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 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 EDTAsymptotic expansion of Warlimont functions on Wright semigroupshttps://projecteuclid.org/euclid.involve/1572055221<strong>Marco Aldi</strong>, <strong>Hanqiu Tan</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 7, 1081--1098.</p><p><strong>Abstract:</strong><br/>
We calculate full asymptotic expansions of prime-independent multiplicative functions on additive arithmetic semigroups that satisfy a strong form of Knopfmacher’s axioms. When applied to the semigroup of unlabeled graphs, our method yields detailed asymptotic information on how graphs decompose into connected components. As a second class of examples, we discuss polynomials in several variables over a finite field.
</p>projecteuclid.org/euclid.involve/1572055221_20191025220031Fri, 25 Oct 2019 22:00 EDTA systematic development of Jeans' criterion with rotation for gravitational instabilitieshttps://projecteuclid.org/euclid.involve/1572055222<strong>Kohl Gill</strong>, <strong>David J. Wollkind</strong>, <strong>Bonni J. Dichone</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 7, 1099--1108.</p><p><strong>Abstract:</strong><br/>
An inviscid fluid model of a self-gravitating infinite expanse of a uniformly rotating adiabatic gas cloud consisting of the continuity, Euler’s, and Poisson’s equations for that situation is considered. There exists a static homogeneous density solution to this model relating that equilibrium density to the uniform rotation. A systematic linear stability analysis of this exact solution then yields a gravitational instability criterion equivalent to that developed by Sir James Jeans in the absence of rotation instead of the slightly more complicated stability behavior deduced by Subrahmanyan Chandrasekhar for this model with rotation, both of which suffered from the same deficiency in that neither of them actually examined whether their perturbation analysis was of an exact solution. For the former case, it was not and, for the latter, the equilibrium density and uniform rotation were erroneously assumed to be independent instead of related to each other. Then this gravitational instability criterion is employed in the form of Jeans’ length to show that there is very good agreement between this theoretical prediction and the actual mean distance of separation of stars formed in the outer arms of the spiral galaxy Andromeda M31. Further, the uniform rotation determined from the exact solution relation to equilibrium density and the corresponding rotational velocity for a reference radial distance are consistent with the spectroscopic measurements of Andromeda and the observational data of the spiral Milky Way galaxy.
</p>projecteuclid.org/euclid.involve/1572055222_20191025220031Fri, 25 Oct 2019 22:00 EDTThe linking-unlinking gamehttps://projecteuclid.org/euclid.involve/1572055223<strong>Adam Giambrone</strong>, <strong>Jake Murphy</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 7, 1109--1141.</p><p><strong>Abstract:</strong><br/>
Combinatorial two-player games have recently been applied to knot theory. Examples of this include the knotting-unknotting game and the region unknotting game, both of which are played on knot shadows. These are turn-based games played by two players, where each player has a separate goal to achieve in order to win the game. In this paper, we introduce the linking-unlinking game which is played on two-component link shadows. We then present winning strategies for the linking-unlinking game played on all shadows of two-component rational tangle closures and played on a large family of general two-component link shadows.
</p>projecteuclid.org/euclid.involve/1572055223_20191025220031Fri, 25 Oct 2019 22:00 EDTOn generalizing happy numbers to fractional-base number systemshttps://projecteuclid.org/euclid.involve/1572055224<strong>Enrique Treviño</strong>, <strong>Mikita Zhylinski</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 7, 1143--1151.</p><p><strong>Abstract:</strong><br/>
Let [math] be a positive integer and [math] be the sum of the squares of its decimal digits. When there exists a positive integer [math] such that the [math] -th iterate of [math] on [math] is 1, i.e., [math] , then [math] is called a happy number. The notion of happy numbers has been generalized to different bases, different powers and even negative bases. In this article we consider generalizations to fractional number bases. Let [math] be the sum of the [math] -th powers of the digits of [math] base [math] . Let [math] be the smallest nonnegative integer for which there exists a positive integer [math] satisfying [math] . We prove that such a [math] , called the height of [math] , exists for all [math] , and that, if [math] or [math] , then [math] can be arbitrarily large.
