Topological degree of a family of convex functions and localization of nash equilibrium points

We define the degree of a family of convex functionsf _x, wherex , and prove that, if this degree is different from zero, one has at least one Nash equilibrium point in . This is a criterion of localization of such equilibrium points.     

Self-bounded controlled invariant subspaces: A straightforward approach to constrained controllability

A lattice-type structure is shown to exist in a particular subset of the set of all (A, )-controlled invariants contained in and containing , whereA denotes a linear map inR ⁿ ; , are arbitrary subspaces ofR ⁿ ; andD is an arbitrary subspace ofJ, the maximum (A, )-controlled invariant contained in . In linear system theory, this property can be used for a more direct theoretical and algorithmic approach to constrained controllability and disturbance rejection problems.     

A Generalization of the Hausdorff-Young Theorem

Considering mixed-norm sequence spaces lp,q, p, q 1, C. N. Kellogg proved the following theorem: if 1 < p 2 then lp,2 and lp,2 , where 1/p + 1/p = 1. This result extends the Hausdorff-Young Theorem.We introduce here multiple mixed-norm sequence spaces , examine their properties and characterize the multipliers of spaces of the form lp,[s;n],q, with the index s repeated n times. By an interpolation-type argument we prove that (l,[2;n],2, lp,[1;n],1) for 1 < p 2. Using these results we obtain a further generalization of the Hausdorff-Young Theorem: if 1 < p 2 then lp,[2;n] and lp,[2;n] for each n = 0, 1, 2, ¨. The spaces lp,[2;n] decrease and lp,[2;n] increase properly with n for 1 < p < 2 and 1/p + 1/p = 1. We also extend a theorem of J. H. Hedlund on multiplers of Hardy spaces and deduce other results.     

Über die Bestimmung der Anzahl dreiachsiger Ellipsoide aus der Anzahl ihrer Schnittellipsen in zufälligen Schnittebenen

Summary A diagram is given for reading off the shape-factor oftri-axial ellipsoids ( = average distance between parallel tangent planes over all orientations).The knowledge of permits the calculation of the number of ellipsoids present per unit volume from the number of elliptical sections observed per unit area.     

Approximations for optimal Laplace transform inversion

A function (p) of the Laplace transform operatorp is approximated by a finite linear combination of functions (p+ _r ), where (p) is a specific function ofp having a known analytic inverse (t), and is chosen in accordance with various considerations. Then parameters _r ,r=1, 2,...,n, and then corresponding coefficientsA _r of the (p + _r ) are determined by a least-square procedure. Then, the corresponding approximation to the inversef(t) of (p) is given by analytic inversion of _r=1 ⁿ A _r (p+ _r ). The method represents a generalization of a method of best rational function approximation due to the author [which corresponds to the particular choice (t)1], but is capable of yielding considerably greater accuracy for givenn.The computations for this paper were carried out on the CDC-6600 computer at the Computation Center of Tel-Aviv University. The author is grateful to Dr. H. Jarosch of the Weizmann Institute of Science Computer Center for use of their Powell minimization subroutine (Ref. 1).     

On the embedding of core-free projective images of normal subgroups

Sunto Siano G e due gruppi, : GG una proiettività e N un sottogruppo normale di G. Vengono provati alcuni risultati sulle proprietà di immersions in e G rispettivamente, del gruppi , la chiusura normale di N in modulo il nocciolo di N in , e .

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Supported by a C.N.R. grant.     

Direct Theorem about Approximation on a Family of Two Segments

Introduce the notation: , is the union of two segments [-1,1] and [-1 + ,1+ ], is a noninteger number, is the Hölder class with exponent on The following result announced by the authors in [J. Math. Sci. 117 (2003), No. 3] is proved. There exist numbers a ₁ ( ) , b ₁ ( ) 0 depending only on such that for any there exists a polynomial , such that . Bibliography: 11 titles.     

Lattice points in a circle and divisors in arithmetic progressions

Let A(x) denote the number of lattice points in the circle u²+v²x and define as the infimum of all reals for which . The objective of this paper is to show that 35/108 which improves upon all previously known results. This estimate is an immediate consequence of a surprisingly easy generalization of KOLESNIK's work on Dirichlet's divisor problem to divisor functions with respect to arithmetic progressions.     

Über Abstand 1 erhaltende Abbildungen in Minkowski-Ebenen

By using a computer the following theorem is proved: Consider K=GF(q), q {32,64,81,128}, :K² K² bijective such that P,Q K². Then is a semi-isometry. The assumption bijective can be dropped if q {32,128}.     

