Showing papers in "Rendiconti del Seminario Matematico della Università di Padova in 2004"

An Interpolation Inequality in Exterior Domains.

Abstract: This note is concerned with some interpolation inequalities of Gagliardo-Nirenberg type. In [4, 13] it is proved that if V%R is a bounded domain having the cone property or if V4R , then any function u belonging to L q (V) with derivatives D a u , a multi-index, belonging to L p (V), q , pF1, satisfies the following inequality: for m4NaN and j40, R , m21 ND j uNL r (V) GC1 ND m uNL p (V) a NuNL q (V) 12a 1C2 NuNL q (V) () , (1.1)

Erratum to : “The finite free extension of artinian $K$-algebras with the strong Lefschetz property” [Rend. Sem. Mat. Univ. Padova 110 (2003), 119–146]

Abstract: It has come to our attention that Proposition 21 in the paper cited in the heading is in error. It affects the proof of Main Theorem. The Main Theorem itself, however, can be proved as follows. With the notation of Main Theorem, write A and B as homomorphic images of polynomial rings: A4R/I and B4S/J, with R4K[v1 , R , vn ] &I and S4K[u1 , R , um , v1 , R , vn ]&J. Put U4(u1 , R , um) and V4(v1 , R , vn). Let D be a monomial order with the following properties:

Differential and Geometric Structure for the Tangent Bundle of a Projective Limit Manifold.

Abstract: The tangent bundle of a wide class of Frechet manifolds is studied he- re. A vector bundle structure is obtained with structural group a topological subgroup of the general linear group of the fiber type. Moreover, basic geo- metric results, known form the classical case of finite dimensional manifolds, are recovered here: Connections can be defined and are characterized by a generalized type of Christoffel symbols while, at the same time, parallel di- splacements of curves are possible despite the problems concerning differen- tial equations in Frechet spaces.

On Calibrations for Lawson's Cones.

Abstract: In this paper a calibration method is recalled and applied to Lawson's cones to prove their minimality. The original proof of Bombieri, De Giorgi and Giusti is reinterpreted and made simpler.

Some results for q-functions of many variables

Abstract: We will give some new formulas and expansions for q{Appell-, q{Lauricella- and q{Kamp e de F eriet functions. This is an area which has been explored by Jackson, R.P. Agarwal, W.A Al-Salam, Andrews, Kalnins, Miller, Manocha, Jain, Srivastava, and Van der Jeugt. The multiple q{hypergeometric functions are dened by the q{shifted factorial and the tilde operator. By a method invented by the author (20){ (23), which also involves the Ward-Alsalam q{addition and the Jackson-Hahn q{ addition, we are able to nd q{analogues of corresponding formu- las for the multiple hypergeometric case. In the process we give a new denition of q{hypergeometric series, illustrated by some ex- amples, which elucidate the integration property of q{calculus, and helps to try to make a rst systematic attempt to nd summation- and reduction theorems for multiple basic series. The Jackson q{ function will be frequently used. Two denitions of a generalized q{Kamp e de F eriet function, in the spirit of Karlsson and Srivas- tava (69), which are symmetric in the variables, and which allow q to be a vector are given. Various connections between transformation formulas for multi- ple q{hypergeometric series and Lie algebras & nite groups known from the literature are cited.

Connections on distributional bundles

Abstract: A general approach to the geometry of distributional bundles is pre- sented. In particular, the notion of connection on these bundles is studied. A few examples, relevant to quantum field theory, are discussed.

Fields of cr meromorphic functions

Abstract: Let M be a smooth compact CR manifold of CR dimension n and CR codimension k, which has a certain local extension property E. In particular, if M is pseudoconcave, it has property E. Then the field K(M) of CR meromorphic functions on M has transcendence degree d, with dn + k. If f1, f2, ..., fd is a maximal set of algebraically independent CR meromorphic functions on M, then K(M) is a simple finite algebraic extension of the field C(f1, f2, ..., fd) of rational functions of the f1, f2, ..., fd. When M has a projective embedding, there is an analogue of Chow's theorem, and K(M) is isomorphic to the field R(Y ) of rational functions on an irreducible projective algebraic variety Y , and M has a CR embedding in regY . The equivalence between algebraic dependence and analytic dependence fails when condition E is dropped.