</p>projecteuclid.org/euclid.involve/1572055224_20191025220031Fri, 25 Oct 2019 22:00 EDTOn the Hadwiger number of Kneser graphs and their random subgraphshttps://projecteuclid.org/euclid.involve/1572055225<strong>Arran Hamm</strong>, <strong>Kristen Melton</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 7, 1153--1161.</p><p><strong>Abstract:</strong><br/>
For [math] , let [math] be the usual Kneser graph (whose vertices are [math] -sets of [math] with [math] if and only if [math] ). The Hadwiger number of a graph [math] , denoted by [math] , is [math] , where [math] if [math] is a minor of [math] . Previously, lower bounds have been given on the Hadwiger number of a graph in terms of its average degree. In this paper we give lower bounds on [math] and [math] , where [math] is the binomial random subgraph of [math] with edge probability [math] . Each of these bounds is larger than previous bounds under certain conditions on [math] and [math] .
</p>projecteuclid.org/euclid.involve/1572055225_20191025220031Fri, 25 Oct 2019 22:00 EDTA binary unrelated-question RRT model accounting for untruthful respondinghttps://projecteuclid.org/euclid.involve/1572055226<strong>Amber Young</strong>, <strong>Sat Gupta</strong>, <strong>Ryan Parks</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 7, 1163--1173.</p><p><strong>Abstract:</strong><br/>
Estimating the prevalence of a sensitive trait in a population is not a simple task due to the general tendency among survey respondents to answer sensitive questions in a way that is socially desirable. Use of randomized response techniques (RRT) is one of several approaches for reducing the impact of this tendency. However, despite the additional privacy provided by RRT models, some respondents may still provide an untruthful response. We consider the impact of untruthful responding on binary unrelated-question RRT models and observe that even if only a small number of respondents lie, a significant bias may be introduced to the model. We propose a binary unrelated-question model that accounts for those respondents who may respond untruthfully. This adds an extra layer of precaution to the estimation of the sensitive trait and decreases the importance of presurvey respondent training. Our results are validated using a simulation study.
</p>projecteuclid.org/euclid.involve/1572055226_20191025220031Fri, 25 Oct 2019 22:00 EDTToward a Nordhaus–Gaddum inequality for the number of dominating setshttps://projecteuclid.org/euclid.involve/1572055227<strong>Lauren Keough</strong>, <strong>David Shane</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 7, 1175--1181.</p><p><strong>Abstract:</strong><br/>
A dominating set in a graph [math] is a set [math] of vertices such that every vertex of [math] is either in [math] or is adjacent to a vertex in [math] . Nordhaus–Gaddum inequalities relate a graph [math] to its complement [math] . In this spirit Wagner proved that any graph [math] on [math] vertices satisfies [math] , where [math] is the number of dominating sets in a graph [math] . In the same paper he commented that proving an upper bound for [math] among all graphs on [math] vertices seems to be much more difficult. Here we prove an upper bound on [math] and prove that any graph maximizing this sum has minimum degree at least [math] and maximum degree at most [math] . We conjecture that the complete balanced bipartite graph maximizes [math] and have verified this computationally for all graphs on at most [math] vertices.
</p>projecteuclid.org/euclid.involve/1572055227_20191025220031Fri, 25 Oct 2019 22:00 EDTOn some obstructions of flag vector pairs $(f_1, f_{04})$ of $5$-polytopeshttps://projecteuclid.org/euclid.involve/1572055228<strong>Hye Bin Cho</strong>, <strong>Jin Hong Kim</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 7, 1183--1192.</p><p><strong>Abstract:</strong><br/>
Motivated by the recent work of Sjöberg and Ziegler, who obtained a complete characterization of the pairs [math] of flag numbers for [math] -polytopes, in this paper we give some new results about the possible flag vector pairs [math] of [math] -polytopes.
