Mathematics Commons^™

Numbers Of Generators Of Ideals In Local Rings And A Generalized Pascal's Triangle, Lucia Riderer

Theses Digitization Project

This paper defines generalized binomial coefficients and shows that they can be used to generate generalized Pascal's Triangles and have properties analogous to binomial coefficients. It uses the generalized binomial coefficients to compute the Dilworth number and the Sperner number of certain rings.

Where Does It All End? Boundaries Beyond Euclidean Space, Jonathan Thompson

Inquiry: The University of Arkansas Undergraduate Research Journal

No abstract provided.

Continued Radicals, Jamie Johnson

Masters Theses & Specialist Projects

If a1, a2, . . . , an are nonnegative real numbers and fj(x) = paj + x, then f1o f2o· · · fn(0) is a nested radical with terms a1, . . . , an. If it exists, the limit as n ! 1 of such an expression is a continued radical. We consider the set of real numbers S(M) representable as an infinite nested radical whose terms a1, a2, . . . are all from a finite set M. We give conditions on the set M for S(M) to be (a) an interval, and (b) homeomorphic to the …

Inner Matrices And Darlington Synthesis, Stephan Ramon Garcia

Pomona Faculty Publications and Research

We describe and parameterize the solutions of the scalar valued Darlington synthesis problem. In the case of rational data we derive a simple procedure for producing all possible solutions.

Reduction Of The Gibbs Phenomenon Via Interpolation Using Chebyshev Polynomials, Filtering And Chebyshev-Pade' Approximations, Rob-Roy L. Mace

Theses, Dissertations and Capstones

In this manuscript, we will examine several methods of interpolation, with an emphasis on Chebyshev polynomials and the removal of the Gibbs Phenomenon. Included as an appendix are the author’s Mat- Lab implementations of Lagrange, Chebyshev, and rational interpolation methods.

Erratum: “Uniqueness Theorems In Bioluminescence Tomography” [Med. Phys. 31, 2289–2299 (2004)], Ge Wang, Yi Li, Ming Jiang

Mathematics and Statistics Faculty Publications

In this Erratum, we present a correction to our proof of Theorem D.4 in Ref. 1.

Associative Structures Based Upon A Categorical Braiding, Stefan Forcey

Mathematical Sciences Faculty Research

It is well known that the existence of a braiding in a monoidal category V allows many structures to be built upon that foundation. These include a monoidal 2-category V-Cat of enriched categories and functors over V, a monoidal bicategory V-Mod of enriched categories and modules, a category of operads in V and a 2-fold monoidal category structure on V. We will begin by focusing our exposition on the first and last in this list due to their ability to shed light on a new question. We ask, given a braiding on V, what non-equal structures of a given kind …

Combinatorial Construction Of Toric Residues, Amit Khetan, Ivan Soprunov

Mathematics and Statistics Faculty Publications

In this paper we investigate the problem of finding an explicit element whose toric residue is equal to one. Such an element is shown to exist if and only if the associated polytopes are essential. We reduce the problem to finding a collection of partitions of the lattice points in the polytopes satisfying a certain combinatorial property. We use this description to solve the problem when n=2 and for any n when the polytopes of the divisors share a complete flag of faces. The latter generalizes earlier results when the divisors were all ample

Gap Labeling, Claude Schochet

Mathematics Faculty Research Publications

This joint review covers:

Moulay-Tahar Benameur and Hervé Oyono-Oyono, Gap-labelling for quasi-crystals (proving a conjecture by J. Bellissard). (English summary) Operator algebras and mathematical physics, (Constanţa, 2001), 11-22, Theta, Bucharest, 2003.

Jerome Kaminker and Ian Putnam, A proof of the gap labeling conjecture, Michigan Mathematical Journal 51(3) (2003), 537-546.

Jean Bellisard, Riccardo Benedetti and Jean-Marc Gambaudo, Spaces of tilings, finite telescopic approximations, and gap-labeling, Communications in Mathematical Physics 261(1) (2006), 1-41.

The gap labeling theorem was originally conjectured by Bellissard [in From number theory to physics (Les Houches, 1989), 538-630, Springer, Berlin, 1992; MR1221111 (94e:46120)]. …

Disko Solution In Braille, Jeremiah Farrell

Scholarship and Professional Work - LAS

A copy of a 4x4 DISKO solution in Braille. Constructed by students at the Indiana School for the Blind on magnetized squares on a "toasted", i.e. raised grid.

