes to distribute proportionally.     

Canonical forms and invariants of some classes of control systems

We study the canonical forms and invariants of linear and bilinear control systems and the properties of orbits of these classes of systems under similarity transformations, that is, the action of the group G = GL _n (a change of coordinates in the state space) on the spaces of systems.     

CHARACTERISTIC MATRICES AND SIMILARITY OF TIME-VARYING LINEAR DIFFERENTIAL SYSTEMS

1IntroductionLineardial'~rentialsystemsa.rethesimplestmod(}lsinmotiolls,tijoyarcwidelyusedincontroltheory,Inechanlcsandengineel'illg.Invest;igationsoiltileasymptoticpropertiesoftinea,rSystclllshay')oss(3ntialsense.Fbi'tilelin(3arsy'stemswithconstantcoeffi…     

Hankel Operators on Weighted Fock Spaces

We describe a general method to construct completely bounded idempotent mappings on operator spaces, starting from amenable semigroups of completely bounded mappings. We then explore several applications of that method to injective operator spaces, fixed points of completely contractive mappings, Toeplitz operators, dynamical systems and similarity orbits of group representations.     

On the transitivity of a parametric family of cardinality-based similarity measures

We introduce a parametric family of cardinality-based similarity measures for ordinary sets (on a finite universe) harbouring numerous well-known similarity measures. We characterize the ?ukasiewicz-transitive and product-transitive members of this family. Their importance derives from their one-to-one correspondence with pseudo-metrics. Fuzzification schemes based on a commutative quasi-copula are then used to transform these similarity measures for ordinary sets into similarity measures for fuzzy sets, rendering them applicable on graded feature set representations of objects. The main result of this paper is that transitivity, and hence also the corresponding dual metric interpretation, is preserved along this fuzzification process.     

Simulation for lattice-valued doubly labeled transition systems

During the last decades, a large amount of multi-valued transition systems, whose transitions or states are labeled with specific weights, have been proposed to analyze quantitative behaviors of reactive systems. To set up a unified framework to model and analyze systems with quantitative information, in this paper, we present an extension of doubly labeled transition systems in the framework of residuated lattices, which we will refer to as lattice-valued doubly labeled transition systems (LDLTSs). Our model can be specialized to fuzzy automata over complete residuated lattices, fuzzy transition systems, and multi-valued Kripke structures. In contrast to the traditional yes/no approach to similarity, we then introduce lattice-valued similarity between LDLTSs to measure the degree of closeness of two systems, which is a value from a residuated lattice. Further, we explore the properties of robustness and compositionality of the lattice-valued similarity. Finally, we extend the Hennessy–Milner logic to the residuate lattice-valued setting and show that the obtained logic is adequate and expressive with lattice-valued similarity.     

The discrete Korteweg-de Vries equation

We review the different aspects of integrable discretizations in space and time of the Korteweg-de Vries equation, including Miura transformations to related integrable difference equations, connections to integrable mappings, similarity reductions and discrete versions of Painlevé equations as well as connections to Volterra systems.     

Analytic equivalence and similarity of operators

The analytic equivalence of two operators is a generalization of similarity. We prove that under some conditions the analytic equivalence between two Hilbert space operatorsT andR implies the similarity of their restrictions on generalized ranges. We also prove that, in certain cases, the similarity ofT to a contraction implies that ofR. An improvement of a well-known criterion of similarity to an isometry due to Sz.-Nagy is given and an extension of a result of Apostol is obtained.     

The application of Hermite interpolation to the analysis of non-linear diffusive initial-boundary value problems

In this paper we examine the feasibility of using two-pointHermite interpolation as a systematic tool in the analysis ofinitial-boundary value problems for non-linear diffusion equations.We do this by considering a series of examples for the porousmedium equation involving both fixed and moving boundaries.Essentially, the idea is to construct polynomials which fitthe known and unknown function values and their derivativesat the two end points of a given interval. Systems of ordinarydifferential equations are then obtained for the unknown endpoint functions of time which need to be determined in orderto specify the polynomial representation—the initial conditionsfor such systems are related to the initial data for the originalproblem. As well as constructing approximate solutions, it emergesthat the method is particularly useful in identifying steadystates and similarity solutions together with their stabilityand other asymptotic properties. We believe that the techniqueprovides scientists and applied mathematicians with a valuablestrategy in the analysis of the qualitative and quantitativefeatures of solutions to initial-boundary value problems involvingnon-linear diffusion and related equations.     

Construction of unital matrix polynomials with mutually distinct characteristic roots

We investigate the construction of unital matrix polynomials with mutually distinct characteristic roots, namely, their similarity and reducibility by the similarity transformation to block-triangular, block-diagonal, and, in particular, to triangular and diagonal forms. We also study the problem of extracting linear factors.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 1, pp. 69–77, January, 1993.     

V-variable fractals: Fractals with partial self similarity

We establish properties of a new type of fractal which has partial self similarity at all scales. For any collection of iterated functions systems with an associated probability distribution and any positive integer V there is a corresponding class of V-variable fractal sets or measures. These V-variable fractals can also be obtained from the points on the attractor of a single deterministic iterated function system. Existence, uniqueness and approximation results are established under average contractive assumptions. We also obtain extensions of some basic results concerning iterated function systems.     

