Bloch wave homogenization of a non-homogeneous Neumann problem

In this paper, we use the Bloch wave method to study the asymptotic behavior of the solution of the Laplace equation in a periodically perforated domain, under a non-homogeneous Neumann condition on the boundary of the holes, as the size of the holes goes to zero more rapidly than the domain period. This method allows to prove that, when the hole size exceeds a given threshold, the non-homogeneous boundary condition generates an additional term in the homogenized problem, commonly referred to as “the strange term” in the literature.     

Bloch wave homogenization of a non-homogeneous Neumann problem

In this paper, we use the Bloch wave method to study the asymptotic behavior of the solution of the Laplace equation in a periodically perforated domain, under a non-homogeneous Neumann condition on the boundary of the holes, as the size of the holes goes to zero more rapidly than the domain period. This method allows to prove that, when the hole size exceeds a given threshold, the non-homogeneous boundary condition generates an additional term in the homogenized problem, commonly referred to as “the strange term” in the literature.     

On a generalization of one game control problem in the class of finitely additive measures

We consider a terminal game control problem for a linear system with discontinuous control coefficients subject to impulse constraints. We construct a generalized game control problem in the class of finitely additive measures with the property of the weak absolute continuity with respect to the restriction of the Lebesgue measure to some “sufficient” measurable structure.     

On a size-structured two-phase population model with infinite states-at-birth

In this work, we introduce and analyze a linear size-structured population model with infinite states-at-birth. We model the dynamics of a population in which individuals have two distinct life-stages: an “active” phase when individuals grow, reproduce and die and a second “resting” phase when individuals only grow. Transition between these two phases depends on individuals’ size. First we show that the problem is governed by a positive quasicontractive semigroup on the biologically relevant state space. Then, we investigate, in the framework of the spectral theory of linear operators, the asymptotic behavior of solutions of the model. We prove that the associated semigroup has, under biologically plausible assumptions, the property of asynchronous exponential growth.     

“Miniaturized” Linearizations for Quadratic 0/1 Problems

In order to solve a quadratic 0/1 problem, some techniques, consisting in deriving a linear integer formulation, are used. Those techniques, called “linearization”, usually involve a huge number of additional variables. As a consequence, the exact resolution of the linear model is, in general, very difficult. Our aim, in this paper, is to propose “economical” linear models. Starting from an existing linearization (typically the so-called “classical linearization”), we find a new linearization with fewer variables. The resulting model is called “Miniaturized” linearization. Based on this approach, we propose a new linearization scheme for which numerical tests have been performed.     

Convex programming with single separable constraint and bounded variables

In this paper a minimization problem with convex objective function subject to a separable convex inequality constraint “≤” and bounded variables (box constraints) is considered. We propose an iterative algorithm for solving this problem based on line search and convergence of this algorithm is proved. At each iteration, a separable convex programming problem with the same constraint set is solved using Karush-Kuhn-Tucker conditions. Convex minimization problems subject to linear equality/ linear inequality “≥” constraint and bounds on the variables are also considered. Numerical illustration is included in support of theory.     

A continuous approach to combinatorial optimization: application of water system pump operations

In this paper, we have suggested a penalty method to modify the combinatorial optimization problem with the linear constraints to a global optimization problem with linear constraints. It also deals with a topic of vital significance of pump operation optimization in a water system. In this connection we have done a lot of work to formulate a model based on a simplified flow volume balance to resolve the problem of optimal pump operation settings of switching “ON” and “OFF” with the reduced gradient method. This global solution approach incorporates some benefits for practical application to a real system as is shown in the case study.     

Asymptotics of the optimal time in a singular perturbed linear time-optimal problem

We consider an abstract attainability problem under constraints of asymptotic character and describe a general approach to constructing “nonsequential” attraction sets in the space of generalized elements formalized as finitely additive measures. We also study the existence and structure of an asymptotic formula universal in the range of “asymptotic constraints” and topologies of the space of generalized elements in the case when the space of ordinary solutions is not necessarily compactifiable.     

Linear stationary control systems over a Boolean semiring: Geometric properties and the isomorphism problem

In the present paper, we consider linear stationary dynamical systems over a Boolean semiring B. We analyze the complete observability, identifiability, reachability, and controllability of such systems. We define the notion of a “graph of modules” of completely controllable, completely reachable Boolean linear stationary systems by analogy with the spaces of modules in the case of systems over fields. We give a graph-theoretic interpretation of systems of this class. We solve the isomorphism problem in this class of systems.     

On the nonresonance property of linear differential-algebraic systems

We consider a system of linear ordinary differential equations in which the coefficient matrix multiplying the derivative of the unknown vector function is identically singular. For systems with constant and variable coefficients, we obtain nonresonance criteria (criteria for bounded-input bounded-output stability). For single-input control systems, we consider the problem of synthesizing a nonresonant system in the stationary and nonstationary cases. An arbitrarily high unsolvability index is admitted. The analysis is carried out under assumptions providing the existence of a so-called “equivalent form” with separated “algebraic” and “differential” components.     

A class of semi-linear equations of evolution

In this paper we prove existence, and study the asymptotic behavior of mild solutions of a class of semi-linear equations of evolution which are characterized by the fact that the associated homogeneous linear problem “generates” a strongly continuous semi-group of compact operators. We also prove a regularity result for which the associated homogeneous linear problem generates a holomorphic semi-group.     

