Optimal constants in the exceptional case of Sobolev inequalities on Riemannian manifolds

Let be a Riemannian compact -manifold. We know that for any , there exists such that for any , , being the smallest constant possible such that the inequality remains true for any . We call the ``first best constant'. We prove in this paper that it is possible to choose and keep a finite constant. In other words we prove the existence of a ``second best constant' in the exceptional case of Sobolev inequalities on compact Riemannian manifolds.

     

Invariant subspaces for Banach space operators with an annular spectral set

Consider an annulus for some , and let be a bounded invertible linear operator on a Banach space whose spectrum contains . Assume there exists a constant such that and for all polynomials . Then there exists a nontrivial common invariant subspace for and .

     

Universal Toda brackets of ring spectra

We construct and examine the universal Toda bracket of a highly structured ring spectrum . This invariant of is a cohomology class in the Mac Lane cohomology of the graded ring of homotopy groups of which carries information about and the category of -module spectra. It determines for example all triple Toda brackets of and the first obstruction to realizing a module over the homotopy groups of by an -module spectrum.

For periodic ring spectra, we study the corresponding theory of higher universal Toda brackets. The real and complex -theory spectra serve as our main examples.

     

A solution to the Baer splitting problem

Let be a commutative domain. We prove that an -module is projective if and only if for any torsion module . This answers in the affirmative a question raised by Kaplansky in 1962.

     

Local and global -regularity

A bounded domain is called -regular if the plurisubharmonic envelope of every continuous function on extends continuously to . We show using Gauthier's Fusion Lemma that a domain is locally -regular if and only if it is -regular.

     

The Brown-Peterson cohomology of the classifying spaces of the projective unitary groups and exceptional Lie groups

For a fixed prime , we compute the Brown-Peterson cohomologies of classifying spaces of and exceptional Lie groups by using the Adams spectral sequence. In particular, we see that and are even dimensionally generated.

     

Higher order PDE's and iterated processes

We introduce a class of stochastic processes based on symmetric -stable processes, for . These are obtained by taking Markov processes and replacing the time parameter with the modulus of a symmetric -stable process. We call them -time processes. They generalize Brownian time processes studied in Allouba and Zheng (2001), Allouba (2002), (2003), and they introduce new interesting examples. We establish the connection of -time processes to some higher order PDE's for rational. We also obtain the PDE connection of subordinate killed Brownian motion in bounded domains of regular boundary.

     

On the polynomial representation for the number of partitions with fixed length

In this paper, it is shown that the number of partitions of a nonnegative integer with parts can be described by a set of polynomials of degree in , where denotes the least common multiple of the integers and denotes the quotient of when divided by . In addition, the sets of the polynomials are obtained and shown explicitly for and .

     

Finite rank Toeplitz operators on the Bergman space

Given a complex Borel measure with compact support in the complex plane the sesquilinear form defined on analytic polynomials and by , determines an operator from the space of such polynomials to the space of linear functionals on . This operator is called the Toeplitz operator with symbol . We show that has finite rank if and only if is a finite linear combination of point masses. Application to Toeplitz operators on the Bergman space is immediate.

     

Norming algebras and automatic complete boundedness of isomorphisms of operator algebras

We combine the notion of norming algebra introduced by Pop, Sinclair and Smith with a result of Pisier to show that if and are operator algebras, then any bounded epimorphism of onto is completely bounded provided that contains a norming -subalgebra. We use this result to give some insights into Kadison's Similarity Problem: we show that every faithful bounded homomorphism of a -algebra on a Hilbert space has completely bounded inverse, and show that a bounded representation of a -algebra is similar to a -representation precisely when the image operator algebra -norms itself. We give two applications to isometric isomorphisms of certain operator algebras. The first is an extension of a result of Davidson and Power on isometric isomorphisms of CSL algebras. Secondly, we show that an isometric isomorphism between subalgebras of -diagonals () satisfying extends uniquely to a -isomorphism of the -algebras generated by and ; this generalizes results of Muhly-Qiu-Solel and Donsig-Pitts.

     

Codes over rings of size four, Hermitian lattices, and corresponding theta functions

Let be an imaginary quadratic field with ring of integers , where is a square free integer such that , and let is a linear code defined over . The level theta function of is defined on the lattice , where is the natural projection. In this paper, we prove that:

i) for any such that , and have the same coefficients up to ,

ii) for , determines the code uniquely,

iii) for , there is a positive dimensional family of symmetrized weight enumerator polynomials corresponding to .

