Residually finite dimensional and AF-embeddable -algebras

We show that every separable nuclear residually finite dimensional -algebras satisfying the Universal Coefficient Theorem can be embedded into a unital separable simple AF-algebra.

     

Higher dimensional aposyndetic decompositions

Let be a homogeneous, decomposable continuum that is not aposyndetic. The Aposyndetic Decomposition Theorem yields a cell-like decomposition of into homogeneous continua with quotient space being an aposyndetic, homogeneous continuum.

Assume the dimension of is greater than one. About 20 years ago the author asked the following questions:

Can this aposyndetic decomposition raise dimension? Can it lower dimension? We answer these questions by proving the following theorem.

Theorem. The dimension of the quotient space is one.

     

Open 3-manifolds whose fundamental groups have infinite center, and a torus theorem for 3-orbifolds

Our main result is a characterization of open Seifert fibered -manifolds in terms of the fundamental group and large-scale geometric properties of a triangulation. As an application, we extend the Seifert Fiber Space Theorem and the Torus Theorem to a class of -orbifolds.

     

Duals and topological center of a class of matrix algebras with applications

We characterize the topological center of a class of matrix algebras, which are called -Munn algebras. This involves a characterization of Arens regular semigroup algebras for a large class of semigroups, which is an extension of Young's Theorem for semigroups. We show by some counter examples that only up to a certain extent Young's Theorem can be generalized.

     

Asymptotic properties of the vector Carleson embedding theorem

The dyadic Carleson embedding operator acting on -valued functions has norm at least . Thus the Carleson Embedding Theorem fails for Hilbert space valued functions.

     

Sturm theorems for degenerate elliptic equations

We employ a version of the Picone Identity, suitable for the -Laplacean, to obtain a Sturm Comparison Theorem for generalized solutions. This answers a question posed by Dunninger in 1995.

     

On possible non-homeomorphic substructures of the real line

We consider the problem, raised by Kunen and Tall, of whether the real continuum can have non-homeomorphic versions in different submodels of the universe of all sets. This requires large cardinals, and we obtain an exact consistency strength:

Theorem 1. The following are equiconsistent:

(i) a Jónsson cardinal;

(ii) a sufficiently elementary submodel of the universe of sets with not homeomorphic to

The reverse direction is a corollary to:

Theorem 2. is Jónsson hereditarily separable, hereditarily Lindelöf, with .

We further consider the large cardinal consequences of the existence of a topological space with a proper substructure homeomorphic to Baire space.

     

Krull-Schmidt fails for serial modules

We answer a question posed by Warfield in 1975: the Krull-Schmidt Theorem does not hold for serial modules, as we show via an example. Nevertheless we prove a weak form of the Krull-Schmidt Theorem for serial modules (Theorem 1.9). And we show that the Grothendieck group of the class of serial modules of finite Goldie dimension over a fixed ring is a free abelian group.

     

The proof of Tchakaloff's Theorem

We provide a simple proof of Tchakaloff's Theorem on the existence of cubature formulas of degree for Borel measures with moments up to order . The result improves known results for non-compact support, since we do not need conditions on st moments. In fact, we reduce the classical assertion of Tchakaloff's Theorem to a well-known statement going back to F. Riesz.

     

A nonstandard Riemann existence theorem

We study elementary extensions of compact complex spaces and deduce that every complete type of dimension is internal to projective space. This amounts to a nonstandard version of the Riemann Existence Theorem, and answers a question posed by Anand Pillay.

     

The curve of ``Prym canonical' Gauss divisors on a Prym theta divisor

The Gauss linear system on the theta divisor of the Jacobian of a nonhyperelliptic curve has two striking properties:

1) the branch divisor of the Gauss map on the theta divisor is dual to the canonical model of the curve;

2) those divisors in the Gauss system parametrized by the canonical curve are reducible.

