О предельном распределении для сумм независимых слууайных велиуин на бикомпактных коммутативных топологиуеских группах

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Optimal trajectories associated with a solution of the contingent Hamilton-Jacobi equation

In this paper we study the existence of optimal trajectories associated with a generalized solution to the Hamilton-Jacobi-Bellman equation arising in optimal control. In general, we cannot expect such solutions to be differentiable. But, in a way analogous to the use of distributions in PDE, we replace the usual derivatives with contingent epiderivatives and the Hamilton-Jacobi equation by two contingent Hamilton-Jacobi inequalities. We show that the value function of an optimal control problem verifies these contingent inequalities.Our approach allows the following three results: (a) The upper semicontinuous solutions to contingent inequalities are monotone along the trajectories of the dynamical system. (b) With every continuous solutionV of the contingent inequalities, we can associate an optimal trajectory along whichV is constant. (c) For such solutions, we can construct optimal trajectories through the corresponding optimal feedback.They are also viscosity solutions of a Hamilton-Jacobi equation. Finally, we prove a relationship between superdifferentials of solutions introduced by Crandallet al. [10] and the Pontryagin principle and discuss the link of viscosity solutions with Clarke's approach to the Hamilton-Jacobi equation.     

Some inequalities about the covering radius of Reed-Muller codes

Let R(r, m) be the rth order Reed-Muller code of length 2 ^m , and let (r, m) be its covering radius. We prove that if 2 k m - r - 1, then (r + k, m + k) (r, m + 2(k - 1). We also prove that if m - r 4, 2 k m - r - 1, and R(r, m) has a coset with minimal weight (r, m) which does not contain any vector of weight (r, m) + 2, then (r + k, m + k) (r, m) + 2k(. These inequalities improve repeated use of the known result (r + 1, m + 1) (r, m).This work was supported by a grant from the Research Council of Wright State University.     

On Bahadur asymptotic efficiency in a semiparametric regression model

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On minimal factorizations of rational matrix functions

The theory of factorization of rational matrix functions W() = = C(I-A)^–1B + D, as presented in the book Bart-Gohberg-Kaashoek [1], is extended here to the case where D = W() is not invertible, and applied to factorizations of monic matrix polynomials.     

A constructive approach to the quantum (cosh ϕ)2 model. I. The method of the Gel'fand-Levitan-Marchenko equations

In a Hubert , with the aid of the postulated Gel'fand-Levitan-Marchenko quantum equations, one introduces the fields ₁(x) and ₂(x), which are the quantum analogues of the classical fields cosh (x) and sinh (x) in the sinh-Gordon model. It is shown that the fields _j(x) satisfy the Wightman axioms, including the invariance relative to reflections of space-time and mutual local commutativity. In addition, one proves the asymptotic completeness of the theory and one computes explicitly the scattering operator. In the developed approach, no cut-offs are used and, therefore, there are no renormalization effects.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 146, pp. 147–190, 1985.     

Notes on Interpolation. X (Two Boundary Value Problems)

In this paper we use (0, 2) interpolational polynomials to give an approximate solution of the differential equation y(x) + A(x)y(x) = F(x), x I := [-1, 1] j in case when the boundary values are y(-1) = and y(1) = , , R.     

Totally umbilical manifolds inG(2,n). I

It is well known that every totally umbilical submanifold of a space of constant curvature is either a small sphere or is totally geodesic. B.-Y. Chen has classified totally umbilical submanifolds of compact, rank one, symmetric spaces ([4], [5]): in particular, they are all extrinsic spheres, that is, they have a parallel mean curvature vector H (or are totally geodesic). In this paper totally umbilical submanifolds F^l of dimension l 3 are classified in the irreducible symmetric space that is "next in complexity": Grassmann manifold G(2, n). Such submanifolds are either 1) totally geodesic [3] or 2) extrinsic spheres [small spheres in totally geodesic spheres; their position in G(2, n) is described here] or 3) essentially totally umbilical (H 0, H 0). If the submanifold is of type 3), then it is either a) an umbilical hypersurface of nonconstant mean curvature in totally geodesic S¹ × S¹ G(2, n) or b) an "oblique diagonal," a diagonal of the product of two small spheres of different radii in totally geodesic S^l+1 × S^l+1 G(2, n) (it has constant mean and sectional curvatures). Submanifolds 3a) and 3b) are described completely. The latter of the two negates two of Chen's conjectures. It is shown that submanifold F^l E^l+2 (l 3) with a totally umbilical Grassmannian image has a totally geodesic Grassmannian image and is classifiable [11].Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 83–98, 1991.     

On Upper and Lower Almost β-continuous Multifunctions

In this paper, the authors define a multifunction F : X Y to be upper (lower) almost -continuous if F+(V) (F- (V)) is -open in X for every regular open set V of Y. They obtain some characterizations and several properties concerning upper (lower) almost -continuous multifunctions.     

Construction of singular solutions to the p-harmonic equation and its limit equation for p=∞

Here, all solutions of the form u=r^kf() to the p-harmonic equation, div(|u|^p–2u)=0, (p>2) in the plane are determined. One main result is a representation formula for such solutions. Further, solutions with an isolated singularity at the origin are constructed (Theorem 1). Graphical illustrations are given at the end of the paper. Finally, all solutions u=r^kf() of the limit equation for p=, u _x ² u_xx+2u_xu_yu_xy+u _y ² u_yy=2, are constructed, some of which have a strong singularity at the origin (Theorem 2).     

