Generalized Quasi-Variational Inequalities Without Continuities

Given a nonempty set and two multifunctions , we consider the following generalized quasi-variational inequality problem associated with X, : Find such that . We prove several existence results in which the multifunction is not supposed to have any continuity property. Among others, we extend the results obtained in Ref. 1 for the case (x(X.     

A note on the optimal quadrature inH p

Summary The existence of optimal nodes with preassigned multiplicities is proved for the Hardy spacesH ^p (1<p<). This is then used to show that the exact order of convergence for the optimal qudrature formula withN nodes (including multiplicity) is where 1/p+1/q=1 and 1p.     

Bergman-Vekua theory in differential form

We develop Bergman-Vekua integral operator theory on the basis of a new kind of series suggested by polynomial operators (classP _j operators); in a sense, this is an analog of the Weierstrass approach as opposed to the Cauchy-Riemann approach in classical complex analysis, which also has advantages over the original form in boundary value problems and other applications.Dedicated to the memory of my friend, Peter Henrici     

The Whitehead Group of the Novikov Ring

The Bass–Heller–Swan–Farrell–Hsiang–Siebenmann decomposition of the Whitehead group K ₁(A[z,z^-1]) of a twisted Laurent polynomial extension A[z,z^-1] of a ring A is generalized to a decomposition of the Whitehead group K ₁(A((z))) of a twisted Novikov ring of power series A((z))=A[[z]][z^-1]. The decomposition involves a summand W₁(A, ) which is an Abelian quotient of the multiplicative group W(A,) of Witt vectors 1+a₁z+a₂z²+ ··· A[[z]]. An example is constructed to show that in general the natural surjection W(A, )^ab W₁(A, ) is not an isomorphism.     

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.     

On the concentration of stresses due to a small elliptic inclusion on the neutral axis of a deep beam under constant bending moment

Zusammenfassung Es wird die Spannungsverteilung untersucht, die sich in einem breiten Balken mit konstanter Höhe unter einem konstanten Biegemoment ausbildet, wenn er eine kleine elliptische Einschliessung mit Zentrum auf der Neutralachse enthält. Insbesondere werden die Fälle eines sehr starren Einschlusses sowie eines elliptischen Loches im Detail diskutiert.

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Nomenclature x, y Cartesian coordinates - , elliptic coordinates - u, v (u ,u )=components of displacements - , unit elongations in -and -directions - shearing strain - , normal stress components in elliptical coordinates - shearing stress in elliptic coordinates - _x , _y normal stress in Cartesian coordinates - _xy shearing stress in Cartesian coordinates - E Young's modulus for the beam - v Poisson's ratio for the beam - 1/h ₁, 1/h ₂ stretch ratios - e _x + _y dilatation - 2 rotation - M bending moment     

A General Decomposition Construction for Incomplete Secret Sharing Schemes

A secret sharing scheme for an incomplete access structure (,) is a method of distributing information about a secret among a group of participants in such a way that sets of participants in can reconstruct the secret and sets of participants in can not obtain any new information about the secret. In this paper we present a more precise definition of secret sharing schemes in terms of information theory, and a new decomposition theorem. This theorem generalizes previous decomposition theorems and also works for a more general class of access structures. We demonstrate some applications of the theorem.     

Convex independent sets and 7-holes in restricted planar point sets

For a finite setA of points in the plane, letq(A) denote the ratio of the maximum distance of any pair of points ofA to the minimum distance of any pair of points ofA. Fork>0 letc (k) denote the largest integerc such that any setA ofk points in general position in the plane, satisfying for fixed , contains at leastc convex independent points. We determine the exact asymptotic behavior ofc (k), proving that there are two positive constants=(), such thatk ^1/3c (k)k ^1/3. To establish the upper bound ofc (k) we construct a set, which also solves (affirmatively) the problem of Alonet al. [1] about the existence of a setA ofk points in general position without a 7-hole (i.e., vertices of a convex 7-gon containing no other points fromA), satisfying . The construction uses Horton sets, which generalize sets without 7-holes constructed by Horton and which have some interesting properties.     

Large Tandem Queueing Networks with Blocking

Systems consisting of many queues in series have been considered by Glynn and Whitt (1991) and Baccelli, Borovkov and Mairesse (2000). We extend their results to apply to situations where the queues have finite capacity and so various types of blocking can occur. The models correspond to max-plus type recursions, of simple form but in infinitely many dimensions; they are related to percolation problems of finding paths of maximum weight through a 2-dimensional lattice with random weights at the vertices. Topics treated include: laws of large numbers for the speed of customers progressing through the system; stationary behaviour for systems with external arrival processes; a functional central limit theorem describing the behaviour of the front of the wave progressing through a system which starts empty; stochastic orderings for waiting times of customers at successive queues. Several open problems are noted.     

Нормаляный поридок аддитявных арифметичыских функций на множестве дсдвинутых” пностых чясел

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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.     

Positive Solutions for a Singular Nonlinear Problem on a Bounded Domain in R 2

For a bounded regular Jordan domain in R ², we introduce and study a new class of functions K() related on its Green function G. We exploit the properties of this class to prove the existence and the uniqueness of a positive solution for the singular nonlinear elliptic equation u+(x,u)=0, in D(), with u=0 on and uC(), where is a nonnegative Borel measurable function in ×(0,) that belongs to a convex cone which contains, in particular, all functions (x,t)=q(x)t ^–,>0 with nonnegative functions qK(). Some estimates on the solution are also given.     

