Factorization of transfer functions. II. The minimality of passive scattering systems under a step-by-step combination

By the method of functional models of operators and systems one proves necessary and sufficient conditions in order that a step-by-step combination of n minimal passive scattering systems be a minimal system.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 4, pp. 488–494, April, 1990.     

Topological conjugacy to given constant length substitution minimal systems

We find necessary and sufficient conditions for a symbolic dynamical system to be topologically conjugate to any given constant length substitution minimal system, thus extending the results in Coven et al. (2008) for the Morse and Toeplitz substitutions.     

On tricyclic graphs of a given diameter with minimal energy

The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. Let G(n,d) be the class of tricyclic graphs G on n vertices with diameter d and containing no vertex disjoint odd cycles C_p,C_q of lengths p and q with p+q≡2(mod4). In this paper, we characterize the graphs with minimal energy in G(n,d).     

To the theory of passive resistance systems with losses of scattering channels

Passive resistance systems with losses of scattering channels are studied. We pay special attention to analysis of passive resistance systems whose basic operators belong to the class C₀ of contractions in a Hilbert space that have a minimal function in the sense of Sz.-Nagy and Foiaş. Bibliography: 20 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 355, 2008, pp. 37–71.     

Hyperbolic polygons of minimal perimeter with given angles

We prove that, among all convex hyperbolic polygons with given angles, the perimeter is minimized by the unique polygon with an inscribed circle. The proof relies on work of Schlenker (Trans Am Math Soc 359(5): 2155–2189, 2007).     

Windows of given area with minimal heat diffusion

For a bounded Lipschitz domain , we show the existence of a measurable set of given area such that the first eigenvalue of the Laplacian with Dirichlet conditions on and Neumann conditions on becomes minimal. If is a ball, will be a spherical cap.

     

Differential games with a given value function

The set of solutions of a differential game with a terminal payoff functional is investigated. A method is obtained that allows us to establish whether a given function is a value of some differential game with a terminal payoff functional. The condition obtained is in fact the condition for the given function to be a minimax (viscosity) solution of some Hamilton-Jacobi equation with Hamiltonian homogeneous in the third variable. We also obtain a sufficient condition for a function to belong to the set of values of differential games with a terminal payoff function.     

Differential games with a given value function

For an integral equation describing stationary distributions of a biological community, we point out conditions on its parameters under which this equation has a unique solution that satisfies necessary requirements for such a distribution.     

Generalized Maxwell formula for the length of a minimal tree with a given topology

The classic Maxwell formula calculates the length of a planar locally minimal binary tree in terms of coordinates of its boundary vertices and directions of incoming edges. However, if an extreme tree with a given topology and a boundary has degenerate edges, then the classic Maxwell formula cannot be applied directly. In order to calculate the length of the extreme tree in this case, it is necessary to know which edges are degenerate. In this paper we generalize the Maxwell formula to arbitrary extreme trees in a Euclidean space of arbitrary dimension. Now, to calculate the length of such a tree, there is no need to know either what edges are degenerate, or the directions of nondegenerate boundary edges. The answer is the maximum of some special linear function on the corresponding compact convex subset of the Euclidean space coinciding with the intersection of some cylinders.     

A note on minimal directed graphs with given girth

Caccetta and Häggkvist conjectured that the minimum order of a directed graph with girth g and outdegree not less than r is r(g − 1) + 1. They proved this conjecture for r = 2. We prove it for r = 3.     

A unification of unitary similarity transforms to compressed representations

In this paper a new framework for transforming arbitrary matrices to compressed representations is presented. The framework provides a generic way of transforming a matrix via unitary similarity transformations to, e.g., Hessenberg, Hessenberg-like form and combinations of both. The new algorithms are deduced, based on the QR-factorization of the original matrix. Relying on manipulations with rotations, all the algorithms consist of eliminating the correct set of rotations, resulting in a matrix obeying the desired structural constraints. Based on this new reduction procedure we investigate further correspondences such as irreducibility, uniqueness of the reduction procedure and the link with (rational) Krylov methods. The unitary similarity transform to Hessenberg-like form as presented here, differs significantly from the one presented in earlier work. Not only does it use less rotations to obtain the desired structure, also the convergence to rational Ritz-values is not observed in the conventional approach.     

A unitary similarity transform of a normal matrix to complex symmetric form

In this work a new unitary similarity transformation of a normal matrix to complex symmetric form will be discussed. A constructive proof as well as some properties and examples will be given.     

Uniqueness of varieties of minimal degree containing a given scheme

We prove that if has dimension and it is -Buchsbaum with \max{(\operatorname{codim}{X}-k,0)}$">, then is contained in at most one variety of minimal degree and dimension .

     

Outgoing Cuntz scattering system for a coisometric lifting and transfer function

We study a coisometry that intertwines Popescu’s presentations of minimal isometric dilations of a given operator tuple and of a coisometric lifting of the tuple. Using this we develop an outgoing Cuntz scattering system which gives rise to an input–output formalism. A transfer function is introduced for the system. We also compare the transfer function and the characteristic function for the associated lifting.     

The semigroup generated by a similarity orbit or a unitary orbit of an operator

Let be an invertible operator that is not a scalar modulo the ideal of compact operators. We show that the multiplicative semigroup generated by the similarity orbit of is the group of all invertible operators. If, in addition, is a unitary operator, then the multiplicative semigroup generated by the unitary orbit of is the group of all unitary operators.