Decomposable form equations without the finiteness property

Let be a finitely generated (but not necessarily algebraic) extension field of . Let be a form (homogeneous polynomial) in variables with coefficients in , and suppose that is decomposable (i.e., that it factorizes into linear factors over some finite extension of ). We say that has the finiteness property over if for every (here denotes the set of non-zero elements in ) and for every subring of which is finitely generated over , the equation

has only finitely many solutions. This paper proves the following result: Let be a decomposable form in variables with coefficients in , which factorizes into linear factors over . Let denote a maximal set of pairwise linearly independent linear factors of . If has the finiteness property over , then 2(m-1)$">.

     

On a conjecture about MRA Riesz wavelet bases

Let be a compactly supported refinable function in such that the shifts of are stable and for a -periodic trigonometric polynomial . A wavelet function can be derived from by . If is an orthogonal refinable function, then it is well known that generates an orthonormal wavelet basis in . Recently, it has been shown in the literature that if is a -spline or pseudo-spline refinable function, then always generates a Riesz wavelet basis in . It was an open problem whether can always generate a Riesz wavelet basis in for any compactly supported refinable function in with stable shifts. In this paper, we settle this problem by proving that for a family of arbitrarily smooth refinable functions with stable shifts, the derived wavelet function does not generate a Riesz wavelet basis in . Our proof is based on some necessary and sufficient conditions on the -periodic functions and in such that the wavelet function , defined by , generates a Riesz wavelet basis in .

     

Counting cusps of subgroups of

Let be a number field with real places and complex places, and let be the ring of integers of . The quotient has cusps, where is the class number of . We show that under the assumption of the generalized Riemann hypothesis that if is not or an imaginary quadratic field and if , then has infinitely many maximal subgroups with cusps. A key element in the proof is a connection to Artin's Primitive Root Conjecture.

     

Real -flats tangent to quadrics in

Let and denote the dimension and the degree of the Grassmannian , respectively. For each there are (a priori complex) -planes in tangent to general quadratic hypersurfaces in . We show that this class of enumerative problems is fully real, i.e., for there exists a configuration of real quadrics in (affine) real space so that all the mutually tangent -flats are real.

     

Basic characters of the unitriangular group (for arbitrary primes)

Let denote the (upper) unitriangular group of degree over the finite field with elements. In this paper we consider the basic (complex) characters of and we prove that every irreducible (complex) character of is a constituent of a unique basic character. This result extends a previous result which was proved by the author under the assumption , where is the characteristic of the field .

     

Irreducible polynomials which are locally reducible everywhere

For any positive integer , there exist polynomials of degree which are irreducible over and reducible over for all primes if and only if is composite. In fact, this result holds over arbitrary global fields.

     

The homological determinant of quantum groups of type

Let be a Hecke symmetry depending algebraically on a parameter . We show that the homology of the Koszul complex associated with is one-dimensional when is not a root of unity. A generator of this homology group then induces the homological determinant of the quantum group associated with .

     

A -closed subalgebra of the Smirnov class

We study real Smirnov functions and investigate a certain -closed subalgebra of the Smirnov class containing them. Motivated by a result of Aleksandrov, we provide an explicit representation for the space . This leads to a natural analog of the Riesz projection on a certain quotient space of for . We also study a Herglotz-like integral transform for singular measures on the unit circle .

     

On restricted weak type : The continuous case

Let denote the unit circle. An example of a sublinear translation-invariant operator acting on is given such that is of restricted weak type but not of weak type .

     

On a conjectured noncommutative Beals-Cordes-type characterization

Given a skew-symmetric matrix we prove that a bounded operator on for which is smooth, and which commutes with all pseudodifferential operators is of the form with possessing bounded derivatives of all orders on Here, and denote the translation and the gauge representations of This was conjectured by Rieffel (1993) and is an application of the well-known Cordes' characterization of the the Heisenberg-smooth operators as pseudodifferential operators.

     

A sharp result on -covers

Let be a finite system of residue classes which forms an -cover of (i.e., every integer belongs to at least members of ). In this paper we show the following sharp result: For any positive integers and , if there is such that the fractional part of is , then there are at least such subsets of . This extends an earlier result of M. Z. Zhang and an extension by Z. W. Sun. Also, we generalize the above result to -covers of the integral ring of any algebraic number field with a power integral basis.

     

Artinianess of graded local cohomology modules

Let be a Noetherian homogeneous ring with local base ring and let be a finitely generated graded -module. Let be the largest integer such that is not Artinian. We will prove that are Artinian for all and there exists a polynomial of degree less than such that for all . Let be the first integer such that the local cohomology module is not cofinite. We will show that for all the graded module is Artinian.

     

Revisiting two theorems of Curto and Fialkow on moment matrices

We revisit two results of Curto and Fialkow on moment matrices. The first result asserts that every sequence whose moment matrix is positive semidefinite and has finite rank is the sequence of moments of an -atomic nonnegative measure on . We give an alternative proof for this result, using algebraic tools (the Nullstellensatz) in place of the functional analytic tools used in the original proof of Curto and Fialkow. An easy observation is the existence of interpolation polynomials at the atoms of the measure having degree at most if the principal submatrix of (indexed by all monomials of degree ) has full rank . This observation enables us to shortcut the proof of the following result. Consider a basic closed semialgebraic set , where and . If is positive semidefinite and has a flat extension such that all localizing matrices are positive semidefinite, then has an atomic representing measure supported by . We also review an application of this result to the problem of minimizing a polynomial over the set .

     

A remark on irregularity of the -Neumann problem on non-smooth domains

It is an observation due to J. J. Kohn that for a smooth bounded pseudoconvex domain in there exists such that the -Neumann operator on maps (the space of -forms with coefficient functions in -Sobolev space of order ) into itself continuously. We show that this conclusion does not hold without the smoothness assumption by constructing a bounded pseudoconvex domain in , smooth except at one point, whose -Neumann operator is not bounded on for any .

     

A note on the support of a Sobolev function on a -cell

It is shown that a -cell (the homeomorphic image of a closed ball in ) in , , cannot support a function in if [\frac{k+1}{2}]$">, the greatest integer in .

     

The linear space of generalized Brownian motions with applications

In this paper, we define, motivated by recent works of Chang and Skoug, stochastic integrals for a generalized Brownian motion ( ) and then use it to study the representation problem on the linear space spanned by . We next establish a translation theorem for -functionals of , , and then use this translation to establish an integration by parts formula for -functionals of .

     

Tauberian type theorem for operators with interpolation spectrum for Hölder classes

We consider an invertible operator on a Banach space whose spectrum is an interpolating set for Hölder classes. We show that if , , with and , then for all , assuming that satisfies suitable regularity conditions. When is a Hilbert space and (i.e. is a contraction), we show that under the same assumptions, is unitary and this is sharp.

     

Homogeneous polynomials on strictly convex domains

We consider a circular, bounded, strictly convex domain with boundary of class . For any compact subset of we construct a sequence of homogeneous polynomials on which are big at each point of . As an application for any circular subset of type we construct a holomorphic function which is square integrable on and such that where denotes unit disc in .

     

Stable constant mean curvature hypersurfaces

Let be a Riemannian manifold with sectional curvatures uniformly bounded from below. When we prove that there are no complete (strongly) stable -hypersurfaces, without boundary, provided is large enough. In particular, we prove that there are no complete strongly stable -hypersurfaces in without boundary,