In contrast, Beauville and Debarre prove on a general Prym theta divisor of dimension all Gauss divisors are irreducible and normal. One is led to ask whether properties 1) and 2) may characterize the Gauss system of the theta divisor of a Jacobian. Since for a Prym theta divisor, the most distinguished curve in the Gauss system is the Prym canonical curve, the natural analog of the canonical curve for a Jacobian, in the present paper we analyze whether the analogs of properties 1) or 2) can ever hold for the Prym canonical curve. We note that both those properties would imply that the general Prym canonical Gauss divisor would be nonnormal. Then we find an explicit geometric model for the Prym canonical Gauss divisors and prove the following results using Beauville's singularities criterion for special subvarieties of Prym varieties:

Theorem. For all smooth doubly covered nonhyperelliptic curves of genus , the general Prym canonical Gauss divisor is normal and irreducible.

Corollary. For all smooth doubly covered nonhyperelliptic curves of genus , the Prym canonical curve is not dual to the branch divisor of the Gauss map.

     

An Ahlfors Islands Theorem for non-archimedean meromorphic functions

We present a -adic and non-archimedean version of Ahlfors' Five Islands Theorem for meromorphic functions, extending an earlier theorem of the author for holomorphic functions. In the non-archimedean setting, the theorem requires only four islands, with explicit constants. We present examples to show that the constants are sharp and that other hypotheses of the theorem cannot be removed.

     

Some remarks on totally imperfect sets

We prove the following two theorems.

Theorem 1. Let be a strongly meager subset of . Then it is dual Ramsey null.

We will say that a -ideal of subsets of satisfies the condition iff for every , if

then .

Theorem 2. The -ideals of perfectly meager sets, universally meager sets and perfectly meager sets in the transitive sense satisfy the condition .

     

estimates for a class of oscillatory integrals

We obtain a simpler proof of Theorem 3.1 of The complete mapping properties of some oscillatory integrals in several dimensions, by G. Sampson and P. Szeptycki (Canad. Math. J. 53 (5) (2001), 1031-1056).

     

Only `free' measures are admissable on is incomplete

Using elementary arguments and without having to recall the Gleason Theorem, we prove that the existence of a nonsingular measure on the lattice of orthogonally closed subspaces of an inner product space is a sufficient (and of course, a necessary) condition for to be a Hilbert space.

     

Free summands of conormal modules and central elements in homotopy Lie algebras of local rings

If is a surjective local homomorphism with kernel , such that and the conormal module has a free summand of rank , then the degree central subspace of the homotopy Lie algebra of has dimension greater than or equal to . This is a corollary of the Main Theorem of this note. The techniques involved provide new proofs of some well known results concerning the conormal module.

     

Irrational rotation numbers and unboundedness of solutions of the second order differential equations with asymmetric nonlinearities

In this paper, we study the dynamics of the mappings

where is a irrational rotation number. We prove the existence of orbits that go to infinity in the future or in the past by using the well-known Birkhoff Ergodic Theorem. Applying this conclusion, we deal with the unboundedness of solutions of Liénard equations with asymmetric nonlinearities.

     

Uniqueness of travelling waves for nonlocal monostable equations

We consider a nonlocal analogue of the Fisher-KPP equation

0 ~{\rm on}~(0,1),\end{displaymath}">

and its discrete counterpart , , and show that travelling wave solutions of these equations that are bounded between and are unique up to translation. Our proof requires finding exact a priori asymptotics of a travelling wave. This we accomplish with the help of Ikehara's Theorem (which is a Tauberian theorem for Laplace transforms).

     

An analogue of Hajós' Theorem for the circular chromatic number

This paper designs a set of graph operations and proves that starting from , by repeatedly applying these operations, one can construct all graphs with (for ). This can be viewed as an analogue of Hajós' Theorem for the circular chromatic number.

     

The existence of positive solutions for the one-dimensional -Laplacian

In this paper we study the existence of positive solutions of the equation , where , , subject to nonlinear boundary conditions. We show the existence of at least one positive solution by a simple application of a Fixed Point Theorem in cones and the Arzela-Ascoli Theorem.