Physical Sciences and Mathematics Commons^™

Efficient Domination In Knights Graphs, Anne Sinko, Peter J. Slater

Mathematics Faculty Publications

The influence of a vertex set S ⊆V(G) is I(S) = Σ_v∈S(1 + deg(v)) = Σ_v∈S |N[v]|, which is the total amount of domination done by the vertices in S. The efficient domination number F(G) of a graph G is equal to the maximum influence of a packing, that is, F(G) is the maximum number of vertices one can dominate under the restriction that no vertex gets dominated more than once.

In …

Balanced Biorthogonal Scaling Vectors Using Fractal Function Macroelements On [0,1], Bruce Kessler

Mathematics Faculty Publications

Geronimo, Hardin, et al have previously constructed orthogonal and biorthogonal scaling vectors by extending a spline scaling vector with functions supported on $[0,1]$. Many of these constructions occurred before the concept of balanced scaling vectors was introduced. This paper will show that adding functions on $[0,1]$ is insufficient for extending spline scaling vectors to scaling vectors that are both orthogonal and balanced. We are able, however, to use this technique to extend spline scaling vectors to balanced, biorthogonal scaling vectors, and we provide two large classes of this type of scaling vector, with approximation order two and three, respectively, with …

An Exceptional Exponential Function, Branko Ćurgus

Mathematics Faculty Publications

We show that there is a link between a standard calculus problem of finding the best view of a painting and special tangent lines to the graphs of exponential functions. Surprisingly, the exponential function with the "best view" is not the one with the base e. A similar link is established for families of functions obtained by composing exponential functions with a fixed linear function. The key tool in the proof is the Lambert W function.

On T-Pure And Almost Pure Exact Sequences Of Lca Groups, Peter Loth

Mathematics Faculty Publications

A proper short exact sequence in the category of locally compact abelian groups is said to be t-pure if φ(A) is a topologically pure subgroup of B, that is, if for all positive integers n. We establish conditions under which t-pure exact sequences split and determine those locally compact abelian groups K ⊕ D (where K is compactly generated and D is discrete) which are t-pure injective or t-pure projective. Calling the extension (*) almost pure if for all positive integers n, we obtain a complete description of the almost pure injectives and almost pure projectives in the category of …

A Classification Of Certain Maximal Subgroups Of Symmetric Groups, Benjamin Newton, Bret Benesh

Mathematics Faculty Publications

Problem 12.82 of the Kourovka Notebook asks for all ordered pairs (n,m) such that the symmetric group Sn embeds in Sm as a maximal subgroup. One family of such pairs is obtained when m=n+1. Kalužnin and Klin [L.A. Kalužnin, M.H. Klin, Certain maximal subgroups of symmetric and alternating groups, Math. Sb. 87 (1972) 91–121] and Halberstadt [E. Halberstadt, On certain maximal subgroups of symmetric or alternating groups, Math. Z. 151 (1976) 117–125] provided an additional infinite family. This paper answers the Kourovka question by producing a third infinite family of ordered …

Composition Operators With Maximal Norm On Weighted Bergman Spaces, Brent J. Carswell, Christopher Hammond

Mathematics Faculty Publications

We prove that any composition operator with maximal norm on one of the weighted Bergman spaces is induced by a disk automorphism or a map that fixes the origin. This result demonstrates a major difference between the weighted Bergman spaces and the Hardy space H², where every inner function induces a composition operator with maximal norm.

Congruences For The Coefficients Of Weakly Holomorphic Modular Forms, Stephanie Treneer

Mathematics Faculty Publications

Recent works have used the theory of modular forms to establish linear congruences for the partition function and for traces of singular moduli. We show that this type of phenomenon is completely general, by finding similar congruences for the coefficients of any weakly holomorphic modular form on any congruence subgroup Γ₀ (N). In particular, we give congruences for a wide class of partition functions and for traces of CM values of arbitrary modular functions on certain congruence subgroups of prime level.

N–Localization Property, Andrzej Rosłanowski

Mathematics Faculty Publications

This paper is concerned with n–localization property introduced by Newelski and Roslanowski in [10] and getting it for CS iterations of forcing notions.

