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.     

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.     

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.     

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.     

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.     

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.     

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.     

Estimates of operator polynomials in the space Lp with respect to the multiplier norm

The paper is devoted to the determination of an analog of J, von Neumann's inequality for the space L^p.The fundamental result of the paper is: If T is an absolute contraction in the space L^p(X,F,μ) (i.e., |T|_L1≤1) and |T|_L∞≤1) then for every polynomial ϕ one has where S is the shift operator in the space. On the basis of this theorem, one finds a theorem on substitutions in the space of multipliers. One gives applications of the inequality (1) to the weighted shift operators in the space It turns out that under some natural restrictions on the weight, inequality (1) becomes an equality for such operators. One also presents a proof of J. von Neumann's inequality based on the approximation of a contraction in a Hilbert space by unitary operators. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 65, pp. 133–148, 1976. The author wishes to express his thanks to N. K. Nikol'skii for suggesting the problem and for his interest in the paper.     

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.     

On the oscillatory and asymptotic behavior of the bounded solutions of differential equations with deviating arguments

Summary This paper is dealing with the oscillatory and asymptotic behavior of the bounded solutions of n-th order (n>1) differential equations with deviating arguments involving the so called r-derivatives D _r ^(i)x (i=0, 1, ..., n) of the unknown function x defined by , where r_i (i=1, 2, ..., n−1) are positive continuous functions on the interval [t ₀ , ∞). The fundamental purpose is to find a necessary and sufficient condition in order to have at least one (bounded nonoscillatory) solution whose the limit at ∞ exists inR−{0}. Entrata in Redazione il 29 giugno 1977. This paper is a part of the author's Doctoral Thesis submitted to the School of Physics and Mathematics of the University of Ioannina.     

Interpolation in Logiken monotoner systeme

A monotone structure ( ;μ) consists of a structure and a monotone systemμ over the domain of .L(Q ⁿ ) is , enlarged by a newn-ary quantifierQ ⁿ . says in ( ;μ) that there isU ∈μ such thatϕ[ā] is valid in ( ;μ) for allā ∈U ⁿ . If is a class of monotone structures, means thatϕ is valid in all expansions of monotone structures in . We show for the class of all ultrafilters that interpolation with respect to holds forL(Q ⁿ ) exactly in casen=1. Then we prove for a large class of (e.g. the class of topological groups) thatL(Q ⁿ ) satisfies interpolation with respect to for alln ≧ 1. Counterexamples indicate that the class of is sharp in some sense. Finally the results are carried over to certain topological structures and the interior quantifiersI ⁿ instead ofQ ⁿ , thereby generalizing results of Makowsky/Ziegler and Sgro, and to a multidimensional type of monotone structures including uniform spaces.     

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.     

Decompositions into nonoverlapping domains and extremal properties of univalent functions

We apply the method of extremal metrics and certain symmetrization approaches to study problems on conformal mappings of a disk and circular annulus. For instance, we solve the problem on the maximal conformal module in the family of all doubly-connected domains of the form \(E₁∪E₂) with E₁∩E₂=0, r₁E₁, 0≤r₁·r₁≤∞, and diam E₂∩{z:|z|<1}. This generalizes the classical result by A. Mori. We also give a new solution of a problem by P. M. Tamrazov, which was initially solved by V. A. Shlyk. Some new theorems on the covering of a regular system of n rays are obtained for certain classes of convex mappings. Bibliography: 22 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 212, 1994, pp. 139–163. Translated by A. Yu. Solynin.     

A Schwarz lemma for a domain related to μ-synthesis

We prove a Schwarz lemma for a domain in ℂ³ that arises in connection with a problem in H^∞ control theory. We describe a class of automorphisms of and determine the distinguished boundary of We apply our Schwarz lemma to a special case of the μ-synthesis problem.     

Inductive definitions,models of comprehension and invariant definability

The connections between inductive definability and models of comprehension are studied. Let = 〈A, R _l, ...,R _n 〉 be an infinite structure and letI _φ be a set inductively defined by a formulaφ of the second order language . We prove that if is a model of Δ ₁ ¹ -Comprehension relativized toφ, andφ is -absolute, then for everyη smaller than the height of (h( )),I _φ is in . If is aβ-structure which satisfies Σ ₁ ¹ -Comprehension relativized toφ and WF(X), and φ is -absolute, thenI _φ is in and ‖φ| <h ( ). These results imply that Barwise-Grilliot theorem is false in the case of uncountable acceptable structures. We also study the notion of invariant definability over models₁ of Δ ₁ ¹ -Comprehension. This paper is registered as Report ZW 69/76 of the Mathematical Centre.     

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.     

Una generalizzazione,per le serie,del metodo di sommazione di Nikola Obrechkoff nella teoria del prolungamento analitico

Sunto In questa Memoria viene posto un nuovo procedimento di sommazione, per ogni a raggio di convergenza ≠ 0, della serie Σ a_nzⁿ determinando il campo (includente il cerchio di convergenza) in cui è, essendo λ intero l'ordine di sommazione , uniformemente (p₁, p₂, ... ps_n sono tutti i numeri dispari positivi ≤ n; q₁, q₂, ... qr_n i numeri positivi pari ≤ n, con q₁=0). Questo campo di sommabilità tende per λ → ∞, ampliandosi, ad un campo limite che abbraccia completamente il campo limite a cui ampliandosi tende il campo di sommabilità, quando λ → ∞, relativo al metodo semplice di N. Obrechkoff; con questo ultimo si somma la ogni qualvolta è .     

Integrals involving Whittaker functions

Summary Integrals involving products of two Whittaker functions and Bessel functions are evaluated in §§ 3, 4. Also the integrals are evaluated in § 5 while in § 6 integrals involving the product of three Whittaker functions are established.