Feldtheorie in der Variationsrechnung Mehrfacher Integrale

The theory of geodesic fields of VAGNER [12] and LIESEN [10] evidently yields the best results in comparison with the field theories of the LEPAGEan bundle. We try to elucidate the theory form a FINSLER-geometric point of view and to make it better applicable. A regularity condition analogous to the HADAMARD sufficient Legendre-condition is deduced, which is less restrictive than the conditions of the original CARATHEODORY and de DONDER-WEYL field theories. Local geodesic fields are constructed. We give a sufficient Legendre-and Weierstraß-condition and finally discuss the scope of field theory.

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Herrn Professor Dr. Dr. Otto Volk zum 80. Geburtstag am 13. Juli 1972 gewidmet     

On nonlinear plane vibration of a moving threadline

We consider plane vibrations of a moving threadline described by the model equations obtained in [1]. We study the propagation of weak discontinuities and their interaction with shock waves. We show that the only possible shocks satisfying the Lax conditions [3] and the evolutionary condition of Brun [9] are the characteristic shocks.

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Riassunto Vengono considerate le vibrazioni piane di un filo flessibile descritte dalle equazioni ottenute in [1], Viene studiata la propagazione di onde di discontinuita' e la loro interazione con un'onda d'urto. Viene anche mostrato che gli unici urti possibili che soddisfano le condizioni di Lax [3] e la condizione di evoluzione di Brun [9] sono gli urti caratteristici.

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This work was supported by C.N.R. through G.N.F.M. and by M.P.I, through Fondi del 40% e 60% per la Ricerca Scientifica.     

Sull'esistenza delle soluzioni delle equazioni differenziali ordinarie in forma implicita negli spazi di Banach

Summary This paper deals with the existence of solutions for the implicit Cauchy problem F(t, x, x)=_B, x(t₀)=x₀ in a Banach space B. By using the Kuratowski and the Hausdorff measure of non compactness, we prove an existence theorem for the previous problem (Teorema 1.1) and its extension to non compact intervals (Teorema 2.1). These results generalize the previous ones by R.Conti [1] (in the case B=R), G.Pulvirenti [2] and T. Dominguez Benavides [3], [4] (in the general case). In particular, we relax a Lipschitz condition assumed by all of the abovementioned authors. Some applications of Teorema 2.1 are presented.

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Lavoro eseguito nell'ambito del G.N.A.F.A. del C.N.R.     

On the so-called comparison theorem in the linear theory of elasticity

Summary The comparison theorems in [3] and [4] are refined to their sharpest possible common form.

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Zusammenfassung Die Vergleichstheoreme in [3] und [4] werden zu ihrer schärfsten gemeinsamen Form verfeinert.

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Dedicated to Professor Henry Görtler on the occasion of his forthcoming 70th birthday     

ON THE VITALI-HAHN-SAKS-NIKODYM THEOREM

Abstract

The classical Vitali-Hahn-Saks-Nikodym Theorem [5, Thm. I.4.8] gives a limit criterion for when a sequence of strongly additive vector measures on a σ-field of sets having their range in a Banach space can be expected to be uniformly strongly additive. In [16, Cor. 8], Saeki proved that the limit condition on the sequence of vector measures could be substantially weakened as long as the Banach space in play is “good enough”. Saeki's result was based upon his work on a class of set functions too large to have Rosenthal's Lemma at his disposal. In Section 2, we prove Saeki's result with Rosenthal's Lemma at the basis of our work and then augment our characterization of Banach spaces enjoying Saeki's result in [1] with another natural equivalent condition. In Section 3 we extend Saeki's result to Boolean algebras having the Subsequential Interpolation property.     

On the strong convergence of sequences of Halpern type in Hilbert spaces

ABSTRACT

In this paper, we introduce a concept of A-sequences of Halpern type where A is an averaging infinite matrix. If A is the identity matrix, this notion become the well-know sequence generated by Halpern's iteration. A necessary and sufficient condition for the strong convergence of A-sequences of Halpern type is given whenever the matrix A satisfies some certain concentrating conditions. This class of matrices includes two interesting classes of matrices considered by Combettes and Pennanen [J. Math. Anal. Appl. 2002;275:521–536]. We deduce all the convergence theorems studied by Cianciaruso et al. [Optimization. 2016;65:1259–1275] and Muglia et al. [J. Nonlinear Convex Anal. 2016;17:2071–2082] from our result. Moreover, these results are established under the weaker assumptions. We also show that the same conclusion remains true under a new condition.     

