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

As a first application of a general method of construction, a class of representations of the Lie algebra of the Poincaré group , which includes Wigner’s representations with spin zero and positive mass, is obtained by forming suitable direct integrals of representations of class I of the principal series of SL(2, C) and extending them to by means of four additional momentum operators which are determined in explicit form. Lavoro eseguito nell’ambito dell’attività di ricerca del Gruppo Nazionale per la Fisica Matematica e per le Applicazioni della Matematica alla Fisica ed all’Ingegneria, C.N.R. Entrata in Redazione il 9 marzo 1971.     

State-constrained control problems with neither coercivity norL 1 bounds on the controls

Summary A state-constrained, nonlinear, minimum problem is considered with dynamics depending sublinearly on a control which is not bounded in theL ¹ norm. Because of the lack of coercivity, the value map fails to be continuous, even in the unconstrained case. However, we prove that under suitable assumptions—which guarantee the continuity of the value maps of the problems withL ¹-bounded controls—the value map is upper semicontinuous and solves a Bellman equation with a continuous Hamiltonian. Moreover, the map obtained by by replacing its values at the horizon t=T with the values of the cost function turns out to be the maximal subsolution of the corresponding value problem. Entrata in Redazione il 31 dicembre 1997.     

On a property of the moebius group

Summary It is shown that the group of all Moebius transformations (M. T.) contains a subgroup such that every non-involutory M. T. not in can be transformed into each of its conjugates outside the subgroup by a unique element of this subgroup. Hence every non-integral and non-involutory M. T. can be reduced into a normal form by means of a unique element of . Also a subgroup *, can be determined such that elements of * reduce all non-involutory M. T. not in *, into their classical (integral) normal forms. There is also a discussion of the question as to other fields over which the results remain valid. To Enrico Bompiani on his scientific Jubilee.     

Sopra le estremaloidi relative ad integrali curvilinei dello spazio in forma parametrica

Sunto Nella presente Memoria l'Autore si occupa delle estremaloidi relative agli integrali curvilinei dello spazio in forma parametrica dimostrando una condizione sufficiente affinchè una curva ordinaria minimante sia un'estrematoide e rilevando alcune proprietà di esse.     

Sopra le estremali relative ad integrali curvilinei dello spazio in forma parametrica

Sunto Nel presente lavoro l' A. stabilisce le equazioni di Eulero relative agli integrali curvilinei e rileva alcune proprietà delle estremali relative alla funzione F, occupandosi in particolare dell'esistenza delle derivate successive delle funzioni da cui le estremali sono definite.     

Sul contatto di terz’ordine di due superficie algebriche lungo curve

Sunto Si dànno delle condizioni necessarie affinchè due superficie algebriche F, G d’una V ₃ algebrica non singolare possano presentare contatto d’ordine assegnato q −1 lungo una curva priva di punti multipli. Si approfondisce sopratuito il caso q=3, con applicazione al problema di determinare le superficie algebriche G_n di S ₃ che presentano contatto del 3^o ordine lungo una ⁿ con una superficie quartica F ₄ .     

Large sample theory for distributions on the hypersphere with rotational symmetries

Summary A distribution on the unit sphere inℝ ^q with a densityf(‖x _v ‖) is considered where is ans(<q) dimensional subspace andx _v is the part ofx in . For a large sample the estimation of , a test that and a test for rotational symmetry within is given. For several samples with possibly different subspaces but the samef, a test that is given. For all tests power functions for contiguous alternatives are given. For the special density proportional to expk‖x _v ‖ ², additional results are given. Research supported in part by a Contract with the Office of Naval Research N00014-81-K-0146 awarded to Princeton University, Princeton, New Jersey 08544.     

Aspetti geometrici delle forme esterne in uno spazio vettoriale quaternionale

Riassunto Si considera l'algebra ℚ degli ordinari quaternioni e si valutano i prodotti simmetrizzato e alternato di elementi di ℚ. Uno tra i risultati è che il prodotto alternato di quattro o più quaternioni è zero. Preso poi in esame uno spazio vettoriale destro _n su ℚ ne conseguono varie identità per le potenze esterne delle forme definite su _n: in particolare, sono nulle le potenze superiori alla terza di tutte le forme a grado dispari. Si considerano infine una varietà ℳ_4n a struttura quaternionale generalizzata e lo spazio vettoriale V_4n tangente ad ℳ_4n in un suo punto. Si trova una relazione tra la deviazione caratteristica di un qualsiasi V₄ ⊂ V_4n e il valore assunto dalla 4-forma reale Ω su un 4-vettore di V₄.

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Summary Let ℚ the algebra of quaternions over the real number field; is an hermitian n-dimensional vector space over ℚ. We establish that the alternated product of four, five, ..., quaternions vanish. Consequently exterior powers A^tθ_p of every form θ_p over vanish if t≥4 and p is odd. Further results are obtained for t=2, 3 and if p is even. Finally, we consider a generalized quaternion manifold ℳ_4n; let V_4n the tangent vector space in O ∈ ℳ_4n. Relations are obtained among the characteristic deviation of every V₄⊂V_4n and the value which the real 4-form Ω assumes upon a 4-vector over V₄.

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A BeniaminoSegre nel suo 70^mo compleanno

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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.

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Entrata in Redazione il 26 febbraio 1973.     

