On the computability-theoretic complexity of trivial, strongly minimal models

We show the existence of a trivial, strongly minimal (and thus uncountably categorical) theory for which the prime model is computable and each of the other countable models computes . This result shows that the result of Goncharov/Harizanov/Laskowski/Lempp/McCoy (2003) is best possible for trivial strongly minimal theories in terms of computable model theory. We conclude with some remarks about axiomatizability.

     

Explicit quantization of dynamical r-matrices for finite dimensional semisimple Lie algebras

We provide an explicit quantization of dynamical r-matrices for semisimple Lie algebras, classified earlier by the third author, which includes the Belavin-Drinfeld r-matrices. We do so by constructing an appropriate (dynamical) twist in the tensor square of the Drinfeld-Jimbo quantum group , which twists the R-matrix of into the desired quantization. The construction of this twist is based on the method stemming from the work of Jimbo-Konno-Odake-Shiraishi and Arnaudon-Buffenoir-Ragoucy-Roche, i.e. on defining the twist as a unique solution of a suitable difference equation. This yields a simple closed formula for the twist.

This construction allows one to confirm the alternate version of the Gerstenhaber-Giaquinto-Schack conjecture (about quantization of Belavin-Drinfeld r-matrices for in the vector representation), which was stated earlier by the second author on the basis of computer evidence. It also allows one to define new quantum groups associated to semisimple Lie algebras. We expect them to have a rich structure and interesting representation theory.

     

Simple Bratteli diagrams with a Gödel-incomplete C*-equivalence problem

An abstract simplicial complex is a finite family of subsets of a finite set, closed under subsets. Every abstract simplicial complex naturally determines a Bratteli diagram and a stable AF-algebra . Consider the following problem:

INPUT: a pair of abstract simplicial complexes and ;

QUESTION: is isomorphic to ?

We show that this problem is Gödel incomplete, i.e., it is recursively enumerable but not decidable. This result is in sharp contrast with the recent decidability result by Bratteli, Jorgensen, Kim and Roush, for the isomorphism problem of stable AF-algebras arising from the iteration of the same positive integer matrix. For the proof we use a combinatorial variant of the De Concini-Procesi theorem for toric varieties, together with the Baker-Beynon duality theory for lattice-ordered abelian groups, Markov's undecidability result, and Elliott's classification theory for AF-algebras.

     

Level one representations of

We construct a level one representation of the quantum affine algebra by vertex operators from bosonic fields.

     

Trivial, strongly minimal theories are model complete after naming constants

We prove that if is any model of a trivial, strongly minimal theory, then the elementary diagram is a model complete -theory. We conclude that all countable models of a trivial, strongly minimal theory with at least one computable model are -decidable, and that the spectrum of computable models of any trivial, strongly minimal theory is .

     

Degrees, kernels and quasikernels of monolithic characters

Theorem 5 yields the condition sufficient for a group to be a direct product of a -group and an abelian -group. We also obtain characterizations of nilpotent groups, prime power groups, -nilpotent and -closed groups in the language of characters. Proofs of some results depend on the classification of finite simple groups. Some problems are posed and discussed.

     

Instability of statistical factor analysis

Factor analysis, a popular method for interpreting multivariate data, models the covariance among variables as being due to a small number (, ) of hidden variables. A factor analysis of can be thought of as an ordered or unordered collection, , of linearly independent lines in . Let be the collection of data sets for which is defined. The ``singularities' of are those data sets, , in the closure, , at which the limit, , does not exist. is unstable near its singularities.

Let be the direct sum of the lines in . determines a -plane bundle, , over a subset, , of . If 1$"> and is rich enough, ordered or, at least if or 3, unordered, must have a singularity at some data set in . The proofs are applications of algebraic topology. Examples are provided.

     

On Bohman's conjecture related to a sum packing problem of Erdos

Motivated by a sum packing problem of Erdos, Bohman discussed an extremal geometric problem which seems to have an independent interest. Let be a hyperplane in such that . The problem is to determine

Bohman (1996) conjectured that

We show that for some constants we have --disproving the conjecture. We also consider a more general question of the estimation of , when , k>1$">.