     

Fractal Hamilton-Jacobi-KPZ equations

Nonlinear and nonlocal evolution equations of the form , where is a pseudodifferential operator representing the infinitesimal generator of a Lévy stochastic process, have been derived as models for growing interfaces in the case when the continuous Brownian diffusion surface transport is augmented by a random hopping mechanism. The goal of this paper is to study properties of solutions to this equation resulting from the interplay between the strengths of the ``diffusive' linear and ``hyperbolic' nonlinear terms, posed in the whole space , and supplemented with nonnegative, bounded, and sufficiently regular initial conditions.

     

Invariant decomposition of functions with respect to commuting invertible transformations

Consider arbitrary elements. We characterize those functions that decompose into the sum of -periodic functions, i.e., with . We show that has such a decomposition if and only if for all partitions with consisting of commensurable elements with least common multiples one has .

Actually, we prove a more general result for periodic decompositions of functions defined on an Abelian group ; in fact, we even consider invariant decompositions of functions with respect to commuting, invertible self-mappings of some abstract set .

We also extend our results to functions between torsion free Abelian groups. As a corollary we also obtain that on a torsion free Abelian group the existence of a real-valued periodic decomposition of an integer-valued function implies the existence of an integer-valued periodic decomposition with the same periods.

     

Noether's problem and and its subgroups

We study Noether's problem for various subgroups of the normalizer of a group generated by an -cycle in , the symmetric group of degree , in three aspects according to the way they act on rational function fields, i.e., , and . We prove that it has affirmative answers for those containing properly and derive a -generic polynomial with four parameters for each . On the other hand, it is known in connection to the negative answer to the same problem for that there does not exist a -generic polynomial for . This leads us to the question whether and how one can describe, for a given field of characteristic zero, the set of -extensions . One of the main results of this paper gives an answer to this question.

     

Convex solids with planar midsurfaces

We show that the boundary of an -dimensional closed convex set , possibly unbounded, is a convex quadric surface if and only if the middle points of every family of parallel chords of lie in a hyperplane. To prove this statement, we show that the boundary of is a convex quadric surface if and only if there is a point such that all sections of by 2-dimensional planes through are convex quadric curves. Generalizations of these statements that involve boundedly polyhedral sets are given.

     

Hypersurfaces whose tangent geodesics do not cover the ambient space

Let be an immersion of an -dimensional connected manifold in an -dimensional connected complete Riemannian manifold without conjugate points. Assume that the union of geodesics tangent to does not cover . Under these hypotheses we have two results. The first one states that is simply connected provided that the universal covering of is compact. The second result says that if is a proper embedding and is simply connected, then is a normal graph over an open subset of a geodesic sphere. Furthermore, there exists an open star-shaped set such that is a manifold with the boundary .

     

Strong integrality of quantum invariants of 3-manifolds

We prove that the quantum -invariant of an arbitrary 3-manifold is always an algebraic integer if the order of the quantum parameter is co-prime with the order of the torsion part of . An even stronger integrality, known as cyclotomic integrality, was established by Habiro for integral homology 3-spheres. Here we also generalize Habiro's result to all rational homology 3-spheres.

     

Convexity and the Exterior Inverse Problem of Potential Theory

Let and be bounded solid domains such that their associated volume potentials agree outside . Under the assumption that one of the domains is convex, it is deduced that .

     

Homogeneous Hilbert scheme

Let be a locally noetherian scheme and an -graded -algebra of finite type. We say that is a homogeneous variety over . In this paper we prove that the functor

is representable by an -scheme that is a disjoint union of locally projective schemes over . The proof is very simple, and it only makes use of the theory of graded modules and standard flatness criteria. From this, one obtains an elementary construction (which does not make use of cohomology) of the ordinary Hilbert scheme of a locally projective -scheme.

     

Primitive central idempotents of finite group rings of symmetric groups

Let be a prime. We denote by the symmetric group of degree , by the alternating group of degree and by the field with elements. An important concept of modular representation theory of a finite group is the notion of a block. The blocks are in one-to-one correspondence with block idempotents, which are the primitive central idempotents of the group ring , where is a prime power. Here, we describe a new method to compute the primitive central idempotents of for arbitrary prime powers and arbitrary finite groups . For the group rings of the symmetric group, we show how to derive the primitive central idempotents of from the idempotents of . Improving the theorem of Osima for symmetric groups we exhibit a new subalgebra of which contains the primitive central idempotents. The described results are most efficient for . In an appendix we display all primitive central idempotents of and for which we computed by this method.