Embeddings of Sobolev spaces on unbounded domains

Summary Piecewise polynomial and Fourier approximation of functions in the Sobolev spaces on unbounded domains Θ ⊂ Rⁿ are applied to the study of the type of compact embeddings into appropriate Lebesgue and Orlicz spaces. It is shown that if Θ and s satisfy certain conditions, the embeddings , m/n+1/q−1/p>0 and , Φ being an Orlicz function subordinate to both φ(t)=|t|^p exp |t|^n/(n−m) and Φ^σ(t)=exp |t|^σ−1, σ ⩾ 1, m/n>1/p, are of type l^s. One result dealing with multiplications maps from into L^q(Θ) is also obtained. Entrata in Redazione il 14 ottobre 1976.     

Some intersection theorems on two-valued functions

Let be a family of two-valued functions defined on ann-element set in which each pair of functions in satisfy a given intersection condition. For certain intersection conditions we determine the maximal value of .     

Solutions of the cauchy problem for the polyharmonic equation (uniqueness and approximation)

One finds conditions which ensure the possibility of weighted mean-square approximation of a vector-function defined on the boundary of an n-dimensional domain by vector-functions of the form , where u is, the solution of the equation Δ^m _u=0 in while∂/∂v denotes differentiation along the normal. The weight function is continuous and positive everywhere on with the point whose relative neighborhood is contained in some (n-1)-dimensional plane. The solution of this approximation problem is closely related with a certain uniqueness theorem for the solution of the Cauchy problem for the polyharmonic equation, also proved in the paper. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Institute im. V. A. Steklova AN SSSR, Vol. 65, pp. 164–171, 1976.     

Regularity results for positive solutions of a semilinear elliptic system

Let , N≧3 be an open set. In this paper we study weak positive solutions of the following semilinear system where p≧1, q≧1, with pq>1, and u∈L^q(Ω), v∈L^p(Ω), and we give in particular some regularity results. Furthermore, we give some applications to the biharmonic equation. Entrata in Redazione il 25 febbraio 1999.     

The Banach envelopes of Besov and Triebel-Lizorkin spaces and applications to partial differential equations

With “hat” denoting the Banach envelope (of a quasi-Banach space) we prove that if 0<p<1, 0<q<1, ℝ, while if 0<p<1, 1≤q<+∞, ∝, and if 1≤p<+∞, 0<q<1, ℝ. Applications to questions regarding the global interior regularity of solutions to Poisson type problems for the three-dimensional Lamé system in Lipschitz domains are presented.     

Existence of minimizers for a class of anisotropic free discontinuity problems

Summary We prove the existence of minimizing pairs (K, u), K compact set ofR ^N and u∈W^{1, p} (Ω/K), for the functional when the integrand f(x, z) is convex with respect to z, |z|^p≤f(x, z)≤L|z|^p, p>1, and satisfies suitable assumptions of uniform continuity in x with respect to z. Entrata in Redazione il 10 luglio 1998.     

Intersections of curve systems and the crossing number ofC 5 ×C 5

It and are two families of pairwise disjoint simple closed curves in the plane such that each curve in intersects each curve in , then the total number of points of intersection in is at least 2(m−1)n, where , and this bound is best possible. We use this to show that the cartesian product of two 5-cycles has crossing number 15.     

On bands in which every finitely generated subband is an endomorphic image

A semigroup S is calledE-semigroup ( -semigroup) if every (finitely generated) subsemigroup of S is an endomorphic image of S and Ē-semigroup ( -semigroup) if every subsemigroup of S is an E-semigroup -semigroup. All classes X ε {Ē, , E, } are distinct even in the case of semilattices. It is established when a free band (semilattice) is an X-semigroup. -, - and Ē-chains and E-chains with identity or zero, Ē-and , X-bands with identity and X-semigroups with identity and zero are found.     

The orbit method and Gelfand pairs, associated with nilpotent Lie groups

Let K be a compact Lie group acting by automorphisms on a nilpotent Lie group N. One calls (K, N) a Gelfand pair when the integrable K-invariant functions on N form a commutative algebra under convolution. We prove that in this case the coadjoint orbits for G:= K × N which meet the annihilator of the Lie algebra of K do so in single K-orbits. This generalizes a result of the authors and R. Lipsman concerning Gelfand pairs associated with Heisenberg groups.     

Sull'apertura di luoghi negli spettri affine e projettivo di un dominio graduato noetheriano

Riassunto Sia una proprietà degli anelli discende per fedele piattezza, ascende quando le fibre sono buone, di carattere locale e per cui vale il criterio di Nagata. SeS è un dominio graduato noetheriano valgono alcune implicazioni per l'apertura del in SpecS, ProjS e altri schemi collegati adS.

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Summary Let be a property meaningful for rings which descends by faithful flatness, ascends when there are good fibers, with local character and for which Nagata's criterion holds. IfS is any graded noetherian domain some implications hold for the openness of the in SpecS, ProjS and other schemes related toS.

     

Structures of polyhedra determined by submodular functions on crossing families

The present paper shows that for any submodular functionf on a crossing family with , if the polyhedron is nonempty, then there exist a unique distributive lattice with and a unique submodular function with such thatB(f) coincides with the base polyhedron associated with the submodular system . Here, iff is integer-valued, thenf ₁ is also integer-valued. Based on this fact, we also show the relationship between the independent-flow problem considered by the author and the minimum cost flow problem considered by J. Edmonds and R. Giles.     

Elementary equivalence and relatively free products of lattices

It is shown that if is a non-trivial variety of lattices, then there existA, B, C ∈ such thatB≡C but notA*B≡A*C. Except in the case when is the variety of all distributive lattices,A can be taken to consist of just one element. For the varietyD of all distributive lattices, it is shown that for anyB, C and any finiteA, B≡C if and only ifA*B≡A*C. The work of the first author was supported in part by NSF Grant GP-29129, and the work of the second author by a grant from the N.R.C. of Canada. Presented by C. C. Chang.