Mathematics Faculty Works

A New Proof of a Theorem of Phan

Curtis Bennett et al. — Tue, 07 Feb 2012 14:41:21 PST

We apply diagram geometry and amalgam techniques to give a new proof of a theorem of K.-W. Phan, characterizing the special unitary group as a group generated by certain systems of subgroups SU(2, q(2)).

Sufficient Conditions for an Operator-Valued Feynman-Kac Formula

Michael D. Grady — Tue, 07 Feb 2012 14:41:20 PST

Let E be a locally compact, second countable Hausdorff space and let X(t) be a Markov process with state space E. Sufficient conditions are given for the existence of a solution to the initial value problem, ∂u/∂t,=Au + V(x) * u, u(0) = f, where A is the infinitesimal generator of the process X on a certain Banach space and for each x ∈ E, V(x) is the infinitesimal generator of a C₀ contraction semigroup on another Banach space.

Pseudoprime L-Ideals in a Class of F-Rings

Suzanne Larson — Tue, 07 Feb 2012 14:41:19 PST

In a commutative f-ring, an l-ideal I is called pseudoprime if ab = 0 implies a ∈ I or b ∈ I, and is called square dominated if for every a ∈ I, |a| ≤ x² for some x ∈ A such that x² ∈ I. Several characterizations of pseudoprime l-ideals are given in the class of commutative semiprime f-rings in which minimal prime l-ideals are square dominated. It is shown that the hypothesis imposed on the f-rings, that minimal prime l-ideals are square dominated, cannot be omitted or generalized.

Sums of Semiprime, Z, and D L-Ideals in a Class of F-Rings

Suzanne Larson — Tue, 07 Feb 2012 14:41:17 PST

In this paper it is shown that there is a large class of f-rings in which the sum of any two semiprime i-ideals is semiprime. This result is used to give a class of commutative f-rings with identity element in which the sum of any two z-ideals which are i-ideals is a z-ideal and the sum of any two d-ideals is a d-ideal.

Lattice-Ordered Algebras That Are Subdirect Products of Valuation Domains

Melvin Henriksen et al. — Tue, 07 Feb 2012 14:41:16 PST

An f-ring (i.e., a lattice-ordered ring that is a subdirect product pf totally ordered rings) A is called an SV-ring if AIP is a valuation domain for every prime ideal P of A. If M is a maximal e-ideal of A,then the rank of A at M is the number of minimal prime ideals of A contained in M, rank of A is the sup of the ranks of A at each of its maximal e-ideals. If the latter is a positive integer, then A is said to have finite rank, and if A = C(X) is the ring of all real-valued continuous functions on a Tychonoff space, the rank of X is defined to be the rank of the f-ring C(X),and X is called an SV-space if C(X) is an SV-ring. X has finite rank k iff k is the maximal number of painwise disjoint cozero sets with a point common to all of their closures. In general f-rings these two concepts are unrelated, but if ii is uniformly complete (in particular, if A = C(X)) then if A is an SV-ring then it has finite rank. Showing that this latter holds makes use of the theory of finite-valued lattice-ordered (abelian) groups. These two kinds of rings are investigated with an emphasis on the uniformly complete case. Fairly powerful machinery seems to have to be used, and even then, we do not know if there is a compact space X of finite rank that fails to be an SV-space.

Cohomology of Polynomials Under an Irrational Rotation

Lawrence W. Baggett et al. — Tue, 07 Feb 2012 14:41:15 PST

A new description of cohomology of functions under an irrational rotation is given in terms of symmetry properties of the functions on subintervals of [0, 1]. This description yields a method for passing information about the cohomology classes for a given irrational to the cohomology classes for an equivalent irrational.

On Functions That Are Trivial Cocycles for a Set of Irrationals. II

Lawrence W. Baggett et al. — Tue, 07 Feb 2012 14:41:14 PST

Two results are obtained about the topological size of the set of irrationals for which a given function is a trivial cocycle. An example of a continuous function which is a coboundary with non-L(1) cobounding function is constructed.

The Growth of Valuations on Rational Function Fields in Two Variables

Edward Mosteig et al. — Tue, 07 Feb 2012 14:41:13 PST

Given a valuation on the function field k( x; y), we examine the set of images of nonzero elements of the underlying polynomial ring k[ x; y] under this valuation. For an arbitrary field k, a Noetherian power series is a map z : Q --> k that has Noetherian (i.e., reverse well-ordered) support. Each Noetherian power series induces a natural valuation on k( x; y). Although the value groups corresponding to such valuations are well-understood, the restrictions of the valuations to underlying polynomial rings have yet to be characterized. Let Lambda(n) denote the images under the valuation v of all nonzero polynomials f is an element of k[ x; y] of at most degree n in the variable y. We construct a bound for the growth of Lambda(n) with respect to n for arbitrary valuations, and then specialize to valuations that arise from Noetherian power series. We provide a sufficient condition for this bound to be tight.

A Map on the Space of Rational Functions

G. Boros et al. — Tue, 07 Feb 2012 14:41:12 PST

We describe dynamical properties of a map defined on the space of rational functions. The fixed points of F are classified and the long time behavior of a subclass is described in terms of Eulerian polynomials.

Computing Boundary Slopes of 2-Bridge Links

Jim Hoste et al. — Tue, 07 Feb 2012 14:41:10 PST

We describe an algorithm for computing boundary slopes of 2-bridge links. As an example, we work out the slopes of the links obtained by 1/k surgery on one component of the Borromean rings. A table of all boundary slopes of all 2-bridge links with 10 or less crossings is also included.

A Multiple-Precision Division Algorithm

David M. Smith — Tue, 07 Feb 2012 14:41:08 PST

The classical algorithm for multiple-precision division normalizes digits during each step and sometimes makes correction steps when the initial guess for the quotient digit turns out to be wrong. A method is presented that runs faster by skipping most of the intermediate normalization and recovers from wrong guesses without separate correction steps.

Efficient Multiple-Precision Evaluation of Elementary Functions

David M. Smith — Tue, 07 Feb 2012 14:41:07 PST

Let M(t) denote the time required to multiply two t-digit numbers using base b arithmetic. Methods are presented for computing the elementary functions in O(t^1/3 M(t)) time.