The cohomology of invariant variational bicomplexes

Let: E M be a fiber bundle and let be an infinitesimal Lie transformation group acting onE. We announce various new results concerning the cohomology of the invariant variational bicomplex ( ^*,* (J(E)), d_H, d_V) and the associated invariant Euler-Lagrange complex. As one application of our general theory, we completely solve the local invariant inverse problem of the calculus of variations for finite-dimensional infinitesimal Lie transformation groups.     

A normalization problem for a class of singular integral operators with carleman shift and unbounded coefficients

The solution to a normalization problem for singular integral operators with Carleman shift and degenerate and unbounded coefficients inL _p () is obtained, where is either the unit circle or the real line. The approach followed consists mainly in two steps: the reduction to a singular integral operator with bounded coefficients and the use of the solution to an abstract normalization problem.This research was supported by JNICT under the grant PBIC/C/CEN/1040/92.     

Properties of certain integral operators

Two integral operatorsP andQ for analytic functions in the open unit disk are introduced. The object of the present paper is to derive some properties of integral operatorsP andQ .     

On the intrinsic definition of local observables

In a theory of local field algebras satisfying the split property, we introduce an algebra generated by the local energy-momentum operators obtained from the canonical local implementation of translations, and possibly by the local charges operators. We discuss the relations of this algebra to the given algebra of local observables and take some steps toward the characterization of theories where they coincide. In the presence of spontaneously broken symmetries, we present a no-go theorem.     

Cosmological principle and primordial spinor fields

An exact solution of Dirac's equation in an open Robertson-Walker universe is constructed such that the corresponding energy-momentum densityT _µv obeys the cosmological principle. However, in anopen universe, this principlemust always be violated by the other physical densities of the cosmological solution (e.g. pseudoscalar, (axial-) current and polarization). If the current densityj _µ = isrequired to obey the cosmological principle, then the universe must beflat.     

q-transformation theory

We obtain the complete set of states of theq-oscillator in both configuration space and momentum space as well as the transformation between these spaces. The states as well as the matrix elements lie in the SU _q (2) algebra. To obtain transition probabilities, one must take the Woronowicz square.     

Arbres,espaces ultramétriques et bases de structure uniforme

Résumé On décrit une bijection canonique qui associe à un bon espace ultramétrique, localement compact, non compact, un arbre pointé par un bout: cet espace s'identifie à l'ensemble des autres bouts.

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We describe a canonical bijection, which associates a tree with a chosen end to a nice ultrametric, locally compact, non-compact space: this space is identified with the set of other ends.

     

New lower semicontinuity results for polyconvex integrals

Summary We study integral functionals of the formF(u, )= f(u)dx, defined foru C¹(;R ^k), R ⁿ . The functionf is assumed to be polyconvex and to satisfy the inequalityf(A) c₀¦(A)¦ for a suitable constant c₀ > 0, where (A) is then-vector whose components are the determinants of all minors of thek×n matrixA. We prove thatF is lower semicontinuous onC ¹(;R ^k) with respect to the strong topology ofL ¹(;R ^k). Then we consider the relaxed functional , defined as the greatest lower semicontinuous functional onL ¹(;R ^k ) which is less than or equal toF on C¹(;R ^k). For everyu BV(;R ^k) we prove that (u,) f(u)dx+c₀¦D^su¦(), whereDu=u dx+D^su is the Lebesgue decomposition of the Radon measureDu. Moreover, under suitable growth conditions onf, we show that (u,)= f(u)dx for everyu W^1,p(;R ^k), withp min{n,k}. We prove also that the functional (u, ) can not be represented by an inte- gral for an arbitrary functionu BV_loc(R ⁿ;R ^k). In fact, two examples show that, in general, the set function (u, ) is not subadditive whenu BV_loc(R ⁿ;R ^k), even ifu W _loc ^1,p (R ⁿ;R ^k) for everyp < min{n,k}. Finally, we examine in detail the properties of the functionsu BV(;R ^k) such that (u, )= f(u)dx, particularly in the model casef(A)=¦(A)¦.     

Sur le comportement asymptotique des potentiels d'une chaîne de Markov sur ℝ d

Résumé Dans cet article j'étudie le comportement à l'infini des potentiels des chaînes de Markov sur ^d (d3) proches du mouvement brownien, tout spécialement le cas des marches aléatoires, ainsi que des critères de transience et de récurrence inspirés de la méthode utilisée.

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We study the asymptotic behaviour of potentials of Markov chains on ^d (d3), closed to Brownian motion, and particularly the case of random walks. Following a similar approach, we give transience and recurrence criteria.