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 EDTErdős–Szekeres theorem for cyclic permutationshttps://projecteuclid.org/euclid.involve/1540432925<strong>Éva Czabarka</strong>, <strong>Zhiyu Wang</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 2, 351--360.</p><p><strong>Abstract:</strong><br/>
We provide a cyclic permutation analogue of the Erdős–Szekeres theorem. In particular, we show that every cyclic permutation of length [math] has either an increasing cyclic subpermutation of length [math] or a decreasing cyclic subpermutation of length [math] , and we show that the result is tight. We also characterize all maximum-length cyclic permutations that do not have an increasing cyclic subpermutation of length [math] or a decreasing cyclic subpermutation of length [math] .
</p>projecteuclid.org/euclid.involve/1540432925_20181024220224Wed, 24 Oct 2018 22:02 EDTOptimal transportation with constant constrainthttps://projecteuclid.org/euclid.involve/1540519225<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 12, Number 1, 1--12.</p><p><strong>Abstract:</strong><br/>
We consider optimal transportation with constraint , as did Korman and McCann (2013, 2015), provide simplifications and generalizations of their examples and results, and provide some new examples and results.
</p>projecteuclid.org/euclid.involve/1540519225_20181025220124Thu, 25 Oct 2018 22:01 EDTFair choice sequenceshttps://projecteuclid.org/euclid.involve/1540519228<strong>William J. Keith</strong>, <strong>Sean Grindatti</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 1, 13--30.</p><p><strong>Abstract:</strong><br/>
We consider turn sequences used to allocate of a set of indivisible items between two players who take turns choosing their most desired element of the set, with the goal of minimizing the advantage of the first player. Balanced alternation, while not usually optimal, is fairer than alternation. Strategies for seeking the fairest choice sequence are discussed. We show an unexpected combinatorial connection between partition dominance and fairness, suggesting a new avenue for future investigations in this subject, and conjecture a connection to a previously studied optimality criterion. Several intriguing questions are open at multiple levels of accessibility.
</p>projecteuclid.org/euclid.involve/1540519228_20181025220124Thu, 25 Oct 2018 22:01 EDTIntersecting geodesics and centrality in graphshttps://projecteuclid.org/euclid.involve/1540519229<strong>Emily Carter</strong>, <strong>Bryan Ek</strong>, <strong>Danielle Gonzalez</strong>, <strong>Rigoberto Flórez</strong>, <strong>Darren A. Narayan</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 1, 31--44.</p><p><strong>Abstract:</strong><br/>
In a graph, vertices that are more central are often placed at the intersection of geodesics between other pairs of vertices. This model can be applied to organizational networks, where we assume the flow of information follows shortest paths of communication and there is a required action (i.e., signature or approval) by each person located on these paths. The number of actions a person must perform is linked to both the topology of the network as well as their location within it. The number of expected actions that a person must perform can be quantified by betweenness centrality . The betweenness centrality of a vertex [math] is the ratio of shortest paths between all other pairs of vertices [math] and [math] in which [math] appears to the total number of shortest paths from [math] to [math] . We precisely compute the betweenness centrality for vertices in several families of graphs motivated by different organizational networks.
</p>projecteuclid.org/euclid.involve/1540519229_20181025220124Thu, 25 Oct 2018 22:01 EDTThe length spectrum of the sub-Riemannian three-spherehttps://projecteuclid.org/euclid.involve/1540519230<strong>David Klapheck</strong>, <strong>Michael VanValkenburgh</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 1, 45--61.</p><p><strong>Abstract:</strong><br/>
We determine the lengths of all closed sub-Riemannian geodesics on the three-sphere [math] . Our methods are elementary and allow us to avoid using explicit formulas for the sub-Riemannian geodesics.
</p>projecteuclid.org/euclid.involve/1540519230_20181025220124Thu, 25 Oct 2018 22:01 EDTStatistics for fixed points of the self-power maphttps://projecteuclid.org/euclid.involve/1540519234<strong>Matthew Friedrichsen</strong>, <strong>Joshua Holden</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 1, 63--78.</p><p><strong>Abstract:</strong><br/>
The map [math] modulo [math] is related to a variation of the ElGamal digital signature scheme in a similar way as the discrete exponentiation map, but it has received much less study. We explore the number of fixed points of this map by a statistical analysis of experimental data. In particular, the number of fixed points can in many cases be modeled by a binomial distribution. We discuss the many cases where this has been successful, and also the cases where a good model may not yet have been found.
</p>projecteuclid.org/euclid.involve/1540519234_20181025220124Thu, 25 Oct 2018 22:01 EDTAnalytical solution of a one-dimensional thermistor problem with Robin boundary conditionhttps://projecteuclid.org/euclid.involve/1540519235<strong>Volodymyr Hrynkiv</strong>, <strong>Alice Turchaninova</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 1, 79--88.</p><p><strong>Abstract:</strong><br/>
A one-dimensional nonlinear heat conduction equation of steady-state Joule heating in the presence of an electric field in a metal with temperature-dependent conductivities is considered. A technique developed by Young (1986) is adapted and used to derive an analytical solution for the problem with a Robin boundary condition.
</p>projecteuclid.org/euclid.involve/1540519235_20181025220124Thu, 25 Oct 2018 22:01 EDTOn the covering number of $S_{14}$https://projecteuclid.org/euclid.involve/1540519236<strong>Ryan Oppenheim</strong>, <strong>Eric Swartz</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 1, 89--96.</p><p><strong>Abstract:</strong><br/>
If all elements of a group [math] are contained in the set-theoretic union of proper subgroups [math] , then we define this collection to be a cover of [math] . When such a cover exists, the cardinality of the smallest possible cover is called the covering number of [math] , denoted by [math] . Maróti determined [math] for odd [math] and provided an estimate for even [math] . The second author later determined [math] for [math] when [math] , while joint work of the second author with Kappe and Nikolova-Popova also verified that Maróti’s rule holds for [math] and established the covering numbers of [math] for various other small [math] . Currently, [math] is the smallest value for which [math] is unknown. In this paper, we prove the covering number of [math] is [math] .