Convergence Of Algorithms For Reconstructing Convex Bodies And Directional Measures, Richard J. Gardner, Markus Kiderlen, Peyman Milanfar

Mathematics Faculty Publications

We investigate algorithms for reconstructing a convex body K in Rⁿ from noisy measurements of its support function or its brightness function in k directions u₁, . . . , u_k. The key idea of these algorithms is to construct a convex polytope P_k whose support function (or brightness function) best approximates the given measurements in the directions u₁, . . . , u_k (in the least squares sense). The measurement errors are assumed to be stochastically independent and Gaussian. It is shown that this procedure is (strongly) consistent, meaning that, …

Multiscale Dynamics Of Biological Cells With Chemotactic Interactions: From A Discrete Stochastic Model To A Continuous Description, Mark Alber, Nan Chen, Tilmann Glimm, Pavel M. Lushnikov

Mathematics Faculty Publications

The Cellular Potts Model (CPM) has been used for simulating various biological phenomena such as differential adhesion, fruiting body formation of the slime mold Dictyostelium discoideum, angiogenesis, cancer invasion, chondrogenesis in embryonic vertebrate limbs, and many others. In this paper, we derive continuous limit of discrete one dimensional CPM with the chemotactic interactions between cells in the form of a Fokker-Planck equation for the evolution of the cell probability density function. This equation is then reduced to the classical macroscopic Keller-Segel model. In particular, all coefficients of the Keller-Segel model are obtained from parameters of the CPM. Theoretical results are …

Multiscale Dynamics Of Biological Cells With Chemotactic Interactions: From A Discrete Stochastic Model To A Continuous Description, Mark Alber, Nan Chen, Tilmann Glimm, Pavel M. Lushnikov

Mathematics Faculty Publications

The cellular Potts model (CPM) has been used for simulating various biological phenomena such as differential adhesion, fruiting body formation of the slime mold Dictyostelium discoideum, angiogenesis, cancer invasion, chondrogenesis in embryonic vertebrate limbs, and many others. We derive a continuous limit of a discrete one-dimensional CPM with the chemotactic interactions between cells in the form of a Fokker-Planck equation for the evolution of the cell probability density function. This equation is then reduced to the classical macroscopic Keller-Segel model. In particular, all coefficients of the Keller-Segel model are obtained from parameters of the CPM. Theoretical results are verified …

Asynchronous Random Boolean Network Model With Variable Number Of Parents Based On Elementary Cellular Automata Rule 126, Mihaela Teodora Matache

Mathematics Faculty Publications

A Boolean network with N nodes, each node’s state at time t being determined by a certain number of parent nodes, which can vary from one node to another is considered. This is a generalization of previous results obtained for a constant number of parent nodes, by Matache and Heidel in Asynchronous random Boolean network model based on elementary cellular automata rule 126, Phys. Rev. E 71, 026232, 2005. The nodes, with randomly assigned neighborhoods, are updated based on various asynchronous schemes. The Boolean rule is a generalization of rule 126 of elementary cellular automata, and is assumed to be …

Pure Extensions Of Locally Compact Abelian Groups, Peter Loth

Mathematics Faculty Publications

In this paper, we study the group Pext(C,A) for locally compact abelian (LCA) groups A and C. Sufficient conditions are established for Pext(C,A) to coincide with the first Ulm subgroup of Ext(C,A). Some structural information on pure injectives in the category of LCA groups is obtained. Letting K denote the class of LCA groups which can be written as the topological direct sum of a compactly generated group and a discrete group, we determine the groups G in K which are pure injective in the category of LCA groups. Finally we describe those groups G in K such that every …

The Integral Cohomology Of The Group Of Loops, Craig Jensen, Jon Mccammond, John Meier

Mathematics Faculty Publications

No abstract provided.

Best Constants For Certain Multilinear Integral Operators, Árpád Bényi, Tadahiro Oh

Mathematics Faculty Publications

We provide explicit formulas in terms of the special function gamma for the best constants in nontensorial multilinear extensions of some classical integral inequalities due to Hilbert, Hardy, and Hardy-Littlewood-Polya.

Undergraduates' Use Of Mathematics Textbooks, Bret Benesh, Tim Boester, Aaron Weinberg, Eimilie Wiesner

Mathematics Faculty Publications

No abstract provided.

A Qualitative Analysis On Nonconstant Graininess Of The Adaptive Grids Via Time Scales, Paul W. Eloe, Stefan Hilger, Qin Sheng

Mathematics Faculty Publications

Calculus on time scales plays a crucial role in unifying the continuous and discrete calculus. In this paper, we apply the time scales calculus methods to study qualitatively properties of the numerical solution of second order ordinary differential equations via different finite difference schemes. The properties become particularly interesting in the case when the computational grids are nonuniform, on which the finite difference operators do not commute. To investigate the solution properties, we introduce the graininess function, and express the numerical solution as functions of the variable grid steps, that is, functions of the graininess and its dynamic derivatives implemented …