Farrell's Spider, Jeremiah Farrell, Ivan Moscovich

Scholarship and Professional Work - LAS

Puzzle game featured in Ivan Moscovich's magnetic puzzle pack:

Place the 18 discs on the web so that the sum of the numbers on each of the three hexagons and on each of the three ribs equals 57.

Srt Division Algorithms As Dynamical Systems, Mark Mccann, Nicholas Pippenger

All HMC Faculty Publications and Research

Sweeney--Robertson--Tocher (SRT) division, as it was discovered in the late 1950s, represented an important improvement in the speed of division algorithms for computers at the time. A variant of SRT division is still commonly implemented in computers today. Although some bounds on the performance of the original SRT division method were obtained, a great many questions remained unanswered. In this paper, the original version of SRT division is described as a dynamical system. This enables us to bring modern dynamical systems theory, a relatively new development in mathematics, to bear on an older problem. In doing so, we are able …

The Closed Topological Vertex Via The Cremona Transform, Jim Bryan, Dagan Karp

All HMC Faculty Publications and Research

We compute the local Gromov-Witten invariants of the "closed vertex", that is, a configuration of three rational curves meeting in a single triple point in a Calabi-Yau threefold. The method is to express the local invariants of the vertex in terms of ordinary Gromov-Witten invariants of a certain blowup of CP^3 and then to compute those invariants via the geometry of the Cremona transformation.

The Convergence Of Difference Boxes, Christopher Kribs, Antonio Behn, Vadim Ponomarenko

Mathematics Faculty Publications - Archive

We consider an elementary mathematical puzzle known as a "difference box" in terms of a discrete map from R⁴ to R⁴ or , canonically, from a subset of the first R² into itself. We identify the map's unique canonical fixed point and answer more generally the question of how many interactions a given "difference box" takes to reach zero. (The number is finite except for boxes corresponding to the fixed point.)

Toric Residue And Combinatorial Degree, Ivan Soprunov

Mathematics and Statistics Faculty Publications

Consider an -dimensional projective toric variety defined by a convex lattice polytope . David Cox introduced the toric residue map given by a collection of divisors on . In the case when the are -invariant divisors whose sum is , the toric residue map is the multiplication by an integer number. We show that this number is the degree of a certain map from the boundary of the polytope to the boundary of a simplex. This degree can be computed combinatorially. We also study radical monomial ideals of the homogeneous coordinate ring of . We give a necessary and sufficient …

Multidimensional Kolmogorov-Petrovsky Test For The Boundary Regularity And Irregularity Of Solutions To The Heat Equation, Ugur G. Abdulla

Mathematics and System Engineering Faculty Publications

This paper establishes necessary and sufficient condition for the regularity of a characteristic top boundary point of an arbitrary open subset of ( ) for the diffusion (or heat) equation. The result implies asymptotic probability law for the standard -dimensional Brownian motion.

Multiple Positive Solutions Of Singular Discrete P-Laplacian Problems Via Variational Methods, Ravi P. Agarwal

Mathematics and System Engineering Faculty Publications

We obtain multiple positive solutions of singular discrete p-Laplacian problems using variational methods.

Unpredictable Binary Strings, Richard Low, Mark Stamp, R. Craigen, G. Faucher

Faculty Publications, Computer Science

We examine a class of binary strings arising from considerations about stream cipher encryption: to what degree can one guarantee that the number of pairs of entries distance k apart that disagree is equal to the number that agree, for all small k? In a certain sense, a keystream with such a property achieves a degree of unpredictability. The problem is also restated combinatorially in terms of seating arrangements. We examine sequences s of length 2n in which this property holds for all k ≤ Mn, where Mn is the largest number for which this is possible among strings of …

Minimal Modules Over Valuation Domains, Brendan Goldsmith, P. Zanardo

Articles

Let R be a valuation domain. We say that a torsion-free R-module is minimal if it is isomorphic to all its submodules of finite index. Here, the usual concept of finite index for groups is replaced by the more appropriate (for module theory) definition: a submodule H of the module G is said to be of finite index in G if the quotient G/H is a finitely presented torsion module. We investigate minimality in various settings and show inter alia that over a maximal valuation domain, all torsion-free modules are minimal. Constructions of non-minimal modules are given by utilizing realization …

Conformal Properties And Baecklund Transform For The Associated Camassa-Holm Equation, Rossen Ivanov

Articles

Integrable equations exhibit interesting conformal properties and can be written in terms of the so-called conformal invariants. The most basic and important example is the KdV equation and the corresponding Schwarz-KdV equation. Other examples, including the Camassa-Holm equation and the associated Camassa-Holm equation are investigated in this paper. It is shown that the B¨acklund transform is related to the conformal properties of these equations. Some particular solutions of the Associated Camassa-Holm Equation are discussed also.

