Involve: A Journal of Mathematics Articles (Project Euclid)
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The latest articles from Involve: A Journal of Mathematics on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2017 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 19 Oct 2017 13:11 EDTThu, 19 Oct 2017 13:11 EDThttps://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Algorithms for finding knight's tours on Aztec diamonds
https://projecteuclid.org/euclid.involve/1508433088
<strong>Samantha Davies</strong>, <strong>Chenxiao Xue</strong>, <strong>Carl Yerger</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 10, Number 5, 721--734.</p><p><strong>Abstract:</strong><br/> A knight’s tour is a sequence of knight’s moves such that each square on the board is visited exactly once. An Aztec diamond is a square board of size [math] where triangular regions of side length [math] have been removed from all four corners. We show that the existence of knight’s tours on Aztec diamonds cannot be proved inductively via smaller Aztec diamonds, and explain why a divide-and-conquer approach is also not promising. We then describe two algorithms that aim to efficiently find knight’s tours on Aztec diamonds. The first is based on random walks, a straightforward but limited technique that yielded tours on Aztec diamonds for all [math] apart from [math] . The second is a path-conversion algorithm that finds a solution for all [math] . We then apply the path-conversion algorithm to random graphs to test the robustness of our algorithm. Online supplements provide source code, output and more details about these algorithms. </p>projecteuclid.org/euclid.involve/1508433088_20171019131139Thu, 19 Oct 2017 13:11 EDTTotal variation based denoising methods for speckle noise imageshttps://projecteuclid.org/euclid.involve/1513135634<strong>Arundhati Bagchi Misra</strong>, <strong>Ethan Lockhart</strong>, <strong>Hyeona Lim</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 10, Number 2, 327--344.</p><p><strong>Abstract:</strong><br/>
In this paper, we introduce a new algorithm based on total variation for denoising speckle noise images. Total variation was introduced by Rudin, Osher, and Fatemi in 1992 for regularizing images. Chambolle proposed a faster algorithm based on the duality of convex functions for minimizing the total variation, but his algorithm was built for Gaussian noise removal. Unlike Gaussian noise, which is additive, speckle noise is multiplicative. We modify the original Chambolle algorithm for speckle noise images using the first noise equation for speckle denoising, proposed by Krissian, Kikinis, Westin and Vosburgh in 2005. We apply the Chambolle algorithm to the Krissian et al. speckle denoising model to develop a faster algorithm for speckle noise images.
</p>projecteuclid.org/euclid.involve/1513135634_20171212222716Tue, 12 Dec 2017 22:27 ESTA new look at Apollonian circle packingshttps://projecteuclid.org/euclid.involve/1513135635<strong>Isabel Corona</strong>, <strong>Carolynn Johnson</strong>, <strong>Lon Mitchell</strong>, <strong>Dylan O’Connell</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 10, Number 2, 345--360.</p><p><strong>Abstract:</strong><br/>
We define an abstract Apollonian supergasket using the solution set of a certain Diophantine equation, showing that the solutions are in bijective correspondence with the circles of any concrete supergasket. Properties of the solution set translate directly to geometric and algebraic properties of Apollonian gaskets, facilitating their study. In particular, curvatures of individual circles are explored and geometric relationships among multiple circles are given simple algebraic expressions. All results can be applied to a concrete gasket using the curvature-center coordinates of its four defining circles. These techniques can also be applied to other types of circle packings and higher-dimensional analogs.
</p>projecteuclid.org/euclid.involve/1513135635_20171212222716Tue, 12 Dec 2017 22:27 ESTOn halving-edges graphshttps://projecteuclid.org/euclid.involve/1513775038<strong>Tanya Khovanova</strong>, <strong>Dai Yang</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 1, 1--11.</p><p><strong>Abstract:</strong><br/>
In this paper we study halving-edges graphs corresponding to a set of halving lines. Particularly, we study the vertex degrees, path, cycles and cliques of such graphs. In doing so, we study a vertex-partition of said graph called chains which are equipped with interesting properties.
</p>projecteuclid.org/euclid.involve/1513775038_20171220080402Wed, 20 Dec 2017 08:04 ESTKnot mosaic tabulationhttps://projecteuclid.org/euclid.involve/1513775039<strong>Hwa Jeong Lee</strong>, <strong>Lewis Ludwig</strong>, <strong>Joseph Paat</strong>, <strong>Amanda Peiffer</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 1, 13--26.</p><p><strong>Abstract:</strong><br/>
In 2008, Lomonaco and Kauffman introduced a knot mosaic system to define a quantum knot system. A quantum knot is used to describe a physical quantum system such as the topology or status of vortexing that occurs in liquid helium II for example. Kuriya and Shehab proved that knot mosaic type is a complete invariant of tame knots. In this article, we consider the mosaic number of a knot, which is a natural and fundamental knot invariant defined in the knot mosaic system. We determine the mosaic number for all eight-crossing or fewer prime knots. This work is written at an introductory level to encourage other undergraduates to understand and explore this topic. No prior knowledge of knot theory is assumed or required.
</p>projecteuclid.org/euclid.involve/1513775039_20171220080402Wed, 20 Dec 2017 08:04 ESTExtending hypothesis testing with persistent homology to three or more groupshttps://projecteuclid.org/euclid.involve/1513775040<strong>Christopher Cericola</strong>, <strong>Inga Jo Johnson</strong>, <strong>Joshua Kiers</strong>, <strong>Mitchell Krock</strong>, <strong>Jordan Purdy</strong>, <strong>Johanna Torrence</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 1, 27--51.</p><p><strong>Abstract:</strong><br/>
We extend the work of Robinson and Turner to use hypothesis testing with persistent homology to test for measurable differences in shape between the spaces of three or more groups. We conduct a large-scale simulation study to validate our proposed extension, considering various combinations of groups, sample sizes and measurement errors. For each such combination, the percentage of p-values below an [math] -level of 0.05 is provided. Additionally, we apply our method to a cardiotocography data set and find statistically significant evidence of measurable differences in shape between the spaces corresponding to normal, suspect and pathologic health status groups.
</p>projecteuclid.org/euclid.involve/1513775040_20171220080402Wed, 20 Dec 2017 08:04 ESTMerging peg solitaire on graphshttps://projecteuclid.org/euclid.involve/1513775041<strong>John Engbers</strong>, <strong>Ryan Weber</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 1, 53--66.</p><p><strong>Abstract:</strong><br/>
Peg solitaire has recently been generalized to graphs. Here, pegs start on all but one of the vertices in a graph. A move takes pegs on adjacent vertices [math] and [math] , with [math] also adjacent to a hole on vertex [math] , and jumps the peg on [math] over the peg on [math] to [math] , removing the peg on [math] . The goal of the game is to reduce the number of pegs to one.
We introduce the game merging peg solitaire on graphs , where a move takes pegs on vertices [math] and [math] (with a hole on [math] ) and merges them to a single peg on [math] . When can a configuration on a graph, consisting of pegs on all vertices but one, be reduced to a configuration with only a single peg? We give results for a number of graph classes, including stars, paths, cycles, complete bipartite graphs, and some caterpillars.
