Translation Invariant Positive Definite Hermitian Bilinear Ultradistributions

Every translation invariant positive definite Hermitian bilinear functional on the Gel'fand-Shilov space sMpMp(n×nK) of general type S is of the form B(,) = (x)(x)d(x), , sMpMp (n), where is a positive {M}-tempered measure, i.e., for every > 0 exp[-M(|x|)] d(x) < . To prove this we prove Schwartz kernel theorem for {M}-tempered ultradistributions and need Bochner-Schwartz theorem for {M}-tempered ultradistributions. Our result includes most of the quasianalytic cases. Also, we obtain parallel results for the case of Beurling type (Mp.     

Zur gruppentheoretischen Darstellung der projektiv-metrischen Geometrien

The extended reflection group of a metric vector space was introduced by Nolte [10] to get a group theoretical representation of the corresponding protective metric geometry (in the sense of Schröder [12]). Nolte characterizes the extended. reflection, group among all representing groups if the characteristic of the underlying field is 2. We give a new proof of Nolte's result which does not depend on the characteristic. As a consequence we get that the generated protective Clifford group (see [12]) is isomorphic to the extended reflection group. Finally, we give examples of other representing groups.     

Two-plane sub-bundles of nonorientable real vector-bundles

Let be a nonorientable m-plane bundle over a CW complex X of dimension m or less. Given a 2-plane bundle over X, we wish to know whether can be embedded as a sub-bundle of . The bundle need not be orientable. When is even-dimensional there is the added complication of twisted coefficients. In that case, we use Postnikov decomposition of certain nonsimple fibrations in order to describe the obstructions for the embedding problem. Emery Thomas [11] and [12] treated this problem for and both orientable. The results found here are applied to the tangent bundle of a closed, connected, nonorientable smooth manifold, as a special case.The writing of this paper was partially supported by CNPq grant     

Existence and uniqueness of global solutions for wave equations with scalar nonlinearities

We prove an existence and uniqueness theorem of global solutions for wave equations with scalar nonlinearities. Our paper is a generalization of the work of D. Kremer [4].

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Zusammenfassung Es wird ein Existenz- und Eindeutigkeitssatz für globale Lösungen von Wellengleichungen mit skalaren Nichtlinearitäten bewiesen. Die Arbeit stellt eine Verallgemeinerung der Arbeit von D. Kremer [4] dar.

     

The intuitionistic propositional calculus with quantifiers

Let L be the language of the intuitionistic propositional calculus J completed by the quantifiers and , and let calculus 2J in language L contain, besides the axioms of J, the axioms xB (x) B(y) and B(y) xB (x). A Kripke semantics is constructed for 2J and a completeness theorem is proven. A result of D. Gabbay is generalized concerning the undecidability of C2J⁺-extension of 2J by schemes x (x B) and x(A B(x))A xB (x) specificially: the undecidability is proven of each T theory in language L such that [2J]T [C2J+] ([2J] ([2J] denotes the set of all theorems of calculus 2J).Translated from Matematicheskie Zametki, Vol. 22, No. 1, pp. 69–76, July, 1977.     

Convolution semigroups and central limit theorem associated with a dual convolution structure

We consider hypergroups associated with Jacobi functions ⁽⁾ (x), (–1/2). We prove the existence of a dual convolution structure on [0,+[i(]0,s ₀]{{) =++1,s ₀=min(,–+1). Next we establish a Lévy-Khintchine type formula which permits to characterize the semigroup and the infinitely divisible probabilities associated with this dual convolution, finally we prove a central limit theorem.     

Open maps of toposes

We present a concept of openness for geometric morphisms between toposes, which extends the usual notion of open map of spaces, and investigate some of its properties. Our two main results are that open maps are precisely those whose inverse image functors preserve full first-order logic, and that open maps and open surjections are stable under bounded pullback in Top. These two results extend earlier work of C. J. Mikkelsen [11] and M. Coste [3] respectively; they are also closely related to recent work of A. Joyal and M. Tierney [9].     

Harmoniques Sphériques et Représentations Irréductibles de O ( ∞ )

We build an irreducible unitary representation of SO( ) from the usual one of SO( n ) in the space of harmonic homogeneous polynomials of degree m of ⁿ . We give a characterization of these new representations which extends in a natural way the finite dimensional characterization. In the particular case of SO( ), we thus get some results of Olshanskii (cf. [12]). This leads to a new proof of McKean conjecture about irreducible representations of ( infin ) (cf. [10]).     

Résolution numérique des grands systèmes différentiels linéaires

Sommaire La solution stricte d'un système différentiel linéaire à coefficients constants [d /d t] = [A] [] + [f (t) ] est donnée par: [ (t)]= [e^At] [ (0) ] + f [e^A(t–)] [f (T) ] d .Cette relation, utilisée dans une méthode de pas à pas, permet le calcul de [(t+u)] en fonction de [(t)]. La mise en oeuvre numérique de cette formule nécessite le calcul de [e^A] et de l'intégrale de matrice du second membre.Le sujet de cette étude est la mise au point de techniques d'approximation permettant le calcul effectif de [e ^Aµ] et de l'intégrale de matrice par des méthodes qui peuvent s'adapter en particulier aux systèmes différentiels à très grand nombre d'inconnues, qui apparaissent par exemple dans l'approximation par discrétisation enx ety, de l'équation aux dérivées partielles, dite de la chaleur.     