Eigenvalues of an elliptic system

We describe the spectrum of a non-self-adjoint elliptic system on a finite interval. Under certain conditions we find that the eigenvalues form a discrete set and converge asymptotically at infinity to one of several straight lines. The eigenfunctions need not generate a basis of the relevant Hilbert space, and the larger eigenvalues are extremely sensitive to small perturbations of the operator. We show that the leading term in the spectral asymptotics is closely related to a certain convex polygon, and that the spectrum does not determine the operator up to similarity. Two elliptic systems which only differ in their boundary conditions may have entirely different spectral asymptotics. While our study makes no claim to generality, the results obtained will have to be incorporated into any future general theory. Received: 15 August 2001 / in final form: 11 February 2002 / Published online: 24 February 2003     

Bäcklund transformations for new fourth Painlevé hierarchies

We consider a system of equations defined using the Hamiltonian operator of the Boussinesq hierarchy, as well as two successive modifications thereof. We are able to reduce the order of these three systems and give Bäcklund transformations between the integrated equations. We also give auto-Bäcklund transformations for the two modified systems.Particular cases of two of the three equations considered correspond to generalized fourth Painlevé hierarchies and are new; these are particular cases of the two modified systems. Thus we obtain auto-Bäcklund transformations for these new fourth Painlevé hierarchies, as well as Bäcklund transformations between our hierarchies. Our results on reduction of order are also applicable in this special case, and include as a particular example a reduction of order for the scaling similarity reduction of the Boussinesq equation, a result which, remarkably, seems not to have been given previously.     

On a homeomorphism between orbit spaces of linear systems and matrix polynomials

The orbit space of controllable systems under system similarity and the orbit space of matrix polynomials with determinant degree equal to the order of the state matrix under right equivalence are proved to be homeomorphic when the quotient compact–open topology is considered in the latter. As a consequence, the variation of the finite and left Wiener–Hopf structures under small perturbations of matrix polynomials with fixed degree for their determinants is described.     

Vibration Analysis of Gyroscopic Systems

Mechanical structures which can vibrate about a state of mean rotation or translation may constitute gyroscopic dynamic systems that are related by the mathematical similarity of their governing equations of motion. Considering as examples a rotating circular ring and an axially travelling string, the analogy of those systems is demonstrated. The linear vibrations obtained by superposing small perturbations on the stationary solution are analyzed. Assuming time-varying speeds, the effect of non-constant stress resultants on the dynamic stability of gyroscopic systems is investigated and the results are compared to those found in literature where the stress resultants of axially moving continua are commonly taken as constant. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Similarity of operator blocks and canonical forms. I. General results,feedback equivalence and kronecker indices

In the present paper the problem of classifying blocks of matrices up to similarity is considered. The notion of block similarity used here is a natural generalization of similarity for matrices. The invariants are described and canonical forms are given. This theory of block-similarity provides a general framework, which includes the state feedback theory for systems, the theory of Kronecker equivalence and a similarity theory for non-everywhere defined operators. New applications, in particular to factorization problems, are also obtained.     

Enhanced FEM-based modeling of brain shift deformation in Image-Guided Neurosurgery

We consider the problem of improving outcomes for neurosurgery patients by enhancing intraoperative navigation and guidance. Current navigation systems do not accurately account for intraoperative brain deformation. We focus on the brain shift deformation that occurs just after the opening of the skull and dura. The heart of our system is a nonrigid registration technique using a biomechanical model. We specifically work on two axes: the representation of the structures in the biomechanical model and the evaluation of the surface landmark displacement fields between intraoperative MR images. Using the modified Hausdorff distance as an image similarity measure, we demonstrate that our approach significantly improves the alignment of the intraoperative images.     

Tractable approximate knowledge fusion using the Horn fragment of serial propositional dynamic logic

In this paper we investigate a technique for fusing approximate knowledge obtained from distributed, heterogeneous information sources. This issue is substantial, e.g., in modeling multiagent systems, where a group of loosely coupled heterogeneous agents cooperate in achieving a common goal. Information exchange, leading ultimately to knowledge fusion, is a natural and vital ingredient of this process. We use a generalization of rough sets and relations [30], which depends on allowing arbitrary similarity relations.The starting point of this research is [6], where a framework for knowledge fusion in multiagent systems is introduced. Agents’ individual perceptual capabilities are represented by similarity relations, further aggregated to express joint capabilities of teams. This aggregation, expressing a shift from individual to social level of agents’ activity, has been formalized by means of dynamic logic. The approach of Doherty et al. (2007) [6] uses the full propositional dynamic logic, which does not guarantee tractability of reasoning. Our idea is to adapt the techniques of Nguyen [26], [27] and [28] to provide an engine for tractable approximate database querying restricted to a Horn fragment of serial dynamic logic. We also show that the obtained formalism is quite powerful in applications.     

Eigenvalues and Quasi-Eigenvalues of Banded Toeplitz Matrices: Some Properties and Applications

In this paper we consider the spectrum and quasi-eigenvalues of a family of banded Toeplitz matrices and define their extensions to the generalized eigenvalue problem. A diagonal similarity transformation on such matrices that allows a suitable modification of the region containing the quasi-eigenvalues is reported. Two kind of applications have been analyzed: the computation of the eigenvalues and the asymptotic spectra of Toeplitz matrices and the solution of block banded quasi-Toeplitz linear systems that arise after the discretization of an ODE using a boundary value method.