Resultants and discriminants of Chebyshev and related polynomials

We show that the resultants with respect to of certain linear forms in Chebyshev polynomials with argument are again linear forms in Chebyshev polynomials. Their coefficients and arguments are certain rational functions of the coefficients of the original forms. We apply this to establish several related results involving resultants and discriminants of polynomials, including certain self-reciprocal quadrinomials.

     

Elliptic Apostol sums and their reciprocity laws

We introduce an elliptic analogue of the Apostol sums, which we call elliptic Apostol sums. These sums are defined by means of certain elliptic functions with a complex parameter having positive imaginary part. When , these elliptic Apostol sums represent the well-known Apostol generalized Dedekind sums. Also these elliptic Apostol sums are modular forms in the variable . We obtain a reciprocity law for these sums, which gives rise to new relations between certain modular forms (of one variable).

     

Upper bounds for the number of solutions of a Diophantine equation

We give upper bound estimates for the number of solutions of a certain diophantine equation. Our results can be applied to obtain new lower bound estimates for the -norm of certain exponential sums.

     

Thue's theorem and the diophantine equation

A constructive version of a theorem of Thue is used to provide representations of certain integers as , where .

     

Divergent Cesàro and Riesz means of Jacobi and Laguerre expansions

We show that for below certain critical indices there are functions whose Jacobi or Laguerre expansions have almost everywhere divergent Cesàro and Riesz means of order .

     

Remarks on a result of Khalil and Saleh

We give a short proof of a recent result that describes onto isometries of for certain pairs of Banach spaces .

     

Operator semigroups with quasinilpotent commutators

It is shown that a multiplicative semigroup of operators is triangularizable if is quasinilpotent for every pair in the semigroup and certain other hypotheses are satisfied.

     

Napoleon in isolation

Napoleon's theorem in elementary geometry describes how certain linear operations on plane polygons of arbitrary shape always produce regular polygons. More generally, certain triangulations of a polygon that tiles admit deformations which keep fixed the symmetry group of the tiling. This gives rise to isolation phenomena in cusped hyperbolic -manifolds, where hyperbolic Dehn surgeries on some collection of cusps leave the geometric structure at some other collection of cusps unchanged.

     

Decomposability of graph -algebras

We give conditions on an arbitrary directed graph for the associated Cuntz-Krieger algebra to be decomposable as a direct sum. We describe the direct summands as certain graph algebras.

     

An approximate universal coefficient theorem

An approximate Universal Coefficient Theorem (AUCT) for certain -algebras is established. We present a proof that Kirchberg-Phillips's classification theorem for separable nuclear purely infinite simple -algebras is valid for -algebras satisfying the AUCT instead of the UCT. It is proved that two versions of AUCT are in fact the same. We also show that -algebras that are locally approximated by -algebras satisfying the AUCT satisfy the AUCT. As an application, we prove that certain simple -algebras which are locally type I are in fact isomorphic to simple AH-algebras. As another application, we show that a sequence of residually finite-dimensional -algebras which are asymptotically nuclear and which asymptotically satisfies the AUCT can be embedded into the same simple AF-algebra.

     

On boundedness of solutions of reaction-diffusion equations with nonlinear boundary conditions

We give conditions on the nonlinearities of a reaction-diffusion equation with nonlinear boundary conditions that guarantee that any solution starting at bounded initial data is bounded locally around a certain point of the boundary, uniformly for all positive time. The conditions imposed are of a local nature and need only to hold in a small neighborhood of the point .

     

Positive eigenvalues of second order boundary value problems and a theorem of M. G. Krein

Conditions are given which guarantee that the least real eigenvalue is positive for certain boundary value problems for the vector-matrix equation . This leads to conditions which guarantee the stable boundedness, according to Krein, for solutions of with certain real values of . As a consequence, a result first stated by Krein is proven.

     

Humbert surfaces and the Kummer plane

A Humbert surface is a hypersurface of the moduli space of principally polarized abelian surfaces defined by an equation of the form with integers . We give geometric characterizations of such Humbert surfaces in terms of the presence of certain curves on the associated Kummer plane. Intriguingly this shows that a certain plane configuration of lines and curves already carries all information about principally polarized abelian surfaces admitting a symmetric endomorphism with given discriminant.

     

A symmetry theorem revisited

We show that if harmonic measure and Hausdorff measure are equal on the boundary of certain domains in Euclidean -space, then these domains are necessarily balls.

     

Strongly almost disjoint sets and weakly uniform bases

A combinatorial principle CECA is formulated and its equivalence with GCH + certain weakenings of for singular is proved. CECA is used to show that certain ``almost point-' families can be refined to point- families by removing a small set from each member of the family. This theorem in turn is used to show the consistency of ``every first countable -space with a weakly uniform base has a point-countable base.'

     

On the asymptotic linearization of acoustic waves

The initial value problem of a certain generalization of the nonlinear, dispersive wave equations with dissipation is rigorously studied. The solutions of the equations can be found exactly up to in certain norms. The essential use is made of the fact that this equation is asymptotically linearizable to i.e., the equations can be mapped to an equation which differs from a linearizable equation only in terms which are of An application of the equations to unidirectional small amplitude acoustic waves is discussed. The general methodology used here can also be applied to other asymptotically linearizable equations.

     

An elementary invariant problem and general linear group cohomology restricted to the diagonal subgroup

Conjecturally, for an odd prime and a certain ring of -integers, the stable general linear group and the étale model for its classifying space have isomorphic mod cohomology rings. In particular, these two cohomology rings should have the same image with respect to the restriction map to the diagonal subgroup. We show that a strong unstable version of this last property holds for any rank if is regular and certain homology classes for vanish. We check that this criterion is satisfied for as evidence for the conjecture.

     

On Iwasawa -invariants of cyclic cubic fields of prime conductor

For certain cyclic cubic fields , we verified that Iwasawa invariants vanished by calculating units of abelian number field of degree 27. Our method is based on the explicit representation of a system of cyclotomic units of those fields.

     

A note on finite abelian gerbes over toric Deligne-Mumford stacks

Any toric Deligne-Mumford stack is a -gerbe over the underlying toric orbifold for a finite abelian group . In this paper we give a sufficient condition so that certain kinds of gerbes over a toric Deligne-Mumford stack are again toric Deligne-Mumford stacks.