where is an ordered sequence of intervals on the right half line (that is, b_{n}$">). Assume that the lengths of the intervals are bounded and that the spaces between consecutive intervals are bounded and bounded away from zero. Let . Let and denote respectively the cone of bounded, positive harmonic functions in and the cone of positive harmonic functions in which satisfy the Dirichlet boundary condition on and the Neumann boundary condition on .
Letting , the main result of this paper, under a modest assumption on the sequence , may be summarized as follows when :
1. If , then and are both one-dimensional (as in the case of the Neumann boundary condition on the entire boundary). In particular, this occurs if with 2$">.
2. If and , then and is one-dimensional. In particular, this occurs if .
3. If , then and the set of minimal elements generating is isomorphic to (as in the case of the Dirichlet boundary condition on the entire boundary). In particular, this occurs if with .
When , as soon as there is at least one interval of Dirichlet boundary condition. The dichotomy for is as above.
Let be a group definable in an o-minimal structure . A subset of is -definable if is definable in the structure (while definable means definable in the structure ). Assume has no -definable proper subgroup of finite index. In this paper we prove that if has no nontrivial abelian normal subgroup, then is the direct product of -definable subgroups such that each is definably isomorphic to a semialgebraic linear group over a definable real closed field. As a corollary we obtain an o-minimal analogue of Cherlin's conjecture.
Sufficient conditions for the convergence in distribution of an infinite convolution product of measures on a connected Lie group with respect to left invariant Haar measure are derived. These conditions are used to construct distributions that satisfy where is a refinement operator constructed from a measure and a dilation automorphism . The existence of implies is nilpotent and simply connected and the exponential map is an analytic homeomorphism. Furthermore, there exists a unique minimal compact subset such that for any open set containing and for any distribution on with compact support, there exists an integer such that implies If is supported on an -invariant uniform subgroup then is related, by an intertwining operator, to a transition operator on Necessary and sufficient conditions for to converge to , and for the -translates of to be orthogonal or to form a Riesz basis, are characterized in terms of the spectrum of the restriction of to functions supported on
RÉSUMÉ. On considère dans un ouvert borné de , à bord régulier, le problème de Dirichlet
où , est positive et s'annule sur un ensemble fini de points de . On démontre alors sous certaines hypothèses sur et si est assez petit, que le problème (1) possède une solution convexe unique .
ABSTRACT. We consider in a bounded open set of , with regular boundary, the Dirichlet problem
where , is positive and vanishes on , a finite set of points in . We prove, under some hypothesis on and if is sufficiently small, that the problem (1) has a unique convex solution .
Genus zero Willmore surfaces immersed in the three-sphere correspond via the stereographic projection to minimal surfaces in Euclidean three-space with finite total curvature and embedded planar ends. The critical values of the Willmore functional are , where , with . When the ambient space is the four-sphere , the regular homotopy class of immersions of the two-sphere is determined by the self-intersection number ; here we shall prove that the possible critical values are , where . Moreover, if , the corresponding immersion, or its antipodal, is obtained, via the twistor Penrose fibration , from a rational curve in and, if , via stereographic projection, from a minimal surface in with finite total curvature and embedded planar ends. An immersion lies in both families when the rational curve is contained in some or (equivalently) when the minimal surface of is complex with respect to a suitable complex structure of .
Let be an -primary ideal in a Gorenstein local ring (, ) with , and assume that contains a parameter ideal in as a reduction. We say that is a good ideal in if is a Gorenstein ring with . The associated graded ring of is a Gorenstein ring with if and only if . Hence good ideals in our sense are good ones next to the parameter ideals in . A basic theory of good ideals is developed in this paper. We have that is a good ideal in if and only if and . First a criterion for finite-dimensional Gorenstein graded algebras over fields to have nonempty sets of good ideals will be given. Second in the case where we will give a correspondence theorem between the set and the set of certain overrings of . A characterization of good ideals in the case where will be given in terms of the goodness in their powers. Thanks to Kato's Riemann-Roch theorem, we are able to classify the good ideals in two-dimensional Gorenstein rational local rings. As a conclusion we will show that the structure of the set of good ideals in heavily depends on . The set may be empty if , while is necessarily infinite if and contains a field. To analyze this phenomenon we shall explore monomial good ideals in the polynomial ring in three variables over a field . Examples are given to illustrate the theorems.
Let be a complete discrete valuation domain with the unique maximal ideal . We suppose that is an algebra over an algebraically closed field and . Subamalgam -suborders of a tiled -order are studied in the paper by means of the integral Tits quadratic form . A criterion for a subamalgam -order to be of tame lattice type is given in terms of the Tits quadratic form and a forbidden list of minor -suborders of presented in the tables.
Let , be finite-dimensional Lie algebras over a field of characteristic zero. Regard and , the dual Lie coalgebra of , as Lie bialgebras with zero cobracket and zero bracket, respectively. Suppose that a matched pair of Lie bialgebras is given, which has structure maps . Then it induces a matched pair of Hopf algebras, where is the universal envelope of and is the Hopf dual of . We show that the group of cleft Hopf algebra extensions associated with is naturally isomorphic to the group of Lie bialgebra extensions associated with . An exact sequence involving either of these groups is obtained, which is a variation of the exact sequence due to G.I. Kac. If , there follows a bijection between the set of all cleft Hopf algebra extensions of by and the set of all Lie bialgebra extensions of by .
Let be the Bessel operator with matricial coefficients defined on by
where is a diagonal matrix and let be an matrix-valued function. In this work, we prove that there exists an isomorphism on the space of even , -valued functions which transmutes and . This allows us to define generalized translation operators and to develop harmonic analysis associated with . By use of the Riemann method, we provide an integral representation and we deduce more precise information on these operators.