On the continuity of midpoint hull convex set-valued functions

Summary The concept of hull convexity (midpoint hull convexity) for set-valued functions in vector spaces is examined. This concept, introduced by A. V. Fiacco and J. Kyparisis (Journal of Optimization Theory and Applications,43 (1986), 95–126), is weaker than one of convexity (midpoint convexity).The main result is a sufficient condition for a midpoint hull convex set-valued function to be continuous. This theorem improves a result obtained by K. Nikodem (Bulletin of the Polish Academy of Sciences, Mathematics,34 (1986), 393–399).     

Über eine Art von gemischter Topologie und einen Satz von A. Grothendieck über (DF)-Räume

Let E be a real or complex linear space, s and t linear topologies on E, ts. We define some kind of mixed topology m(s, t) such that tm(s, t)s. The linear topology m(s, t) is characterized by a theorem of Banach-Steinhaus type and we show that the condition t=m(s, t) covers various concepts of barrelledness. Secondly we show that s and m(s, t) induce the same uniformity on certain subsets of E, if there is a base of s-neighbourhoods of O which are t-closed; this theorem will be a generalization of a well known theorem of A. Grothendieck on (DF)-spaces.     

Some functional inequalities and their Baire category properties

Summary In the paper we consider, from a topological point of view, the set of all continuous functionsf:I I for which the unique continuous solution:I – [0, ) of(f(x)) (x, (x)) and(x, (x)) (f(x)) (x, (x)), respectively, is the zero function. We obtain also some corollaries on the qualitative theory of the functional equation(f(x)) = g(x, (x)). No assumption on the iterative behaviour off is imposed.     

Inclusion statements for operator equations by a continuity principle

The paper is concerned with Range-Domain Implications MvCvK, where M is a given operator and C,K denote given sets. Sufficient conditions are derived by a very general continuity principle. Various special cases are considered such as inverse-positivity, MvMwvw, and a generalization H(,[,])MvH(,[,]) v, where Mu=H(u,u) and [,] denotes an order interval. These results are applied to differential operators related to boundary or initial value problems. The goal is to furnish a simple uniform approach, to explain its application, and to provide a kind of survey on what problems have been treated in this way.     

Piecewise polynomial approximation

For best piecewise polynomial approximation _n=_n (f; [0, 1]) of a functionf, which is continuous on the interval [0, 1] and admits a bounded analytic continuation onto the disk K=z:¦z–1¦<, the relation _n=o[ _f (e ^–n )] is valid.Translated from Matematicheskie Zametki, Vol. 11, No. 2, pp. 129–134, February, 1972.     

Lagrangian Spheres in the Complex Euclidean Space Satisfying a Geometric Equality

For a Lagrangian submanifold of Cⁿ with scalar curvature and mean curvature vector H, the inequality ( n²(n-1)/n+2 |H|²) holds, and the equality is given only in open sets of the Lagrangian subspaces of ⁿ or of the Whitney sphere (cf. [RU] and also [BCM]). In this paper, a one-parameter family of Lagrangian spheres including the Whitney sphere is constructed. They satisfy a geometric equality of type = |H|², with >0, and they are globally characterized inside the family of compact Lagrangian submanifolds with null first Betti number in Cⁿ.     

Spline polynomials with a prescribed sequence of extrema

In the present note a theorem about strong suitability of the space of algebraic polynomials of degree n in C[a,b] (Theorem A in [1]) is generalized to the space of spline polynomials _{[a, b} ^{]n, k} (n2, 0) in C[a, b]. Namely, it is shown that the following theorem is valid: for arbitrary numbers ₀, ₁, ..., _n+k, satisfying the conditions (_i–_i–1) (_{i+1{ i}< 0(i=1, ..., n +k–1), there is a unique polynomials _n,k (t) _{[a, b} ^]/n,k and pointsa=₀,<₁<...< _n+k– 1< _n+k = b (₁1 <_n, ..., _kk<_n+k–1), such that s_n,k(_i) = _i(i=0, ..., n + k), s_n,k(_i)=0 (i=1, ..., n + k–1).Translated from Matematicheskii Zametki, Vol. 11, No. 3, pp. 251–258, March, 1972.     

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.     

Existence theorems for nonlinear evolution inclusions

Summary In this paper we obtain an existence theorem for the abstract Cauchy problem for multivalued differential equations of the form u–^– f(u)+G(u), u(O)=x₀, where ^– f is the Fréchet subdifferential of a functionf defined on an open subset of a real separable Hilbert space H, taking its values in R {+} and G is a multifunction from C([0, T], ) into the nonempty subsets of L²([0, T], H). As an application we obtain an existence theorem for the multivalued perturbed problem x–^– f(x)+F(t, x), x(0)=x₀, where F:[0, T]×(H) is a multifunction satisfying some regularity assumptions.     

Parabolic Differential Equations in Ordered Banach Spaces of Bounded Functions

Let A be a set, and let E be the Banach space of bounded functions : A R, equipped with its natural order. With a rectangle R = (a,b) × (0,T] let F(x,t,) : R × E E be a bounded, continuous function satisfying a local Hölder condition and being quasimonotone increasing with respect to . Then there exists a solution u: [a,b] × [0,T] E of the problem ut(x,t) – uxx(x,t) = F(x,t,u(x,t)) ((x,t) R), u(x,t) = 0 ((x,t) R R).     

Transferring estimates for zonal convolution operators

_p - , . Ad- .