</p>projecteuclid.org/euclid.involve/1540519236_20181025220124Thu, 25 Oct 2018 22:01 EDTUpper and lower bounds on the speed of a one-dimensional excited random walkhttps://projecteuclid.org/euclid.involve/1540519237<strong>Erin Madden</strong>, <strong>Brian Kidd</strong>, <strong>Owen Levin</strong>, <strong>Jonathon Peterson</strong>, <strong>Jacob Smith</strong>, <strong>Kevin M. Stangl</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 1, 97--115.</p><p><strong>Abstract:</strong><br/>
An excited random walk (ERW) is a self-interacting non-Markovian random walk in which the future behavior of the walk is influenced by the number of times the walk has previously visited its current site. We study the speed of the walk, defined as [math] , where [math] is the state of the walk at time [math] . While results exist that indicate when the speed is nonzero, there exists no explicit formula for the speed. It is difficult to solve for the speed directly due to complex dependencies in the walk since the next step of the walker depends on how many times the walker has reached the current site. We derive the first nontrivial upper and lower bounds for the speed of the walk. In certain cases these upper and lower bounds are remarkably close together.
</p>projecteuclid.org/euclid.involve/1540519237_20181025220124Thu, 25 Oct 2018 22:01 EDTClassifying linear operators over the octonionshttps://projecteuclid.org/euclid.involve/1540519238<strong>Alex Putnam</strong>, <strong>Tevian Dray</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 1, 117--124.</p><p><strong>Abstract:</strong><br/>
We classify linear operators over the octonions and relate them to linear equations with octonionic coefficients and octonionic variables. Along the way, we also classify linear operators over the quaternions, and show how to relate quaternionic and octonionic operators to real matrices. In each case, we construct an explicit basis of linear operators that maps to the canonical (real) matrix basis; in contrast to the complex case, these maps are surjective. Since higher-order polynomials can be reduced to compositions of linear operators, our construction implies that the ring of polynomials in one variable over the octonions is isomorphic to the product of eight copies of the ring of real polynomials in eight variables.
</p>projecteuclid.org/euclid.involve/1540519238_20181025220124Thu, 25 Oct 2018 22:01 EDTSpectrum of the Kohn Laplacian on the Rossi spherehttps://projecteuclid.org/euclid.involve/1540519239<strong>Tawfik Abbas</strong>, <strong>Madelyne M. Brown</strong>, <strong>Ravikumar Ramasami</strong>, <strong>Yunus E. Zeytuncu</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 1, 125--140.</p><p><strong>Abstract:</strong><br/>
We study the spectrum of the Kohn Laplacian [math] on the Rossi example [math] . In particular we show that [math] is in the essential spectrum of [math] , which yields another proof of the global nonembeddability of the Rossi example.
</p>projecteuclid.org/euclid.involve/1540519239_20181025220124Thu, 25 Oct 2018 22:01 EDTOn the complexity of detecting positive eigenvectors of nonlinear cone mapshttps://projecteuclid.org/euclid.involve/1540519240<strong>Bas Lemmens</strong>, <strong>Lewis White</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 1, 141--150.</p><p><strong>Abstract:</strong><br/>
In recent work with Lins and Nussbaum, the first author gave an algorithm that can detect the existence of a positive eigenvector for order-preserving homogeneous maps on the standard positive cone. The main goal of this paper is to determine the minimum number of iterations this algorithm requires. It is known that this number is equal to the illumination number of the unit ball [math] of the variation norm, [math] on [math] . In this paper we show that the illumination number of [math] is equal to [math] , and hence provide a sharp lower bound for the running time of the algorithm.
</p>projecteuclid.org/euclid.involve/1540519240_20181025220124Thu, 25 Oct 2018 22:01 EDTAntiderivatives and linear differential equations using matriceshttps://projecteuclid.org/euclid.involve/1540519241<strong>Yotsanan Meemark</strong>, <strong>Songpon Sriwongsa</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 1, 151--156.</p><p><strong>Abstract:</strong><br/>
We show how to find the closed-form solutions for antiderivatives of [math] and [math] for all [math] and [math] with [math] by using an idea of Rogers, who suggested using the inverse of the matrix for the differential operator. Additionally, we use the matrix to illustrate the method to find the particular solution for a nonhomogeneous linear differential equation with constant coefficients and forcing terms involving [math] or [math] .
</p>projecteuclid.org/euclid.involve/1540519241_20181025220124Thu, 25 Oct 2018 22:01 EDTPatterns in colored circular permutationshttps://projecteuclid.org/euclid.involve/1540519242<strong>Daniel Gray</strong>, <strong>Charles Lanning</strong>, <strong>Hua Wang</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 1, 157--169.</p><p><strong>Abstract:</strong><br/>
Pattern containment and avoidance have been extensively studied in permutations. Recently, analogous questions have been examined for colored permutations and circular permutations. In this note, we explore these problems in colored circular permutations. We present some interesting observations, some of which are direct generalizations of previously established results. We also raise some questions and propose directions for future study.
</p>projecteuclid.org/euclid.involve/1540519242_20181025220124Thu, 25 Oct 2018 22:01 EDTSolutions of boundary value problems at resonance with periodic and antiperiodic boundary conditionshttps://projecteuclid.org/euclid.involve/1540519243<strong>Aldo E. Garcia</strong>, <strong>Jeffrey T. Neugebauer</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 1, 171--180.</p><p><strong>Abstract:</strong><br/>
We study the existence of solutions of the second-order boundary value problem at resonance [math] satisfying the boundary conditions [math] , [math] , or [math] , [math] . We employ a shift method, making a substitution for the nonlinear term in the differential equation so that these problems are no longer at resonance. Existence of solutions of equivalent boundary value problems is obtained, and these solutions give the existence of solutions of the original boundary value problems.
</p>projecteuclid.org/euclid.involve/1540519243_20181025220124Thu, 25 Oct 2018 22:01 EDTDarboux calculushttps://projecteuclid.org/euclid.involve/1549335627<strong>Marco Aldi</strong>, <strong>Alexander McCleary</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 3, 361--380.</p><p><strong>Abstract:</strong><br/>
We introduce a formalism to analyze partially defined functions between ordered sets. We show that our construction provides a uniform and conceptual approach to all the main definitions encountered in elementary real analysis including Dedekind cuts, limits and continuity.
