An elementary and constructive solution to Hilbert's 17th Problem for matrices

We give a short and elementary proof of a theorem of Procesi, Schacher and (independently) Gondard, Ribenboim that generalizes a famous result of Artin. Let be an symmetric matrix with entries in the polynomial ring . The result is that if is positive semidefinite for all substitutions , then can be expressed as a sum of squares of symmetric matrices with entries in . Moreover, our proof is constructive and gives explicit representations modulo the scalar case.

     

Gradient ranges of bumps on the plane

For a -smooth bump function we show that the gradient range is the closure of its interior, provided that admits a modulus of continuity satisfying as . The result is a consequence of a more general result about gradient ranges of bump functions of the same degree of smoothness. For such bump functions we show that for open sets , either the intersection is empty or its topological dimension is at least two. The proof relies on a new Morse-Sard type result where the smoothness hypothesis is independent of the dimension of the space.

     

Uniqueness implies existence and uniqueness criterion for nonlocal boundary value problems for third order differential equations

For the third order differential equation, we consider uniqueness implies existence results for solutions satisfying the nonlocal -point boundary conditions, Uniqueness of solutions of such boundary value problems is intimately related to solutions of the third order equation satisfying certain nonlocal -point boundary conditions. These relationships are investigated as well.

     

Bordism groups of special generic mappings

The Pontrjagin-Thom construction expresses a relation between the oriented bordism groups of framed immersions , and the stable homotopy groups of spheres. We apply the Pontrjagin-Thom construction to the oriented bordism groups of mappings n$">, with mildest singularities. Recently, O. Saeki showed that for , the group is isomorphic to the group of smooth structures on the sphere of dimension . Generalizing, we prove that is isomorphic to the -th stable homotopy group , , where is the group of oriented auto-diffeomorphisms of the sphere and is the group of rotations of .

     

Covering by graphs of -ary functions and long linear orderings of Turing degrees

A point is covered by a function iff there is a permutation of such that .

By a theorem of Kuratowski, for every infinite cardinal exactly -ary functions are needed to cover all of . We show that for arbitrarily large uncountable it is consistent that the size of the continuum is and is covered by -ary continuous functions.

We study other cardinal invariants of the -ideal on generated by continuous -ary functions and finally relate the question of how many continuous functions are necessary to cover to the least size of a set of parameters such that the Turing degrees relative to this set of parameters are linearly ordered.

     

On diophantine approximation along algebraic curves

Let be a quadratic form such that the associated algebraic curve contains a rational point. Here we show that there exists a domain such that for almost all , there exists an infinite sequence of nonzero integer triples satisfying the following two properties: (i) For each , is an excellent rational approximation to , in the sense that

and (ii) is a rational point on the curve . In addition, we give explicit values of for which both (i) and (ii) hold, and produce a similar result for a certain class of cubic curves.

     

Radical and cyclotomic extensions of the rational numbers

A radical extension of the rational numbers is a field generated by an element having a power in , and a cyclotomic extension is an extension generated by a root of unity. We show that a radical extension that is almost Galois over is almost cyclotomic. More precisely, we prove that if is radical with Galois closure , then contains a cyclotomic field such that the degree is bounded above by an almost linear function of . In particular, if is Galois, it contains a cyclotomic field such that .

     

Isolated invariant curves of a foliation

We bound the equisingularity type of the set of isolated separatrices of a holomorphic foliation of in terms of the Milnor number of . This result gives a bound for the degree of an algebraic invariant curve of a foliation of in terms of the degree of , provided that all the branches of are isolated separatrices.

     

Avoiding the projective hierarchy in expansions of the real field by sequences

Some necessary conditions are given on infinitely oscillating real functions and infinite discrete sets of real numbers so that first-order expansions of the field of real numbers by such functions or sets do not define . In particular, let be such that , as for some , is o-minimal, and the expansion of by the set does not define . Then there exist 0$"> and such that as .

