Phase Condition for the Grover Algorithm

For the Grover algorithm, we derive the exact formula of the norm of the amplitude in the marked state in a sine-function form and use this formula to derive the necessary and sufficient phase condition sin Δ ≤ |β| for this algorithm with arbitrary phase rotations. We show that the condition of identical rotation angles θ = φ, which is a special case of our condition, is a sufficient but not necessary phase condition.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 3, pp. 472–483, September, 2005.     

Markov Uniqueness and Rademacher Theorem for Smooth Measures on an Infinite-Dimensional Space under Successful-Filtration Condition

For a smooth measure on an infinite-dimensional space, a “successful-filtration” condition is introduced and the Markov uniqueness and Rademacher theorem for measures satisfying this condition are proved. Some sufficient conditions, such as the well-known Hoegh-Krohn condition, are also considered. Examples demonstrating connections between these conditions and applications to convex measures are given.__________Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 2, pp. 170–186, February, 2005.     

A Practical Optimality Condition Without Constraint Qualifications for Nonlinear Programming

A new optimality condition for minimization with general constraints is introduced. Unlike the KKT conditions, the new condition is satisfied by local minimizers of nonlinear programming problems, independently of constraint qualifications. The new condition is strictly stronger than and implies the Fritz–John optimality conditions. Sufficiency for convex programming is proved.     

On Groups with Minimality Condition for Noninvariant Abelian pd-Subgroups

We study properties and the structure of non-Abelian groups with minimality condition for non-invariant Abelian pd-subgroups in the case where they do not satisfy the minimality condition for Abelian pd-subgroups. We prove the solvability of these groups and establish relations with non-Abelian groups in which all infinite Abelian pd-subgroups are invariant.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 2, pp. 265–270, February, 2005.     

Weight Hierarchies of Linear Codes Satisfying the Chain Condition

The weight hierarchy of a linear [n,k;q] code C over GF(q) is the sequence (d₁,d₂,...,d_k) where d_r is the smallest support of an r–dimensional subcode of C. By explicit construction, it is shown that if a sequence (a₁,a₂,...,a_k) satisfies certain conditions, then it is the weight hierarchy of a code satisfying the chain condition.     

A Condition for the Existence of a Unique Green–Samoilenko Function for the Problem of Invariant Torus

Under the assumption that a linear homogeneous system defined on the direct product of a torus and a Euclidean space is exponentially dichotomous on the semiaxes, we obtain a condition for the existence of a unique Green–Samoilenko function for the problem of invariant torus. We find an expression for this function in terms of projectors that determine the dichotomy on the semiaxes.     

On blow up of generalized Kolmogorov–Petrovskii–Piskunov equation

In this paper we obtain sufficient condition of blow up of weak solution of generalized Kolmogorov–Petrovskii–Piskunov equation. Moreover, we obtain some sufficient condition of global solvability in the weak sense.     

Solvability of Semilinear Hemivariational Inequalities at Resonance Using Generalized Landesman–Lazer Conditions

In this paper we examine semilinear hemivariational inequalities at resonance at any eigenvalue of the negative Laplacian with Dirichlet boundary condition. Our approach is variational based on the Nonsmooth Critical Point Theory and it uses a generalized Landesman–Lazer condition. We prove two existence theorems using two different forms of the generalized Landesman–Lazer condition. In the last section we show that our Landesman-Lazer condition is more general than the ones existing in the literature.     

Characterizations by Automorphism Groups of Some Rank 3 Buildings, III. Moufang-Like Conditions

In this paper, we introduce the root-Moufang condition and the p-adic Moufang condition. We show that affine buildings of type Ã2 satisfying the root-Moufang condition are Bruhat–Tits buildings. Also, every rank 3 affine building satisfying the p-adic Moufang condition is a Bruhat-Tits building. We motivate the introduction of the new conditions by showing that all Bruhat– Tits Ã2-buildings satisfy the root-Moufang condition, and that the Ã2-buildings over a p-adic field also satisfy the p-adic Moufang condition. Another application of the p-adic Moufang condition is given in Part IV of this paper.     

Solvability of Semilinear Hemivariational Inequalities at Resonance Using Generalized Landesman–Lazer Conditions

In this paper we examine semilinear hemivariational inequalities at resonance at any eigenvalue of the negative Laplacian with Dirichlet boundary condition. Our approach is variational based on the Nonsmooth Critical Point Theory and it uses a generalized Landesman–Lazer condition. We prove two existence theorems using two different forms of the generalized Landesman–Lazer condition. In the last section we show that our Landesman-Lazer condition is more general than the ones existing in the literature.     

Genericity of Well-Posedness, Perturbations and Smooth Variational Principles

We show that three important topics in nonlinear analysis and optimization are intimately related: the theory of perturbations, the notion of well-posedness and variational principles in the sense of Ekeland, Borwein–Preiss and Deville–Godefroy–Zizler. The concept of genericity and the new notion of flexible perturbation play a key role in these connections. This notion enables one to consider topologies on spaces of functions which have been introduced recently. A link between the Asplund and Ekeland–Lebourg methods and the Palais–Smale condition, another important topic in nonlinear analysis, is pointed out.     

