Properties of Solutions to Second Order Evolution Control Systems. II

We continue the research of the first part of the article. We mainly study codensity for the set of admissible trajectory-control pairs of a system with nonconvex constraints in the set of admissible trajectory-control pairs of the system with convexified constraints. We state necessary and sufficient conditions for the set of admissible trajectory-control pairs of a system with nonconvex constraints to be closed in the corresponding function spaces. Using an example of a control hyperbolic system, we give an interpretation of the abstract results obtained. As application we consider the minimization problem for an integral functional on solutions of a control system.     

Characterization of regular singular linear systems of difference equations

This paper deals with linear systems of difference equations whose coefficients admit generalized factorial series representations atz=. We are concerned with the behavior of solutions near the pointz= (the only fixed singularity for difference equations). It is important to know whether a system of linear difference equations has a regular singularity or an irregular singularity. To a given system () we can assign a number , called the Moser's invariant of (), so that the system is regular singular if and only if 1. We shall develop an algorithm, implementable in a computer algebra system, which reduces in a finite number of steps the system of difference equations to an irreducible form. The computation ot the number can be done explicitly from this irreducible form.     

Polynomial Wavelet-Type Expansions on the Sphere

We present a polynomial wavelet-type system on S ^d such that any continuous function can be expanded with respect to these wavelets. The order of the growth of the degrees of the polynomials is optimal. The coefficients in the expansion are the inner products of the function and the corresponding element of a dual wavelet system. The dual wavelets system is also a polynomial system with the same growth of degrees of polynomials. The system is redundant. A construction of a polynomial basis is also presented. In contrast to our wavelet-type system, this basis is not suitable for implementation, because, first, there are no explicit formulas for the coefficient functionals and, second, the growth of the degrees of polynomials is too rapid.     

Point Code Minimum Steiner Triple Systems

The point code of a Steiner triple system uniquely determines the system when the number of vectors whose weight equals the replication number agrees with the number of points. The existence of a Steiner triple system with this minimum point code property is established for all v 1,3 (mod 6) with v 15.     

On partial synchronization of nonlinear oscillations of two Berger plates coupled by internal subdomains

The problem of nonlinear oscillations of two Berger plates occupying bounded domains Ω in different parallel planes and coupled by internal subdomains Ω₁⊂Ω is considered. A dynamical system generated by the problem in the space is studied. The long-time behavior of the trajectories of the system and its dependence on the value of the coupling parameter γ is described in terms of the system global attractor. In particular, we prove a synchronization phenomenon at the level of attractor for the system. Namely, we consider a (limiting) dynamical system generated by a suitable second order in time evolution equation in the space consisting of the elements from H with coordinates equal for the values of the spatial variable x from the closed set : , and prove that the attractor of the system describing oscillations of two partially coupled Berger plates approaches the attractor of the limiting system as γ tends to the infinity.     

On stability of the trivial solution of a nonautonomous quasilinear system whose characteristic equation has multiple roots

We obtain sufficient conditions for the Lyapunov stability of the trivial solution of a nonautonomous differential system of a special form ast , + . For this system, the coefficient matrix of a differential system of the first approximation has almost Jordan form with triangular blocks. We indicate methods that enable one to reduce certain classes of differential systems of the general form to special differential systems.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 8, pp. 1072–1079, August, 1994.     

Existence of a universal attractor for a fully hyperbolic phase-field system

Here we study a nonlinear hyperbolic integrodifferential system which was proposed by H.G. Rotstein et al. to describe certain peculiar phase transition phenomena. This system governs the evolution of the (relative) temperature and the order parameter (or phase-field) . We first consider an initial and boundary value problem associated with the system and we frame it in a history space setting. This is done by introducing two additional variables accounting for the histories of and . Then we show that the reformulated problem generates a dissipative dynamical system in a suitable infinite-dimensional phase space. Finally, we prove the existence of a universal attractor.     