Analysis of 1-server n-unit parallel system

This paper deals with the analysis of 1-server n-unit system. At t = 0, all the units are switched on. The repair times of the units are arbitrarily distributed, while the failure rates of the units are constants. The system is analysed by writing integral equations for the probabilities of the system being found in various states by identifying it at suitable regeneration epochs. These equations are solved using integral transforms. The following system characteristics, namely,

1) Expected duration in [0, t] k units are operating; 0 ≤ k ≤ n

2) Expected duration in [0, t] the server is busy with the repair of the units

3) Expected number of repairs in the interval [0, t]

4) Expected number of times the system enters down-state

5) Expected duration in [0, t] the system is in down-state are evaluated to carry out the analysis of the system     

Comatrix corings and invertible bimodules

We extend Masuoka's Theorem [11] concerning the isomorphism between the group of invertible bimodules in a non-commutative ring extension and the group of automorphisms of the associated Sweedler's canonical coring, to the class of finite comatrix corings introduced in [6].

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Sunto Estendiamo il Teorema di Masuoka [11] riguardante l'isomorfismo tra il gruppo dei bimoduli invertibili su un'estensione di anelli non commutativa e il gruppo di automorfismi del coanello canonico associato di Sweedler, alla classe dei coanelli di comatrici finiti introdotta in [6].

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Research supported by the grant BFM2001-3141 from the Ministerio de Ciencia y Tecnología of Spain and FEDER.

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Supported by the grant SB2003-0064 from the Ministerio de Educación, Cultura y Deporte of Spain.     

Further observations on the anharmonic motion equation

Peters [1] discussed some theoretical aspects of the ‘ringing’ sound associated with barbershop quartet singing. These combination tones were first investigated theoretically by Helmholtz [2] who showed that the transient displacement of the ear membrane is governed by the anharmonic motion equation. Rayleigh [3] modelled the situation as an initial value problem and this is now referred to as the Rayleigh problem. Following [1] there has been a renewed interest in the problem (see [4‐8]). The intention of the present paper is to:

• illustrate the qualitative behaviour of the solutions of the Rayleigh problem by a phase plane analysis;

• describe how the method of multiple scales provides a framework within which previous analyses can be seen;

• describe an improvement to the method of multiple scales, which may also prove useful in other applications.

     

A note on stress strain relations of the linear theory of shells

Zusammenfassung Es wird gezeigt, wie man gewisse bekannte Elastizitätsgleichungen der linearen Schalentheorie [2] umschreiben kann mit Hilfe von verallgemeinerten Dehnungskomponenten und verallgemeinerten Energiefunktionen, welche von dem erstgenannten Verfasser [3] eingeführt worden sind.

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Supported by the Office of Naval Research under a contract with Massachusetts Institute of Technology.     

A Two-Sided Contracting Stefan Problem

Abstract

We study a novel two-sided Stefan problem—motivated by the study of certain 2D interfaces—in which boundaries at both sides of the sample encroach into the bulk with rate equal to the boundary value of the gradient. Here the density is in [0, 1] and takes the two extreme values at the two free boundaries. It is noted that the problem is borderline ill-posed: densities in excess of unity liable to cause catastrophic behavior. We provide a general proof of existence and uniqueness for these systems under the condition that the initial data is in [0, 1] and with some mild conditions near the boundaries. Applications to 2D shapes are provided, in particular motion by weighted mean curvature for the relevant interfaces is established.     

Topologically defined classes of commutative rings

Sunto In questo lavoro viene studiata l'operazione di somma amalgamata [13] di spazi spettrali [24] e vengono esaminate in dettaglio alcune proprietà algebriche degli anelli che inter_i vengono in tale operazione. Dei risultati ottenuti vengono poi fornite numerose applicazionalla teoria dei « D + m » domini diGilmer [19], a quella della seminormalizzazione diTra verso [38] e a quella delle CPI-estensioni nel senso diBoisen-Sheldon [5].

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Work performed under the auspices of C.N.R. (Gruppo Nazionale per le Strutture Algebriche, Geometriche e loro Applicazioni).     

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     

Existence and uniqueness of positive radial solutions for a class of quasilinear elliptic equations

The existence and uniqueness of positive radial solutions of the equations of the type [IML0001] in B_R, p>1 with Dirichlet condition are proved for λ large enough and f satisfying a condition[IML0002] is non-decreasing on [IML0003] It is also proved that all the positive solutions in C₁ ⁰(B^R) of the above equations are radially symmetric solutions for f satisfying [IML0004] and λ large enough.     

CHARACTERIZING THE NIL RADICAL OF A GAMMA RING

ABSTRACT

The nil radical, N(M) of a Γ-ring M was defined by Coppage and Luh [3], and shown by Groenewald [4] to be a special radical. We define s-prime ideals of M and show that N(M) is equal to the intersection of the s-prime ideals of M. If R is a ring, the nil radical of R considered as a Γ-ring with Γ = R is equal to the upper nil radical of R. We also give a sufficient condition for the equality N(R)* = N(M), where R is the right operator ring of M, and N(R) is its upper nil radical.     