Locally nilpotent groups with the min−∞−n condition

The study of locally nilpotent groups with the weak minimality condition for normal subgroups, the min^––n condition, is continued. The following results are obtained.THEOREM 1. A locally nilpotent group with the min^––n condition is countable.THEOREM 2. If G is a locally nilpotent group with the min^––n condition whose periodic part is nilpotent and the orders of the elements of the periodic part are bounded in the aggregate, then G=t(G)A, where the subgroup A is minimax.THEOREM 3. Suppose G is a locally nilpotent group with the min^––n condition and T is its periodic part. If T is nilpotent and G/T is Abelian, then G=TA, where the subgroup A is minimax.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 3, pp. 340–346, March, 1990.     

Covariance of Lax Pairs and Integrability of the Compatibility Condition

We study the joint covariance of Lax pairs (LPs) with respect to Darboux transformations (DT). The scheme is based on comparing general expressions for the transformed coefficients of a LP and its Frechet derivative. We use the compact expressions of the DT via a version of non-Abelian Bell polynomials. We show that the so-called binary version of Bell polynomials forms a convenient basis for specifying the invariant subspaces. Some nonautonomous generalizations of KdV and Boussinesq equations are discussed in this context. We consider a Zakharov–Shabat-like problem to obtain restrictions at a minimal operator level. The subclasses that allow a DT symmetry (covariance at the LP level) are considered from the standpoint of dressing-chain equations. The cases of the classical DT and binary combinations of elementary DTs are considered with possible reduction constraints of the Mikhailov type (generated by an automorphism). Examples of Liouville–von Neumann equations for the density matrix are considered as illustrations.     

Constructing integral matrices with given line sums

D. Gale, in 1957 and H.J. Ryser, in 1963, independently proved the famous Gale–Ryser theorem on the existence of (0, 1)–matrices with prescribed row and column sums. Around the same time, in 1968, Mirsky solved the more general problem of finding conditions for the existence of a nonnegative integral matrix with entries less than or equal to p and prescribed row and column sums. Using the results of Mirsky, Brualdi shows that a modified version of the domination condition of Gale–Ryser is still necessary and sufficient for the existence of a matrix under the same constraints. In this article we prove another extension of Gale–Ryser’s domination condition. Furthermore we present a method to build nonnegative integral matrices with entries less than or equal to p and prescribed row and column sums.     

Unique Solvability of the Inverse Problem with a Time-Dependent Boundary Condition for a Sorption Model

A mixed initial boundary-value problem is considered for nonequilibrium sorption dynamics with inner-diffusion kinetics. The problem allows for convection and longitudinal diffusion and has a time-dependent boundary condition. This condition contains the time derivative of a solution component and constitutes the balance equation for the absorbed mixture near the output boundary of the sorption region—inside the diffusion barrier. Bounds on the solution of the direct problem are obtained: nonnegativity of the solution and its first time derivatives, and boundedness of the solution by known functions. The inverse problem of estimating the nonlinear system parameter—the sorption isotherm—is considered and a uniqueness theorem is proved.     

Sufficient Conditions for Boundedness of Convolution Operators in Rearrangement-Invariant Spaces

We prove equivalence of sufficient conditions for boundedness of convolution integral operators in rearrangement-invariant spaces which were obtained in the articles of S. G. Kren, E. M. Semenov, and the author. The first of these conditions generalizes the Hardy–Littlewood–Sobolev condition and the second is a modification of the Hörmander condition.     

Strong law of large numbers for countable nonhomogeneous Markov chains

The aim of this paper is to establish a strong law of large numbers for the bivariate functions of countable nonhomogeneous Markov chains under the condition of uniform convergence in the Cesàro sense which differs from my previous results. As corollaries, we generalize one of the Liu and Liu’s results for the univariate functions case and obtain another Shannon–McMillan–Breiman theorem for this Markov chains.     

Equivalence and existence of weak Pareto optima for multiobjective optimization problems with cone constraints

In this work, a differentiable multiobjective optimization problem with generalized cone constraints is considered, and the equivalence of weak Pareto solutions for the problem and for its η-approximated problem is established under suitable conditions. Two existence theorems for weak Pareto solutions for this kind of multiobjective optimization problem are proved by using a Karush–Kuhn–Tucker type optimality condition and the F-KKM theorem.     

Groups with maximum condition for nonabelian subgroups

We study locally soluble-by-finite groups with the maximum condition for nonabelian subgroups. These groups do not necessarily satisfy the maximum condition for all subgroups. But they are finitely generated and metabelian-byfinite.Deceased.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 925–930, July–August, 1991.     

Estimates of Deviations for Generalized Newtonian Fluids

The paper is concerned with estimates of deviations from exact solutions for stationary models of viscous incompressible fluids. It is shown that if a function compared with exact solution is subject to the incompressibility condition, then the deviation majorant consists of terms that penalize the inaccuracy in the equilibrium equation and the rheological relation defined by a ceratin dissipative potential. If such a function does not satisfy the incompressibility condition, then the majorant includes an additional term. The factor of this term depends on the constant in the Ladyzhenskaya–Babuka–Brezzi condition. Bibliography: 27 titles.