Orthogonal vector measures

This paper introduces the concept of orthogonal vector measures, and gives the Yosida-Hewittdecomposition theorem for this kind of vector measures. The major results are(a) Any orthogonal vector measure can gain it countable additivity by enlarging its domain;(b) Every orthogonal vector measure can be represented as the sum of two orthogonal vectormeasures, one of which is countably additive, and the other is purely finitely additive. Furthermore,these vector measures are completely perpendicular to each other.     

Some results on surface measures in calculus of variations

We study some relations between the concepts of perimeter, Hausdorff measure, and Minkowsky content, when R ^N is endowed with a convex Finsler metric depending in a continuous way on the position. We show some connections with the theory of -convergence and with the anisotropic motion of a smooth hypersurface by mean curvature.This work was partially supported by NSF Grant DMS-9008999, and by MURST (Progetto Nazionale «Equazioni di Evoluzione e Applicazioni Fisico-Matematiche» and «Analisi Numerica e Matematica Computazionale») and CNR (IAN and Contracts 92.00833.01, 93.00564.01) of Italy.     

Algebraic Riccati equations with non-smoothing observation arising in hyperbolic and Euler-Bernoulli boundary control problems

Summary This paper considers the optimal quadratic cost problem (regulator problem) for a class of abstract differential equations with unbounded operators which, under the same unified framework, model in particular «concrete» boundary control problems for partial differential equations defined on a bounded open domain of any dimension, including: second order hyperbolic scalar equations with control in the Dirichlet or in the Neumann boundary conditions; first order hyperbolic systems with boundary control; and Euler-Bernoulli (plate) equations with (for instance) control(s) in the Dirichlet and/or Neumann boundary conditions. The observation operator in the quadratic cost functional is assumed to be non-smoothing (in particular, it may be the identity operator), a case which introduces technical difficulties due to the low regularity of the solutions. The paper studies existence and uniqueness of the resulting algebraic (operator) Riccati equation, as well as the relationship between exact controllability and the property that the Riccati operator be an isomorphism, a distinctive feature of the dynamics in question (emphatically not true for, say, parabolic boundary control problems). This isomorphism allows one to introduce a «dual» Riccati equation, corresponding to a «dual» optimal control problem. Properties between the original and the «dual» problem are also investigated.Research partially supported by the National Science Foundation under Grant NSF-DMS-8301668 and by the Air Force Office of Scientific Research under Grant AFOSR-84-0365.     

Bifurcation from 2-dimensional to 3-dimensional invariant tori

Summary The existence of 2 -dimensional invariant tori and their bifurcation in 3-dimensional invariant tori are investigated for a family of (non- hamiltonian) differential sistems in R ⁴.Techniques inspired to the « K.A.M. theory » are used to identify « paths of bifurcation » in the parameters space.Work performed under the auspices of the Italian Council of Research (C.N.R.) and of Ministero della Pubblica Istruzione (M.P.I.).     

Disjointness in Partially Ordered Vector Spaces

A notion of disjointness in arbitrary partially ordered vector spaces is introduced by calling two elements x and y disjoint if the set of all upper bounds of x + y and −x − y equals the set of all upper bounds of x − y and −x + y. Several elementary properties are easily observed. The question whether the disjoint complement of a subset is a linear subspace appears to be more difficult. It is shown that in directed Archimedean spaces disjoint complements are always subspaces. The proof relies on theory on order dense embedding in vector lattices. In a non-Archimedean directed space even the disjoint complement of a singleton may fail to be a subspace. According notions of disjointness preserving operator, band, and band preserving operator are defined and some of their basic properties are studied.     

A mathematical analysis of quantum transport in three-dimensional crystals

Summary We present an existence and uniqueness result for a quantum transport model in three dimensional crystals. The model consists of a quantum transport (Wigner) equation posed on the phase space consisting of a discrete position variable and a «continuous» wave vector, which is restricted to a bounded domain inR ³ (first Brillouin zone of the crystal). The potential is modeled self-consistently by a discrete Poisson equation (Coulomb interaction). Also we investigate the limits of solutions of this model as the grid spacing tends to zero and show that they converge to the solution of a quantum transport model posed on the «fully continuous» phase space. The transport model derived by this limiting procedure treats the band diagram of the crystal in a semi-classical way and the potential energy term quantum mechanically.     

Annulation de la cohomologie pour des fibrés à courbure de signe variable sur une variété kählerienne

Summary We prove a vanishing theorem for the cohomology of an holomorphic vector bundle over a kählerian compact manifold when the curvature form can change its sign. The technique is an adaptation of Malliavin's method in the riemannian case.

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Ce travail a été effectué lors d'une invitation du premier auteur par la SFB 170 « Geometrie und analysis » à Göttingen et lors de séjours du second auteur à l'Université de Florence.     

