On solving the second four-element Riemann-type boundary-value problem on a disk

The paper is devoted to solving one of the main four-element Riemann-type boundary-value problems in classes of piecewise-bianalytic functions with unit circumference as jump line. We prove a theorem on reduction of solving the stated problem to solving two vector-matrix Riemann problems with respect to piecewise-analytic vector-functions. Using it, we obtain an algorithm for solving the problem and conditions under which one can get a constructive and explicit solution in terms of Cauchy-type integrals. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 46, No. 3, pp. 377–385, July–September, 2006.     

Bartle-Graves Theorem Revisited

We take a fresh look at the Bartle-Graves theorem pointing out the main differences with the standard implicit function theorem. We then present a set-valued version of this theorem which generalizes some recent results. Applications to variational inequalities and differential inclusions are also given.

     

Ein einfacher Beweis des Satzes von Alexandroff-Lester

By presenting counterexamples and a new proof, we determine all metric vector spaces, for which the theorem of ALEXANDROFF-LESTER holds. In this context, the theorem will be extended to certain nonregular metric vector spaces of characteristic 2.

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Dedicated to Professor Giuseppe Tallini on the occasion of his 60th birthday

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Herrn H. Lenz danke ich für mehrere Vorschläge zur durchsichtigeren Formulierung der Beweise.     

Un teorema di esistenza per campi di orientori e applicazioni al problema libero del Calcolo delle Variazioni in dimensione infinita

Summary We prove a compactness theorem for a sequence of functions which take values in an infinite dimensional Hilbert space and which satisfy a differential inclusion, related to a map Q. Then we give sufficient conditions for the existence of an optimal solution of a free Lagrange problem, always in the infinite dimensional case, applying the theorem previously proved to a minimizing sequence.

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Entrata in Redazione il 19 maggio 1978.     

Effective Universality Theorem: A Survey

In 1975, S.M. Voronin proved the universality theorem for the Riemann zeta-function. This famous theorem is ineffective. Here we survey results related to the effectivization of Voronin’s theorem.

     

An elementary proof of the Loomis–Whitney theorem

Loomis and Whitney proved an inequality between volume and areas of projections of an open set in n-dimensional space related to the isoperimetric inequality. They reduced the problem to a combinatorial theorem proved by a repeated use of Hölder inequality. In this paper we prove a general inequality between real numbers which easily implies the combinatorial theorem of Loomis and Whitney.

     

Ein Beweis von H. Karzel und I. Pieper

An incidence space of dimension at least 2 and order at least 4 in which all planes are affine planes is an affine space ([1 ] and [2]). Karzel's and Pieper's proof of this theorem is represented here lightened from some material relevant to the report [2] but not to this theorem.

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Herrn Professor Dr. Adriano Barlotti zum 65. Geburtstag     

A Characterization of the Standard Tori in C2 as Compact Lagrangian ξ-Submanifolds*

In this paper, the authors give a characterization theorem for the standard tori S¹(a) × S¹(b), a, b > 0, as the compact Lagrangian ξ-submanifolds in the two-dimensional complex Euclidean space C² , and obtain the best version of a former rigidity theorem for compact Lagrangian ξ-submanifold in C² . Furthermore, their argument in this paper also proves a new rigidity theorem which is a direct generalization of a rigidity theorem by Li and Wang for Lagrangian self-shrinkers in C².     

Exterior square gamma factors for cuspidal representations of $$\mathrm {GL}_n$$ GL n : simple supercuspidal representations

We compute the local twisted exterior square gamma factors for simple supercuspidal representations, using which we prove a local converse theorem for simple supercuspidal representations.

     

A Note on a Unicity Theorem of N. Terglane

In this article, we prove a unicity theorem of meromorphic functions sharing three values, which is an improvement of the theorem given by Terglane. Some examples show that the result in this article is best possible.     

Sul teorema di Riesz-Thorin

Summary We give an extension of the Riesz-Thorin theorem to linear operators which map Morrey spaces in spaces of the same or similar type.

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A Bruno Finzi nel suo 70^mo compleann

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Entrata in Redazione il 10 dicembre 1969.     