Ideals with Maximal Local Cohomology Modules.

Abstract: This paper finds its motivation in the pursuit of ideals whose local cohomology modules have maximal Hilbert functions. In [11], [12] we proved that the lexicographic (resp. squarefree lexicographic) ideal of the family of graded (resp. squarefree) ideals with assigned Hilbert function provides sharp upper bounds for the local cohomology modules of any ideal of the family. More precisely, the Hilbert series of the local cohomology modules of any ideal of the family are smaller than or equal to those of the lexicographic ideal. Moreover these bounds are determined explicitly in terms of the Hilbert function, which is the specified starting data. In the present paper a characterization of the class of ideals with the property of having maximal local cohomology modules in the sense explained above is accomplished. Let us set some notation to be used henceforth. Let Ru K[X1 , R , Xn ] denote the polynomial ring in n variables over a field K of characteristic 0 with its standard grading, Yu (X1 , R , Xn ) the maximal homogeneous ideal of R. We set X1 D X2 D R D Xn and consider the lexicographic order induced by this assignment on Md, the set of all monomials of R of degree d, for all d. A lex-segment of degree d is

Dynamics of Granular Fluids.

Abstract: We investigate the consequences of a proposal for the balance equa- tions of a continuum where local moment of momentum has a crucial role. We explore, in particular, applications to the dynamics of some granular flows and advance some, perhaps controversial, suggestions on related thermal concepts.

On the Semi-Simplicity of Galois Actions.

Abstract: If K is a finite field, then Conjecture 2 implies Conjecture 1. This is well-known and was written-up in [8] and [4], manuscript notes distributed at the 1991 Seattle conference on motives. Strangely, this is the only result of op. cit. that was not reproduced in [10]. We propose here a simpler proof than those in [8] and [4], which does not involve Jordan blocks, representations of SL2 or the Lefschetz trace formula. We also show that Conjecture 2 for K finite implies Conjecture 1 for any K of positive characteristic. The proof is exactly similar to that in [3, pp. 212-213], except that it relies on Deligne’s geometric semi-simplicity theorem [2, cor. 3.4.13]; I am grateful to Yves André for explaining it to me. This gives a rather simple proof of Zarhin’s semi-simplicity theorem

Complements of the socle in almost simple groups

Abstract: Assume that a finite group H has a unique minimal normal subgroup, say N , and that N has a complement in H . We want to bound the number of conjugacy classes of complements of N in H ; in particular we are looking for a bound which depends on the order of N . When N4soc H is abelian, the conjugacy classes of complements of N in H are in bijective correspondence with the elements of the first cohomology group H (H/N , N). Using the classification of finite simple groups, Aschbacher and Guralnick [1] proved that NH (H/N , N)NENNN ; therefore, when soc H4N is abelian, there are at most NNN conjugacy classes of complements of N in H . To study the case when N4soc H is nonabelian we can employ a result proved by Gross and Kovács ([6], Theorem 1): there exists a finite group K containing a (non necessarily unique) minimal normal subgroup S which is simple and nonabelian (indeed S is isomorphic to a composition factor of N) and there is a correspondence between conjugacy classes of complements of N in H and conjugacy classes of complements of S in K . Using this result it is not difficult to prove that there exists an absolute constant cG4 such that the number of conjuga-

On Infinite-Dimensional Grassmannians and their Quantum Deformations.

Abstract: An algebraic approach is developed to define and study infinite dimensional grassmannians. Using this approach a quantum deformation is obtained for both the ind-variety union of all finite dimensional grassmannians, and the Sato grassmannian introduced by Sato. They are both quantized as homogeneous spaces, that is together with a coaction of a quantum infinite dimensional group. At the end, an infinite dimensional version of the first theorem of invariant theory is discussed for both the infinite dimensional special linear group and its quantization.

On Lie powers of regular modules in characteristic 2

Abstract: We study Lie powers of regular modules for finite groups over a field of characteristic 2 . First we prove two rather general reduction theorems, and then we apply them to Lie powers of the regular module for the Klein four group. For the latter, we solve the decomposition problem for the Lie power in degree 8, a module of dimension 8160. It has been known that of the infinitely many possible indecomposables, only four occur as direct summands in Lie powers of degree not divisible by 4, but that a fifth makes its appearance in the Lie power of degree 4. It is quite a surprise that no new indecomposables appear among the direct summands in degree 8.