Asymptotic Methods In The Spectral Analysis Of Sturm-Liouville Operators, Daphne Gilbert

Articles

Gilbert, D.: Asymptotic Methods in the Spectral Analysis of Sturm-Lioville Operators.Sturm-Liouville theory : Past and Present, 2005, pp 121-136.

Factors Of Dickson Polynomials Over Finite Fields, Robert W. Fitzgerald, Joseph L. Yucas

Articles and Preprints

We give new descriptions of the factors of Dickson polynomials over finite fields.

Highly Degenerate Quadratic Forms Over Finite Fields Of Characteristic 2, Robert W. Fitzgerald

Articles and Preprints

Let K/F be an extension of finite fields of characteristic two. We consider quadratic forms written as the trace of xR(x), where R(x) is a linearized polynomial. We show all quadratic forms can be so written, in an essentially unique way. We classify those R, with coefficients 0 or 1, where the form has a codimension 2 radical. This is applied to maximal Artin-Schreier curves and factorizations of linearized polynomials.

Twistwise Flow Equivalence And Beyond..., Michael C. Sullivan

Articles and Preprints

An expository account of recent progress on twistwise flow equivalence. There is a new result in the appendix. (Appendix joint with Mike Boyle.)

Feedback Classification Of Multi-Input Nonlinear Control Systems, Issa Amadou Tall

Articles and Preprints

We study the feedback group action on multi-input nonlinear control systems with uncontrollable mode. We follow slightly an approach proposed in Kang and Krener [W. Kang and A. J. Krener, SIAM J. Control. Optim., 30 (1992), pp. 1319–1337] which consists of analyzing the system and the feedback group step by step. We construct a normal form which generalizes, on one hand, the results obtained in the single-input case and, on the other hand, those recently obtained by the same author in the controllable case. We illustrate our results by studying the Caltech Multi-Vehicle Wireless Testbed (MVWT) and the prototype …

Knots On A Positive Template Have A Bounded Number Of Prime Factors., Michael C. Sullivan

Articles and Preprints

Templates are branched 2-manifolds with semi-flows used to model "chaotic" hyperbolic invariant sets of flows on 3-manifolds. Knotted orbits on a template correspond to those in the original flow. Birman and Williams conjectured that for any given template the number of prime factors of the knots realized would be bounded. We prove a special case when the template is positive; the general case is now known to be false.

A Symbolic Operator Approach To Several Summation Formulas For Power Series, Tian-Xiao He, Leetsch Hsu, Peter Shiue, D. Torney

Scholarship

This paper deals with the summation problem of power series of the form Sba (f; x) = ∑a ≤ k ≤ b f(k) xk, where 0≤ a < b ≤ ∞, and {f(k)} is a given sequence of numbers with k Є [a, b) or f(t) is a differentiable function defined on [a, b). We present a symbolic summation operator with its various expansions, and construct several summation formulas with estimable remainders for Sba (f; x), by the aid of some classical interpolation series due to Newton, Gauss and Everett, respectively.

On An Extension Of Abel-Gontscharoff's Expansion Formula, Tian-Xiao He, Leetsch Hsu, Peter Shiue

Scholarship

We present a constructive generalization of Abel-Gontscharoff’s series expansion to higher dimensions. A constructive application to a problem of multivariate interpolation is also investigated. In addition, two algorithms for the constructing the basis functions of the interpolants are given.

Characterizations Of Augmented Graded Rings, Mashhoor Refai, Fida A. M. Moh'd

Turkish Journal of Mathematics

In this paper, we introduce some characterizations for augmented graded rings in special cases.

Simplex Codes Over The Ring \Sum_{N=0}^Su^N F_2, Mohammed M. Al-Ashker

Turkish Journal of Mathematics

In this paper, we introduce simplex linear codes over the ring \sum_{n=0}^{n=s}u^n F_2 of types \alpha and \beta, where u^{s+1}=0. And we determine their properties. These codes are an extension and generalization of simplex codes over the ring Z_{2^s}.