Dualität bei konvexen Optimierungs-(Programmierungs-)Aufgaben

Summary For a linear-programming-problem Max {px | Ax b, x 0} the so called dual problem is defined as Min {b y | y A p, y 0}. The two problems are linked by the following: Existence Theorem: If both problems have feasible solutions, then both have optimal solutions; and the Duality Theorem: If one of the problems has an optimal solution, then both have optimal solutions with equal optimal values.In this paper we show that all the different generalisations of these theorems to convex, nonlinear programming problems are special cases of two general theorems, which use the means of the theory of conjugate functions as set forth byFenchel.For easy understanding we develop the necessary properties of conjugate functions and give proofs of these two general theorems, of which the duality theorem has not been proved before. In the second part we specialise these theorems to some non-linear programming problems and obtain all the different duality theorems ofDennis, Dorn, Hanson, Huard, Wolfe as special cases of these general theorems.

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Résumé On dit que les problèmes de programmation linéaire Max {px | Ax b, x 0} et Min {by | y A p, y 0} sont duals. Les deux problèmes sont liés par les théorèmes suivants: Théorème d'existence: «Si les deux problèmes ont des solutions possibles, alors ils ont tout deux des solutions optimales.» Théorème de dualité: «Si un problème a une solution optimale, alors les deux ont des solutions optimales de mêmes valeurs.» Dans ce travail nous montrons que toutes les differentes généralisations de ces théorèmes aux problèmes de programmations convexes non linéaires sont des cas particuliers de deux théorèmes généraux qui utilisent les méthodes de la theorie des fonctions conjuguées comme les poseFenchel.Pour une compréhension plus facile nous développons les propriétés nécessaires des fonctions conjuguées et nous donnons des preuves de ces deux théorèmes généraux dont le théorème de dualité n'a pas été démontré précédemment. Dans la deuxième partie nous particularisons ces théorèmes à quelques problèmes de programmation non linéaire et nous obtenons tous les différents théorèmes de dualité deDennis, Dorn, Hanson, Huard, Wolfe, en tant que cas particulier de ces deux théorèmes généraux.

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Der Deutschen Forschungsgemeinschaft danke ich für finanzielle Unterstützung während der Abfassung dieser Arbeit.

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Vorgel. v.:W. Krelle     

Some remarks on the schemesW d r

Summary Let X be an irreducible smooth projective curve of genus g. Let _d ^r (g) be the Brill-Noether Number. In this paper we prove some results concerning the schemes W _d ^r of special divisors. 1) Suppose dim (W _d–1 ^r )= _{d– 1} ^r (g)0 and _d ^r (g) < g. If W _{d– 1} ^r is a reduced (resp. irreducible) scheme, then W _d ^r is a reduced (resp. irreducible) scheme. 2) Under certain conditions, if Z is a generically reduced irreducible component of W _d–1 ^r then Z W ₁ ⁰ is a generically reduced irreducible component of W _d ^r . For r=1, we obtain some further results in this direction. 3) As an application of it we are able to prove some dimension theorems for the schemes W _d ¹ .     

A Lévy Type Martingale Convergence Theorem for Random Sets with Unbounded Values

Given a nondecreasing sequence ( _n ) of sub--fields and a real or vector valued random variable f, the Lévy Martingale convergence Theorem (LMCT) asserts that E(f/ _n ) converges to E(f/) almost surely and in L ¹, where stands for the -field generated by the _n . In the present paper, we study the validity of the multivalued analog this theorem for a random set F whose values are members of (X), the space of nonempty closed sets of a Banach space X, when (X) is endowed either with the Painlevé–Kuratowski convergence or its infinite dimensional extensions. We deduce epi-convergence results for integrands via the epigraphical multifunctions. As it is known, these results are useful for approximating optimization problems. The method relies on countability supportness hypotheses which are shown to hold when the values of the random set E(F/ _n ) do not contain any line. On the other hand, since the values of F are not assumed to be bounded, conditions involving barrier and asymptotic cones are shown to be necessary. Moreover, we discuss the relations with other multivalued martingale convergence theorems and provide examples showing the role of the hypotheses. Even in the finite dimensional setting, our results are new or subsume already existing ones.     

Connectivity of Level Sets of Quadratic Forms and Hausdorff-Toeplitz Type Theorems

Main theorem: for an arbitrary linear operator A : X X in a complex pre-Hilbert space X, dim X 3, all level sets { x X : = ,x = 1} are connected. This fails if dim X=2 and int W(A) where W(A) is the numerical range. The main theorem implies the known result on convexity of generalized numerical range of three Hermitian operators.     

The Harish-Chandra theorem for the quantum algebrau q (sl (3))

A basis of a quantum universal enveloping algebraU is constructed; the following theorem is proved with the help of this basis: For any nonzero element U, there exists a finite-dimensional representation such that(u) 0.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45 No. 3, pp. 436–439, March, 1993.     

On the Structure of Cimpact Simply Connected Manifolds of Positive Sectional Curvature

We present a first structure theorem for compact simply connected positively curved manifolds with arbitrarily small pinching constants: For each nN and 0<1, there exists a positive number V = V(n,) such that if (M,g) is a compact simply connected n-dimensional Riemannian manifold with sectional curvature 0相似文献   

On a nonlinear eigenvalue problem

In the present paper we consider a selfadjoint and nonsmooth operator-valued function on (c, d)R ¹. We suppose that the equation (L()x, x)=0,x0, has exactly one rootp(x) (c, d) and the functionf()=(L()x, x) is increasing at the pointp(x). We discuss questions of the variational theory of the spectrum. Some theorems on the variational properties of the spectrum are proved.     

On the representation theory of wreath products of finite groups and symmetric groups

Let G S_N be the wreath product of a finite group G and the symmetric group S_N. The aim of this paper is to prove the branching theorem for the increasing sequence of finite groups G S₁ G S₂ ... G S_N ... and the analog of Young's orthogonal form for this case, using the inductive approach invented by A. Vershik and A. Okounkov for the case of symmetric group.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 240, 1997, pp. 229–244.