A Q-Continued Fraction, Douglas Bowman, James Mclaughlin, Nancy Wyshinksi

Mathematics Faculty Publications

Let a, b, c, d be complex numbers with d 6= 0 and |q| < 1. Define H1(a, b, c, d, q) := 1 1 + −abq + c (a + b)q + d + · · · + −abq2n+1 + cqn (a + b)q n+1 + d + · · · . We show that H1(a, b, c, d, q) converges and 1 H1(a, b, c, d, q) − 1 = c − abq d + aq P∞ j=0 (b/d) j (−c/bd)j q j(j+3)/2 (q)j (−aq2/d)j P∞ j=0 (b/d) j (−c/bd)j q j(j+1)/2 (q)j (−aq/d)j . We then use this result to deduce various corollaries, including the following: 1 1 − q 1 + q − q 3 1 + q 2 − q 5 1 + q 3 − · · · − q 2n−1 1 + q n − · · · = (q 2 ; q 3 )∞ (q; q 3)∞ , (−aq)∞ X∞ j=0 (bq) j (−c/b)j q j(j−1)/2 (q)j (−aq)j = (−bq)∞ X∞ j=0 (aq) j (−c/a)j q j(j−1)/2 (q)j (−bq)j , and the Rogers-Ramanujan identities, X∞ n=0 q n 2 (q; q)n = 1 (q; q 5)∞(q 4; q 5)∞ , X∞ n=0 q n 2+n (q; q)n = 1 (q 2; q 5)∞(q 3; q 5)∞.

The Convergence Behavior Of Q-Continued Fractions On The Unit Circle, Douglas Bowman, James Mclaughlin

Mathematics Faculty Publications

In a previous paper, we showed the existence of an uncountable set of points on the unit circle at which the Rogers-Ramanujan continued fraction does not converge to a finite value. In this present paper, we generalise this result to a wider class of qcontinued fractions, a class which includes the Rogers-Ramanujan continued fraction and the three Ramanujan-Selberg continued fractions. We show, for each q-continued fraction, G(q), in this class, that there is an uncountable set of points, YG, on the unit circle such that if y ∈ YG then G(y) does not converge to a finite value. We discuss …

Continued Fractions And Generalizations With Many Limits: A Survey, Douglas Bowman, James Mclaughlin

Mathematics Faculty Publications

There are infinite processes (matrix products, continued fractions, (r, s)-matrix continued fractions, recurrence sequences) which, under certain circumstances, do not converge but instead diverge in a very predictable way. We give a survey of results in this area, focusing on recent results of the authors.

Further Combinatorial Identities Deriving From The N-Th Power Of A 2 X 2 Matrix, James Mclaughlin, Nancy Wyshinski

Mathematics Faculty Publications

In this paper we use a formula for the n-th power of a 2×2 matrix A (in terms of the entries in A) to derive various combinatorial identities. Three examples of our results follow. 1) We show that if m and n are positive integers and s ∈ {0, 1, 2, . . . , b(mn − 1)/2c}, then X i,j,k,t 2 1+2t−mn+n (−1)nk+i(n+1) 1 + δ(m−1)/2, i+k m − 1 − i i ! m − 1 − 2i k ! × n(m − 1 − 2(i + k)) 2j ! j t − n(i + k) ! n …

The Convergence And Divergence Of Q-Continued Fractions Outside The Unit Circle, Douglas Bowman, James Mclaughlin

Mathematics Faculty Publications

We consider two classes of q-continued fraction whose odd and even parts are limit 1-periodic for |q| > 1, and give theorems which guarantee the convergence of the continued fraction, or of its odd- and even parts, at points outside the unit circle.

Monte Carlo Random Walk Simulations Based On Distributed Order Differential Equations With Applications In Cell Biology, Erik Andries, Sabir Umarov, Stanly Steinberg

Mathematics Faculty Publications

In this paper the multi-dimensional random walk models governed by distributed fractional order differential equations and multi-term fractional order differential equations are constructed. The scaling limits of these random walks to a diffusion process in the sense of distributions is proved. Simulations based upon multi-term fractional order differential equations are performed.

Strategies In Maximizing The Use Of Existing Technology In Philippine Schools, Debbie Marie Y. Bautista, Ma. Louise Antonette N. De Las Peñas

Mathematics Faculty Publications

One of the challenges that continue to confront teachers in Philippine schools is the accessibility of technology for the study and learning of mathematics. In this paper, we will look at several situations and actual experiences happening in Philippine schools. Strategies on how existing technological tools are to be maximized will be discussed, including the creation of lesson plans and classroom activities. The use of technology-based manipulatives in mathematics learning as alternatives to unavailable technology will also be looked at.