</p>projecteuclid.org/euclid.involve/1572055228_20191025220031Fri, 25 Oct 2019 22:00 EDTBenford's law beyond independence: tracking Benford behavior in copula modelshttps://projecteuclid.org/euclid.involve/1572055229<strong>Rebecca F. Durst</strong>, <strong>Steven J. Miller</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 7, 1193--1218.</p><p><strong>Abstract:</strong><br/>
Benford’s law describes a common phenomenon among many naturally occurring data sets and distributions in which the leading digits of the data are distributed with the probability of a first digit of [math] base [math] being [math] . As it often successfully detects fraud in medical trials, voting, science and finance, significant effort has been made to understand when and how distributions exhibit Benford behavior. Most of the previous work has been restricted to cases of independent variables, and little is known about situations involving dependence. We use copulas to investigate the Benford behavior of the product of [math] dependent random variables. We develop a method for approximating the Benford behavior of a product of [math] dependent random variables modeled by a copula distribution [math] and quantify and bound a copula distribution’s distance from Benford behavior. We then investigate the Benford behavior of various copulas under varying dependence parameters and number of marginals. Our investigations show that the convergence to Benford behavior seen with independent random variables as the number of variables in the product increases is not necessarily preserved when the variables are dependent and modeled by a copula. Furthermore, there is strong indication that the preservation of Benford behavior of the product of dependent random variables may be linked more to the structure of the copula than to the Benford behavior of the marginal distributions.
</p>projecteuclid.org/euclid.involve/1572055229_20191025220031Fri, 25 Oct 2019 22:00 EDTClosed geodesics on doubled polygonshttps://projecteuclid.org/euclid.involve/1572055230<strong>Ian M. Adelstein</strong>, <strong>Adam Y. W. Fong</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 7, 1219--1227.</p><p><strong>Abstract:</strong><br/>
We study [math] -geodesics, those closed geodesics that minimize on any subinterval of length [math] , where [math] is the length of the geodesic. We investigate the existence and behavior of these curves on doubled polygons and show that every doubled regular [math] -gon admits a [math] -geodesic. For the doubled regular [math] -gons, with [math] an odd prime, we conjecture that [math] is the minimum value for [math] such that the space admits a [math] -geodesic.
</p>projecteuclid.org/euclid.involve/1572055230_20191025220031Fri, 25 Oct 2019 22:00 EDTSign pattern matrices that allow inertia $\mathbb{S}_{n}$https://projecteuclid.org/euclid.involve/1572055231<strong>Adam H. Berliner</strong>, <strong>Derek DeBlieck</strong>, <strong>Deepak Shah</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 7, 1229--1240.</p><p><strong>Abstract:</strong><br/>
Sign pattern matrices of order [math] that allow inertias in the set [math] are considered. All sign patterns of order 3 (up to equivalence) that allow [math] are classified and organized according to their associated directed graphs. Furthermore, a minimal set of such matrices is found. Then, given a pattern of order [math] that allows [math] , a construction is given that generates families of irreducible sign patterns of order [math] that allow [math] .
</p>projecteuclid.org/euclid.involve/1572055231_20191025220031Fri, 25 Oct 2019 22:00 EDTSome combinatorics from Zeckendorf representationshttps://projecteuclid.org/euclid.involve/1572055232<strong>Tyler Ball</strong>, <strong>Rachel Chaiser</strong>, <strong>Dean Dustin</strong>, <strong>Tom Edgar</strong>, <strong>Paul Lagarde</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 7, 1241--1260.</p><p><strong>Abstract:</strong><br/>
We explore some properties of the so-called Zeckendorf representations of integers, where we write an integer as a sum of distinct, nonconsecutive Fibonacci numbers. We examine the combinatorics arising from the arithmetic of these representations, with a particular emphasis on understanding the Zeckendorf tree that encodes them. We introduce some possibly new results related to the tree, allowing us to develop a partial analog to Kummer’s classical theorem about counting the number of “carries” involved in arithmetic. Finally, we finish with some conjectures and possible future projects related to the combinatorics of these representations.
