Semiclassical analysis for highly degenerate potentials

This paper characterizes the semi-classical limit of the fundamental energy,

and ground state of the Schrödinger operator in a bounded domain , in the highly degenerate case when and consists of two components, say and . The main result establishes that

and that approximates in the ground state of in if

     

Unbounded solutions and periodic solutions for second order differential equations with asymmetric nonlinearity

In this paper we will prove the coexistence of unbounded solutions and periodic solutions for the asymmetric oscillator

where and are positive constants satisfying the nonresonant condition

and is periodic in the first variable and bounded.

     

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.

     

Integral representation for a class of vector valued operators

Let be a compact space and let , be a (real, for simplicity) Banach space. We consider the space of all continuous -valued functions on , with the supremum norm .

We prove in this paper a Bochner integral representation theorem for bounded linear operators

which satisfy the following condition:

where is the conjugate space of . In the particular case where , this condition is obviously satisfied by every bounded linear operator

and the result reduces to the classical Riesz representation theorem.

If the dimension of is greater than , we show by a simple example that not every bounded linear admits an integral representation of the type above, proving that the situation is different from the one dimensional case.

Finally we compare our result to another representation theorem where the integration process is performed with respect to an operator valued measure.

     

Twisted higher moments of Kloosterman sums

Let be a nontrivial Dirichlet character modulo an odd prime . Write

We shall prove

and, for complex ,

0, \end{displaymath}">

where is a constant depending only on .

     

A polarized partition relation for cardinals of countable cofinality

We prove that if and , then

for all . This polarized partition relation holds if for every partition either there are and with or there are and with .

     

Multiple solutions for elliptic problems with singular and sublinear potentials

For certain positive numbers and we establish the multiplicity of solutions to the problem

where is a bounded open domain in containing the origin with smooth boundary while is continuous, superlinear at zero and sublinear at infinity.

     

Explicit continued fractions with expected partial quotient growth

For let be the continued fraction expansion of . Write

We construct some numbers 's with

     

Push-forward of degeneracy classes and ampleness

Let be a projective variety and vector bundles on . Suppose is a surjective map onto another variety . Let be any vector bundle map and the 'th degeneracy locus of . We show that the dimension of is at least equal to

under the hypothesis that is an ample vector bundle on .

     

Extended Cesàro operators on mixed norm spaces

We define an extended Cesàro operator with holomorphic symbol in the unit ball of as

where is the radial derivative of . In this paper we characterize those for which is bounded (or compact) on the mixed norm space .

     

Asymptotics of Sobolev embeddings and singular perturbations for the -Laplacian

We consider the best constant for the embedding of into where , . Here with a smooth, bounded domain in and a large positive number. It is proven by the validity of the expansion

as , where is a positive constant depending on and . The behavior of associated extremals, which satisfy an equation involving the -Laplacian operator, is also analyzed.

     

Sharp estimates for the identity minus Hardy operator on the cone of decreasing functions

It is shown that if we restrict the identity minus Hardy operator on the cone of nonnegative decreasing functions in , then we have the sharp estimate

for In other words,

for each and each integer .

It is also shown, via a connection between the operator and Laguerre functions, that

for all .

     

On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent

We consider the nonlinear eigenvalue problem

in , on , where is a bounded open set in with smooth boundary and , are continuous functions on such that , , and for all . The main result of this paper establishes that any sufficiently small is an eigenvalue of the above nonhomogeneous quasilinear problem. The proof relies on simple variational arguments based on Ekeland's variational principle.

     

On a Littlewood-Paley type inequality

The following is proved: If is a function harmonic in the unit ball and if then the inequality

holds, where is the nontangential maximal function of This improves a recent result of Stoll. This inequality holds for polyharmonic and hyperbolically harmonic functions as well.

     

Character sums over shifted smooth numbers

We give nontrivial bounds in various ranges for character sums of the form

where is a nontrivial multiplicative character modulo a prime and is the set of positive integers that are divisible only by primes .

     

On positive unipotent operators on Banach lattices

Let be a positive operator on a complex Banach lattice. We prove that is greater than or equal to the identity operator if

     

On a class of sublinear quasilinear elliptic problems

We establish existence and multiplicity of positive solutions to the quasilinear boundary value problem

where is a bounded domain in with smooth boundary , is continuous and p-sublinear at and is a large parameter.

     

The rational LS-category of -trivial fibrations

We provide new upper and lower bounds for the rational LS-category of a rational fibration of simply connected spaces that depend on a measure of the triviality of which is strictly finer than the vanishing of the higher holonomy actions. In particular, we prove that if is -trivial for some and enjoys Poincaré duality, then

     

-isomorphisms, Jordan isomorphisms, and numerical range preserving maps

Let or , where is the algebra of a bounded linear operator acting on the Hilbert space , and is the set of self-adjoint operators in . Denote the numerical range of by It is shown that a surjective map satisfies

if and only if there is a unitary operator such that has the form

where is the transpose of with respect to a fixed orthonormal basis. In other words, the map or is a -isomorphism on and a Jordan isomorphism on . Moreover, if has finite dimension, then the surjective assumption on can be removed.

     

Detecting completeness from Ext-vanishing

Motivated by work of C. U. Jensen, R.-O. Buchweitz, and H. Flenner, we prove the following result. Let be a commutative noetherian ring and an ideal in the Jacobson radical of . Let be the -adic completion of . If is a finitely generated -module such that for all , then is -adically complete.