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 .     

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.     

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.     

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.     

Symmetric centres of braided monoidal categories

This paper introduces the concept of ‘symmetric centres’ of braided monoidal categories. LetH be a Hopf algebra with bijective antipode over a fieldk. We address the symmetric centre of the Yetter-Drinfel’d module category: and show that a left Yetter-Drinfel’d moduleM belongs to the symmetric centre of and only ifM is trivial. We also study the symmetric centres of categories of representations of quasitriangular Hopf algebras and give a sufficient and necessary condition for the braid of, _Hℳ to induce the braid of , or equivalently, the braid of , whereA is a quantum commutativeH-module algebra     

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.     

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.     

Two methods for avoiding hereditary completeness

A family of vectors of a Hubert space H is said to be hereditarily complete if it posses a biorthogonal family {x_n′;n≥1}((x_n,x_k′)=δ_nk) and if any elementx, xε H can be reconstructed in terms of the component of its Fourier series, i.e., x∈V((x,x′_n)x_n:n≥1),∀x∈H. In the paper we indicate two simple methods for constructing nonhereditary complete minimal families having a total biorthogonal family, which just not long ago has caused well-known difficulties (see Ref. Zh. Mat., 1975, 7B802). The first method consists in the fact that a given pair of biorthogonal families Y, Y′ of the space H′,H′⊂H is represented as the projection of the families of the same type but already complete in H.. Clearly, in this case cannot be hereditarily complete. The second method consists in considering linear deformation _n :n⩾1 of the orthogonal bases_n: n⩾1; here A is an unbounded operator of a special type. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 65, pp. 183–188, 1976.     

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.     

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 .     

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.

     

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.     

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.     

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.     

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.     

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.     

A decomposition of 2-weak vertex-packing polytopes

The 2-weak vertex-packing polytope of a loopless graphG withd vertices is the subset of the unitd-cube satisfyingx _i +x _j ≤1 for every edge (i,j) ofG. The dilation by 2 of this polytope is a polytope with integral vertices. We triangulate with lattice simplices of minimal volume and label the maximal simplices with elements of the hyperoctahedral groupB _d . This labeling gives rise to a shelling of the triangulation of , where theh-vector of (and the Ehrharth ^*-vector of can be computed as a descent statistic on a subset ofB _d defined in terms ofG. A recursive way of computing theh-vector of is also given, and a recursive formula for the volume of . This work was partially supported by grants from the Icelandic Council of Science and the Royal Swedish Academy of Sciences, respectively.     

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.