On the initial boundary value problem for a nonlinear nonhomogeneous equation of Sobolev type

The initial boundary value problem for a nonlinear nonhomogeneous equation of Sobolev type used for modeling nonstationary processes in semiconductors is examined. It is proved that this problem is uniquely solvable at least locally in time. Sufficient conditions for the problem to be solvable globally in time are found, as well as sufficient conditions for the local (but not global) solvability. In the case of only local solvability, upper and lower estimates for the time when a solution exists are determined in the form of either explicit or quadrature formulas.     

Relationships between discrete-time and continuous-time algebraic riccati inequalities

The bilinear transformation is used to establish a direct relationship between a discrete-time algebraic Riccati inequality (DARI) and an associated continuous-time algebraic Riccati inequality (CARI). It is shown that under mild conditions, the DARI is solvable if and only if the corresponding CARI is solvable. The relationship between the DARI and the CARI is then used to translate the general solvability conditions for a CARI given by Scherer into analogous conditions for the DARI. It is shown how such conditions can be applied to determine the solvability of a discrete-time H_∞ control problem whose solution set is characterized by two DARIs.     

Finding adapted solutions of forward–backward stochastic differential equations: method of continuation

Summary. The notion of bridge is introduced for systems of coupled forward–backward stochastic differential equations (FBSDEs, for short). This notion helps us to unify the method of continuation in finding adapted solutions to such FBSDEs over any finite time durations. It is proved that if two FBSDEs are linked by a bridge, then they have the same unique solvability. Consequently, by constructing appropriate bridges, we obtain several classes of uniquely solvable FBSDEs. Received: 23 April 1996 / In revised form: 10 October 1996     

Invertible Linear Maps on Simple Lie Algebras Preserving Solvability

Let 𝔤 be a (finite-dimensional) complex simple Lie algebra of rank l. An invertible linear map ? on 𝔤 is said to preserve solvability in both directions if ?, as well as ?^?1, sends every solvable subalgebra to some solvable one. In this article, it is shown that an invertible linear map ? on 𝔤 preserves solvability in both directions if and only if it can be decomposed into the product of an inner automorphism, a graph automorphism, a scalar multiplication map and a diagonal automorphism.     

带跳的耦合正倒向随机微分方程

本文对带跳的耦合正倒向随机微分方程引入了“桥”的概念,证明了如果两个带跳的耦合正倒向随机微分方程被桥连接着,那么它们有相同的唯一可解性.在此基础上,通过桥的构造,得到一些带跳的正倒向随机微分方程的唯一可解性.     

Generalized linear complementarity problems

It has been shown by Lemke that if a matrix is copositive plus on ⁿ , then feasibility of the corresponding linear complementarity problem implies solvability. In this article we show, under suitable conditions, that feasibility of ageneralized linear complementarity problem (i.e., defined over a more general closed convex cone in a real Hilbert space) implies solvability whenever the operator is copositive plus on that cone. We show that among all closed convex cones in a finite dimensional real Hilbert Space, polyhedral cones are theonly ones with the property that every copositive plus, feasible GLCP is solvable. We also prove a perturbation result for generalized linear complementarity problems.This research has been partially supported by the Air Force Office of Scientific Research under grants #AFOSR-82-0271 and #AFOSR-87-0350.     

Computing test rank for a free solvable group

It is proved that test rank of a free solvable non-Abelian group of finite rank is 1 less than the rank of that group. This gives the answer to Question 14.88 posed in the Kourovka Notebook by Fine and Shpilrain, asking whether or not a free solvable group of rank 2 and solvability index n ≥ 3 has test elements. Supported by RFBR grant No. 05-01-00292. __________ Translated from Algebra i Logika, Vol. 45, No. 4, pp. 447–457, July–August, 2006.     

On solvability of systems of polynomial equations

We study the computational complexity of the solvability problem of systems of polynomial equations over finite algebras. We prove a new dichotomy theorem that extends most of the dichotomy results which have been obtained over different families of finite algebras so far. As a corollary, for example, we get that if \mathbbA{\mathbb{A}} is a finite algebra of finite signature and omits the Hobby-McKenzie type 1, then the problem is solvable in polynomial time whenever \mathbbA{\mathbb{A}} is a reduct of a generalized affine algebra, and NP-complete otherwise.     

The completion of solvable groups

It is proved that every noncyclic solvable free group with solvability length exceeding two has isomorphic completions.Translated from Matematicheskie Zametki, Vol. 9, No. 5, pp. 543–550, May, 1971.     

The Local Embedding of the Hall Group into the Class of Finite Groups

It is proved that the Hall group (a solvable, not residually finite group of solvability length 3) can be locally embedded into the class of finite groups. Bibliography: 2 titles.     