Fefferman–Stein inequalities for the dyadic-like maximal operators

The paper contains a transference theorem which allows to extend a large class of unweighted inequalities for the dyadic maximal operator to their weighted Fefferman–Stein counterparts on general probability spaces.

     

A divergence theorem for Finsler metrics

By means of the general form of Stokes' theorem on manifolds a divergence theorem is derived for hypersurfaces which bound a compact region of ann-dimensional Finsler spaceF _n . In general the integrand of then-fold volume integral will depend on the covariant derivatives of an arbitrary vector field which defines the element of support; certain conditions under which this dependence may be circumvented are discussed. The scalar curvature ofF _n is expressed in terms of the divergence of a certain vector field: forn=2 this formula reduces to a particularly simple form, and its substitution into the aforementioned divergence theorem gives rise to a formula which represents a generalization of the classical Gauss-Bonnet Theorem.

     

Darstellung und Entwicklung des Restgliedes der Gregoryschen Quadraturformel mit Hilfe von Spline-Funktionen

Summary Due to a theorem of Peano, the remainder term of Gregory's quadrature formula may be expressed by means of a polynomial spline-function. An explicite representation of this spline-function is given by a direct method, which does not make use of Peano's theorem. As an application the remainder term obtained is developed in terms of boundary derivations of the integrand, which leads to a general quadrature formula containing the Gregory formula and the Euler-MacLaurin summation formula as special cases.

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Herrn Johannes Weissinger zum 60. Geburtstag am 12. 5. 1973 gewidmet     

Beiträge zur Shannonschen Informationstheorie im Falle nichtstationärer Kanäle

Coding theorem and strong converse of the coding theorem are proved for nonstationary semicontinous channels and for almost periodic channels with unknown phase. More general cases are considered.

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Die Arbeit entstand teilweise im Rahmen eines von der Deutschen Forschungsgemeinschaft finanzierten Forschungsvorhabens.

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Herrn ProfessorJacobs gilt mein besonderer Dank für zahlreiche Anregungen.     

Punktprozesse mit Wechselwirkung

We consider classical, continuous systems of particles in r dimensions described by infinite system equilibrium states which have been defined by Dobrushin [5] and Lanford/Ruelle [24]. For a large class of potentials we prove the theorem of Lee/Yang [43] together with a variational characterizafor these equilibrium states. The main idea stems from Föllmer [9] who showed that in the case of lattice systems, the theorem of Lee/Yang is intimately related to Birkhoff's ergodic theorem and McMillan's theorem (ergodic theorem of information theory). Following this idea we obtain as main results an r-dimensional ergodic theorem for random measures in ^r , limit theorems concerning energy and entropy and an r-dimensional version of Breiman's theorem showing that there is almost sure convergence behind McMillan's theorem.

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Danken möchten wir Klaus Krickeberg, der diese Arbeit durch eine Fülle wertvoller Hinweise und Anregungen gefördert hat.     

Spine Decompositions and Limit Theorems for a Class of Critical Superprocesses

In this paper we first establish a decomposition theorem for size-biased Poisson random measures. As consequences of this decomposition theorem, we get a spine decomposition theorem and a 2-spine decomposition theorem for some critical superprocesses. Then we use these spine decomposition theorems to give probabilistic proofs of the asymptotic behavior of the survival probability and Yaglom’s exponential limit law for critical superprocesses.

     

On Harmonic Diffeomorphisms of the Unit Disc onto a Convex Domain

We prove a theorem for harmonic diffeomorphisms between the unit disc and a convex Jordan domain, which is a generalization of Heinz theorem [E. Heinz (1959). On one-to-one harmonic mappings. Pacific J . Math ., 9 , 101-105] for harmonic diffeomorphisms of the unit disc onto itself. We give a number of corollaries of the theorem we prove.     

Un teorema di esistenza per controlli ottimi di sistemi definiti da equazioni integrali di Urysohn

Summary In this paper an existence theorem is proved for optimal control problems described by Urysohn systems, with pointwise constraints on controls and states. An example of application of the existence theorem is given.

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Entrata in Redazione il 15 marzo 1978.

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Lavoro eseguito nell'ambito della mia attività di ricerca come borsista C.N.R., presso l'Università di Genova.