Construction of representations of the Poincaré Lie algebra by extension of direct integrals of representations ofSL(2,C)

Summary Continuing the program initiated in the first paper of this series, representations of the Lie algebra of the Poincaré group are constructed by forming suitable direct integrals of representations of SL(2, C) and extending them to by means of four additional momentum operators whose matrix elements are derived in a basis adapted to the direct integral structure. The representations obtained here include Wigner’s representations with positive mass and spin and Wigner’s representations with zero mass, spin and helicity ± . Lavoro eseguito con contributo del C.N.R. nell’ambito dell’attività di ricerca del Gruppo Nazionale per la Fisica Matematica e per le applicazioni della Matematica alla Fisica ed all’Ingegneria. Entrata in Redazione il 12 marzo 1972.     

Sui fibrati con struttura quaternionale generalizzata

Summary Quaternion generalized fiber bundles are studied, both isomorphic to global tensorial product ordinary quaternion fiber bundles right and left respectively) and quite general ones. A cohomology class is considered which represents the obstruction in order the fiber bundle be a tensorial product. Several properties and a splitting principle are proved for bundles . On this ground and founding on a convenient bundle B_E → X associated to jaz (that we call Bonan's bundle and for which ɛ( =ɛ(B_E)) relations are stated among Stiefel-Whitney classes of , B_E and the class ɛ.

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Entrata in Redazione il 14 agosto 1974.

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Lavoro eseguito con contributo del C.N.R., nell'ambito del Gruppo Nazionale per le Strutture Algebriche e Geometriche e loro Applicazioni.     

Some topological properties of vector measures with bounded variation and its applications

Summary In this paper, we study the duality between ℜ(T, E′) and , where ℜ(T, E′) is the space of vector measures with bounded variation, defined on a completely regular Suslin space T, with values in the weak dual space E′ of a separable Fréchet space E; is the space of continuous bounded mappings defined on T, with values in a finite dimensional subspace of E. We show that ℜ(T, E′) is a Suslin space when equipped with the weak topology . Some applications are given: a weak closure's theorem for integrable vector mappings (a generalization of a recent result of Olech), a theorem of integral representation, a theorem of existence of conditional expectation, and finally a theorem of density related to the optimal control theory. A new application of these results is the study of comparison of measures on the unit sphere Sⁿ ofR ⁿ, which will be published elsewhere. Entrata in Redazione il 20 ottobre 1976.     

Connessioni adattate a una struttura quasi prodotto

Sunto Un campo di vettori di una varietà riemanniana V_n+1 induce in essa mediante la decomposizione locale dello spazio tangente nel sottospazio parallelo a e nel sottospazio ad esso normale e supplementare una particolare struttura quasi prodotto. Vengono qui studiate le connessioni rispettanti tale struttura (connessioni vincolate) con particolare riguardo alla connessione vincolata canonicamente associata alla connessione riemanniana Γ. Si danno semplici formule che legano la derivazione covariante secondo alla derivazione covariante riemanniana. Lavoro eseguito nell'ambito dell'attività del Gruppo di ricerca matematica N^o 1 del C. N. R.     

Elementary properties of free extensions

For any partial groupoid , let Fr be the free extension of to a total groupoid. We show that implies and that the theory of Fr is uniformly recursive in the theory of . These results fail if “groupoid” is replaced by “semigroup”, “commutative semigroup”, “group”, “abelian group”, “semilattice”, “K-lattice” for any nontrivial varietyK of lattices, or “Boolean algebra”. Research supported in part by NSF Grant MCS78-01867. We thank the referee for his valuable comments. Presented by B. Jónsson.     

Trasporto di neutroni con sezioni d'urto dipendenti dalla temperatura

Riassunto. Si studia un problema non lineare di trasporto di neutroni in un muro omogeneo con sezioni d'urto dipendenti dalla temperatura. Facendo uso di alcune tecniche standard della teoria delle equazioni non lineari di evoluzione, si prova l'esistenza e l'unicità di una soluzione forte u=u(t), per ogni t ∈ [0, ], ove è scelto in modo opportuno. Infine, si indica un procedimento per determinare una funzione continua e non negativa b=b(t), tale che ‖u(t)‖⩽b(t) per ogni t ∈ [0, ].

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Summary We study a non-linear neutron transport problem in a homogeneous slab with temperature feedback. By using some standard techniques from the theory of non-linear evolution equations, we prove existence and uniqueness of a strong solution u=u(t) at any t ∈ [0, ]. Finally, we indicate a procedure to find a non-negative continuous b=b(t), such that ‖u(t)‖⩽b(t), at any t ∈ [0, ].