</p>projecteuclid.org/euclid.involve/1513775041_20171220080402Wed, 20 Dec 2017 08:04 ESTLabeling crossed prisms with a condition at distance twohttps://projecteuclid.org/euclid.involve/1513775042<strong>Matthew Beaudouin-Lafon</strong>, <strong>Serena Chen</strong>, <strong>Nathaniel Karst</strong>, <strong>Jessica Oehrlein</strong>, <strong>Denise Troxell</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 1, 67--80.</p><p><strong>Abstract:</strong><br/>
An L ( 2,1 ) -labeling of a graph is an assignment of nonnegative integers to its vertices such that adjacent vertices are assigned labels at least two apart, and vertices at distance two are assigned labels at least one apart. The [math] -number of a graph is the minimum span of labels over all its L(2,1)-labelings. A generalized Petersen graph (GPG) of order [math] consists of two disjoint cycles on [math] vertices, called the inner and outer cycles , respectively, together with a perfect matching in which each matching edge connects a vertex in the inner cycle to a vertex in the outer cycle. A prism of order [math] is a GPG that is isomorphic to the Cartesian product of a path on two vertices and a cycle on [math] vertices. A crossed prism is a GPG obtained from a prism by crossing two of its matching edges; that is, swapping the two inner cycle vertices on these edges. We show that the [math] -number of a crossed prism is 5, 6, or 7 and provide complete characterizations of crossed prisms attaining each one of these [math] -numbers.
</p>projecteuclid.org/euclid.involve/1513775042_20171220080402Wed, 20 Dec 2017 08:04 ESTNormal forms of endomorphism-valued power serieshttps://projecteuclid.org/euclid.involve/1513775043<strong>Christopher Keane</strong>, <strong>Szilárd Szabó</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 1, 81--94.</p><p><strong>Abstract:</strong><br/>
We show for [math] , and an [math] -dimensional complex vector space [math] that if an element [math] has constant term similar to a Jordan block, then there exists a polynomial gauge transformation [math] such that the first [math] coefficients of [math] have a controlled normal form. Furthermore, we show that this normal form is unique by demonstrating explicit relationships between the first [math] coefficients of the Puiseux series expansion of the eigenvalues of [math] and the entries of the first [math] coefficients of [math] .
</p>projecteuclid.org/euclid.involve/1513775043_20171220080402Wed, 20 Dec 2017 08:04 ESTContinuous dependence and differentiating solutions of a second order boundary value problem with average value conditionhttps://projecteuclid.org/euclid.involve/1513775044<strong>Jeffrey Lyons</strong>, <strong>Samantha Major</strong>, <strong>Kaitlyn Seabrook</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 1, 95--102.</p><p><strong>Abstract:</strong><br/>
Using a few conditions, continuous dependence, and a result regarding smoothness of initial conditions, we show that derivatives of solutions to the second order boundary value problem [math] , [math] , satisfying [math] , [math] , where [math] and [math] with respect to each of the boundary data [math] , [math] , [math] , [math] , [math] solve the associated variational equation with interesting boundary conditions. Of note is the second boundary condition, which is an average value condition.
</p>projecteuclid.org/euclid.involve/1513775044_20171220080402Wed, 20 Dec 2017 08:04 ESTOn uniform large-scale volume growth for the Carnot–Carathéodory metric on unbounded model hypersurfaces in $\mathbb{C}^2$https://projecteuclid.org/euclid.involve/1513775045<strong>Ethan Dlugie</strong>, <strong>Aaron Peterson</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 1, 103--118.</p><p><strong>Abstract:</strong><br/>
We consider the rate of volume growth of large Carnot–Carathéodory metric balls on a class of unbounded model hypersurfaces in [math] . When the hypersurface has a uniform global structure, we show that a metric ball of radius [math] either has volume on the order of [math] or [math] . We also give necessary and sufficient conditions on the hypersurface to display either behavior.
</p>projecteuclid.org/euclid.involve/1513775045_20171220080402Wed, 20 Dec 2017 08:04 ESTVariations of the Greenberg unrelated question binary modelhttps://projecteuclid.org/euclid.involve/1513775046<strong>David P. Suarez</strong>, <strong>Sat Gupta</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 1, 119--126.</p><p><strong>Abstract:</strong><br/>
We explore different variations of the Greenberg unrelated question RRT model for a binary response. In one of the variations, we allow multiple independent responses from each respondent. In another variation, we use inverse sampling. It turns out that both of these variations produce more efficient models, a fact validated by both theoretical comparisons as well as extensive computer simulations.
</p>projecteuclid.org/euclid.involve/1513775046_20171220080402Wed, 20 Dec 2017 08:04 ESTGeneralized exponential sums and the power of computershttps://projecteuclid.org/euclid.involve/1513775047<strong>Francis N. Castro</strong>, <strong>Oscar E. González</strong>, <strong>Luis A. Medina</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 1, 127--142.</p><p><strong>Abstract:</strong><br/>
Today’s era can be characterized by the rise of computer technology. Computers have been, to some extent, responsible for the explosion of the scientific knowledge that we have today. In mathematics, for instance, we have the four color theorem, which is regarded as the first celebrated result to be proved with the assistance of computers. In this article we generalize some fascinating binomial sums that arise in the study of Boolean functions. We study these generalizations from the point of view of integer sequences and bring them to the current computer age of mathematics. The asymptotic behavior of these generalizations is calculated. In particular, we show that a previously known constant that appears in the study of exponential sums of symmetric Boolean functions is universal in the sense that it also emerges in the asymptotic behavior of all of the sequences considered in this work. Finally, in the last section, we use the power of computers and some remarkable algorithms to show that these generalizations are holonomic; i.e., they satisfy homogeneous linear recurrences with polynomial coefficients.
</p>projecteuclid.org/euclid.involve/1513775047_20171220080402Wed, 20 Dec 2017 08:04 ESTCoincidences among skew stable and dual stable Grothendieck polynomialshttps://projecteuclid.org/euclid.involve/1513775048<strong>Ethan Alwaise</strong>, <strong>Shuli Chen</strong>, <strong>Alexander Clifton</strong>, <strong>Rebecca Patrias</strong>, <strong>Rohil Prasad</strong>, <strong>Madeline Shinners</strong>, <strong>Albert Zheng</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 1, 143--167.</p><p><strong>Abstract:</strong><br/>
The question of when two skew Young diagrams produce the same skew Schur function has been well studied. We investigate the same question in the case of stable Grothendieck polynomials, which are the [math] -theoretic analogues of the Schur functions. We prove a necessary condition for two skew shapes to give rise to the same dual stable Grothendieck polynomial. We also provide a necessary and sufficient condition in the case where the two skew shapes are ribbons.
