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.
For an nonnegative matrix , an isomorphism is obtained between the lattice of initial subsets (of ) for and the lattice of -invariant faces of the nonnegative orthant . Motivated by this isomorphism, we generalize some of the known combinatorial spectral results on a nonnegative matrix that are given in terms of its classes to results for a cone-preserving map on a polyhedral cone, formulated in terms of its invariant faces. In particular, we obtain the following extension of the famous Rothblum index theorem for a nonnegative matrix: If leaves invariant a polyhedral cone , then for each distinguished eigenvalue of for , there is a chain of distinct -invariant join-irreducible faces of , each containing in its relative interior a generalized eigenvector of corresponding to (referred to as semi-distinguished -invariant faces associated with ), where is the maximal order of distinguished generalized eigenvectors of corresponding to , but there is no such chain with more than members. We introduce the important new concepts of semi-distinguished -invariant faces, and of spectral pairs of faces associated with a cone-preserving map, and obtain several properties of a cone-preserving map that mostly involve these two concepts, when the underlying cone is polyhedral, perfect, or strictly convex and/or smooth, or is the cone of all real polynomials of degree not exceeding that are nonnegative on a closed interval. Plentiful illustrative examples are provided. Some open problems are posed at the end.
Let be the group of automorphisms of a homogeneous tree , and let be a lattice subgroup of . Let be the tensor product of two spherical irreducible unitary representations of . We give an explicit decomposition of the restriction of to . We also describe the spherical component of explicitly, and this decomposition is interpreted as a multiplication formula for associated orthogonal polynomials.
On the rank of the -class group of . Let be a square-free positive integer and be a prime such that . We set , where or . In this paper, we determine the rank of the -class group of .
RÉSUMÉ. Soit , un corps biquadratique où ou bien un premier et étant un entier positif sans facteurs carrés. Dans ce papier, on détermine le rang du -groupe de classes de .
Let be a bounded symmetric domain in a complex vector space with a real form and be the real bounded symmetric domain in the real vector space . We construct the Berezin kernel and consider the Berezin transform on the -space on . The corresponding representation of is then unitarily equivalent to the restriction to of a scalar holomorphic discrete series of holomorphic functions on and is also called the canonical representation. We find the spectral symbol of the Berezin transform under the irreducible decomposition of the -space.
The main result of this paper is that the variety of presentations of a general cubic form in variables as a sum of cubes is isomorphic to the Fano variety of lines of a cubic -fold , in general different from .
A general surface of genus determines uniquely a pair of cubic -folds: the apolar cubic and the dual Pfaffian cubic (or for simplicity and ). As Beauville and Donagi have shown, the Fano variety of lines on the cubic is isomorphic to the Hilbert scheme of length two subschemes of . The first main result of this paper is that parametrizes the variety of presentations of the cubic form , with , as a sum of cubes, which yields an isomorphism between and . Furthermore, we show that sets up a correspondence between and . The main result follows by a deformation argument.
If is a binary cubic form with integer coefficients such that has at least two distinct complex roots, then the equation possesses at most ten solutions in integers and , nine if has a nontrivial automorphism group. If, further, is reducible over , then this equation has at most solutions, unless is equivalent under -action to either or . The proofs of these results rely upon the method of Thue-Siegel as refined by Evertse, together with lower bounds for linear forms in logarithms of algebraic numbers and techniques from computational Diophantine approximation. Along the way, we completely solve all Thue equations for cubic and irreducible of positive discriminant . As corollaries, we obtain bounds for the number of solutions to more general cubic Thue equations of the form and to Mordell's equation , where and are nonzero integers.
Fix integers with k>0$"> and . Let be an integral projective curve with and a rank torsion free sheaf on which is a flat limit of a family of locally free sheaves on . Here we prove the existence of a rank subsheaf of such that . We show that for every there is an integral projective curve not Gorenstein, and a rank 2 torsion free sheaf on with no rank 1 subsheaf with . We show the existence of torsion free sheaves on non-Gorenstein projective curves with other pathological properties.
In this paper, it is proved that for 2$"> and , if masses are located at fixed points in a plane, then there are only a finite number of -point central configurations that can be generated by positioning a given additional th mass in the same plane. The result is established by proving an equivalent isolation result for planar central configurations of five or more points. Other general properties of central configurations are established in the process. These relate to the amount of centrality lost when a point mass is perturbed and to derivatives associated with central configurations.
Let be a hyperbolic diffeomorphism on a basic set and let be a connected Lie group. Let be Hölder. Assuming that satisfies a natural partial hyperbolicity assumption, we show that if is a measurable solution to a.e., then must in fact be Hölder. Under an additional centre bunching condition on , we show that if assigns `weight' equal to the identity to each periodic orbit of , then for some Hölder . These results extend well-known theorems due to Livsic when is compact or abelian.
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 .
Kadison has shown that local derivations from a von Neumann algebra into any dual bimodule are derivations. In this paper we extend this result to local derivations from any -algebra into any Banach -bimodule . Most of the work is involved with establishing this result when is a commutative -algebra with one self-adjoint generator. A known result of the author about Jordan derivations then completes the argument. We show that these results do not extend to the algebra of continuously differentiable functions on . We also give an automatic continuity result, that is, we show that local derivations on -algebras are continuous even if not assumed a priori to be so.
Let be a Banach function algebra on a compact space , and let be such that for any scalar the element is not a divisor of zero. We show that any complete norm topology on that makes the multiplication by continuous is automatically equivalent to the original norm topology of . Related results for general Banach spaces are also discussed.
Let be an odd prime number and let be an extraspecial -group. The purpose of the paper is to show that has no non-zero essential mod- cohomology (and in fact that is Cohen-Macaulay) if and only if and .
Let be a group with a normal subgroup contained in the upper central subgroup . In this article we study the influence of the quotient group on the lower central subgroup . In particular, for any finite group we give bounds on the order and exponent of . For equal to a dihedral group, or quaternion group, or extra-special group we list all possible groups that can arise as . Our proofs involve: (i) the Baer invariants of , (ii) the Schur multiplier of relative to a normal subgroup , and (iii) the nonabelian tensor product of groups. Some results on the nonabelian tensor product may be of independent interest.