A note on extremality and completeness in financial markets with infinitely many risky assets

Abstract: In this paper we study the interplay existing between completeness of financial markets with infinitely many risky assets and extremality of equivalent martingale measures. In particular, we obtain a version of the Douglas- Naimark Theorem for a dual system ‹X, Y› of locally convex topological real vector spaces equipped with the weak topology σ(X, Y), and we apply it to the space L∞ with the topology σ(L∞, Lp) for p≥1. Thanks to these results, we obtain a condition equivalent to the market completeness and based on the notion of extremality of measures, which allows us to give new and simpler proofs of the second fundamental theorems of asset pricing. Finally, we discuss also the completeness of a slight generalization of the Artzner and Heath example.

Large Integer Polynomials in Several Variables.

Abstract: For every sufficiently large positive integer D , we construct a family of irreducible integer polynomials of degree D in n variables whose Mahler measures are bounded by D and whose values at (1 , R , 1 ) are greater than exp ](1 /9n) D n/(n11)(. This shows that an upper bound for the height of integer irreducible polynomials in terms of their degree and Mahler measure obtained by Amoroso and Mignotte is sharp up to a logarithmic factor.

Endoprimal abelian groups of torsion-free rank 1

Abstract: This paper completely solves the endoprimality problem of (mixed) abelian groups of torsion-free rank 1.

Des domaines de Fatou-Bieberbach à plusieurs feuillets

Abstract: RÉSUMÉ Nous définissons l’idée de bassin d’attraction pour l’itération des germes holomorphes attractifs de (CN , 0 ) et la notion de domaine de RiemannFatou-Bieberbach: c’est un domaine de Riemann R biholomorphe à CN mais recouvrant une region V%CN. Enfin, étant donné un endomorphisme de CN admettant un point fixe répulsif en 0 (satisfaisant une hypothèse technique supplémentaire), nous prouvons que le bassin d’attraction du germe inverse admet un recouvrement par un domaine de Riemann-Fatou-Bieberbach.

A Note on Abelian Varieties Embedded in Quadrics.

Abstract: We show that if A is a d-dimensional abelian variety in a smooth quadric of dimension 2 d then d 4 1 and A is an elliptic curve of bidegree ( 2 , 2 ) on a quadric. This extends a result of Van de Ven which says that A only can be embedded in P 2 d when d 4 1 or 2 .

On a generalization of groups with all subgroups subnormal

Abstract: In this article we prove the following result: Let G be a Fitting p-group. If for every proper subgroup H of G , H GcG and H (n) is hypercentral for a non-negative integer n, then G 8cG.

Characterisation of finitely generated soluble finite-by-nilpotent groups

Abstract: We say that a group G has finite lower central depth if the lower cen- tral series of G stabilises after a finite number of steps, that is, G has finite lower central depth if and only if gk (G) 4 gk 1 1 (G) for some positive integer k . The least integer k such that gk (G) 4 gk 1 1 (G), is called the depth of G. We denote by V the class of groups which has finite lower central depth. If k is a positive integer, we denote by V k the class of all groups having finite lower central depth at most k . Let G be a finitely generated soluble group. In this note we prove, G is finite-by-nilpotent if and only if in every infinite set of ele- ments of G there exist two distinct elements x , y such that ax , ybV , and G is finite by a group in wich every two generator subgroup is nilpotent of class at most k if and only if in every infinite set of elements of G there exist two dis- tinct elements x , y such that ax , ybV k .

Hilbert functions on Cohen-Macaulay ideals with assigned generators’ degrees

Abstract: We give information on the Hilbert function of a Cohen-Macaulay ideal I of the polynomial ring R4k[x0 , x1 , R , xr ] which is minimally generated by t forms of degrees d1 , R , dt . Mainly we deal with the codimension two case in which we show that the Dubreil bound tGd111 is a necessary and sufficient condition to have such an ideal and we give a sharp upper bound and lower bound for the Hilbert function. In codimension greater than two we give a characterization for having such an ideal and in codimension 3 we find an Hilbert function which is maximal for these ideals with d14R4dt4a and we produce a scheme which realizes such a Hilbert function.