</p>projecteuclid.org/euclid.involve/1513775048_20171220080402Wed, 20 Dec 2017 08:04 ESTA probabilistic heuristic for counting components of functional graphs of polynomials over finite fieldshttps://projecteuclid.org/euclid.involve/1513775049<strong>Elisa Bellah</strong>, <strong>Derek Garton</strong>, <strong>Erin Tannenbaum</strong>, <strong>Noah Walton</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 1, 169--179.</p><p><strong>Abstract:</strong><br/>
Flynn and Garton (2014) bounded the average number of components of the functional graphs of polynomials of fixed degree over a finite field. When the fixed degree was large (relative to the size of the finite field), their lower bound matched Kruskal’s asymptotic for random functional graphs. However, when the fixed degree was small, they were unable to match Kruskal’s bound, since they could not (Lagrange) interpolate cycles in functional graphs of length greater than the fixed degree. In our work, we introduce a heuristic for approximating the average number of such cycles of any length. This heuristic is, roughly, that for sets of edges in a functional graph, the quality of being a cycle and the quality of being interpolable are “uncorrelated enough”. We prove that this heuristic implies that the average number of components of the functional graphs of polynomials of fixed degree over a finite field is within a bounded constant of Kruskal’s bound. We also analyze some numerical data comparing implications of this heuristic to some component counts of functional graphs of polynomials over finite fields.
</p>projecteuclid.org/euclid.involve/1513775049_20171220080402Wed, 20 Dec 2017 08:04 ESTFinding cycles in the $k$-th power digraphs over the integers modulo a primehttps://projecteuclid.org/euclid.involve/1513775055<strong>Greg Dresden</strong>, <strong>Wenda Tu</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 2, 181--194.</p><p><strong>Abstract:</strong><br/>
For [math] prime and [math] , let us define [math] to be the digraph whose set of vertices is [math] such that there is a directed edge from a vertex [math] to a vertex [math] if [math] . We find a new way to decide if there is a cycle of a given length in a given graph [math] .
</p>projecteuclid.org/euclid.involve/1513775055_20171220080420Wed, 20 Dec 2017 08:04 ESTEnumerating spherical $n$-linkshttps://projecteuclid.org/euclid.involve/1513775056<strong>Madeleine Burkhart</strong>, <strong>Joel Foisy</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 2, 195--206.</p><p><strong>Abstract:</strong><br/>
We investigate spherical links: that is, disjoint embeddings of 1-spheres and 0-spheres in the 2-sphere, where the notion of a split link is analogous to the usual concept. In the quest to enumerate distinct nonsplit [math] -links for arbitrary [math] , we must consider when it is possible for an embedding of circles and an even number of points to form a nonsplit link. The main result is a set of necessary and sufficient conditions for such an embedding. The final section includes tables of the distinct embeddings that yield nonsplit [math] -links for [math] .
</p>projecteuclid.org/euclid.involve/1513775056_20171220080420Wed, 20 Dec 2017 08:04 ESTDouble bubbles in hyperbolic surfaceshttps://projecteuclid.org/euclid.involve/1513775057<strong>Wyatt Boyer</strong>, <strong>Bryan Brown</strong>, <strong>Alyssa Loving</strong>, <strong>Sarah Tammen</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 2, 207--217.</p><p><strong>Abstract:</strong><br/>
We seek the least-perimeter way to enclose and separate two prescribed areas in certain hyperbolic surfaces.
</p>projecteuclid.org/euclid.involve/1513775057_20171220080420Wed, 20 Dec 2017 08:04 ESTWhat is odd about binary Parseval frames?https://projecteuclid.org/euclid.involve/1513775058<strong>Zachery J. Baker</strong>, <strong>Bernhard G. Bodmann</strong>, <strong>Micah G. Bullock</strong>, <strong>Samantha N. Branum</strong>, <strong>Jacob E. McLaney</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 2, 219--233.</p><p><strong>Abstract:</strong><br/>
This paper examines the construction and properties of binary Parseval frames. We address two questions: When does a binary Parseval frame have a complementary Parseval frame? Which binary symmetric idempotent matrices are Gram matrices of binary Parseval frames? In contrast to the case of real or complex Parseval frames, the answer to these questions is not always affirmative. The key to our understanding comes from an algorithm that constructs binary orthonormal sequences that span a given subspace, whenever possible. Special regard is given to binary frames whose Gram matrices are circulants.
</p>projecteuclid.org/euclid.involve/1513775058_20171220080420Wed, 20 Dec 2017 08:04 ESTNumbers and the heights of their happinesshttps://projecteuclid.org/euclid.involve/1513775059<strong>May Mei</strong>, <strong>Andrew Read-McFarland</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 2, 235--241.</p><p><strong>Abstract:</strong><br/>
A generalized happy function, [math] maps a positive integer to the sum of its base [math] digits raised to the [math] -th power. We say that [math] is a base- [math] , [math] -power, height- [math] , [math] -attracted number if [math] is the smallest positive integer such that [math] . Happy numbers are then base-10, 2-power, 1-attracted numbers of any height. Let [math] denote the smallest height- [math] , [math] -attracted number for a fixed base [math] and exponent [math] and let [math] denote the smallest number such that every integer can be written as [math] for some nonnegative integers [math] . We prove that if [math] is the smallest nonnegative integer such that [math] ,
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and [math] , then [math] .
</p>projecteuclid.org/euclid.involve/1513775059_20171220080420Wed, 20 Dec 2017 08:04 ESTThe truncated and supplemented Pascal matrix and applicationshttps://projecteuclid.org/euclid.involve/1513775060<strong>Michael Hua</strong>, <strong>Steven B. Damelin</strong>, <strong>Jeffrey Sun</strong>, <strong>Mingchao Yu</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 2, 243--251.</p><p><strong>Abstract:</strong><br/>
In this paper, we introduce the [math] (with [math] ) truncated, supplemented Pascal matrix, which has the property that any [math] columns form a linearly independent set. This property is also present in Reed–Solomon codes; however, Reed–Solomon codes are completely dense, whereas the truncated, supplemented Pascal matrix has multiple zeros. If the maximum distance separable code conjecture is correct, then our matrix has the maximal number of columns (with the aforementioned property) that the conjecture allows. This matrix has applications in coding, network coding, and matroid theory.
</p>projecteuclid.org/euclid.involve/1513775060_20171220080420Wed, 20 Dec 2017 08:04 ESTHexatonic systems and dual groups in mathematical music theoryhttps://projecteuclid.org/euclid.involve/1513775061<strong>Cameron Berry</strong>, <strong>Thomas M. Fiore</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 2, 253--270.</p><p><strong>Abstract:</strong><br/>
Motivated by the music-theoretical work of Richard Cohn and David Clampitt on late-nineteenth century harmony, we mathematically prove that the [math] -group of a hexatonic cycle is dual (in the sense of Lewin) to its [math] / [math] -stabilizer. Our points of departure are Cohn’s notions of maximal smoothness and hexatonic cycle, and the symmetry group of the 12-gon; we do not make use of the duality between the [math] / [math] -group and [math] -group. We also discuss how some ideas in the present paper could be used in the proof of [math] / [math] - [math] duality by Crans, Fiore, and Satyendra ( Amer. Math. Monthly 116 :6 (2009), 479–495).