On the problem of output feedback control synthesis under uncertainty and finite-time observers

We consider the problem of controlling a linear system of ordinary differential equations with a linear observable output. The system contains uncertain items (disturbances), for which we know only “hard” pointwise constraints. The problem of synthesizing a control that brings the trajectories of the system into a given target set in finite time is solved under weakened conditions without assuming that the control and the disturbance are of the same type. To this end, we suggest an approach that amounts to constructing an information set and a weakly invariant set with subsequent “aiming” of the first set at the second. Both stages are carried out in a finite-dimensional space, which permits one to use an efficient algorithm for solving the synthesis problem approximately on the basis of the ellipsoidal calculus technique. The results are illustrated by an example in which the control of a linear oscillation system is constructed.     

An algebraic foundation for factoring linear boundary problems

Motivated by boundary problems for linear differential equations, we define an abstract boundary problem as a pair consisting of a surjective linear map (“differential operator”) and an orthogonally closed subspace of the dual space (“boundary conditions”). Defining the composition of boundary problems corresponding to their Green’s operators in reverse order, we characterize and construct all factorizations of a boundary problem from a given factorization of the defining operator. For the case of ordinary differential equations, the main results can be made algorithmic. We conclude with a factorization of a boundary problem for the wave equation. This work was supported by the Austrian Science Fund (FWF) under the SFB grant F1322.     

On a version of the problem of choosing a vector subset

The NP-completeness is proved of the problem of choosing some subset of “similar” vectors. One of the variants of the a posteriori (off-line) noise-proof detection problem of an unknown repeating vector in a numeric sequence can be reduced to this problem in the case of additive noise. An approximation polynomial algorithm with a guaranteed performance bound is suggested for this problem in the case of a fixed space dimension.     

Instanton in the Field of a Pointlike Source of a Euclidean Non-Abelian Field

We consider the behavior of an (anti)instanton in the field of a pointlike source of a Euclidean non-Abelian field and investigate (anti)instanton deformations described by variations of its characteristic parameters. We formulate the variational problem of seeking the corresponding “crumpled” topological configurations and solve it algebraically using the Ritz method (of multipole decomposition of deformation fields). We investigate the domain of parameters specific for the instanton liquid model. We propose a simple method of taking contributions of such configurations in the functional integral into account approximately. In the framework of superposition analysis, we obtain an estimate for the mean energy of the source in the instanton liquid, and the estimate increases linearly with distance. In the case of a dipole in the color-singlet state, the energy is a linear function of the distance between the sources with the “strength” coefficient agreeing well with the known model and lattice estimates. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 2, pp. 267–298, February, 2006.     

The return of the linear search problem

The linear search problem concerns a search on the real line for a point selected at random according to a given probability distribution. The search begins at zero and is made by a continuous motion with constant speed, first in one direction and then the other. The problem is to determine when it is possible to devise a “best” search plan. In former papers the best plan has been selected according to the criterion of minimum expected path length. In this paper we consider a more general, nonlinear criterion for a “best” plan and show that the substantive requirements of the earlier results are not affected by these changes.     

The Polynomial Method for Random Matrices

We define a class of “algebraic” random matrices. These are random matrices for which the Stieltjes transform of the limiting eigenvalue distribution function is algebraic, i.e., it satisfies a (bivariate) polynomial equation. The Wigner and Wishart matrices whose limiting eigenvalue distributions are given by the semicircle law and the Marčenko–Pastur law are special cases. Algebraicity of a random matrix sequence is shown to act as a certificate of the computability of the limiting eigenvalue density function. The limiting moments of algebraic random matrix sequences, when they exist, are shown to satisfy a finite depth linear recursion so that they may often be efficiently enumerated in closed form. In this article, we develop the mathematics of the polynomial method which allows us to describe the class of algebraic matrices by its generators and map the constructive approach we employ when proving algebraicity into a software implementation that is available for download in the form of the RMTool random matrix “calculator” package. Our characterization of the closure of algebraic probability distributions under free additive and multiplicative convolution operations allows us to simultaneously establish a framework for computational (noncommutative) “free probability” theory. We hope that the tools developed allow researchers to finally harness the power of infinite random matrix theory.     

Homological properties of Sklyanin algebras

Summary. To a pair consisting of an elliptic curve and a point on it, Odeskii and Feigin associate certain quadratic algebras (“Sklyanin algebras”), having the Hilbert series of a polynomial algebra. In this paper we show that Sklyanin algebras have good homological properties and we obtain some information about their so-called linear modules. We also show how the construction by Odeskii and Feigin may be generalized so as to yield other “Sklyanin-type” algebras. Oblatum 25-XI-1993     

On subadditive functions and ψ-additive mappings

In [4], assuming among others subadditivity and submultiplicavity of a function ψ: [0, ∞)→[0, ∞), the authors proved a Hyers-Ulam type stability theorem for “ψ-additive” mappings of a normed space into a normed space. In this note we show that the assumed conditions of the function ψ imply that ψ=0 and, consequently, every “ψ-additive” mapping must be additive     

Ols parameter estimation for a linear paired confluent model

We investigate OLS parameter estimation for a linear paired model in the case of a passive experiment with errors in both variables. The explicit form of the OLS estimates is obtained, their equivalence to maximum likelihood estimates is demonstrated in the presence of normal errors, and estimate consistency is proved. The OLS estimates are compared analytically and numerically with known parameter estimates of “direct,” “orthogonal,” and “diagonal” regression models.