</p>projecteuclid.org/euclid.involve/1549335627_20190204220043Mon, 04 Feb 2019 22:00 ESTA countable space with an uncountable fundamental grouphttps://projecteuclid.org/euclid.involve/1549335628<strong>Jeremy Brazas</strong>, <strong>Luis Matos</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 3, 381--394.</p><p><strong>Abstract:</strong><br/>
Traditional examples of spaces that have an uncountable fundamental group (such as the Hawaiian earring space) are path-connected compact metric spaces with uncountably many points. We construct a [math] compact, path-connected, locally path-connected topological space [math] with countably many points but with an uncountable fundamental group. The construction of [math] , which we call the “coarse Hawaiian earring” is based on the construction of the usual Hawaiian earring space [math] where each circle [math] is replaced with a copy of the four-point “finite circle”.
</p>projecteuclid.org/euclid.involve/1549335628_20190204220043Mon, 04 Feb 2019 22:00 ESTToeplitz subshifts with trivial centralizers and positive entropyhttps://projecteuclid.org/euclid.involve/1549335629<strong>Kostya Medynets</strong>, <strong>James P. Talisse</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 3, 395--410.</p><p><strong>Abstract:</strong><br/>
Given a dynamical system [math] , the centralizer [math] denotes the group of all homeomorphisms of [math] which commute with the action of [math] . This group is sometimes called the automorphism group of the dynamical system [math] . We generalize the construction of Bułatek and Kwiatkowski (1992) to [math] -Toeplitz systems by identifying a class of [math] -Toeplitz systems that have trivial centralizers. We show that this class of [math] -Toeplitz systems with trivial centralizers contains systems with positive topological entropy.
</p>projecteuclid.org/euclid.involve/1549335629_20190204220043Mon, 04 Feb 2019 22:00 ESTAssociated primes of $h$-wheelshttps://projecteuclid.org/euclid.involve/1549335630<strong>Corey Brooke</strong>, <strong>Molly Hoch</strong>, <strong>Sabrina Lato</strong>, <strong>Janet Striuli</strong>, <strong>Bryan Wang</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 3, 411--425.</p><p><strong>Abstract:</strong><br/>
We study the associated primes of the powers of the cover ideal of [math] -wheels. The main result generalizes a theorem of Kesting, Pozzi, and Striuli (2011).
</p>projecteuclid.org/euclid.involve/1549335630_20190204220043Mon, 04 Feb 2019 22:00 ESTAn elliptic curve analogue to the Fermat numbershttps://projecteuclid.org/euclid.involve/1549335631<strong>Skye Binegar</strong>, <strong>Randy Dominick</strong>, <strong>Meagan Kenney</strong>, <strong>Jeremy Rouse</strong>, <strong>Alex Walsh</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 3, 427--449.</p><p><strong>Abstract:</strong><br/>
The Fermat numbers have many notable properties, including order universality, coprimality, and definition by a recurrence relation. We use rational points of infinite order on elliptic curves to generate sequences that are analogous to the Fermat numbers. We demonstrate that these sequences have many of the same properties as the Fermat numbers, and we discuss results about the prime factors of sequences generated by specific curves and points.
</p>projecteuclid.org/euclid.involve/1549335631_20190204220043Mon, 04 Feb 2019 22:00 ESTNilpotent orbits for Borel subgroups of $\mathrm{SO}_{5}(k)$https://projecteuclid.org/euclid.involve/1549335632<strong>Madeleine Burkhart</strong>, <strong>David Vella</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 3, 451--462.</p><p><strong>Abstract:</strong><br/>
Let [math] be a quasisimple algebraic group defined over an algebraically closed field [math] and [math] a Borel subgroup of [math] acting on the nilradical [math] of its Lie algebra [math] via the adjoint representation. It is known that [math] has only finitely many orbits in only five cases: when [math] is type [math] for [math] , and when [math] is type [math] . We elaborate on this work in the case when [math] (type [math] ) by finding the defining equations of each orbit. We use these equations to determine the dimension of the orbits and the closure ordering on the set of orbits. The other four cases, when $G$ is type $A_n$, can be approached the same way and are treated in a separate paper.
</p>projecteuclid.org/euclid.involve/1549335632_20190204220043Mon, 04 Feb 2019 22:00 ESTHomophonic quotients of linguistic free groups: German, Korean, and Turkishhttps://projecteuclid.org/euclid.involve/1549335633<strong>Herbert Gangl</strong>, <strong>Gizem Karaali</strong>, <strong>Woohyung Lee</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 3, 463--474.</p><p><strong>Abstract:</strong><br/>
The homophonic quotient groups for French and English (i.e., the quotient of the free group generated by the French/English alphabet determined by relations representing standard pronunciation rules) were explicitly characterized by Mestre et al. (1993). We apply the same methodology to three different language systems: German, Korean, and Turkish. We argue that our results point to some interesting differences between these three languages (or at least their current script systems).
</p>projecteuclid.org/euclid.involve/1549335633_20190204220043Mon, 04 Feb 2019 22:00 ESTEffective moments of Dirichlet $L$-functions in Galois orbitshttps://projecteuclid.org/euclid.involve/1549335634<strong>Rizwanur Khan</strong>, <strong>Ruoyun Lei</strong>, <strong>Djordje Milićević</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 3, 475--490.</p><p><strong>Abstract:</strong><br/>
Khan, Milićević, and Ngo evaluated the second moment of [math] -functions associated to certain Galois orbits of primitive Dirichlet characters to modulus a large power of any fixed odd prime [math] . Their results depend on [math] -adic Diophantine approximation and are ineffective, in the sense of computability. We obtain an effective asymptotic for this second moment in the case of [math] .