</p>projecteuclid.org/euclid.involve/1513775061_20171220080420Wed, 20 Dec 2017 08:04 ESTOn computable classes of equidistant sets: finite focal setshttps://projecteuclid.org/euclid.involve/1513775062<strong>Csaba Vincze</strong>, <strong>Adrienn Varga</strong>, <strong>Márk Oláh</strong>, <strong>László Fórián</strong>, <strong>Sándor Lőrinc</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 2, 271--282.</p><p><strong>Abstract:</strong><br/>
The equidistant set of two nonempty subsets [math] and [math] in the Euclidean plane is the set of all points that have the same distance from [math] and [math] . Since the classical conics can be also given in this way, equidistant sets can be considered as one of their generalizations: [math] and [math] are called the focal sets. The points of an equidistant set are difficult to determine in general because there are no simple formulas to compute the distance between a point and a set. As a simplification of the general problem, we are going to investigate equidistant sets with finite focal sets. The main result is the characterization of the equidistant points in terms of computable constants and parametrization. The process is presented by a Maple algorithm. Its motivation is a kind of continuity property of equidistant sets. Therefore we can approximate the equidistant points of [math] and [math] with the equidistant points of finite subsets [math] and [math] . Such an approximation can be applied to the computer simulation, as some examples show in the last section.
</p>projecteuclid.org/euclid.involve/1513775062_20171220080420Wed, 20 Dec 2017 08:04 ESTZero divisor graphs of commutative graded ringshttps://projecteuclid.org/euclid.involve/1513775063<strong>Katherine Cooper</strong>, <strong>Brian Johnson</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 2, 283--295.</p><p><strong>Abstract:</strong><br/>
We study a natural generalization of the zero divisor graph introduced by Anderson and Livingston to commutative rings graded by abelian groups, considering only homogeneous zero divisors. We develop a basic theory for graded zero divisor graphs and present many examples. Finally, we examine classes of graphs that are realizable as graded zero divisor graphs and close with some open questions.
</p>projecteuclid.org/euclid.involve/1513775063_20171220080420Wed, 20 Dec 2017 08:04 ESTThe behavior of a population interaction-diffusion equation in its subcritical regimehttps://projecteuclid.org/euclid.involve/1513775064<strong>Mitchell G. Davis</strong>, <strong>David J. Wollkind</strong>, <strong>Richard A. Cangelosi</strong>, <strong>Bonni J. Kealy-Dichone</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 2, 297--309.</p><p><strong>Abstract:</strong><br/>
A model interaction-diffusion equation for population density originally analyzed through terms of third-order in its supercritical parameter range is extended through terms of fifth-order to examine the behavior in its subcritical regime. It is shown that under the proper conditions the two subcritical cases behave in exactly the same manner as the two supercritical ones unlike the outcome for the truncated system. Further, there also exists a region of metastability allowing for the possibility of population outbreaks. These results are then used to offer an explanation for the occurrence of isolated vegetative patches and sparse homogeneous distributions in the relevant ecological parameter range where there is subcriticality for a plant-groundwater model system, as opposed to periodic patterns and dense homogeneous distributions occurring in its supercritical regime.
</p>projecteuclid.org/euclid.involve/1513775064_20171220080420Wed, 20 Dec 2017 08:04 ESTForbidden subgraphs of coloring graphshttps://projecteuclid.org/euclid.involve/1513775065<strong>Francisco Alvarado</strong>, <strong>Ashley Butts</strong>, <strong>Lauren Farquhar</strong>, <strong>Heather M. Russell</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 2, 311--324.</p><p><strong>Abstract:</strong><br/>
Given a graph [math] , its [math] -coloring graph has vertex set given by the proper [math] -colorings of the vertices of [math] with two [math] -colorings adjacent if and only if they differ at exactly one vertex. Beier et al. ( Discrete Math. 339 :8 (2016), 2100–2112) give various characterizations of coloring graphs, including finding graphs which never arise as induced subgraphs of coloring graphs. These are called forbidden subgraphs, and if no proper subgraph of a forbidden subgraph is forbidden, it is called minimal forbidden. In this paper, we construct a finite collection of minimal forbidden subgraphs that come from modifying theta graphs. We also construct an infinite family of minimal forbidden subgraphs similar to the infinite family found by Beier et al.
</p>projecteuclid.org/euclid.involve/1513775065_20171220080420Wed, 20 Dec 2017 08:04 ESTComputing indicators of Radford algebrashttps://projecteuclid.org/euclid.involve/1513775066<strong>Hao Hu</strong>, <strong>Xinyi Hu</strong>, <strong>Linhong Wang</strong>, <strong>Xingting Wang</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 2, 325--334.</p><p><strong>Abstract:</strong><br/>
We compute higher Frobenius–Schur indicators of Radford algebras in positive characteristic and find minimal polynomials of these linearly recursive sequences. As a result of the work of Kashina, Montgomery and Ng, we obtain gauge invariants for the monoidal categories of representations of Radford algebras.
</p>projecteuclid.org/euclid.involve/1513775066_20171220080420Wed, 20 Dec 2017 08:04 ESTUnlinking numbers of links with crossing number 10https://projecteuclid.org/euclid.involve/1513775067<strong>Lavinia Bulai</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 2, 335--353.</p><p><strong>Abstract:</strong><br/>
We investigate the unlinking numbers of 10-crossing links. We make use of various link invariants and explore their behaviour when crossings are changed. The methods we describe have been used previously to compute unlinking numbers of links with crossing number at most 9. Ultimately, we find the unlinking numbers of all but two of the 287 prime, nonsplit links with crossing number 10.
</p>projecteuclid.org/euclid.involve/1513775067_20171220080420Wed, 20 Dec 2017 08:04 ESTOn a connection between local rings and their associated graded algebrashttps://projecteuclid.org/euclid.involve/1513775068<strong>Justin Hoffmeier</strong>, <strong>Jiyoon Lee</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 2, 355--359.</p><p><strong>Abstract:</strong><br/>
We study a class of local rings and a local adaptation of the homogeneous property for graded rings. While the rings of interest satisfy the property in the local case, we show that their associated graded [math] -algebras do not satisfy the property in the graded case.
</p>projecteuclid.org/euclid.involve/1513775068_20171220080420Wed, 20 Dec 2017 08:04 ESTA mathematical model of treatment of cancer stem cells with immunotherapyhttps://projecteuclid.org/euclid.involve/1513775073<strong>Zachary J. Abernathy</strong>, <strong>Gabrielle Epelle</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 3, 361--382.</p><p><strong>Abstract:</strong><br/>
Using the work of Shelby Wilson and Doron Levy (2012), we develop a mathematical model to study the growth and responsiveness of cancerous tumors to various immunotherapy treatments. We use numerical simulations and stability analysis to predict long-term behavior of passive and aggressive tumors with a range of antigenicities. For high antigenicity aggressive tumors, we show that remission is only achieved after combination treatment with TGF- [math] inhibitors and a peptide vaccine. Additionally, we show that combination treatment has limited effectiveness on low antigenicity aggressive tumors and that using TGF- [math] inhibition or vaccine treatment alone proves generally ineffective for all tumor types considered. A key feature of our model is the identification of separate cancer stem cell and tumor cell populations. Our model predicts that even with combination treatment, failure to completely eliminate the cancer stem cell population leads to cancer recurrence.