Asymptotic expansions obtained by a center manifold theorem

Sunto Viene presentato un nuovo metodo per la determinazione degli sviluppi asintotici della soluzione esterna di sistemi di equazioni differenziali ordinarie singolarmente perturbati. Il metodo proposto, basato sulla teoria geometrica delle perturbazioni singolari e in particolare su un teorema di esistenza di varietà centrale, permette di ottenere le equazioni differenziali che definiscono le variabili « lente » senza la preventiva conoscenza dei corrispondenti sviluppi per le variabili « veloci ». Inoltre, se i sistemi vengono dati con condizioni iniziali, alcune formule che esprimono le corrette condizioni iniziali da assegnare alle equazioni differenziali trovate — formule già note nel « caso stabile » — vengono estese al « caso condizionalmente stabile »; il procedimento qui usato risulta anche più sintetico rispetto a quelli precedentemente proposti. Infine viene studiata un'applicazione ad una classe assai generale di equazioni derivanti dalla cinetica delle reazioni enzimatiche.

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Lavoro eseguito nell'ambito dei programmi del gruppo di ricerca « Equazioni di Evoluzione e Applicazioni », M.P.I., e del Gruppo Nazionale Fisica-Matematica del C.N.R.     

Nontrivial solutions of elliptic boundary value problems with resonance at zero

Summary We consider a general nonlinear parabolic BVP (P) on a bounded and smooth domain R_n, the nonlinearity being given by a functionf: . We impose various hypotheses on f: « nonresonance » (with respect to the linearized BVP) at infinity, « nonresonance » or «resonance» at zero. Using an extension of Conley's index theory to noncompact spaces, we prove the existence of equilibria of (P) (i.e. solutions of a corresponding elliptic equation), as well as trajectories joining some of these equilibria. The results obtained generalize earlier results of Amann and Zehnder (who were the first to apply the Conley index to elliptic equations), of Peitgen and Schmitt, and of this author.Dedicated to Professor Jack K. Hale on his 55-th birthdayThis research was supported, in part, by a grant from the Deutsche Forschungsgemeinschaft (D.F.G.).     

Régularisation S.C.I. et γ-convergence: Approximations inf-convolutives associées à un référentiel

Summary The well-known Moreau-Yosida approximation is inoperative for nonconvex functionals without an a priori inequality of quadratic type. We point out an inf-convolution approximation (generally locally lipschitz) by a convex function (called « referential ») connected to the «infinite negative growth» of the approximated function. This method is easily extended to problems of - convergence: epi- convergence, epi-hypo-convergence ... Several applications are given to the case of integral functionals and to conditional expectations of integrands.

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article dédié à J. J.Moreau     

Some variations on the notion of connection

Distributions on manifolds are studied in terms of jets of submanifolds and are interpreted as «pre-connections» or «almost-fibrings»; the associated differential calculus is developed in detail. A comparison with connections on fibred manifolds is analysed. Moreover, «higher order pre-connections», defined as pre-connections dependent on jets of arbitrary order, are introduced and studied. It is shown that infinite jets play an essential role in the associated differential calculus.This work has been performed the visits of Prof. A. M.Vinogradov at the Department of Applied Mathematics, supported by Gruppo Nazionale per la Fisica Matematica of CNR (1989, 1990).This work has been partially supported by funds (40% and 60%) of MURST.     

Some dimension-free features of vector-valued martingales

Summary Given any local maringaleM inR ^d orl ², there exists a local martingaleN inR ², such that |M|=|N|, [M]=[N], and «M»=«N». It follows in particular that any inequality for martingales inR ² which involves only the processes |M|, [M] and «M» remains true in arbitrary dimension. WhenM is continuous, the processes |M|² and |M| satisfy certain SDE's which are independent of dimension and yield information about the growth rate ofM. This leads in particular to tail estimates of the same order as in one dimension. The paper concludes with some new maximal inequalities in continuous time.Research supported by NSF grant DMS-9002732 and by AFOSR Contract F49620 85C 0144     

Grassmann bundles

Summary In the classical theory of the Grassmann Variety there are three principal results. The Basis Theorem asserts that the Chow ring has a selfdual linear basis of classes. Determinantal Formulawhich expresses any basic class as a determinant in the special classes. Finally the ring structure is elucidated by Pieri's Formulawhich expresses the intersection of a basic class and a special class in terms of the basic classes. Here we show how all these results can be established also for the Chow ring of a Grassmann bundle. There are however some differences. In the classical case the basic classes are Schubert classes: this is impossible in the general case as there need not be enough Schubert classes to provide a basis and in the general case there is a pair of dual bases which both reduce to the Schubert basis in the classical case. In addition to these generalizations of the classical results we also enlarge on the theory of Schubert classes developed in the important paper of Kempf and Laksov [4].Following them we shall henceforth use the phrase « determinantal formula » to mean their formula for Schubert classes and our generalization of it to « improper » Schubert classes.     