     

Inverse spectral theory of finite Jacobi matrices

We solve the following physically motivated problem: to determine all finite Jacobi matrices and corresponding indices such that the Green's function

is proportional to an arbitrary prescribed function . Our approach is via probability distributions and orthogonal polynomials.

We introduce what we call the auxiliary polynomial of a solution in order to factor the map

(where square brackets denote the equivalence class consisting of scalar multiples). This enables us to construct the solution set as a fibration over a connected, semi-algebraic coordinate base. The end result is a wealth of explicit constructions for Jacobi matrices. These reveal precise geometric information about the solution set, and provide the basis for new existence theorems.

     

Dehn surgeries on strongly invertible knots which yield lens spaces

In this article we show no Dehn surgery on nontrivial strongly invertible knots can yield the lens space for any integer . In order to do that, we determine band attaches to -torus links producing the trivial knot.

     

On the number of generators of the torsion module of differentials

In this paper we study the (minimum) global number of generators of the torsion module of differentials of affine hypersurfaces with only isolated singularities. We show that for reduced plane curves the torsion module of differentials can be generated by at most two elements, whereas for higher codimensions there is no universal upper bound. We then proceed to give explicit examples. In particular (when ) , we give examples of a reduced hypersurface with a single isolated singularity at the origin in that require

generators for the torsion module, Torsion .

     

Jacobi polynomials from compatibility conditions

We revisit the ladder operators for orthogonal polynomials and re-interpret two supplementary conditions as compatibility conditions of two linear over-determined systems; one involves the variation of the polynomials with respect to the variable (spectral parameter) and the other a recurrence relation in (the lattice variable). For the Jacobi weight

we show how to use the compatibility conditions to explicitly determine the recurrence coefficients of the monic Jacobi polynomials.

     

The quartile operator and pointwise convergence of Walsh series

The bilinear Hilbert transform is given by

It satisfies estimates of the type

In this paper we prove such estimates for a discrete model of the bilinear Hilbert transform involving the Walsh Fourier transform. We also reprove the well-known fact that the Walsh Fourier series of a function in , with converges pointwise almost everywhere. The purpose of this exposition is to clarify the connection between these two results and to present an easy approach to recent methods of time-frequency analysis.

     

Solution of a question of Knarr

The Moufang condition is one of the central group theoretical conditions in Incidence Geometry, and was introduced by Jacques Tits in his famous lecture notes (1974).

About ten years ago, Norbert Knarr studied generalized quadrangles (buildings of Type ) which satisfy one of the Moufang conditions locally at one point. He then posed the fundamental question whether the group generated by the root-elations with its root containing that point is always a sharply transitive group on the points opposite this point, that is, whether this group is an elation group.

In this paper, we solve the question and a more general version affirmatively for finite generalized quadrangles.

Moreover, we show that this group is necessarily nilpotent (which was only known up till now when both Moufang conditions are satisfied for all points and lines).

In fact, as a corollary, we will prove that these groups always have to be -groups for some prime .

     

The Laplace transform of the digamma function: An integral due to Glasser, Manna and Oloa

The definite integral

is related to the Laplace transform of the digamma function

by when . Certain analytic expressions for in the complementary range, , are also provided.

     

Regularized orbital integrals for representations of

Given a finite-dimensional representation of , on a vector space defined over a local field of characteristic zero, we produce a regularization of orbital integrals and determine when the resulting distribution is non-trivial.

     

On Cartan matrices and lower defect groups for covering groups of symmetric groups

We determine the elementary divisors of the Cartan matrices of spin -blocks of the covering groups of the symmetric groups when is an odd prime. As a consequence, we also compute the determinants of these Cartan matrices, and in particular we confirm a conjecture by Brundan and Kleshchev that these determinants depend only on the weight but not on the sign of the block.

     

Estimates for viscosity solutions of parabolic equations with Dirichlet boundary conditions

It is shown how one can get upper bounds for when and are the (viscosity) solutions of

respectively, in with Dirichlet boundary conditions. Similar results are obtained for some other parabolic equations as well, including certain equations in divergence form.