</p>projecteuclid.org/euclid.involve/1513775073_20171220080435Wed, 20 Dec 2017 08:04 ESTRNA, local moves on plane trees, and transpositions on tableauxhttps://projecteuclid.org/euclid.involve/1513775074<strong>Laura Del Duca</strong>, <strong>Jennifer Tripp</strong>, <strong>Julianna Tymoczko</strong>, <strong>Judy Wang</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 3, 383--411.</p><p><strong>Abstract:</strong><br/>
We define a collection of functions [math] on the set of plane trees (or standard Young tableaux). The functions are adapted from transpositions in the representation theory of the symmetric group and almost form a group action. They were motivated by local moves in combinatorial biology, which are maps that represent a certain unfolding and refolding of RNA strands. One main result of this study identifies a subset of local moves that we call [math] -local moves, and proves that [math] -local moves correspond to the maps [math] acting on standard Young tableaux. We also prove that the graph of [math] -local moves is a connected, graded poset with unique minimal and maximal elements. We then extend this discussion to functions [math] that mimic reflections in the Weyl group of type [math] . The corresponding graph is no longer connected, but we prove it has two connected components, one of symmetric plane trees and the other of asymmetric plane trees. We give open questions and possible biological interpretations.
</p>projecteuclid.org/euclid.involve/1513775074_20171220080435Wed, 20 Dec 2017 08:04 ESTSix variations on a theme: almost planar graphshttps://projecteuclid.org/euclid.involve/1513775075<strong>Max Lipton</strong>, <strong>Eoin Mackall</strong>, <strong>Thomas W. Mattman</strong>, <strong>Mike Pierce</strong>, <strong>Samantha Robinson</strong>, <strong>Jeremy Thomas</strong>, <strong>Ilan Weinschelbaum</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 3, 413--448.</p><p><strong>Abstract:</strong><br/>
A graph is apex if it can be made planar by deleting a vertex, that is, there exists [math] such that [math] is planar. We also define several related notions; a graph is edge apex if there exists [math] such that [math] is planar, and contraction apex if there exists [math] such that [math] is planar. Additionally we define the analogues with a universal quantifier: for all [math] , [math] is planar; for all [math] , [math] is planar; and for all [math] , [math] is planar. The graph minor theorem of Robertson and Seymour ensures that each of these six notions gives rise to a finite set of obstruction graphs. For the three definitions with universal quantifiers we determine this set. For the remaining properties, apex, edge apex, and contraction apex, we show there are at least 36, 55, and 82 obstruction graphs respectively. We give two similar approaches to almost nonplanar (there exists [math] such that [math] is nonplanar, and for all [math] , [math] is nonplanar) and determine the corresponding minor minimal graphs.
</p>projecteuclid.org/euclid.involve/1513775075_20171220080435Wed, 20 Dec 2017 08:04 ESTNested Frobenius extensions of graded superringshttps://projecteuclid.org/euclid.involve/1513775076<strong>Edward Poon</strong>, <strong>Alistair Savage</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 3, 449--461.</p><p><strong>Abstract:</strong><br/>
We prove a nesting phenomenon for twisted Frobenius extensions. Namely, suppose [math] are graded superrings such that [math] and [math] are both twisted Frobenius extensions of [math] , [math] is contained in the center of [math] , and [math] is projective over [math] . Our main result is that, under these assumptions, [math] is a twisted Frobenius extension of [math] . This generalizes a result of Pike and the second author, which considered the case where [math] is a field.
</p>projecteuclid.org/euclid.involve/1513775076_20171220080435Wed, 20 Dec 2017 08:04 ESTOn $G$-graphs of certain finite groupshttps://projecteuclid.org/euclid.involve/1513775077<strong>Mohammad Reza Darafsheh</strong>, <strong>Safoora Madady Moghadam</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 3, 463--476.</p><p><strong>Abstract:</strong><br/>
The notion of [math] -graph was introduced by Bretto et al. and has interesting properties. This graph is related to a group [math] and a set of generators [math] of [math] and is denoted by [math] . In this paper, we consider several types of groups [math] and study the existence of Hamiltonian and Eulerian paths and circuits in [math] .
</p>projecteuclid.org/euclid.involve/1513775077_20171220080435Wed, 20 Dec 2017 08:04 ESTThe tropical semiring in higher dimensionshttps://projecteuclid.org/euclid.involve/1513775078<strong>John Norton</strong>, <strong>Sandra Spiroff</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 3, 477--488.</p><p><strong>Abstract:</strong><br/>
We discuss the generalization, in higher dimensions, of the tropical semiring, whose two binary operations on the set of real numbers together with infinity are defined to be the minimum and the sum of a pair, respectively. In particular, our objects are closed convex sets, and for any pair, we take the convex hull of their union and their Minkowski sum, respectively, as the binary operations. We consider the semiring in several different cases, determined by a recession cone.
</p>projecteuclid.org/euclid.involve/1513775078_20171220080435Wed, 20 Dec 2017 08:04 ESTA tale of two circles: geometry of a class of quartic polynomialshttps://projecteuclid.org/euclid.involve/1513775079<strong>Christopher Frayer</strong>, <strong>Landon Gauthier</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 3, 489--500.</p><p><strong>Abstract:</strong><br/>
Let [math] be the family of complex-valued polynomials of the form [math] with [math] . The Gauss–Lucas theorem guarantees that the critical points of [math] will lie within the unit disk. This paper further explores the location and structure of these critical points. For example, the unit disk contains two “desert” regions, the open disk [math] and the interior of [math] , in which critical points of [math] cannot occur. Furthermore, each [math] inside the unit disk and outside of the two desert regions is the critical point of at most two polynomials in [math] .
</p>projecteuclid.org/euclid.involve/1513775079_20171220080435Wed, 20 Dec 2017 08:04 ESTZeros of polynomials with four-term recurrencehttps://projecteuclid.org/euclid.involve/1513775080<strong>Khang Tran</strong>, <strong>Andres Zumba</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 3, 501--518.</p><p><strong>Abstract:</strong><br/>
Given real numbers [math] , we form the sequence of polynomials [math] satisfying the four-term recurrence
H
m
(
z
)
+
c
H
m
−
1
(
z
)
+
b
H
m
−
2
(
z
)
+
z
H
m
−
3
(
z
)
=
0
,
m
≥
1
,
with the initial conditions [math] and [math] . We find necessary and sufficient conditions on [math] and [math] under which the zeros of [math] are real for all [math] , and provide an explicit real interval on which [math] is dense, where [math] is the set of zeros of [math] .