On the global unique solvability of some two-dimensional problems for the water solutions of polymers

The global unique solvability of the two-dimensional initial-boundary-value problem with some slip-boundary conditions for a quasilinear system describing the flow of weak water solutions of polymers is proved. It is noted that the global unique solvability of the Cauchy problem and the initial-boundary-value problem with periodic boundary conditions are proved in a similar way. Biblography 14 titles. Dewdicated to the memory of A. P. Oskolkov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 243, 1997, pp. 138–152. Translated by O. A. Ladyzhenskaya.     

Forward-backward doubly stochastic differential equations and related stochastic partial differential equations

The notion of bridge is introduced for systems of coupled forward-backward doubly stochastic differential equations (FBDSDEs). It is proved that if two FBDSDEs are linked by a bridge, then they have the same unique solvability. Consequently, by constructing appropriate bridges, we obtain several classes of uniquely solvable FBDSDEs. Finally, the probabilistic interpretation for the solutions to a class of quasilinear stochastic partial differential equations (SPDEs) combined with algebra equations is given. One distinctive character of this result is that the forward component of the FBDSDEs is coupled with the backward variable.     

Universally concordant Galois extensions

A Galois extension is called universally concordant of period q, if for any imbedding problem of this extension whose kernel is an Abelian group of period q the concordance condition is satisifed. A necessary and sufficient condition is given for the imbeddability of one universally concordant extension into another. For a universally concordant extension of period of an algebraic number field containing no roots of 1 of degrees p₁, ..., p_m the solvability of any imbedding problem with solvable kernel of period q is proved.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 71, pp. 133–152, 1977.     

Number representations of satisfiability

Propositional formulas are represented by real-valued matrices so that their unsatisfiability is proved to be equivalent to the stable nonnegative solvability of linear algebraic systems. The complete calculus for the generation of all nonnegatively solvable linear homogeneous systems is presented as a consequence. Solvability of systems of linear inequalities with real coefficients and Boolean variables is reduced to the satisfiability of some formula. The alternatives theorem is proved concerning the solvability of such systems. The technique of implicit enumeration in the search for solutions both in the systems considered and in a wide range of discrete optimization problems is described. Bibliography: 21 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol 241, 1997, pp. 72–96 Translated by E. A. Hirsch.     

Problem without Boundary Conditions for a Hyperbolic Equation

We consider a mixed problem for a hyperbolic equation with classical initial conditions and two nonlocal integral conditions and prove that it is uniquely solvable. This problem is not self-adjoint; this is an obstruction to applying the usual methods for studying the existence, uniqueness, and stability of its solutions. We reduce this problem to a system whose solvability can be proved by using the contraction mapping principle.__________Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 10, Suzdal Conference-4, 2003.     

On some classes of coefficient inverse problems for parabolic systems of equations

We examine the question on solvability in the Sobolev spaces of coefficient inverse problems for parabolic systems of equations with the overdetermination conditions on a collection of surfaces. Under certain conditions on the geometry of the domain and the boundary operators, the local solvability of the problem is proven. It is demonstrated that the conditions on the boundary operators are sharp and that, in some cases, the problem is not unconditionally solvable.     

On self-normality and abnormality in the alternating groups

It is known that in a finite solvable group G, a subgroup H is abnormal if and only if every subgroup of G containing H is self-normalizing in G. Although, in general, the assumption of solvability cannot be dropped, in this paper we prove the theorem for the special case G = A_n and H a second maximal intransitive subgroup of A_n.Received: 1 July 2003     

Local moment problem

The work is devoted to the local moment problem, which consists in finding of non-decreasing functions on the real axis having given first 2n + 1, n ≥ 0, power moments on the whole axis and also 2m + 1 first power moments on a certain finite axis interval. Considering the local moment problem as a combination of the Hausdorff and Hamburger truncated moment problems we obtain the conditions of its solvability and describe the class of its solutions with minimal number of growth points if the problem is solvable. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Interval systems of max-separable linear equations

An interval system of equations is weakly solvable if at least one of its subsystems is solvable, and it is strongly solvable if all its subsystems are solvable. We give necessary and sufficient conditions enabling efficient testing of weak and strong solvability over max-plus and max-min algebras.     

Inverse problem for N × N hyperbolic systems on the plane and the N-wave interactions

We study regiorously the solvability of the direct and inverse problems associated with Ψ_x–JΨ_y = QΨ,(x,y) ∈ ?², where (i) Ψ is an N × N-matrix-valued function on ?² (N ≦ 2), (ii) J is a constant, real, diagonal N × N matrix with entries, J₁ > J₂ > …? > J_N and (iii) Q is off-diagonal with rapidly decreasing (Schwartz) component functions. In particular we show that the direct problem is always solvable and give a small norm condition for the solvability of the inverse problem. In the particular case that Q is skew Hermitian the inverse problem is solvable without the small norm assumption. Furthermore we show how these results can be used to solve certain Cauchy problems for the associated nonlinear evolution equations. For concreteness we consider the N-wave interactions and show that if a certain norm of Q(x, y, 0) is smallor if Q(x, y, 0) is skew Hermitian the N-wave interations equation has a unique global solution.