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A Dario Graffi nel suo 70^o compleanno

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Entrata in Redazione il 15 marzo 1975.     

Nichtlineares invariantes Differential

Zusammenfassung Das invariante Differential eines Vektors ist selbst ein Vektor, der vom Orte xⁱ, von dxⁱ, und von selbst abh?ngig ist. Es wird nun infolge dieser Definition in einigen F?llen die m?gliche Form des invarianten Differentials bestimmt. Enrico Bompiani zu seinem wissenschaftlichen Jubil?um.     

Inequalities of Paley type for noncommutative martingales

Summary The purpose of this paper is to study the validity of the Paley inequality on square function, for noncommutative martingales. Let be a regular gage space, and a sequence of von-Neumann algebras such that we prove that for every , where ɛ_n(F) is the conditional expectation of F with respect to the subalgebra : We also consider the case of a martingale arising in the context of harmonic analysis on noncommutative discrete groups, in analogy to the theorem of R.E.A.C. Paley on Fourier-Walsh series. Entrata in Redazione il 26 gennaio 1977. Partially sponsored by C.N.R.     

Generalized interval exchanges and the 2–3 conjecture

We introduce the notion of a generalized interval exchange induced by a measurable k-partition of [0,1). can be viewed as the corresponding restriction of a nondecreasing function on ℝ with . A is called λ-dense if λ(A _i ∩(a, b))>0 for each i and any 0≤ a< b≤1. We show that the 2–3 Furstenberg conjecture is invalid if and only if there are 2 and 3 λ-dense partitions A and B of [0,1), such that . We give necessary and sufficient conditions for this equality to hold. We show that for each integer m≥2, such that 3∤2m+1, there exist 2 and 3 non λ-dense partitions A and B of [0,1), corresponding to the interval exchanges on 2m intervals, for which and commute.     

Congruence theorems for compact hypersurfaces of a riemannian manifold

Summary Let M^m, ^m be two m-dimensional compact oriented hypersurfaces of class C³ immersed in a Riemannian manifold R^m+1 of constant sectional curvature. Suppose that R^m+1 admits a one-parameter continuous group G of conformal transformations satisfying a certain condition (which holds automatically when G is a group of isometric transformations). Suppose further that there is a1 − 1 transformation T_τ ∈ G between M^m and ^m such that for each P ∈ M^m and each ^m. If the r-th mean curvature for any r, 1 ⩽ r ⩽ m, of M^m at each point P ∈ M^m is equal to that of ^m at the corresponding point , together with other conditions, then M^m and ^m are congruent mod G. This is a generalization of a joint theorem ofH. Hopf andY. Katsurada [5] in which G is a group of isometric transformations. Entrata in Redazione il 13 Giugno 1975. The first author was partially supported by the National Science Foundation grant GP-33944.     

Special finite-dimensional monocomposition algebras

Special monocomposition algebras introduced in [1] are studied. Using their properties, we prove that any nondegenerate monocomposition algebra ,dim ≥ 3, with unity contains no proper ideal of dimension ≤8. This implies that if 3≤dim ≤ 9, then is a central simple algebra. Translated fromAlgebra i Logika, Vol. 35, No. 2, pp. 125–144, March–April, 1996.     

The spectrum of compact hypersurface in sphere

Let M be a compact minimal hypersurface of sphere Sn 1(1). Let (M) be H (r)-torus of sphere Sn 1 (1).Assume they have the same constant mean curvature H, the result in [1] is that ifSpec0(M, g) =Spec0((M), g),then for 3≤ n ≤ 6, r2≤n-1/n or n ≥ 6, r2 ≥ n-1, then M is isometric to (M). We improved the result and prove that: if Spec0(M,g) =Spec0((M),g), then M is isometric to (M). Generally, if Specp(M,g) =Specp((M),g), here p is fixed and satisfies that n(n - 1) ≠ 6p(n - p), then M is isometric to (M).     

Sur les anneaux tels que tout produit de copies d'un module quasi-injectif soit un module quasi injectif. — II

Résumé On étudie dans cet article les anneaux noethériens commutatifs tels que tout produit de copies d'un module quasi injectif soit un module quasi-injectif, un tel anneau est produit fini d'anneaux locaux vérifiant certaines propriétés. On étudie également des anneaux noethériens commutatifs un peu plus généraux: les C1-anneaux, qui sont caractérisés comme étant des produits finis d'anneaux locaux A_i,1≤i≤n tels que tout idéal ′ du complété R(A_i)-adique ?_i de A_i vérifie ′ = ( ′∩A_i)?_i. On donne des exemples de tels anneaux. Entrata in Redazione il 12 febbraio 1975.     