</p>projecteuclid.org/euclid.involve/1549335634_20190204220043Mon, 04 Feb 2019 22:00 ESTOn the preservation of properties by piecewise affine maps of locally compact groupshttps://projecteuclid.org/euclid.involve/1549335635<strong>Serina Camungol</strong>, <strong>Matthew Morison</strong>, <strong>Skylar Nicol</strong>, <strong>Ross Stokke</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 3, 491--502.</p><p><strong>Abstract:</strong><br/>
As shown by Cohen (1960) and Ilie and Spronk (2005), for locally compact groups [math] and [math] , there is a one-to-one correspondence between the completely bounded homomorphisms of their respective Fourier and Fourier–Stieltjes algebras [math] and piecewise affine continuous maps [math] . Using elementary arguments, we show that several (locally compact) group-theoretic properties, including amenability, are preserved by certain continuous piecewise affine maps. We discuss these results in relation to Fourier algebra homomorphisms.
</p>projecteuclid.org/euclid.involve/1549335635_20190204220043Mon, 04 Feb 2019 22:00 ESTBin decompositionshttps://projecteuclid.org/euclid.involve/1549335636<strong>Daniel Gotshall</strong>, <strong>Pamela E. Harris</strong>, <strong>Dawn Nelson</strong>, <strong>Maria D. Vega</strong>, <strong>Cameron Voigt</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 3, 503--519.</p><p><strong>Abstract:</strong><br/>
It is well known that every positive integer can be expressed as a sum of nonconsecutive Fibonacci numbers provided the Fibonacci numbers satisfy [math] for [math] , [math] and [math] . For any [math] we create a sequence called the [math] -bin sequence with which we can define a notion of a legal decomposition for every positive integer. These sequences are not always positive linear recurrences, which have been studied in the literature, yet we prove, that like positive linear recurrences, these decompositions exist and are unique. Moreover, our main result proves that the distribution of the number of summands used in the [math] -bin legal decompositions displays Gaussian behavior.
</p>projecteuclid.org/euclid.involve/1549335636_20190204220043Mon, 04 Feb 2019 22:00 ESTRigidity of Ulam sets and sequenceshttps://projecteuclid.org/euclid.involve/1549335637<strong>Joshua Hinman</strong>, <strong>Borys Kuca</strong>, <strong>Alexander Schlesinger</strong>, <strong>Arseniy Sheydvasser</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 3, 521--539.</p><p><strong>Abstract:</strong><br/>
We give a number of results about families of Ulam sequences and sets, further exploring recent work on rigidity phenomena. For Ulam sequences, using elementary methods we give an upper bound on the density and prove regularity for various families of sequences. For Ulam sets, we consider extensions of classification work done by Kravitz and Steinerberger.
</p>projecteuclid.org/euclid.involve/1549335637_20190204220043Mon, 04 Feb 2019 22:00 ESTOrbigraphs: a graph-theoretic analog to Riemannian orbifoldshttps://projecteuclid.org/euclid.involve/1559095401<strong>Kathleen Daly</strong>, <strong>Colin Gavin</strong>, <strong>Gabriel Montes de Oca</strong>, <strong>Diana Ochoa</strong>, <strong>Elizabeth Stanhope</strong>, <strong>Sam Stewart</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 5, 721--736.</p><p><strong>Abstract:</strong><br/>
A Riemannian orbifold is a mildly singular generalization of a Riemannian manifold that is locally modeled on [math] modulo the action of a finite group. Orbifolds have proven interesting in a variety of settings. Spectral geometers have examined the link between the Laplace spectrum of an orbifold and the singularities of the orbifold. One open question in this field is whether or not a singular orbifold and a manifold can be Laplace isospectral. Motivated by the connection between spectral geometry and spectral graph theory, we define a graph-theoretic analog of an orbifold called an orbigraph. We obtain results about the relationship between an orbigraph and the spectrum of its adjacency matrix. We prove that the number of singular vertices present in an orbigraph is bounded above and below by spectrally determined quantities, and show that an orbigraph with a singular point and a regular graph cannot be cospectral. We also provide a lower bound on the Cheeger constant of an orbigraph.
</p>projecteuclid.org/euclid.involve/1559095401_20190528220326Tue, 28 May 2019 22:03 EDTSparse neural codes and convexityhttps://projecteuclid.org/euclid.involve/1559095402<strong>R. Amzi Jeffs</strong>, <strong>Mohamed Omar</strong>, <strong>Natchanon Suaysom</strong>, <strong>Aleina Wachtel</strong>, <strong>Nora Youngs</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 5, 737--754.</p><p><strong>Abstract:</strong><br/>
Determining how the brain stores information is one of the most pressing problems in neuroscience. In many instances, the collection of stimuli for a given neuron can be modeled by a convex set in [math] . Combinatorial objects known as neural codes can then be used to extract features of the space covered by these convex regions. We apply results from convex geometry to determine which neural codes can be realized by arrangements of open convex sets. We restrict our attention primarily to sparse codes in low dimensions. We find that intersection-completeness characterizes realizable 2-sparse codes, and show that any realizable 2-sparse code has embedding dimension at most [math] . Furthermore, we prove that in [math] and [math] , realizations of 2-sparse codes using closed sets are equivalent to those with open sets, and this allows us to provide some preliminary results on distinguishing which 2-sparse codes have embedding dimension at most [math] .
</p>projecteuclid.org/euclid.involve/1559095402_20190528220326Tue, 28 May 2019 22:03 EDTThe number of rational points of hyperelliptic curves over subsets of finite fieldshttps://projecteuclid.org/euclid.involve/1559095403<strong>Kristina Nelson</strong>, <strong>József Solymosi</strong>, <strong>Foster Tom</strong>, <strong>Ching Wong</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 5, 755--765.</p><p><strong>Abstract:</strong><br/>
We prove two related concentration inequalities concerning the number of rational points of hyperelliptic curves over subsets of a finite field. In particular, we investigate the probability of a large discrepancy between the numbers of quadratic residues and nonresidues in the image of such subsets over uniformly random hyperelliptic curves of given degrees. We find a constant probability of such a high difference and show the existence of sets with an exceptionally large discrepancy.