     

Multiplicatively spectrum-preserving maps of function algebras

Let be a compact Hausdorff space and a function algebra. Assume that is the maximal ideal space of . Denoting by the spectrum of an , which in this case coincides with the range of , a result of Molnár is generalized by our Main Theorem: If is a surjective map with the property for every pair of functions , then there exists a homeomorphism such that

for every and every with .

     

A note on Gabor orthonormal bases

The study of Gabor bases of the form for has interested many mathematicians in recent years. Alex Losevich and Steen Pedersen in 1998, Jeffery C. Lagarias, James A. Reeds and Yang Wang in 2000 independently proved that, for any fixed positive integer , is an orthonormal basis for if and only if is a tiling of . Palle E. T. Jorgensen and Steen Pedersen in 1999 gave an explicit characterization of such for , , . Inspired by their work, this paper addresses Gabor orthonormal bases of the form for and some other related problems, where is as above. For a fixed , the generating function of a Gabor orthonormal basis for corresponding to the above is characterized explicitly provided that , which is new even if ; a Shannon type sampling theorem about such is derived when , ; for an arbitrary positive integer , an explicit expression of the with being an orthonormal basis for is obtained under the condition that .

     

Convolution operators induced by approximate identities and pointwise convergence in spaces

Given a sequence of kernels for which the operators converge a.e. in all spaces, , a perturbation method is provided with the property that the modified convolution operators converge pointwise only in selective spaces.

     

Approximation of holomorphic maps with a lower bound on the rank

Let be a closed polydisc or ball in , and let be a quasi-projective algebraic manifold which is Zariski locally equivalent to , or a complement of an algebraic subvariety of codimension in such a manifold. If is an integer satisfying , then every holomorphic map from a neighborhood of to with rank at every point of can be approximated uniformly on by entire maps with rank at every point of .

     

On sums of Darboux and nowhere constant continuous functions

We show that the property

(P)

for every Darboux function there exists a continuous nowhere constant function such that is Darboux

follows from the following two propositions:

(A)

for every subset of of cardinality there exists a uniformly continuous function such that ,

(B)

for an arbitrary function whose image contains a non-trivial interval there exists an of cardinality such that the restriction of to is uniformly continuous,

which hold in the iterated perfect set model.

     

Associated primes of local cohomology modules

Let be an ideal of a commutative Noetherian ring and a finitely generated -module. Let be a natural integer. It is shown that there is a finite subset of , such that is contained in union with the union of the sets , where and . As an immediate consequence, we deduce that the first non- -cofinite local cohomology module of with respect to has only finitely many associated prime ideals.

     

Combinatorics of rank jumps in simplicial hypergeometric systems

Let be an integer matrix, and assume that the convex hull of its columns is a simplex of dimension  not containing the origin. It is known that the semigroup ring is Cohen-Macaulay if and only if the rank of the GKZ hypergeometric system equals the normalized volume of for all complex parameters (Saito, 2002). Our refinement here shows that has rank strictly larger than the volume of if and only if lies in the Zariski closure (in  ) of all -graded degrees where the local cohomology is nonzero. We conjecture that the same statement holds even when is not a simplex.

     

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 .

     

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.

     

Polynomial detection of matrix subalgebras

The double Capelli polynomial of total degree is

It was proved by Giambruno-Sehgal and Chang that the double Capelli polynomial of total degree is a polynomial identity for . (Here, is a field and is the algebra of matrices over .) Using a strengthened version of this result obtained by Domokos, we show that the double Capelli polynomial of total degree is a polynomial identity for any proper -subalgebra of . Subsequently, we present a similar result for nonsplit inequivalent extensions of full matrix algebras.

     

A character of the gradient estimate for diffusion semigroups

Let be the semigroup of the diffusion process generated by on . It is proved that there exists and an -valued function such that holds for all 0$"> and all if and only if satisfies the formula for all