Little-Hölder continuous functions and elliptic systems

Summary We give an integral characterization of Little-H?lder continuous functions; by means of it, we prove Little-H?lder continuity of first derivatives of weak solutions to elliptic systems in divergence form.

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Riassunto In questo lavoro diamo una caratterizzazione integrale delle funzioni piccolo-h?lderiane; per mezzo di essa proviamo un teorema di regolarità piccolo-h?lderiana per le soluzioni deboli di sistemi ellittici in forma divergenza.

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The author is member of G.N.A.F.A.; this work is supported by the Ministero Pubblica Istruzione.     

Dérivées de Lie des spineurs

Résumé On étudie le problème de la transformation d'un champ de spineurs sur une variété spinorielle par un groupe à un paramètre de difféomorphismes de la variété, en généralisant aux champs de spineurs la notion de dérivation de Lie. On peut alors étudier l'invariance d'un système d'équations aux dérivées partielles sur les spineurs (équations de Dirac) avec des méthodes analogues à celles qui sont valables dans le cas des tenseurs (équations de Maxwell).

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Summary The notion of a Lie derivative is generalized to spinor fields on a Spin manifold and the problem of transforming a spinor field under a one-parameter group of diffeomorphisms of the manifold is studied. The invariance of a differential system acting on spinors (Dirac's equations) is studied using methods similar to those used in the case of tensors (Maxwell's equations).

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Actuellement: Université des Sciences et Techniques de Lille.

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Entrata in Redazione il 19 aprile 1971.     

A memoir on the projective geometry of spinors

Summary The classical methods of projective geometry are applied to a number of questions in general relativity, by using the Van der Waerden spinor analysis. These include a new geometric theory of spinors, refinements in the spinor calculus, the classification of electromagnetic and gravitational fields, Weyl-Maxwell fields, a classification of the Einstein spinor, and the projective geometry of the Bel-Petrov types. Dedicated to the memory of my teacher and friend Professor Dr.Vaclav Hlaváty (1894–1969) Entrata in Redazione il 14 marzo 1975.     

Riferimenti isotropi in relatività generale

We consider anew approach for a relativistic frame of reference in the polar sense [1] (two-congruences) and the related non-holonomic techniques for anull congruence (light flux); 1° order characteristic tensors, Ricci rotation coefficients, longitudinal and transversal covariant derivatives are deduced in general form. Thus, the method of a standard frame of reference is extended to thenull case.

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Sunto Si consideraun nuovo approccio per un sistema di riferimento relativistico nel senso polare [1] (bicongruenza), adattando le tecniche non-olonome connesse al caso in cuiuna delle due congruenze sia nulla (flusso di luce); si derivano, in forma generale, i tensori caratteristici del 1° ordine, i coefficienti di rotazione di Ricci, le derivate covarianti longitudinali e trasversali. I metodi dei sistemi di riferimento standard vengono così estesi alcaso nullo, determinando unaalternativa tensoriale al cosidetto metodo (non reale) delle tetradi nulle [15].

     

Barriers on cones for uniformly elliptic operators

Summary On every right circular cone and for every uniformly elliptic operator in nondivergence form there exists a barrier, that is, a supersolution which is zero at the vertex and positive elsewhere on the closed cone. These barriers are applied to the Dirichlet problem. Other applications are deferred to a later paper on ? extremal barriers ?

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Sommario Si considerano operatori uniformemente ellittici del secondo ordine, ma in forma di nondivergenza, a coefficienti misurabili in un cono circolare retto; si prova che esiste una funzione barriera, vale a dire una supersoluzione positiva nel cono chiuso eccettuato il vertice ove si annulla. Queste barriere sono impiegate per lo studio del problema di Dirichlet. Altre applicazioni sono rinviate ad una successiva pubblicazione su le ? barriere estremali ?.

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This work was supported by the Air Force Office of Scientific Research and the National Academy of Sciences through a Postdoctoral Research Fellowship for 1964–1965 at the University of Genova and also in the writeup stage by AFOSR grant number 553–64.     

Integral Equation for the Spinor Amplitude of a Dirac Particle in a Curved Space–Time

We obtain a system of integral equations for the spinor amplitude of a wave packet describing a massive neutral Dirac particle in a curved space–time with an arbitrary geometry. This equation permits describing the spin dynamics of fermions in gravitational fields adequately to the quantum nature of spin. We consider a specific example of the Kerr–Schild metric. We also discuss the problem of massive neutrino oscillations in an external gravitational field.     