Capacity analysis of an automated kit transportation system

In this paper, we present a capacity analysis of an automated transportation system in a flexible assembly factory. The transportation system, together with the workstations, is modeled as a network of queues with multiple job classes. Due to its complex nature, the steadystate behavior of this network is not described by a productform solution. Therefore, we present an approximate method to determine the capacity of the network. We first study a number of key elements of the system separately and subsequently combine the results of this analysis in an Approximate Mean Value Analysis (AMVA) algorithm. The key elements are a buffer/transfer system (the bottleneck of the system), modeled as a preemptiverepeat priority queue with identical deterministic service times for the different job classes, a set of elevators, modeled as vacation servers, a number of work cells, modeled as multiserver queues, and several nonaccumulating conveyor belts, modeled as ample servers. The AMVA algorithm exploits the property that the initial multiclass queueing network can be decomposed into a sequence of singleclass queueing networks and hence is very efficient. Comparison of numerical results of the AMVA algorithm for the throughputs for the different job classes to simulation results shows that the AMVA algorithm is also accurate. For several series of instances, the maximum relative error that we found was only 4.0%.     

On conditions for the discreteness of the spectrum of a non-self-adjoint system of differential equations of the Dirac type

A first-order non-self-adjoint system of differential equations with 2n unknown functions and with certain conditions on the coefficients is considered on the entire axis. The system is similar to the well-known Dirac system. An operator , associated with the considered system, is introduced in the space L²(– , ) of 2n-component vector-valued functions. Conditions for the compactness of the resolvent of the operator are found, yielding sufficient conditions for the discreteness of its spectrum.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 16, pp. 3–24, 1992.     

β-Shifts have unique maximal measure

It is proved that -shifts have unique measure with maximal entropy by constructing an isomorphism of the -shift with another topological dynamical system and proving it for this system.     

Two-magnon states in a one-dimensional non-Heisenberg ferromagnet of spins = 1 with interactions up to third neighbors

We consider a two-magnon system in a one-dimensional non-Heisenberg ferromagnet of spin s = 1 with interactions of nearest, second, and third neighbors. We prove that for = and J = J₁, the system has a unique two-magnon bound state (TMBS), while for = and J = 2J₁, the system has three bound states. If = and J J₁, J 2J₁, the system has at most five TMBS. The energies of these bound states are evaluated.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 2, pp. 262–268, May, 1996.Translated by A. M. Semikhatov.     

Boundary-value problems for two-dimensional canonical systems

The two-dimensional canonical systemJy=–Hy where the nonnegative Hamiltonian matrix functionH(x) is trace-normed on (0, ) has been studied in a function-theoretic way by L. de Branges in [5]–[8]. We show that the Hamiltonian system induces a closed symmetric relation which can be reduced to a, not necessarily densely defined, symmetric operator by means of Kac' indivisible intervals; of. [33], [34]. The formal defect numbers related to the system are the defect numbers of this reduced minimal symmetric operator. By using de Branges' one-to-one correspondence between the class of Nevanlinna functions and such canonical systems we extend our canonical system from (0, ) to a trace-normed system on which is in the limit-point case at ±. This allows us to study all possible selfadjoint realizations of the original system by means of a boundaryvalue problem for the extended canonical system involving an interface condition at 0.     

Sufficient conditions for the existence of at least n or exactly n limit cycles for the Liénard differential systems

In this paper we study the limit cycles of the Liénard differential system of the form , or its equivalent system , . We provide sufficient conditions in order that the system exhibits at least n or exactly n limit cycles.     

New transformations for Painlevé's third transcendent

We present transformations relating the third transcendent of Painlevé with parameter sets located at the corners of the Weyl chamber for the symmetry group of the system, the affine Weyl group of the root system , to those at the origin. This transformation entails a scaling of the independent variable, and implies additive identities for the canonical Hamiltonians and product identities for the -functions with these parameter sets.

     

Irregular hypergeometric systems associated with a singular monomial curve

In this paper we study irregular hypergeometric systems defined by one row. Specifically, we calculate slopes of such systems. In the case of reduced semigroups, we generalize the case studied by Castro and Takayama. In all the cases we find that there always exists a slope with respect to a hyperplane of this system. Only in the case of an irregular system defined by a integer matrix we might need a change of coordinates to study slopes at infinity. In the other cases slopes are always at the origin, defined with respect to a hyperplane. We also compute all the -characteristic varieties of the system, so we have a section of the Gröbner fan of the module defined by the hypergeometric system.