</p>projecteuclid.org/euclid.involve/1559095403_20190528220326Tue, 28 May 2019 22:03 EDTSpace-efficient knot mosaics for prime knots with mosaic number 6https://projecteuclid.org/euclid.involve/1559095404<strong>Aaron Heap</strong>, <strong>Douglas Knowles</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 5, 767--789.</p><p><strong>Abstract:</strong><br/>
In 2008, Kauffman and Lomonaco introduced the concepts of a knot mosaic and the mosaic number of a knot or link [math] , the smallest integer [math] such that [math] can be represented on an [math] -mosaic. In 2018, the authors of this paper introduced and explored space-efficient knot mosaics and the tile number of [math] , the smallest number of nonblank tiles necessary to depict [math] on a knot mosaic. They determine bounds for the tile number in terms of the mosaic number. In this paper, we focus specifically on prime knots with mosaic number 6. We determine a complete list of these knots, provide a minimal, space-efficient knot mosaic for each of them, and determine the tile number (or minimal mosaic tile number) of each of them.
</p>projecteuclid.org/euclid.involve/1559095404_20190528220327Tue, 28 May 2019 22:03 EDTShabat polynomials and monodromy groups of trees uniquely determined by ramification typehttps://projecteuclid.org/euclid.involve/1559095405<strong>Naiomi Cameron</strong>, <strong>Mary Kemp</strong>, <strong>Susan Maslak</strong>, <strong>Gabrielle Melamed</strong>, <strong>Richard A. Moy</strong>, <strong>Jonathan Pham</strong>, <strong>Austin Wei</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 5, 791--812.</p><p><strong>Abstract:</strong><br/>
A dessin d’enfant or dessin is a bicolored graph embedded into a Riemann surface. Acyclic dessins can be described analytically by preimages of Shabat polynomials and algebraically by their monodromy groups. We determine the Shabat polynomials and monodromy groups of planar acyclic dessins that are uniquely determined by their ramification types.
</p>projecteuclid.org/euclid.involve/1559095405_20190528220327Tue, 28 May 2019 22:03 EDTOn some edge Folkman numbers, small and largehttps://projecteuclid.org/euclid.involve/1559095406<strong>Jenny M. Kaufmann</strong>, <strong>Henry J. Wickus</strong>, <strong>Stanisław P. Radziszowski</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 5, 813--822.</p><p><strong>Abstract:</strong><br/>
Edge Folkman numbers [math] can be viewed as a generalization of more commonly studied Ramsey numbers. [math] is defined as the smallest order of any [math] -free graph [math] such that any red-blue coloring of the edges of [math] contains either a red [math] or a blue [math] . In this note, first we discuss edge Folkman numbers involving graphs [math] , including the results [math] , [math] , and [math] . Our modification of computational methods used previously in the study of classical Folkman numbers is applied to obtain upper bounds on [math] for all [math] .
</p>projecteuclid.org/euclid.involve/1559095406_20190528220327Tue, 28 May 2019 22:03 EDTWeighted persistent homologyhttps://projecteuclid.org/euclid.involve/1559095407<strong>Gregory Bell</strong>, <strong>Austin Lawson</strong>, <strong>Joshua Martin</strong>, <strong>James Rudzinski</strong>, <strong>Clifford Smyth</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 5, 823--837.</p><p><strong>Abstract:</strong><br/>
We introduce weighted versions of the classical Čech and Vietoris–Rips complexes. We show that a version of the Vietoris–Rips lemma holds for these weighted complexes and that they enjoy appropriate stability properties. We also give some preliminary applications of these weighted complexes.
</p>projecteuclid.org/euclid.involve/1559095407_20190528220327Tue, 28 May 2019 22:03 EDTLeibniz algebras with low-dimensional maximal Lie quotientshttps://projecteuclid.org/euclid.involve/1559095408<strong>William J. Cook</strong>, <strong>John Hall</strong>, <strong>Vicky W. Klima</strong>, <strong>Carter Murray</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 5, 839--853.</p><p><strong>Abstract:</strong><br/>
Every Leibniz algebra has a maximal homomorphic image that is a Lie algebra. We classify cyclic Leibniz algebras over an arbitrary field. Such algebras have the 1-dimensional abelian Lie algebra as their maximal Lie quotient. We then give examples of Leibniz algebras whose associated maximal Lie quotients exhaust all 2-dimensional possibilities.
</p>projecteuclid.org/euclid.involve/1559095408_20190528220327Tue, 28 May 2019 22:03 EDTSpectra of Kohn Laplacians on sphereshttps://projecteuclid.org/euclid.involve/1559095409<strong>John Ahn</strong>, <strong>Mohit Bansil</strong>, <strong>Garrett Brown</strong>, <strong>Emilee Cardin</strong>, <strong>Yunus E. Zeytuncu</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 5, 855--869.</p><p><strong>Abstract:</strong><br/>
We study the spectrum of the Kohn Laplacian on the unit spheres in [math] and revisit Folland’s classical eigenvalue computation. We also look at the growth rate of the eigenvalue counting function in this context. Finally, we consider the growth rate of the eigenvalues of the perturbed Kohn Laplacian on the Rossi sphere in [math] .
</p>projecteuclid.org/euclid.involve/1559095409_20190528220327Tue, 28 May 2019 22:03 EDTPairwise compatibility graphs: complete characterization for wheelshttps://projecteuclid.org/euclid.involve/1559095410<strong>Matthew Beaudouin-Lafon</strong>, <strong>Serena Chen</strong>, <strong>Nathaniel Karst</strong>, <strong>Denise Sakai Troxell</strong>, <strong>Xudong Zheng</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 5, 871--882.</p><p><strong>Abstract:</strong><br/>
A simple graph [math] is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree [math] with positive weights and nonnegative numbers [math] and [math] such that the leaves of [math] are exactly the vertices of [math] , and [math] is an edge in [math] if and only if the sum of weights of edges on the unique path between [math] and [math] in [math] is at least [math] and at most [math] . We show that a wheel on [math] vertices is a PCG if and only if [math] , settling an open problem proposed by Calamoneri and Sinaimeri ( SIAM Review 58 :3 (2016), 445–460). Our approach is based on unavoidable binary classifications of the edges in the complement of wheels that are PCGs. (Note: during the review process of our work, we learned that the same result has been obtained independently with an alternative proof.)