Mixed-norm estimates for a class of integral operators

(, ) — R ^m ×R ⁿ . f R ^m ×R ⁿ f_p,q, f L _p (R ^m) x y, L^q(Rⁿ). ׃ _q,r cƒ _p,r , ׃ R ^m ×R ⁿ , , , q r . , ( ¦¦) K ₀ (y); p, g r , K ₀.     

Non removable edges in 3-connected cubic graphs

LetG be a cyclicallyk-edge-connected cubic graph withk 3. Lete be an edge ofG. LetG be the cubic graph obtained fromG by deletinge and its end vertices. The edgee is said to bek-removable ifG is also cyclicallyk-edge-connected. Let us denote by S _k (G) the graph induced by thek-removable edges and by N _k (G) the graph induced by the non 3-removable edges ofG. In a previous paper [7], we have proved that N ₃(G) is empty if and only ifG is cyclically 4-edge connected and that if N ₃(G) is not empty then it is a forest containing at least three trees. Andersen, Fleischner and Jackson [1] and, independently, McCuaig [11] studied N ₄(G). Here, we study the structure of N _k (G) fork 5 and we give some constructions of graphs such thatN _k (G) = E(G). We note that the main result of this paper (Theorem 5) has been announced independently by McCuaig [11].

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Résumé SoitG un graphe cubique cyliquementk-arête-connexe, aveck 3. Soite une arête deG et soitG le graphe cubique obtenu à partir deG en supprimante et ses extrémités. L'arêtee est ditek-suppressible siG est aussi cycliquementk-arête-connexe. Désignons par S _k (G) le graphe induit par les arêtesk-suppressibles et par N _k (G) celui induit par les arêtes nonk-suppressibles. Dans un précédent article [7], nous avons montré que N ₃(G) est vide si et seulement siG est cycliquement 4-arête-connexe et que si N ₃(G) n'est pas vide alors c'est une forêt possédant au moins trois arbres. Andersen, Fleischner and Jackson [1] et, indépendemment, McCuaig [11] ont étudié N ₄(G). Ici, nous étudions la structure de N _k (G) pourk 5 et nous donnons des constructions de graphes pour lesquelsN _k (G) = E(G). Nous signalons que le résultat principal de cet article (Théorème 5) a été annoncé indépendamment par McCuaig [11].

     

On the number of non conjugate subgroups

We shall prove that every group of cardinality ₁ has at least ₁ non conjugate subgroups, and we shall generalize this theorem to many more uncountable cardinalities. For example underGCH for every uncountable cardinal and every groupG of cardinality ,G has at least non conjugate subgroups.Presented by W. Taylor.I would like to thank Rami Grossberg for writing and rewriting this paper, and Wilfrid Hodges for removing many errors and suggesting improvements in presentation; many facts are proved only due to his explicit request.This research was supported by grant (No. 1110) from the United States-Israel Binational Science Foundation.     

Sets of uniqueness and closed subgroups in Vilenkin groups

, ( ) . , : , , .

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This research was partially supported by National Science Foundation under grant INT-8400708.     

Galerkin methods for parabolic equations with nonlinear boundary conditions

A variety of Galerkin methods are studied for the parabolic equationu _t =(a(x) u),x ⁿ ,t (O,T], subject to the nonlinear boundary conditionu _v =g(x,t,u),x,t (O,T] and the usual initial condition. Optimal order error estimates are derived both inL ² () andH ¹ () norms for all methods treated, including several that produce linear computational procedures.The authors were partially supported by The National Science Foundation during the preparation of this paper.     

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.     

Bijectivity of -endomorphisms of ℬ(H) and the unilateral shift

LetH be a separable infinite-dimensional complex Hilbert space. We prove that if : (H)(H) is a^*-preserving ring homomorphism whose range contains a rank-one operator and an operator with dense range, then is an isometric linear or conjugate-linear algebra automorphism of (H). In particular, if the unilateral shift is contained in the range of a^*-endomorphism of (H), then is bijective.Research partially supported by the Hungarian National Research Science Foundation, Operating Grant Number OTKA 1652 and K&H Bank Ltd., Universitas Foundation.     

Embeddings of Lorentz—Marcinkiewicz spaces with mixed norms

X(Y) f -:X(Y)={fM(×): f_{X(Y)=f(x,.)YX<} . =(0, ), M (×) — , ×, X, Y, Z— . X(Y) Z(×).     

Phragmén-Lindelöf theorems for second-order quasilinear elliptic equations

Analogues are formulated of the well-known, in the theory of analytic functions, Phragmen-Lindelöf theorem for the gradients of solutions of a broad class of quasilinear equations of elliptic type. Examples are given illustrating the accuracy of the results obtained for the gradients of solutions of the equations of the form div(|U|^–2u)=f(x, u, u), where f(x, u, u) is a function locally bounded in ²ⁿ⁺¹. f(x, 0, u)=0, uf(x, u, u) c¦u¦1+q(1+ ¦u|), > 1, c > 0, q > 0, is an arbitrary real number, and n >- 2. The basic role in the technique employed in the paper is played by the apparatus of capacitary characteristics.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 10, pp. 1376–1381, October, 1992.The author sincerely appreciates E. M. Landis's permanent attention and numerous useful discussions.     

A splitting technique for Harris recurrent Markov chains

Summary A technique is presented, which enables the state space of a Harris recurrent Markov chain to be split in a way, which introduces into the split state space an atom. Hence the full force of renewal theory can be used in the analysis of Markov chains on a general state space. As a first illustration of the method we show how Derman's construction for the invariant measure works in the general state space. The Splitting Technique is also applied to the study of sums of transition probabilities.