Let be a lattice with and . An endomorphism of is a -endomorphism, if it satisfies and . The -endomorphisms of form a monoid. In 1970, the authors proved that every monoid can be represented as the -endomorphism monoid of a suitable lattice with and . In this paper, we prove the stronger result that the lattice with a given -endomorphism monoid can be constructed as a uniquely complemented lattice; moreover, if is finite, then can be chosen as a finite complemented lattice.
We investigate the interrelationships between the dynamical properties of commuting continuous maps of a compact metric space. Let be a compact metric space.
First we show the following. If is an expansive onto continuous map with the pseudo-orbit tracing property (POTP) and if there is a topologically mixing continuous map with , then is topologically mixing. If and are commuting expansive onto continuous maps with POTP and if is topologically transitive with period , then for some dividing , , where the , , are the basic sets of with such that all have period , and the dynamical systems are a factor of each other, and in particular they are conjugate if is a homeomorphism.
Then we prove an extension of a basic result in symbolic dynamics. Using this and many techniques in symbolic dynamics, we prove the following. If is a topologically transitive, positively expansive onto continuous map having POTP, and is a positively expansive onto continuous map with , then has POTP. If is a topologically transitive, expansive homeomorphism having POTP, and is a positively expansive onto continuous map with , then has POTP and is constant-to-one.
Further we define `essentially LR endomorphisms' for systems of expansive onto continuous maps of compact metric spaces, and prove that if is an expansive homeomorphism with canonical coordinates and is an essentially LR automorphism of , then has canonical coordinates. We add some discussions on basic properties of the essentially LR endomorphisms.
We show that the expressive power of first-order logic over finite models embedded in a model is determined by stability-theoretic properties of . In particular, we show that if is stable, then every class of finite structures that can be defined by embedding the structures in , can be defined in pure first-order logic. We also show that if does not have the independence property, then any class of finite structures that can be defined by embedding the structures in , can be defined in first-order logic over a dense linear order. This extends known results on the definability of classes of finite structures and ordered finite structures in the setting of embedded finite models. These results depend on several results in infinite model theory. Let be a set of indiscernibles in a model and suppose is elementarily equivalent to where is -saturated. If is stable and is saturated, then every permutation of extends to an automorphism of and the theory of is stable. Let be a sequence of -indiscernibles in a model , which does not have the independence property, and suppose is elementarily equivalent to where is a complete dense linear order and is -saturated. Then -types over are order-definable and if is -saturated, every order preserving permutation of can be extended to a back-and-forth system.
For a given convex (semi-convex) function , defined on a nonempty open convex set , we establish a local Steiner type formula, the coefficients of which are nonnegative (signed) Borel measures. We also determine explicit integral representations for these coefficient measures, which are similar to the integral representations for the curvature measures of convex bodies (and, more generally, of sets with positive reach). We prove that, for , the -th coefficient measure of the local Steiner formula for , restricted to the set of -singular points of , is absolutely continuous with respect to the -dimensional Hausdorff measure, and that its density is the -dimensional Hausdorff measure of the subgradient of .
As an application, under the assumptions that is convex and Lipschitz, and is bounded, we get sharp estimates for certain weighted Hausdorff measures of the sets of -singular points of . Such estimates depend on the Lipschitz constant of and on the quermassintegrals of the topological closure of .
The Bryant-Ferry-Mio-Weinberger surgery exact sequence for compact homology manifolds of dimension is used to obtain transversality, splitting and bordism results for homology manifolds, generalizing previous work of Johnston.
First, we establish homology manifold transversality for submanifolds of dimension : if is a map from an -dimensional homology manifold to a space , and is a subspace with a topological -block bundle neighborhood, and , then is homology manifold -cobordant to a map which is transverse to , with an -dimensional homology submanifold.
Second, we obtain a codimension splitting obstruction in the Wall -group for a simple homotopy equivalence from an -dimensional homology manifold to an -dimensional Poincaré space with a codimension Poincaré subspace with a topological normal bundle, such that if (and for only if) splits at up to homology manifold -cobordism.
Third, we obtain the multiplicative structure of the homology manifold bordism groups .
We prove that every continuum of weight is a continuous image of the Cech-Stone-remainder of the real line. It follows that under the remainder of the half line is universal among the continua of weight -- universal in the `mapping onto' sense.
We complement this result by showing that 1) under every continuum of weight less than is a continuous image of , 2) in the Cohen model the long segment of length is not a continuous image of , and 3) implies that is not a continuous image of , whenever is a -saturated ultrafilter.
We also show that a universal continuum can be gotten from a -saturated ultrafilter on , and that it is consistent that there is no universal continuum of weight .
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 .
Let denote a sequence of complex numbers ( 0, \gamma _{ij}=\bar{\gamma}_{ji}$">), and let denote a closed subset of the complex plane . The Truncated Complex -Moment Problem for entails determining whether there exists a positive Borel measure on such that ( ) and . For a semi-algebraic set determined by a collection of complex polynomials , we characterize the existence of a finitely atomic representing measure with the fewest possible atoms in terms of positivity and extension properties of the moment matrix and the localizing matrices . We prove that there exists a -atomic representing measure for supported in if and only if and there is some rank-preserving extension for which , where or .
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
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.
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 a closed submanifold of a complete smooth Riemannian manifold and the total space of the unit normal bundle of . For each , let denote the distance from to the cut point of on the geodesic with the velocity vector The continuity of the function on is well known. In this paper we prove that is locally Lipschitz on which is bounded; in particular, if and are compact, then is globally Lipschitz on . Therefore, the canonical interior metric may be introduced on each connected component of the cut locus of and this metric space becomes a locally compact and complete length space.