Sul criterio dell'energia per la stabilitá di un sistema termomeccanico

Summary The energy criterion for mechanical stability asserts that the stable configurations are those that minimize the potential energy. Recent studies have shown that the energy criterion can be extended to stability of thermomechanical systems under suitable environment conditions, provided that the «stored energy» is interpreted as the equilibrium free-energy at the environmental temperature ^e. The aim of this paper is to provide a contribution to a general theory of thermomechanical stability. Essentially we have restated the theory for general materials introduced by Gurtin with a new framework in the light of recent theories of Noll and Coleman-Owen on simple materials and on thermodynamical potentials. We define a «thermomechanical system» which posseses two main features: i) state space has a «natural topology» depending on the thermodynamical behaviour of system; ii) internal energy E and entropy S are not supposed to exist but are expressely obtained with their smoothness properties.

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Lavoro eseguito nell'ambito del G.N.F.M. del C.N.R,     

Cohesive categories and manifolds

Sunto Le strutture ottenibili per incollamento di «spazi elementari», come le varietà, i fibrati, le varietà fogliettate, possono essere definite da «atlanti di incollamento» e, formalmente, come categorie arricchite su opportune categorie ordinate.

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Work partially supported by M.P.I. Research Projects.     

On the minimal free resolution of some projective curves

Summary Here we study the minimal free resolution of non linearly normal embedded curves of high degree and the homogeneous ideal of general enough curves (for general k-gonal curves and, in a much under range, for curves with general moduli). Then we consider the surjectivity of the Gaussian maps (or «Wahl maps») for non complete linear systems.     

Measure theory of statistical convergence

The question of establishing measure theory for statistical convergence has been moving closer to center stage, since a kind of reasonable theory is not only fundamental for unifying various kinds of statistical convergence, but also a bridge linking the studies of statistical convergence across measure theory, integration theory, probability and statistics. For this reason, this paper, in terms of subdifferential, first shows a representation theorem for all finitely additive probability measures defined on the σ-algebra of all subsets of N, and proves that every such measure can be uniquely decomposed into a convex combination of a countably additive probability measure and a statistical measure (i.e. a finitely additive probability measure μ with μ(k) = 0 for all singletons {k}). This paper also shows that classical statistical measures have many nice properties, such as: The set of all such measures endowed with the topology of point-wise convergence on forms a compact convex Hausdorff space; every classical statistical measure is of continuity type (hence, atomless), and every specific class of statistical measures fits a complementation minimax rule for every subset in N. Finally, this paper shows that every kind of statistical convergence can be unified in convergence of statistical measures. This work was supported by the National Natural Science Foundation of China (Grant Nos. 10771175, 10471114)     

OnBMO regularity for linear elliptic systems

Summary We prove a refinement of Campanato's result on local and global (under Dirichlet boundary conditions) BMO regularity for the gradient of solutions of linear elliptic systems of second order in divergence form: we just need that the coefficients are «small multipliers of BMO()», a class neither containing, nor contained in . We also prove local and global L^p regularity: this result neither implies, nor follows by the classical one by Agmon, Douglis and Nirenberg.Work partially supported by M.P.I.Project 40% «Equazioni di evoluzione e applicazioni fisico-matematiche».     

Ordered representations of spaces of integrable functions

Let X be a Banach space, (Ω,Σ) a measurable space and let m : Σ → X be a (countably additive) vector measure. Consider the corresponding space of integrable functions L¹(m). In this paper we analyze the set of (countably additive) vector measures n satisfying that L¹(n) = L¹(m). In order to do this we define a (quasi) order relation on this set to obtain under adequate requirements the simplest representation of the space L¹(m) associated to downward directed subsets of the set of all the representations. This research has been partially supported by La Junta de Andalucía. The support of D.G.I. under project MTM2006–11690–C02 (M.E.C. Spain) and FEDER is gratefully acknowledged.     

Existence and uniqueness for a reaction-diffusion problem in infiltration

We consider a system of coupled PDE'smodeling the infiltration of a reacting fluid in a soluble porous medium. The system is made of a parabolic equation for the concentration of the dissolved material, an ODE (hyperbolic equation with characteristic x=Const.)for the porosity, and an elliptic equation for the fluid pressure. We prove the existence and uniqueness of a classical solution. The classical solution is global in time in the one-dimensional case. Global existence of a weak solution is proved for the n- dimensional case.The authors would like to acknowledge the M.U.R.S.T. Project 40% «Problemi non lineari...» and the Italia C.N.R. Strategic Project «Metodi matematici per le applicazioni industriali» for partial financial support of this work.