</p>projecteuclid.org/euclid.involve/1513775080_20171220080435Wed, 20 Dec 2017 08:04 ESTBinary frames with prescribed dot products and frame operatorhttps://projecteuclid.org/euclid.involve/1513775081<strong>Veronika Furst</strong>, <strong>Eric P. Smith</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 3, 519--540.</p><p><strong>Abstract:</strong><br/>
This paper extends three results from classical finite frame theory over real or complex numbers to binary frames for the vector space [math] . Without the notion of inner products or order, we provide an analog of the “fundamental inequality” of tight frames. In addition, we prove the binary analog of the characterization of dual frames with given inner products and of general frames with prescribed norms and frame operator.
</p>projecteuclid.org/euclid.involve/1513775081_20171220080435Wed, 20 Dec 2017 08:04 ESTModeling of breast cancer through evolutionary game theoryhttps://projecteuclid.org/euclid.involve/1522202414<strong>Ke’Yona Barton</strong>, <strong>Corbin Smith</strong>, <strong>Jan Rychtář</strong>, <strong>Tsvetanka Sendova</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 4, 541--548.</p><p><strong>Abstract:</strong><br/>
We present a simple mathematical model of the development and progression of breast cancer based on evolutionary game theory. Four types of cellular populations are considered: stromal (native) cells, macrophages, benign tumor cells, and motile (malignant) tumor cells. Despite the relative simplicity of the model, it provides a way to explore the interactions between the various cell types and suggests potential approaches to managing and treating cancer.
</p>projecteuclid.org/euclid.involve/1522202414_20180327220024Tue, 27 Mar 2018 22:00 EDTThe isoperimetric problem in the plane with the sum of two Gaussian densitieshttps://projecteuclid.org/euclid.involve/1522202415<strong>John Berry</strong>, <strong>Matthew Dannenberg</strong>, <strong>Jason Liang</strong>, <strong>Yingyi Zeng</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 4, 549--567.</p><p><strong>Abstract:</strong><br/>
We consider the isoperimetric problem for the sum of two Gaussian densities in the line and the plane. We prove that the double Gaussian isoperimetric regions in the line are rays and that if the double Gaussian isoperimetric regions in the plane are half-spaces, then they must be bounded by vertical lines.
</p>projecteuclid.org/euclid.involve/1522202415_20180327220024Tue, 27 Mar 2018 22:00 EDTFiniteness of homological filling functionshttps://projecteuclid.org/euclid.involve/1522202416<strong>Joshua W. Fleming</strong>, <strong>Eduardo Martínez-Pedroza</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 4, 569--583.</p><p><strong>Abstract:</strong><br/>
Let [math] be a group. For any [math] -module [math] and any integer [math] , we define a function [math] generalizing the notion of [math] -dimensional filling function of a group. We prove that this function takes only finite values if [math] is of type [math] and [math] , and remark that the asymptotic growth class of this function is an invariant of [math] . In the particular case that [math] is a group of type [math] , our main result implies that its [math] -dimensional homological filling function takes only finite values.
</p>projecteuclid.org/euclid.involve/1522202416_20180327220024Tue, 27 Mar 2018 22:00 EDTExplicit representations of 3-dimensional Sklyanin algebras associated to a point of order 2https://projecteuclid.org/euclid.involve/1522202417<strong>Daniel J. Reich</strong>, <strong>Chelsea Walton</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 4, 585--608.</p><p><strong>Abstract:</strong><br/>
The representation theory of a 3-dimensional Sklyanin algebra [math] depends on its (noncommutative projective algebro-) geometric data: an elliptic curve [math] in [math] , and an automorphism [math] of [math] given by translation by a point. Indeed, by a result of Artin, Tate, and van den Bergh, we have that [math] is module-finite over its center if and only if [math] has finite order. In this case, all irreducible representations of [math] are finite-dimensional and of at most dimension [math] .
In this work, we provide an algorithm in Maple to directly compute all irreducible representations of [math] associated to [math] of order 2, up to equivalence. Using this algorithm, we compute and list these representations. To illustrate how the algorithm developed in this paper can be applied to other algebras, we use it to recover well-known results about irreducible representations of the skew polynomial ring [math] .
</p>projecteuclid.org/euclid.involve/1522202417_20180327220024Tue, 27 Mar 2018 22:00 EDTA classification of Klein links as torus linkshttps://projecteuclid.org/euclid.involve/1522202418<strong>Steven Beres</strong>, <strong>Vesta Coufal</strong>, <strong>Kaia Hlavacek</strong>, <strong>M. Kate Kearney</strong>, <strong>Ryan Lattanzi</strong>, <strong>Hayley Olson</strong>, <strong>Joel Pereira</strong>, <strong>Bryan Strub</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 4, 609--624.</p><p><strong>Abstract:</strong><br/>
We classify Klein links. In particular, we calculate the number and types of components in a [math] Klein link. We completely determine which Klein links are equivalent to a torus link, and which are not.
</p>projecteuclid.org/euclid.involve/1522202418_20180327220024Tue, 27 Mar 2018 22:00 EDTInterpolation on Gauss hypergeometric functions with an applicationhttps://projecteuclid.org/euclid.involve/1522202419<strong>Hina Manoj Arora</strong>, <strong>Swadesh Kumar Sahoo</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 4, 625--641.</p><p><strong>Abstract:</strong><br/>
We use some standard numerical techniques to approximate the hypergeometric function
[math]
for a range of parameter triples [math] on the interval [math] . Some of the familiar hypergeometric functional identities and asymptotic behavior of the hypergeometric function at [math] play crucial roles in deriving the formula for such approximations. We also focus on error analysis of the numerical approximations leading to monotone properties of quotients of gamma functions in parameter triples [math] . Finally, an application to continued fractions of Gauss is discussed followed by concluding remarks consisting of recent works on related problems.
</p>projecteuclid.org/euclid.involve/1522202419_20180327220024Tue, 27 Mar 2018 22:00 EDTProperties of sets of nontransitive dice with few sideshttps://projecteuclid.org/euclid.involve/1522202420<strong>Levi Angel</strong>, <strong>Matt Davis</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 4, 643--659.</p><p><strong>Abstract:</strong><br/>
We define and investigate several properties that sets of nontransitive dice might have. We prove several implications between these properties, which hold in general or for dice with few sides. We also investigate some algorithms for creating sets of 3-sided dice that realize certain tournaments.
</p>projecteuclid.org/euclid.involve/1522202420_20180327220024Tue, 27 Mar 2018 22:00 EDTNumerical studies of serendipity and tensor product elements for eigenvalue problemshttps://projecteuclid.org/euclid.involve/1522202421<strong>Andrew Gillette</strong>, <strong>Craig Gross</strong>, <strong>Ken Plackowski</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 4, 661--678.</p><p><strong>Abstract:</strong><br/>
While the use of finite element methods for the numerical approximation of eigenvalues is a well-studied problem, the use of serendipity elements for this purpose has received little attention in the literature. We show by numerical experiments that serendipity elements, which are defined on a square reference geometry, can attain the same order of accuracy as their tensor product counterparts while using dramatically fewer degrees of freedom. In some cases, the serendipity method uses only 50% as many basis functions as the tensor product method while still producing the same numerical approximation of an eigenvalue. To encourage the further use and study of serendipity elements, we provide a table of serendipity basis functions for low-order cases and a Mathematica file that can be used to generate the basis functions for higher-order cases.