Two Splittings of a Square Matrix

In this note it is shown that any square matrix AC ^n×n can be represented as the sum A= , where is complex symmetric and rank . The corresponding persymmetric result can be used in finding the terms of a small rank perturbed Toeplitz matrix via an O(n ²) computation. This allows one to perform fast matrix–vector products in case n is large.     

Generalized Quasi-Variational Inequalities in Infinite-Dimensional Normed Spaces

In this paper, we deal with the following problem: given a real normed space E with topological dual E*, a closed convex set XE, two multifunctions :X2^X and , find such that We extend to the above problem a result established by Ricceri for the case (x)X, where in particular the multifunction is required only to satisfy the following very general assumption: each set (x) is nonempty, convex, and weakly-star compact, and for each yX–:X the set is compactly closed. Our result also gives a partial affirmative answer to a conjecture raised by Ricceri himself.     

Finitely Valued f-Modules, An Addendum

In an -group M with an appropriate operator set it is shown that the -value set (M) can be embedded in the value set (M). This embedding is an isomorphism if and only if each convex -subgroup is an -subgroup. If (M) has a.c.c. and M is either representable or finitely valued, then the two value sets are identical. More generally, these results hold for two related operator sets ₁ and ₂ and the corresponding -value sets and . If R is a unital -ring, then each unital -module over R is an f-module and has exactly when R is an f-ring in which 1 is a strong order unit.     

An Intersection Theorem on an Unbounded Set and Its Application to the Fair Allocation Problem

We prove the following theorem. Let m and n be any positive integers with mn, and let be a subset of the n-dimensional Euclidean space ⁿ . For each i=1, . . . , m, there is a class of subsets M _i ^j of ^Tn . Assume that for each i=1, . . . , m, that M _i ^j is nonempty and closed for all i, j, and that there exists a real number B(i, j) such that and its jth component _{xjB(i, j)} imply . Then, there exists a partition of {1, . . . , n} such that for all i and We prove this theorem based upon a generalization of a well-known theorem of Birkhoff and von Neumann. Moreover, we apply this theorem to the fair allocation problem of indivisible objects with money and obtain an existence theorem.     

On a Problem of the Theory of Multiply Local Formations

We describe the -closed n-multiply local formations such that the lattice of all -closed n-multiply local formations between and is Boolean.     

Variants of Robinson's essentially undecidable theoryR

Cobham has observed that Raphael Robinson's well known essentially undecidable theoryR remains essentially undecidable if the fifth axiom scheme is omitted. We note that whether the resulting system is in a sense minimal essentially undecidable depends on what the basic constants are taken to be. We give an essentially undecidable theory based on three axiom schemes involving only multiplication and less than or equals.     

Extremal Decompositions of Riemann Surfaces and Quasiconformal Mappings of Special Form

It is shown that the extremal decomposition of a finite Riemann surface into a system of doubly connected domains may be associated with a family of quasiconformal mappings , which are similar to the Teichmüller mappings. In the case , this construction allows us to prove that the extremal value of the functional in the indicated problem on the extremal decomposition is a pluriharmonic function of the coordinates of the distinguished points on . Bibliography: 5 titles.     

A Remark on Subgroup Separability in the Class of Finite π-Groups

We prove that if a group G is residually , then for every -subgroup of the group G, the set of -roots from this subgroup is a -separable -subgroup.     

Two natural metrics and their covariant derivatives on a manifold of embeddings

OnE(M, ⁿ ), the Fréchet-manifold of all smooth embeddings of a smooth, compact, closed, orientable manifoldM (of dimensionn-1) into ⁿ two natural metricsG and are considered. The metric plays a central rôle in elasticity theory. Using a generalised notion of the Fréchet derivative their respective spraysS and and the correspoonding Levi-Civita connections are computed. BothS and are smooth in a well defined sense. In contrast toS the spray turns out to be trivial.     

Lagrangian Globalization Methods for Nonlinear Complementarity Problems

This paper extends the Lagrangian globalization (LG) method to the nonsmooth equation arising from a nonlinear complementarity problem (NCP) and presents a descent algorithm for the LG phase. The aim of this paper is not to present a new method for solving the NCP, but to find such that when the NCP has a solution and is a stationary point but not a solution.     

Conditions of Local Asymptotic Normality for Gaussian Stationary Processes

Let {\bold x}[] be a stationary Gaussian process with zero mean and spectral density f, let be the -algebra induced by the random variables {\bold x}[], D(R¹), and let _t, t > 0, be the -algebra induced by the random variables x[],supp [-t,t]. Denote by (f) the Gaussian measure on generated by {\bold x}. Let _t(f) be the restriction of (f) to _t. Let f and g be nonnegative functions such that the measures _t(f) and _t(g) are absolutely continuous. Put