</p>projecteuclid.org/euclid.involve/1559095410_20190528220327Tue, 28 May 2019 22:03 EDTThe financial value of knowing the distribution of stock prices in discrete market modelshttps://projecteuclid.org/euclid.involve/1559095411<strong>Ayelet Amiran</strong>, <strong>Fabrice Baudoin</strong>, <strong>Skylyn Brock</strong>, <strong>Berend Coster</strong>, <strong>Ryan Craver</strong>, <strong>Ugonna Ezeaka</strong>, <strong>Phanuel Mariano</strong>, <strong>Mary Wishart</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 5, 883--899.</p><p><strong>Abstract:</strong><br/>
An explicit formula is derived for the value of weak information in a discrete-time model that works for a wide range of utility functions, including the logarithmic utility and power utility. We assume a complete market with a finite number of assets and a finite number of possible outcomes. Explicit calculations are performed for a binomial model with two assets.
</p>projecteuclid.org/euclid.involve/1559095411_20190528220327Tue, 28 May 2019 22:03 EDTEuler's formula for the zeta function at the positive even integershttps://projecteuclid.org/euclid.involve/1559181651<strong>Samyukta Krishnamurthy</strong>, <strong>Micah B. Milinovich</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 4, 541--548.</p><p><strong>Abstract:</strong><br/>
We give a new proof of Euler’s formula for the values of the Riemann zeta function at the positive even integers. The proof involves estimating a certain integral of elementary functions two different ways and using a recurrence relation for the Bernoulli polynomials evaluated at [math] .
</p>projecteuclid.org/euclid.involve/1559181651_20190529220103Wed, 29 May 2019 22:01 EDTDescents and des-Wilf equivalence of permutations avoiding certain nonclassical patternshttps://projecteuclid.org/euclid.involve/1559181652<strong>Caden Bielawa</strong>, <strong>Robert Davis</strong>, <strong>Daniel Greeson</strong>, <strong>Qinhan Zhou</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 4, 549--563.</p><p><strong>Abstract:</strong><br/>
A frequent topic in the study of pattern avoidance is identifying when two sets of patterns [math] are Wilf equivalent, that is, when [math] for all [math] . In recent work of Dokos et al. the notion of Wilf equivalence was refined to reflect when avoidance of classical patterns preserves certain statistics. We continue their work by examining des-Wilf equivalence when avoiding certain nonclassical patterns.
</p>projecteuclid.org/euclid.involve/1559181652_20190529220103Wed, 29 May 2019 22:01 EDTThe classification of involutions and symmetric spaces of modular groupshttps://projecteuclid.org/euclid.involve/1559181653<strong>Marc Besson</strong>, <strong>Jennifer Schaefer</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 4, 565--583.</p><p><strong>Abstract:</strong><br/>
The involutions and the symmetric spaces associated to the family of modular groups of order [math] are explored. We begin by analyzing the structure of the automorphism group and by establishing which automorphisms are involutions. We conclude by calculating the fixed-point group and symmetric spaces determined by each involution.
</p>projecteuclid.org/euclid.involve/1559181653_20190529220103Wed, 29 May 2019 22:01 EDTWhen is $a^{n} + 1$ the sum of two squares?https://projecteuclid.org/euclid.involve/1559181654<strong>Greg Dresden</strong>, <strong>Kylie Hess</strong>, <strong>Saimon Islam</strong>, <strong>Jeremy Rouse</strong>, <strong>Aaron Schmitt</strong>, <strong>Emily Stamm</strong>, <strong>Terrin Warren</strong>, <strong>Pan Yue</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 4, 585--605.</p><p><strong>Abstract:</strong><br/>
Using Fermat’s two squares theorem and properties of cyclotomic polynomials, we prove assertions about when numbers of the form [math] can be expressed as the sum of two integer squares. We prove that [math] is the sum of two squares for all [math] if and only if [math] is a square. We also prove that if [math] , [math] is odd, and [math] is the sum of two squares, then [math] is the sum of two squares for all [math] , [math] . Using Aurifeuillian factorization, we show that if [math] is a prime and [math] , then there are either zero or infinitely many odd [math] such that [math] is the sum of two squares. When [math] , we define [math] to be the least positive integer such that [math] is the sum of two squares, and prove that if [math] is the sum of two squares for [math] odd, then [math] , and both [math] and [math] are sums of two squares.
</p>projecteuclid.org/euclid.involve/1559181654_20190529220103Wed, 29 May 2019 22:01 EDTIrreducible character restrictions to maximal subgroups of low-rank classical groups of types $B$ and $C$https://projecteuclid.org/euclid.involve/1559181655<strong>Kempton Albee</strong>, <strong>Mike Barnes</strong>, <strong>Aaron Parker</strong>, <strong>Eric Roon</strong>, <strong>A. A. Schaeffer Fry</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 4, 607--631.</p><p><strong>Abstract:</strong><br/>
Representations are special functions on groups that give us a way to study abstract groups using matrices, which are often easier to understand. In particular, we are often interested in irreducible representations, which can be thought of as the building blocks of all representations. Much of the information about these representations can then be understood by instead looking at the trace of the matrices, which we call the character of the representation. This paper will address restricting characters to subgroups by shrinking the domain of the original representation to just the subgroup. In particular, we will discuss the problem of determining when such restricted characters remain irreducible for certain low-rank classical groups.
</p>projecteuclid.org/euclid.involve/1559181655_20190529220103Wed, 29 May 2019 22:01 EDTPrime labelings of infinite graphshttps://projecteuclid.org/euclid.involve/1559181656<strong>Matthew Kenigsberg</strong>, <strong>Oscar Levin</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 4, 633--646.</p><p><strong>Abstract:</strong><br/>
A finite graph on [math] vertices has a prime labeling provided there is a way to label the vertices with the integers 1 through [math] such that every pair of adjacent vertices has relatively prime labels. We extend the definition of prime labeling to infinite graphs and give a simple necessary and sufficient condition for an infinite graph to have a prime labeling. We then measure the complexity of prime labelings of infinite graphs using techniques from computability theory to verify that our condition is as simple as possible.
</p>projecteuclid.org/euclid.involve/1559181656_20190529220103Wed, 29 May 2019 22:01 EDTPositional strategies in games of best choicehttps://projecteuclid.org/euclid.involve/1559181657<strong>Aaron Fowlkes</strong>, <strong>Brant Jones</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 4, 647--658.</p><p><strong>Abstract:</strong><br/>
We study a variation of the game of best choice (also known as the secretary problem or game of googol) under an additional assumption that the ranks of interview candidates are restricted using permutation pattern-avoidance. We describe the optimal positional strategies and develop formulas for the probability of winning.