</p>projecteuclid.org/euclid.involve/1522202421_20180327220024Tue, 27 Mar 2018 22:00 EDTConnectedness of two-sided group digraphs and graphshttps://projecteuclid.org/euclid.involve/1522202422<strong>Patreck Chikwanda</strong>, <strong>Cathy Kriloff</strong>, <strong>Yun Teck Lee</strong>, <strong>Taylor Sandow</strong>, <strong>Garrett Smith</strong>, <strong>Dmytro Yeroshkin</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 4, 679--699.</p><p><strong>Abstract:</strong><br/>
Two-sided group digraphs and graphs, introduced by Iradmusa and Praeger, provide a generalization of Cayley digraphs and graphs in which arcs are determined by left and right multiplying by elements of two subsets of the group. We characterize when two-sided group digraphs and graphs are weakly and strongly connected and count connected components, using both an explicit elementary perspective and group actions. Our results and examples address four open problems posed by Iradmusa and Praeger that concern connectedness and valency. We pose five new open problems.
</p>projecteuclid.org/euclid.involve/1522202422_20180327220024Tue, 27 Mar 2018 22:00 EDTNonunique factorization over quotients of PIDshttps://projecteuclid.org/euclid.involve/1522202423<strong>Nicholas R. Baeth</strong>, <strong>Brandon J. Burns</strong>, <strong>Joshua M. Covey</strong>, <strong>James R. Mixco</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 4, 701--710.</p><p><strong>Abstract:</strong><br/>
We study factorizations of elements in quotients of commutative principal ideal domains that are endowed with an alternative multiplication. This study generalizes the study of factorizations both in quotients of PIDs and in rings of single-valued matrices. We are able to completely describe the sets of factorization lengths of elements in these rings, as well as compute other finer arithmetical invariants. In addition, we provide the first example of a finite bifurcus ring.
</p>projecteuclid.org/euclid.involve/1522202423_20180327220024Tue, 27 Mar 2018 22:00 EDTLocating trinomial zeroshttps://projecteuclid.org/euclid.involve/1522202424<strong>Russell Howell</strong>, <strong>David Kyle</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 4, 711--720.</p><p><strong>Abstract:</strong><br/>
We derive formulas for the number of interior roots (i.e., zeros with modulus less than 1) and exterior roots (i.e., zeros with modulus greater than 1) for trinomials of the form [math] , where [math] . Combined with earlier work by Brilleslyper and Schaubroeck, who focus on unimodular roots (i.e., zeros that lie on the unit circle), we give a complete count of the location of zeros of these trinomials.
</p>projecteuclid.org/euclid.involve/1522202424_20180327220024Tue, 27 Mar 2018 22:00 EDTOn the minuscule representation of type $B_n$https://projecteuclid.org/euclid.involve/1523498539<strong>William J. Cook</strong>, <strong>Noah A. Hughes</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 5, 721--733.</p><p><strong>Abstract:</strong><br/>
We study the action of the Weyl group of type [math] acting as permutations on the set of weights of the minuscule representation of type [math] (also known as the spin representation). Motivated by a previous work, we seek to determine when cycle structures alone reveal the irreducibility of these minuscule representations. After deriving formulas for the simple reflections viewed as permutations, we perform a series of computer-aided calculations in GAP. We are then able to establish that, for certain ranks, the irreducibility of the minuscule representation cannot be detected by cycle structures alone.
</p>projecteuclid.org/euclid.involve/1523498539_20180411220238Wed, 11 Apr 2018 22:02 EDTPythagorean orthogonality of compact setshttps://projecteuclid.org/euclid.involve/1523498540<strong>Pallavi Aggarwal</strong>, <strong>Steven Schlicker</strong>, <strong>Ryan Swartzentruber</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 5, 735--752.</p><p><strong>Abstract:</strong><br/>
The Hausdorff metric [math] is used to define the distance between two elements of [math] , the hyperspace of all nonempty compact subsets of [math] . The geometry this metric imposes on [math] is an interesting one — it is filled with unexpected results and fascinating connections to number theory and graph theory. Circles and lines are defined in this geometry to make it an extension of the standard Euclidean geometry. However, the behavior of lines and segments in this extended geometry is much different from that of lines and segments in Euclidean geometry. This paper presents surprising results about rays in the geometry of [math] , with a focus on attempting to find well-defined notions of angle and angle measure in [math] .
</p>projecteuclid.org/euclid.involve/1523498540_20180411220238Wed, 11 Apr 2018 22:02 EDTDifferent definitions of conic sections in hyperbolic geometryhttps://projecteuclid.org/euclid.involve/1523498541<strong>Patrick Chao</strong>, <strong>Jonathan Rosenberg</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 5, 753--768.</p><p><strong>Abstract:</strong><br/>
In classical Euclidean geometry, there are several equivalent definitions of conic sections. We show that in the hyperbolic plane, the analogues of these same definitions still make sense, but are no longer equivalent, and we discuss the relationships among them.
</p>projecteuclid.org/euclid.involve/1523498541_20180411220238Wed, 11 Apr 2018 22:02 EDTThe Fibonacci sequence under a modulus: computing all moduli that produce a given periodhttps://projecteuclid.org/euclid.involve/1523498542<strong>Alex Dishong</strong>, <strong>Marc S. Renault</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 5, 769--774.</p><p><strong>Abstract:</strong><br/>
The Fibonacci sequence [math] , when reduced modulo [math] is periodic. For example, [math] . The period of [math] is denoted by [math] , so [math] . In this paper we present an algorithm that, given a period [math] , produces all [math] such that [math] . For efficiency, the algorithm employs key ideas from a 1963 paper by John Vinson on the period of the Fibonacci sequence. We present output from the algorithm and discuss the results.
</p>projecteuclid.org/euclid.involve/1523498542_20180411220238Wed, 11 Apr 2018 22:02 EDTOn the faithfulness of the representation of $\mathrm{GL}(n)$ on the space of curvature tensorshttps://projecteuclid.org/euclid.involve/1523498543<strong>Corey Dunn</strong>, <strong>Darien Elderfield</strong>, <strong>Rory Martin-Hagemeyer</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 5, 775--785.</p><p><strong>Abstract:</strong><br/>
We prove that the standard representation of [math] on the space of algebraic curvature tensors is almost faithful by showing that the kernel of this representation contains only the identity map and its negative. We additionally show that the standard representation of [math] on the space of algebraic covariant derivative curvature tensors is faithful.