Two constructions of affine Lie algebra representations and boson-fermion correspondence in quantum field theory

We establish an isomorphism between the vertex and spinor representations of affine Lie algebras for types D_l⁽¹⁾and D_{l + 1}⁽²⁾. We also study decomposition of spinor representations using the infinite family of Casimir operators and prove that they are either irreducible or have two irreducible components. We show that the vertex and spinor constructions of the representations can be reformulated in the language of two-dimensional quantum field theory. In this physical context, the two constructions yield the generalized sine-Gordon and Thirring models, respectively, already in renormalized form. The isomorphism of representations implies an equivalence of these two models which is known in quantum field theory as the boson-fermion correspondence     

Estimates for normal forms of differential equations near an equilibrium point

We consider the problem of finding a normal form for differential equations in the neighbourhood of an equilibrium point, and produce general explicit estimates for both the normal form at a finite order and the remainder, using the method of Lie transforms. With such technique, the classical Poincaré-Dulac theorems are recovered, and the problem of the stability of a reversible system of coupled harmonic oscillators up to exponentially large times is discussed.

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Riassunto Si considera il problema di porre in forma normale un sistema di equazioni differenziali nell'intomo di un punto di equilibrio, e si danno in generale stime esplicite sia per la forma normale troncata ad un ordine finito che per i resti. Si fa uso dell'algoritmo della trasformata di Lie. Con questo metodo si riottengono i teoremi classici di Poincaré-Dulac, e si discute il problema della stabilità per tempi esponenzialmente lunghi di un sistema reversibile di oscillatori armonici accoppiati.

     

A remark on some degenerate elliptic problems

We prove existence of bounded weak solutions to a quasilinear degenerate elliptic equation in divergence form.

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Sunto Dimostriamo l'esistenza di soluzioni deboli limitate per una equazione ellittica, quasilineare, degenere, in forma di divergenza.

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We acknowledge the support of MURST (60%, 40%) and GNAFA-CNR.     

Un risultato di regolarità per le soluzioni delle equazioni ellittiche degeneri del secondo ordine di tipo non variazionale

Riassunto Si definisce una classe di transformazioniK-quasiconformi su un cerchio del piano, doveK è una funzione illimitata. Per tali transformazioni si dimostra una proprietà di continuità che si riduce alla H?lderianità seK è costante. Come applicazione si dimostra che le soluzioni appartenenti ad un opportuno spazio di Sobolev con peso di una equazione ellittica degenere in forma non variazionale, hanno il gradiente continuo. Si riportano infine degli esempi per far notare che le ipotesi fatte sono essenziali.

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Summary We define a class ofK-quasiconformal mappings on a disk of the plane, whereK is an unbounded function. For such mappings we derive a continuity property similar to those that we have whenK is a constant. As application we obtain that the solutions belonging to a suitable weighed Sobolev space of a second order elliptic degenerate equation in non variational form (in two variables) have the gradient continuous.

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

Principi variazionali nella teoria lineare delle miscele di due solidi elastici

Riassunto In questa nota gli Autori stabiliscono dei principi variazionali che caratterizzano in maniera completa il più generale problema ai limiti sia della statica sia della dinamica di una miscela isotropa di due solidi elastici lineari. Questi principi sono del tutto generali poichè non si richiede agli stati o processi ammissibili di soddisfare ad alcuna delle equazioni di campo e delle condizioni iniziali o al contorno. Tale risultato viene ottenuto esprimendo i problemi considerati in forma operazionale.

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Summary In this paper the Authors establish variational principles which characterize the most general boundary-value problem for statics and dynamics of an isotropic mixture of two linear elastic solids. These principles are completely general because the admissible states or processes are not required to meet any of the field equations, boundary or initial conditions and they are obtained by expressing the problems in operator form.

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Il lavoro è stato eseguito nell’ambito del Gruppo Nazionale per la Fisica Matematica del C.N.R.     

On the variational formulation for linear initial value problems

Summary The paper shows the key place of the choice of the bilinear form in order to give a variational formulation to a given problem. In particular it is shown how the use of a convolution bilinear form makes possible a variational formulation for linear initial value problems. A critical survey of the three main methods that was devised in the past to solve the same problem is done.

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Sunto. La nota mette in evidenza il ruolo fondamentale della scelta di una forma bilineare al fine di dare formulazione variazionale ad un dato problema. In particolare è mostrato come l'uso di una forma bilineare di convoluzione renda possibile la formulazione variazionale dei problemi ai valori iniziali. Si fa un esame critico dei tre principali metodi che sono stati escogitati nel passato per risolvere lo stesso problema.

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This work has been sponsored by the Consiglio Nazionale delle Ricerche.

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Entrata in Redazione il 19 novembre 1971.     

On the symmetric conservative form of Landau's superfluid equations

Summary It is shown how to obtain Landau's equations for a superfluid in a symmetric conservative form. On account of the constraints involved, this case is more complicated than those of the classical theory developed by Godunov and by other authors.