</p>projecteuclid.org/euclid.involve/1559181657_20190529220103Wed, 29 May 2019 22:01 EDTGraphs with at most two trees in a forest-building processhttps://projecteuclid.org/euclid.involve/1559181658<strong>Steve Butler</strong>, <strong>Misa Hamanaka</strong>, <strong>Marie Hardt</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 4, 659--670.</p><p><strong>Abstract:</strong><br/>
Given a graph, we can form a spanning forest by first sorting the edges in a random order, and then only keeping edges incident to a vertex which is not incident to any previous edge. The resulting forest is dependent on the ordering of the edges, and so we can ask, for example, how likely is it for the process to produce a graph with [math] trees.
We look at all graphs which can produce at most two trees in this process and determine the probabilities of having either one or two trees. From this we construct infinite families of graphs which are nonisomorphic but produce the same probabilities.
</p>projecteuclid.org/euclid.involve/1559181658_20190529220103Wed, 29 May 2019 22:01 EDTLog-concavity of Hölder means and an application to geometric inequalitieshttps://projecteuclid.org/euclid.involve/1559181659<strong>Aurel I. Stan</strong>, <strong>Sergio D. Zapeta-Tzul</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 4, 671--686.</p><p><strong>Abstract:</strong><br/>
The log-concavity of the Hölder mean of two numbers, as a function of its index, is presented first. The notion of [math] -cevian of a triangle is introduced next, for any real number [math] . We use this property of the Hölder mean to find the smallest index [math] such that the length of an [math] -cevian of a triangle is less than or equal to the [math] -Hölder mean of the lengths of the two sides of the triangle that are adjacent to that cevian.
</p>projecteuclid.org/euclid.involve/1559181659_20190529220103Wed, 29 May 2019 22:01 EDTApplying prospect theory to multiattribute problems with independence assumptionshttps://projecteuclid.org/euclid.involve/1559181660<strong>Jack Stanley</strong>, <strong>Frank P. A. Coolen</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 4, 687--711.</p><p><strong>Abstract:</strong><br/>
We discuss a descriptive theory of decision making which has received much attention in recent decades: prospect theory. We specifically focus on applying the theory to problems with two attributes, assisted by different independence assumptions. We discuss a process for solving decision problems using the theory before applying it to a real life example of purchasing breakdown cover.
</p>projecteuclid.org/euclid.involve/1559181660_20190529220103Wed, 29 May 2019 22:01 EDTOn weight-one solvable configurations of the Lights Out puzzlehttps://projecteuclid.org/euclid.involve/1559181661<strong>Yuki Hayata</strong>, <strong>Masakazu Yamagishi</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 4, 713--720.</p><p><strong>Abstract:</strong><br/>
We show that the center-one configuration is always solvable in the Lights Out puzzle on a square grid with odd vertices.
</p>projecteuclid.org/euclid.involve/1559181661_20190529220103Wed, 29 May 2019 22:01 EDTOccurrence graphs of patterns in permutationshttps://projecteuclid.org/euclid.involve/1565661763<strong>Bjarni Jens Kristinsson</strong>, <strong>Henning Ulfarsson</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 6, 901--918.</p><p><strong>Abstract:</strong><br/>
We define the occurrence graph [math] of a pattern [math] in a permutation [math] as the graph whose vertices are the occurrences of [math] in [math] , with edges between the vertices if the occurrences differ by exactly one element. We then study properties of these graphs. The main theorem in this paper is that every hereditary property of graphs gives rise to a permutation class.
</p>projecteuclid.org/euclid.involve/1565661763_20190812220256Mon, 12 Aug 2019 22:02 EDTTruncated path algebras and Betti numbers of polynomial growthhttps://projecteuclid.org/euclid.involve/1565661764<strong>Ryan Coopergard</strong>, <strong>Marju Purin</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 6, 919--940.</p><p><strong>Abstract:</strong><br/>
We investigate a class of truncated path algebras in which the Betti numbers of a simple module satisfy a polynomial of arbitrarily large degree. We produce truncated path algebras where the [math] -th Betti number of a simple module [math] is [math] for [math] and provide a result of the existence of algebras where [math] is a polynomial of degree 4 or less with nonnegative integer coefficients. In particular, we prove that this class of truncated path algebras produces Betti numbers corresponding to any polynomial in a certain family.
</p>projecteuclid.org/euclid.involve/1565661764_20190812220256Mon, 12 Aug 2019 22:02 EDTOrbit spaces of linear circle actionshttps://projecteuclid.org/euclid.involve/1565661765<strong>Suzanne Craig</strong>, <strong>Naiche Downey</strong>, <strong>Lucas Goad</strong>, <strong>Michael J. Mahoney</strong>, <strong>Jordan Watts</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 6, 941--959.</p><p><strong>Abstract:</strong><br/>
We show that nonisomorphic effective linear circle actions yield nondiffeomorphic differential structures on the corresponding orbit spaces.
</p>projecteuclid.org/euclid.involve/1565661765_20190812220256Mon, 12 Aug 2019 22:02 EDTOn a theorem of Besicovitch and a problem in ergodic theoryhttps://projecteuclid.org/euclid.involve/1565661766<strong>Ethan Gwaltney</strong>, <strong>Paul Hagelstein</strong>, <strong>Daniel Herden</strong>, <strong>Brian King</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 6, 961--968.</p><p><strong>Abstract:</strong><br/>
In 1935, Besicovitch proved a remarkable theorem indicating that an integrable function [math] on [math] is strongly differentiable if and only if its associated strong maximal function [math] is finite a.e. We consider analogues of Besicovitch’s result in the context of ergodic theory, in particular discussing the problem of whether or not, given a (not necessarily integrable) measurable function [math] on a nonatomic probability space and a measure-preserving transformation [math] on that space, the ergodic averages of [math] with respect to [math] converge a.e. if and only if the associated ergodic maximal function [math] is finite a.e. Of particular relevance to this discussion will be recent results in the field of inhomogeneous diophantine approximation.