Intégration Symplectique des Variétés de Poisson Régulières

Asymplectic integration of a Poisson manifold (M, Λ) is a symplectic groupoid (Γ,η) whichrealizes the given Poisson manifold, i.e. such that the space of units Γ₀ with the induced Poisson structure Λ₀ is isomorphic to (M, Λ). This notion was introduced by A. Weinstein in [28] in order to quantize Poisson manifolds by quantizing their symplectic integration. Any Poisson manifold can be integrated by alocal symplectic groupoid ([4], [13]) but already for regular Poisson manifolds there are obstructions to global integrability ([2], [6], [11], [17], [28]). The aim of this paper is to summarize all the known obstructions and present a sufficient topological condition for integrability of regular Poisson manifolds; we will indeed describe a concrete procedure for this integration. Further our criterion will provide necessary and sufficient if we require Γ to be Hausdorff, which is a suitable condition to proceed to Weinstein’s program of quantization. These integrability results may be interpreted as a generalization of the Cartan-Smith proof of Lie’s third theorem in the infinite dimensional case.

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Recherche supportée par D.G.I.C.Y.T. Espagne (Proyecto PB90-0765) et Xunta de Galicia (Proxecto XUGA20704B90)     

Eine Bemerkung über absolute Ebenen

By separating the compatibility conditions for incidence, order and congruence the axioms for absolute planes in [3] are put in a somewhat more lucid form.

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Die absoluten Ebenen aus [3] entsprechen den Ebenen, die die ebenen Axiome der Verknüpfung, die Axiome der Anordung und die Axiome der Kongruenz aus D.HilbertsGrundlagen der Geometrie erfüllen.

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Werner Leiner zum 60. Geburtstag     

Invariants de Von Neumann des faisceaux analytiques cohérents

In order to study the group of holomorphic sections of the pull-back to the universal covering space of an holomorphic vector bundle on a compact complex manifold, it would be convenient to have a cohomological formalism, generalizing Atiyah's index theorem. In [Eys99], such a formalism is proposed in a restricted context. To each coherent analytic sheaf on a n-dimensionnal smooth projective variety and each Galois infinite unramified covering , whose Galois group is denoted by , cohomology groups denoted by are attached, such that: 1. The underly a cohomological functor on the abelian category of coherent analytic sheaves on X. 2. If is locally free, is the group of holomorphic sections of the pull-back to of the holomorphic vector bundle underlying . 3. belongs to a category of -modules on which a dimension function with real values is defined. 4. Atiyah's index theorem holds [Ati76]: The present work constructs such a formalism in the natural context of complex analytic spaces. Here is a sketch of the main ideas of this construction, which is a Cartan-Serre version of [Ati76]. A major ingredient will be the construction [Farb96] of an abelian category containing every closed -submodule of the left regular representation. In topology, this device enables one to use standard sheaf theoretic methods to study Betti numbers [Ati76] and Novikov-Shubin invariants [NovShu87]. It will play a similar r?le here. We first construct a -cohomology theory () for coherent analytic sheaves on a complex space endowed with a proper action of a group such that conditions 1-2 are fulfilled. The -cohomology on the Galois covering of a coherent analytic sheaf onX is the ordinary cohomology of a sheaf on X obtained by an adequate completion of the tensor product of by the locally constant sheaf on X associated to the left regular representation of the discrete group in the space of functions on . Then, we introduce an homological algebra device, montelian modules, which can be used to calculate the derived category of and are a good model of the Čech complex calculating -cohomology. Using this we prove that , if X is compact. This is stronger than condition 3, since this also yields Novikov-Shubin type invariants. To explain the title of the article, Betti numbers and Novikov-Shubin invariants of are the Von Neumann invariants of the coherent analytic sheaf . We also make the connection with Atiyah's -index theorem [Ati76] thanks to a Leray-Serre spectral sequence. From this, condition 4 is easily deduced.

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Received: 30 October 1998 / Published online: 8 May 2000     

Two valued measure and summability of double sequences in asymmetric context

We extend the ideas of convergence and Cauchy condition of double sequences extended by a two valued measure (called ??-statistical convergence/Cauchy condition and convergence/Cauchy condition in ??-density, studied for real numbers in our recent paper [7]) to a very general structure like an asymmetric (quasi) metric space. In this context it should be noted that the above convergence ideas naturally extend the idea of statistical convergence of double sequences studied by Móricz [15] and Mursaleen and Edely [17]. We also apply the same methods to introduce, for the first time, certain ideas of divergence of double sequences in these abstract spaces. The asymmetry (or rather, absence of symmetry) of asymmetric metric spaces not only makes the whole treatment different from the real case [7] but at the same time, like [3], shows that symmetry is not essential for any result of [7] and in certain cases to get the results, we can replace symmetry by a genuinely asymmetric condition called (AMA).     

SOME SPECTRAL PROPERTIES OF MULTIPLIERS ON SEMI-PRIME BANACH ALGEBRAS

Abstract

We extend to arbitrary semi-prime Banach algebras some results of spectral theory and Fredholm theory obtained in [1] and [2] for multipliers defined in commutative semi-simple Banach algebras.