This paper considers the boundary rigidity problem for a compact convex Riemannian manifold with boundary whose curvature satisfies a general upper bound condition. This includes all nonpositively curved manifolds and all sufficiently small convex domains on any given Riemannian manifold. It is shown that in the space of metrics on there is a -neighborhood of such that is the unique metric with the given boundary distance-function (i.e. the function that assigns to any pair of boundary points their distance -- as measured in ). More precisely, given any metric in this neighborhood with the same boundary distance function there is diffeomorphism which is the identity on such that . There is also a sharp volume comparison result for metrics in this neighborhood in terms of the boundary distance-function.
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 .
Let be the upper half strip with a hole. In this paper, we show there exists a positive higher energy solution of semilinear elliptic equations in and describe the dynamic systems of solutions of equation in various . We also show there exist at least two positive solutions of perturbed semilinear elliptic equations in .
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.
The equation where and are fractional derivatives of order and is studied. It is shown that if , , and are Hölder-continuous and , then there is a solution such that and are Hölder-continuous as well. This is proved by first considering an abstract fractional evolution equation and then applying the results obtained to (). Finally the solution of () with is studied. 相似文献
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.
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.
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
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.
In this article we address the problem of computing the dimension of the space of plane curves of degree with general points of multiplicity . A conjecture of Harbourne and Hirschowitz implies that when , the dimension is equal to the expected dimension given by the Riemann-Roch Theorem. Also, systems for which the dimension is larger than expected should have a fixed part containing a multiple -curve. We reformulate this conjecture by explicitly listing those systems which have unexpected dimension. Then we use a degeneration technique developed to show that the conjecture holds for all .
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 .
In this article, we study dihedral coverings of algebraic surfaces branched along curves with at most simple singularities. A criterion for a reduced curve to be the branch locus of some dihedral covering is given. As an application we have the following:
Let be a reduced plane curve of even degree having only nodes and cusps. If 2d^2 - 6d + 6$">, then is non-abelian.
Note that Nori's result implies that is abelian, provided that .
along rays of representations in a positive Weyl chamber , i.e. for sequences of representations , with . As a corollary we obtain some estimates on the spectral radius of the random walk. We also analyse the fine structure of the spectrum for certain random walks on (for which is essentially a direct sum of Harper operators).