Canonical systems and transfer matrix-functions

The generalised Backlund-Darboux transform and its modifications are applied to the inverse spectral problems and problems on similarity of Hamiltonians for canonical systems.

     

Canonical forms for linear systems

This paper considers canonical forms for the similarity action of Gl(n) on $\text{∑}{}_{\mathrm{n,m}}\text{={(A,B)∈}\text{C}{}^{\mathrm{n}}\text{·}{}^{\mathrm{n}}\text{×}\text{C}{}^{\mathrm{n}}\text{·}{}^{\mathrm{m}}\text{}}$:

$\text{Gl(n×∑}{}_{\mathrm{n,m}}\text{→∑}{}_{\mathrm{n,m}}$

,

$\text{(H,(A,B))?(HAH}{}^{-1}\text{,HB)}$

Those canonical forms are obtained as an application of a more general method to select canonical elements M_c in the orbits $\text{O}{}_{\mathrm{M}}$ of a matrix group G acting on a set of matrices $\text{M}\text{?}\text{C}{}^{\mathrm{l}}\text{·}{}^{\mathrm{p}}$. We define a total order (?) on $\text{C}{}^{\mathrm{l}}\text{·}{}^{\mathrm{p}}$, different from the lexicographic order ^l? [0^l?x ? x <0, but $\text{0?x≠0 for x∈}\text{R}\text{]}$ and consider normalized $\text{O}{}_{\mathrm{M}}$-elements with a minimal number of parameters:

$\text{min{}\text{M}\text{?}\text{∈}\text{O}{}_{\mathrm{M}}\text{:}\text{M}\text{?}\text{normalized}}$

It is shown that the row and column echelon forms, the Jordan canonical form, and “nice” control canonical forms for reachable (A,B)-pairs have a homogeneous interpretation as such (?)-minimal orbit elements. Moreover new canonical forms for the general action (?) are determined via this method.     

Canonical decompositions of symmetric submodular systems

Let E be a finite set, R be the set of real numbers and f: 2^E→R be a symmetric submodular function. The pair (E,f) is called a symmetric submodular system. We examine the structures of symmetric submodular systems and provide a decomposition theory of symmetric submodular systems. The theory is a generalization of the decomposition theory of 2-connected graphs developed by Tutte and can be applied to any (symmetric) submodular systems.     

Canonical forms for positive discrete-time linear control systems

In this paper, the properties of reachability, controllability and essential reachability of positive discrete-time linear control systems are studied. These properties are characterized in terms of the directed graph of the state matrix. From these characterizations canonical forms of those properties are deduced.     

Canonical transformations of the extended phase space and integrable systems

We investigate the explicit construction of a canonical transformation of the time variable and the Hamiltonian whereby a given completely integrable system is mapped into another integrable system. The change of time induces a transformation of the equations of motion and of their solutions, the integrals of motion, the methods of separation of variables, the Lax matrices, and the correspondingr-matrices. For several specific families of integrable systems (Toda chains, Holt systems, and Stäckel-type systems), we construct canonical transformations of time in the extended phase space that preserve the integrability property.     

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.     

Quantum dissipative systems. I. Canonical quantization and quantum Liouville equation

Sedov's variational principle, which is a generalization of the principle of least action to dissipative processes, is used to generalize canonical quantization and the von Neumann equations to dissipative systems. The example of a harmonic oscillator with friction is considered.Institute of Nuclear Physics, State University, Moscow. Translated from Teoreticheskaya i Matematicheskaya Fizika Vol. 100, No. 3, pp. 402–417, September, 1994.     

Fuzzy标准矩阵及Fuzzy强标准矩阵

本文在格L=[0,1]上定义了Fuzzy标准矩阵及Fuzzy强标准矩阵,得出了他们的一些基本性质。证明了Fuzzy标准矩阵是幂收敛阵;给出了Fuzzy强标准矩阵的判别条件及Fuzzy强标准矩阵是幂等的充要条件。最后对Fuzzy强标准矩阵的秩进行了讨论。     

Canonical dual least square method for solving general nonlinear systems of quadratic equations

This paper presents a canonical dual approach for solving general nonlinear algebraic systems. By using least square method, the nonlinear system of m-quadratic equations in n-dimensional space is first formulated as a nonconvex optimization problem. We then proved that, by the canonical duality theory developed by the second author, this nonconvex problem is equivalent to a concave maximization problem in ℝ ^m , which can be solved easily by well-developed convex optimization techniques. Both existence and uniqueness of global optimal solutions are discussed, and several illustrative examples are presented.     

Canonical configurations

Given a finite configuration E in Rⁿ and an arbitrary coloring of Rⁿ (possibly with an infinite number of colors) it is shown that there exists E′ homothetic to E such that E′ is colored in a “special” way. The proof is based on the recent theorem of W. Deuber, R. L. Graham, H. J. Prömel, and B. Voigt, J. Combin. Theory Ser. A.34 (1983), 331–339.     

Canonical semigroups

We define abstract canonical semigroups modeled after the canonical reductive monoids associated with the canonical compactification of a group of adjoint type. It then becomes possible for us to come up with semigroups having some of the algebraic properties of monoids of Lie type (without first starting with a group).     

Canonical B-series

Summary. B-series provide a powerful general tool to express numerical methods for differential equations. Many differential equations are of Hamiltonian form and there has been much recent interest in constructing so-called canonical or symplectic integrators for the Hamiltonian case. In this paper we provide a necessary and sufficient condition for a B-series to correspond to a canonical method. Received March 15, 1993     

Canonical Forest

We propose a new classification ensemble method named Canonical Forest. The new method uses canonical linear discriminant analysis (CLDA) and bootstrapping to obtain accurate and diverse classifiers that constitute an ensemble. We note CLDA serves as a linear transformation tool rather than a dimension reduction tool. Since CLDA will find the transformed space that separates the classes farther in distribution, classifiers built on this space will be more accurate than those on the original space. To further facilitate the diversity of the classifiers in an ensemble, CLDA is applied only on a partial feature space for each bootstrapped data. To compare the performance of Canonical Forest and other widely used ensemble methods, we tested them on 29 real or artificial data sets. Canonical Forest performed significantly better in accuracy than other ensemble methods in most data sets. According to the investigation on the bias and variance decomposition, the success of Canonical Forest can be attributed to the variance reduction.     

Canonical forms and spectral determination for a class of hyperbolic distributed parameter control systems

In this article we examine the effect of linear feedback control in the hyperbolic distributed parameter control system

$\text{?}\text{?t}\text{w}\text{v}\text{?}\text{0}\text{1}\text{1}\text{0}\text{?}\text{?x}\text{w}\text{v}\text{? A(x)}\text{w}\text{v}\text{= g(x) u(t).}$

By means of a reduction to canonical form similar to the one already familiar for finite-dimensional systems we show this system to be equivalent to the controlled difference-delay system

$\text{ζ(t+2)=e}{}^{\mathrm{2\alpha }}\text{ζ(t) +}\text{∫}\text{0}\text{2}\text{p(2?s)}\text{ζ(t+s) ds + u(t).}$

The theory of nonharmonic Fourier series is then employed to investigate the placement of eigenvalues in the closed loop system. Boundary value control and canonical form for observed systems are also studied.