Separable asymptotic cost of evaluating elementary functions

The computational cost, in the bit model of computation, of the evaluation of a real function in a point is analyzed, when the number d of correct digits of the result increases asymptotically. We want to study how the cost depends on also when approaches a critical point for the function f. We investigate the hypotheses under which it is possible to give upper bounds on the cost as functions of “separated variables” d and , that is as products of two functions, each of one variable. We examine in particular the case of elementary functions. This revised version was published online in June 2006 with corrections to the Cover Date.     

Problemi di curvatura su di una varietà quasi hermitiana

Sunto Su di una varietà quasi hermitiana V, la considerazione di alcune medie di curvature semplici e miste, relative a convenienti sistemi di 2-spazi associati ad un 2-spazio α, dà informazioni sulla curvatura semplice K_α, sulla media e sulla curvatura mista ( =Jα). Due delle medie accennate riescono nulle; le altre danno luogo a relazioni di dipendenza lineare. Considerate per la varietà V ipotesi di isotropia in media e di J-isotropia si ottengono teoremi del tipo di Schur. Altri risultati si stabiliscono nel caso k?hleriano.

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Summary Let V be an almost Hermite space, α a tangent 2-dimensional space of V; five systems of tangent 2-spaces of V are considered, suitably related to α. The sectional curvature K_α, the mean value , and the bisectional curvature are expressed in terms of mean values of sectional and bisectional curvatures, evalued over the above mentioned systems. Two of the mean values vanish; linear relationships hold among the other ones. Schur-like theorems are proven under the assumption that V be isotropic in mean or J-isotropic. Further results are obtained for the case that V be a K?hler space.

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Entrata in Redazione il 25 novembre 1971.

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Lavoro eseguito con contributo del C.N.R. nell'ambito del Gruppo Nazionale Strutture Algebriche, Geometriche e loro Applicazioni. I primi risultati sono stati annunciati in una riunione di Gruppo a Bologna (dicembre 1970) e in una conferenza tenuta all'Ist. Mat. Univ. Perugia (aprile 1971).     

Differential equations of infinite order for orthogonal polynomials

Summary Differential equations of infinite order of the form , where M_k is a polynomial of degree ≦k, have polynomial solutions for suitable values of λ. Necessary and sufficient conditions are given in order that these solutions form a set of orthogonal polynomials. The cases where { degree M_k }=1 and 2 are studied in detail. Supported by N.S.F. Grant GP-5311.     

On tensors which are functions of vectors

It has been shown that the most general constant tensor of any rank is the possible algebraic combination of Kronecker tensor and the totally antisymmetric tensor ... μ_n. Further, the tensor of a given rank r, which are functions of vectors p's, are suitable algebraic combinations of the tensor and the two constant tensors with coefficients which are functions of the scalar products formed by the vectors.     

Dérivée et primitive aréolaires

Résumé En définissant la notion de primitive aréolaire, on étudie à ce point de vue les fonctions holomorphes (α) et les fonctions monogènes (α) introduites par l'auteur, ainsi que les sapports entre ces fonctions et les fonctions de la classe et les -primitives selon I. N. Vekua. à Giovanni Sansone pour son 70^me anniversaire.     

Asymptotic properties of the maximum likelihood estimate in the first order autoregressive process

Summary In this paper we obtain an asymptotic expansion of the distribution of the maximum likelihood estimate (MLE) based onT observations from the first order Gaussian process up to the term of orderT ⁻¹. The expansion is used to compare with a generalized estimate including the least square estimate (LSE) , based on the asymptotic probabilities around the true value of the estimates up to the terms of orderT ⁻¹. It is shown that (or the modified MLE ) is better than (or the modified estimate ). Further, we note that does not attain the bound for third order asymptotic median unbiased estimates.     

Equazioni ellittiche del IIo ordine e spazi\mathfrak{L}^{(2,\lambda )} .

Summary Regularity results of Schauder type in the class of H?lderian functions are proved for solutions of second order elliptic equations in the variational form. Morrey and other authors have obtained results of this kind in certain classes of functions . The method of proof followed here makes use of the classes of functions which generalize the classes considered by Morrey, and whose properties in relation to H?lder continuity are studied in some proceeding works.

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Ha parzialmente contribuito finanziariamente alla preparazione di questo lavoro l'Air Force Office of Scientific Research OAR con il Grant AF EOAR 65–42.