</p>projecteuclid.org/euclid.involve/1523498543_20180411220238Wed, 11 Apr 2018 22:02 EDTQuasipositive curvature on a biquotient of Sp$(3)$https://projecteuclid.org/euclid.involve/1523498544<strong>Jason DeVito</strong>, <strong>Wesley Martin</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 5, 787--801.</p><p><strong>Abstract:</strong><br/>
Suppose [math] denotes the unique irreducible complex [math] -dimensional representation of [math] , and consider the two subgroups [math] with [math] and [math] . We show that the biquotient [math] admits a quasipositively curved Riemannian metric.
</p>projecteuclid.org/euclid.involve/1523498544_20180411220238Wed, 11 Apr 2018 22:02 EDTSymmetric numerical ranges of four-by-four matriceshttps://projecteuclid.org/euclid.involve/1523498545<strong>Shelby L. Burnett</strong>, <strong>Ashley Chandler</strong>, <strong>Linda J. Patton</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 5, 803--826.</p><p><strong>Abstract:</strong><br/>
Numerical ranges of matrices with rotational symmetry are studied. Some cases in which symmetry of the numerical range implies symmetry of the spectrum are described. A parametrized class of [math] matrices [math] such that the numerical range [math] has fourfold symmetry about the origin but the generalized numerical range [math] does not have this symmetry is included. In 2011, Tsai and Wu showed that the numerical ranges of weighted shift matrices, which have rotational symmetry about the origin, are also symmetric about certain axes. We show that any [math] matrix whose numerical range has fourfold symmetry about the origin also has the corresponding axis symmetry. The support function used to prove these results is also used to show that the numerical range of a composition operator on Hardy space with automorphic symbol and minimal polynomial [math] is not a disk.
</p>projecteuclid.org/euclid.involve/1523498545_20180411220238Wed, 11 Apr 2018 22:02 EDTCounting eta-quotients of prime levelhttps://projecteuclid.org/euclid.involve/1523498546<strong>Allison Arnold-Roksandich</strong>, <strong>Kevin James</strong>, <strong>Rodney Keaton</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 5, 827--844.</p><p><strong>Abstract:</strong><br/>
It is known that a modular form on [math] can be expressed as a rational function in [math] , [math] and [math] . By using known theorems and calculating the order of vanishing, we can compute the eta-quotients for a given level. Using this count, knowing how many eta-quotients are linearly independent, and using the dimension formula, we can figure out a subspace spanned by the eta-quotients. In this paper, we primarily focus on the case where the level is [math] , a prime. In this case, we will show an explicit count for the number of eta-quotients of level [math] and show that they are linearly independent.
</p>projecteuclid.org/euclid.involve/1523498546_20180411220238Wed, 11 Apr 2018 22:02 EDTThe $k$-diameter component edge connectivity parameterhttps://projecteuclid.org/euclid.involve/1523498547<strong>Nathan Shank</strong>, <strong>Adam Buzzard</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 5, 845--856.</p><p><strong>Abstract:</strong><br/>
We focus on a network reliability measure based on edge failures and considering a network operational if there exists a component with diameter [math] or larger. The [math] -diameter component edge connectivity parameter of a graph is the minimum number of edge failures needed so that no component has diameter [math] or larger. This implies each resulting vertex must not have a [math] -neighbor. We give results for specific graph classes including path graphs, complete graphs, complete bipartite graphs, and a surprising result for perfect [math] -ary trees.
</p>projecteuclid.org/euclid.involve/1523498547_20180411220238Wed, 11 Apr 2018 22:02 EDTTime stopping for Tsirelson's normhttps://projecteuclid.org/euclid.involve/1523498548<strong>Kevin Beanland</strong>, <strong>Noah Duncan</strong>, <strong>Michael Holt</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 5, 857--866.</p><p><strong>Abstract:</strong><br/>
Tsirelson’s norm [math] on [math] is defined as the limit of an increasing sequence of norms [math] . For each [math] let [math] be the smallest integer satisfying [math] for all [math] with [math] . We show that [math] is [math] . This is an improvement of the upper bound of [math] given by P. Casazza and T. Shura in their 1989 monograph on Tsirelson’s space.
</p>projecteuclid.org/euclid.involve/1523498548_20180411220238Wed, 11 Apr 2018 22:02 EDTEnumeration of stacks of sphereshttps://projecteuclid.org/euclid.involve/1523498549<strong>Lauren Endicott</strong>, <strong>Russell May</strong>, <strong>Sienna Shacklette</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 5, 867--875.</p><p><strong>Abstract:</strong><br/>
As a three-dimensional generalization of fountains of coins, we analyze stacks of spheres and enumerate two particular classes, so-called “pyramidal” stacks and “Dominican” stacks. Using the machinery of generating functions, we obtain exact formulas for these types of stacks in terms of the sizes of their bases.
</p>projecteuclid.org/euclid.involve/1523498549_20180411220238Wed, 11 Apr 2018 22:02 EDTRings isomorphic to their nontrivial subringshttps://projecteuclid.org/euclid.involve/1523498550<strong>Jacob Lojewski</strong>, <strong>Greg Oman</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 5, 877--883.</p><p><strong>Abstract:</strong><br/>
Let [math] be a nontrivial group, and assume that [math] for every nontrivial subgroup [math] of [math] . It is a simple matter to prove that [math] or [math] for some prime [math] . In this note, we address the analogous (though harder) question for rings; that is, we find all nontrivial rings [math] for which [math] for every nontrivial subring [math] of [math] .
</p>projecteuclid.org/euclid.involve/1523498550_20180411220238Wed, 11 Apr 2018 22:02 EDTOn generalized MacDonald codeshttps://projecteuclid.org/euclid.involve/1523498551<strong>Padmapani Seneviratne</strong>, <strong>Lauren Melcher</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 5, 885--892.</p><p><strong>Abstract:</strong><br/>
We show that the generalized [math] -ary MacDonald codes [math] with parameters [math] are two-weight codes with nonzero weights [math] , [math] and determine the complete weight enumerator of these codes. This leads to a family of strongly regular graphs with parameters [math] . Further, we show that the codes [math] satisfy the Griesmer bound and are self-orthogonal for [math] .
</p>projecteuclid.org/euclid.involve/1523498551_20180411220238Wed, 11 Apr 2018 22:02 EDTA simple proof characterizing interval orders with interval lengths between 1 and $k$https://projecteuclid.org/euclid.involve/1523498552<strong>Simona Boyadzhiyska</strong>, <strong>Garth Isaak</strong>, <strong>Ann N. Trenk</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 5, 893--900.</p><p><strong>Abstract:</strong><br/>
A poset [math] has an interval representation if each [math] can be assigned a real interval [math] so that [math] in [math] if and only if [math] lies completely to the left of [math] . Such orders are called interval orders . Fishburn (1983, 1985) proved that for any positive integer [math] , an interval order has a representation in which all interval lengths are between [math] and [math] if and only if the order does not contain [math] as an induced poset. In this paper, we give a simple proof of this result using a digraph model.
</p>projecteuclid.org/euclid.involve/1523498552_20180411220238Wed, 11 Apr 2018 22:02 EDT