</p>projecteuclid.org/euclid.involve/1565661766_20190812220256Mon, 12 Aug 2019 22:02 EDTAlgorithms for classifying points in a 2-adic Mandelbrot sethttps://projecteuclid.org/euclid.involve/1565661767<strong>Brandon Bate</strong>, <strong>Kyle Craft</strong>, <strong>Jonathon Yuly</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 6, 969--994.</p><p><strong>Abstract:</strong><br/>
In her Ph.D. thesis, Jacqueline Anderson identified a nonarchimedean set similar in spirit to the Mandelbrot set which appears to exhibit a fractal-like boundary. We continue this research by presenting algorithms for determining when rational points lie in this set. We then prove that certain infinite families of points lie in (or out) of this set, giving greater resolution to the self-similarity present in this set.
</p>projecteuclid.org/euclid.involve/1565661767_20190812220256Mon, 12 Aug 2019 22:02 EDTSidon sets and 2-caps in $\mathbb{F}_3^n$https://projecteuclid.org/euclid.involve/1565661768<strong>Yixuan Huang</strong>, <strong>Michael Tait</strong>, <strong>Robert Won</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 6, 995--1003.</p><p><strong>Abstract:</strong><br/>
For each natural number [math] , we introduce the concept of a [math] -cap in [math] . A set of points in [math] is called a [math] -cap if, for each [math] , no [math] of the points lie on a [math] -dimensional flat. This generalizes the notion of a cap in [math] . We prove that the [math] -caps in [math] are exactly the Sidon sets in [math] and study the problem of determining the size of the largest [math] -cap in [math] .
</p>projecteuclid.org/euclid.involve/1565661768_20190812220256Mon, 12 Aug 2019 22:02 EDTCovering numbers of upper triangular matrix rings over finite fieldshttps://projecteuclid.org/euclid.involve/1565661769<strong>Merrick Cai</strong>, <strong>Nicholas J. Werner</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 6, 1005--1013.</p><p><strong>Abstract:</strong><br/>
A cover of a finite ring [math] is a collection of proper subrings [math] of [math] such that [math] . If such a collection exists, then [math] is called coverable, and the covering number of [math] is the cardinality of the smallest possible cover. We investigate covering numbers for rings of upper triangular matrices with entries from a finite field. Let [math] be the field with [math] elements and let [math] be the ring of [math] upper triangular matrices with entries from [math] . We prove that if [math] , then [math] has covering number [math] , that [math] has covering number 4, and that when [math] is prime, [math] has covering number [math] for all [math] .
</p>projecteuclid.org/euclid.involve/1565661769_20190812220256Mon, 12 Aug 2019 22:02 EDTNonstandard existence proofs for reaction diffusion equationshttps://projecteuclid.org/euclid.involve/1565661770<strong>Connor Olson</strong>, <strong>Marshall Mueller</strong>, <strong>Sigurd B. Angenent</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 6, 1015--1034.</p><p><strong>Abstract:</strong><br/>
We give an existence proof for distribution solutions to a scalar reaction diffusion equation, with the aim of illustrating both the differences and the common ingredients of the nonstandard and standard approaches. In particular, our proof shows how the operation of taking the standard part of a nonstandard real number can replace several different compactness theorems, such as Ascoli’s theorem and the Banach–Alaoglu theorem on weak [math] -compactness of the unit ball in the dual of a Banach space.
</p>projecteuclid.org/euclid.involve/1565661770_20190812220256Mon, 12 Aug 2019 22:02 EDTImproving multilabel classification via heterogeneous ensemble methodshttps://projecteuclid.org/euclid.involve/1565661771<strong>Yujue Wu</strong>, <strong>Qing Wang</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 6, 1035--1050.</p><p><strong>Abstract:</strong><br/>
We consider the task of multilabel classification, where each instance may belong to multiple labels simultaneously. We propose a new method, called multilabel super learner (MLSL), that is built upon the problem transformation approach using the one-vs-all binary relevance method. MLSL is an ensemble model that predicts multilabel responses by integrating the strength of multiple base classifiers, and therefore it is likely to outperform each base learner. The weights in the ensemble classifier are determined by optimization of a loss function via [math] -fold cross-validation. Several loss functions are considered and evaluated numerically. The performance of various realizations of MLSL is compared to existing problem transformation algorithms using three real data sets, spanning applications in biology, music, and image labeling. The numerical results suggest that MLSL outperforms existing methods most of the time evaluated by the commonly used performance metrics in multilabel classification.
</p>projecteuclid.org/euclid.involve/1565661771_20190812220256Mon, 12 Aug 2019 22:02 EDTThe number of fixed points of AND-OR networks with chain topologyhttps://projecteuclid.org/euclid.involve/1565661772<strong>Alan Veliz-Cuba</strong>, <strong>Lauren Geiser</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 6, 1051--1068.</p><p><strong>Abstract:</strong><br/>
AND-OR networks are Boolean networks where each coordinate function is either the AND or OR logical operator. We study the number of fixed points of these Boolean networks in the case that they have a wiring diagram with chain topology. We find closed formulas for subclasses of these networks and recursive formulas in the general case. Our results allow for an effective computation of the number of fixed points in the case that the topology of the Boolean network is an open chain (finite or infinite) or a closed chain. We further explore how our approach could be used in “fractal” chains.
</p>projecteuclid.org/euclid.involve/1565661772_20190812220256Mon, 12 Aug 2019 22:02 EDTPositive solutions to singular second-order boundary value problems for dynamic equationshttps://projecteuclid.org/euclid.involve/1565661773<strong>Curtis Kunkel</strong>, <strong>Alex Lancaster</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 6, 1069--1080.</p><p><strong>Abstract:</strong><br/>
We study singular second-order boundary value problems with mixed boundary conditions on an infinitely discrete time scale. We prove the existence of a positive solution by means of a lower and upper solutions method and the Brouwer fixed-point theorem, in conjunction with perturbation methods used to approximate regular problems.
</p>projecteuclid.org/euclid.involve/1565661773_20190812220256Mon, 12 Aug 2019 22:02 EDT