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Sunto Si dimostra che è possibile porre in forma simmetrica conservativa il sistema di equazioni di Landau del superfluido. A causa dei vincoli a cui é soggetto il moto del sistema la teoria che si usa risulta piú complessa di quella classica sviluppata da Godunov e da altri autori.

     

Ulteriori sviluppi della teoria diametrale delle superficie algebriche

Riassunto Si precisa la forma dell'equazione di una superficie algebrica con punto principale proprio, improprio o indeterminato e si stabilisce inoltre il numero dei piani diametrali principali di una superficie algebrica qualunque determinando i tipi di superficie algebriche che ne possiedono infiniti.

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Résumé On établit la forme de l'équation d'une surface algébrique ayant point principal propre, impropre ou indéterminé et on établit aussi le nombre des plans diamétraux principaux d'une surface algébrique quelconque en déterminant les types des surfaces algébriques qui en possèdent d'infinis.

     

On lattice-like properties of orthoquaternary categories

Summary We point a class of categories for which necessary and sufficient conditions in order they have an orthodox quaternary symmetrization are expressible by means of lattice-like properties of subobjects, these one given by axiomsD1)-5). This implies that canonical isomorphisms between subquotients are composable and that Noether isomorphism theorems hold.

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Riassunto Si individua una classe di categoric per le quali l’esistenza e l’ortodossia della simmetrizazione quaternaria è exprimibile mediante proprietà di tipo reticolare di sottooggetti, enunciate nella forma di assiomiD1)-5). Ciò garantisce la componibilità degli isomorfismi canonici tra subquozienti e la validità dei teoremi di isomorfismo di Noether.

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Written while the authors were members of C.N.R. sec. G.N.S.A.G.A.     

Biconservative surfaces in BCV‐spaces

Biconservative hypersurfaces are hypersurfaces with conservative stress‐energy tensor with respect to the bienergy functional, and form a geometrically interesting family which includes that of biharmonic hypersurfaces. In this paper we study biconservative surfaces in the 3‐dimensional Bianchi–Cartan–Vranceanu spaces, obtaining their characterization in the following cases: when they form a constant angle with the Hopf vector field; when they are SO(2)‐invariant.     

Entropies with the branching property

Summary It is proved that every entropy with the property of branching has the form , where J is a compositive information measure under a regular binary operation. In particular all the classical entropies with the branching property such as that of Shannon have the form .

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Sunto Si dimostra che ogni entropia che goda della proprietà di diramazione ha necessariamente la forma dove J è una misura di informazione compositiva con una operazione binaria regolare. In particolare tutte le entropie classiche che godono della proprietà di diramazione quale quella di Shannon hanno la forma .

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Entrata in Redazione il 9 aprile 1973.     

Isogénie minimale entre modules de Drinfel'd

Résumé. Nous majorons le degré d'une isogénie minimale entre deux modules de Drinfel'd. Il s'agit d'un analogue d'un résultat démontré tout d'abord sur les courbes elliptiques, puis généralisé aux variétés abéliennes par Masser et Wüstholz. Comme dans le cas abélien, la majoration dépend uniquement de la hauteur de l'un des modules et du degré d'un corps de définition commun aux deux modules. Cette dépendance est polyn?miale.

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We give a bound for the degree of a minimal isogeny between two Drinfel'd modules. This result is an anlogue of a theorem first proved on elliptic curves and then extended to abelian varieties by Masser and Wüstholz. This upper bound, as in the abelian case depends only on the height of one of the modules and on the degree of a field over which both modules are defined. We get a polynomial bound.

Received: 4 October 1994 / in final form: 30 July 1998     

FIRST ORDER FIELD EQUATIONS FOR BIANCHI TYPES II—VIIh

Abstract

The automorphism groups of the Bianchi Lie algebras are used to obtain the Einstein Field Equations for Bianchi types II-VII_h in a form which suggests a natural introduction of new variables, in terms of which all equations assume a first order form. The similarity between the equations thus obtained and the first order equations due to Jantzen is Investigated.     

Hypothesis ofP-odd contribution to the gravitational interactions and its consequences

A generalized Lagrangian of spinor particles in a gravitational field containingP-odd terms is proposed. It is used to obtain a quasirelativistic equation of spinors in the field of a gravitational source with nonvanishing intrinsic angular momentum in both an inertial frame and in a frame rotating with the source. For these cases the quasirelativistic Hamiltonian of a test body with nonzero intrinsic angular momentum is obtained in the classical limit; it is shown that the presence in the Hamiltonian ofP-odd terms leads to the appearance of forces that, depending on the orientation of the intrinsic angular momentum of the test body, are attractive or repulsive. It is noted that theC noninvariance of the theory can in principle result in an evolution of the universe in which particles predominate over antiparticles. Other macroscopic consequences whose experimental verification would yield information about the values of the constants introduced in the theory are noted.Moscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 91, No. 2, pp. 309–319, May, 1992.