</p>projecteuclid.org/euclid.involve/1572055232_20191025220031Fri, 25 Oct 2019 22:00 EDTOn the zero-sum group-magicness of cartesian productshttps://projecteuclid.org/euclid.involve/1576119623<strong>Adam Fong</strong>, <strong>John Georges</strong>, <strong>David Mauro</strong>, <strong>Dylan Spagnuolo</strong>, <strong>John Wallace</strong>, <strong>Shufan Wang</strong>, <strong>Kirsti Wash</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 8, 1261--1278.</p><p><strong>Abstract:</strong><br/>
Let [math] be a graph and let [math] be an abelian group with identity 0. Then an [math] -magic labeling of [math] is a function [math] from [math] into [math] such that for some [math] , [math] for every [math] , where [math] is the set of edges incident to [math] . If [math] exists such that [math] , then [math] is zero-sum [math] -magic . Let [math] be the cartesian product of two or more graphs. We establish that [math] is zero-sum [math] -magic and we introduce a graph invariant [math] to explore the zero-sum integer-magic spectrum (or null space) of [math] . For certain [math] , we establish [math] , the set of nontrivial abelian groups for which [math] is zero-sum group-magic. Particular attention is given to [math] for regular [math] , odd/even [math] , and [math] isomorphic to a product of paths.
</p>projecteuclid.org/euclid.involve/1576119623_20191211220046Wed, 11 Dec 2019 22:00 ESTThe variable exponent Bernoulli differential equationhttps://projecteuclid.org/euclid.involve/1576119624<strong>Karen R. Ríos-Soto</strong>, <strong>Carlos E. Seda-Damiani</strong>, <strong>Alejandro Vélez-Santiago</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 8, 1279--1291.</p><p><strong>Abstract:</strong><br/>
We investigate the realization of a Bernoulli-type first-order differential equation with a variable exponent. Using substitution methods, we show the existence of an implicit solution to the Bernoulli problem. Numerical simulations applied to several examples are also provided.
</p>projecteuclid.org/euclid.involve/1576119624_20191211220046Wed, 11 Dec 2019 22:00 ESTThe supersingularity of Hurwitz curveshttps://projecteuclid.org/euclid.involve/1576119625<strong>Erin Dawson</strong>, <strong>Henry Frauenhoff</strong>, <strong>Michael Lynch</strong>, <strong>Amethyst Price</strong>, <strong>Seamus Somerstep</strong>, <strong>Eric Work</strong>, <strong>Dean Bisogno</strong>, <strong>Rachel Pries</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 8, 1293--1306.</p><p><strong>Abstract:</strong><br/>
We study when Hurwitz curves are supersingular. Specifically, we show that the curve [math] , with [math] and [math] relatively prime, is supersingular over the finite field [math] if and only if there exists an integer [math] such that [math] . If this holds, we prove that it is also true that the curve is maximal over [math] . Further, we provide a complete table of supersingular Hurwitz curves of genus less than 5 for characteristic less than 37.
</p>projecteuclid.org/euclid.involve/1576119625_20191211220046Wed, 11 Dec 2019 22:00 ESTMulticast triangular semilattice networkhttps://projecteuclid.org/euclid.involve/1576119626<strong>Angelina Grosso</strong>, <strong>Felice Manganiello</strong>, <strong>Shiwani Varal</strong>, <strong>Emily Zhu</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 8, 1307--1328.</p><p><strong>Abstract:</strong><br/>
We investigate the structure of the code graph of a multicast network that has a characteristic shape of an inverted equilateral triangle. We provide a criterion that determines the validity of a receiver placement within the code graph, present invariance properties of the determinants corresponding to receiver placements under symmetries, and provide a complete study of these networks’ receivers and required field sizes up to a network of four sources. We also improve on various definitions related to code graphs.
</p>projecteuclid.org/euclid.involve/1576119626_20191211220046Wed, 11 Dec 2019 22:00 ESTEdge-transitive graphs and combinatorial designshttps://projecteuclid.org/euclid.involve/1576119627<strong>Heather A. Newman</strong>, <strong>Hector Miranda</strong>, <strong>Adam Gregory</strong>, <strong>Darren A. Narayan</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 8, 1329--1341.</p><p><strong>Abstract:</strong><br/>
A graph is said to be edge-transitive if its automorphism group acts transitively on its edges. It is known that edge-transitive graphs are either vertex-transitive or bipartite. We present a complete classification of all connected edge-transitive graphs on less than or equal to [math] vertices. We investigate biregular bipartite edge-transitive graphs and present connections to combinatorial designs, and we show that the Cartesian products of complements of complete graphs give an additional family of edge-transitive graphs.
</p>projecteuclid.org/euclid.involve/1576119627_20191211220046Wed, 11 Dec 2019 22:00 ESTA logistic two-sex model with mate-finding Allee effecthttps://projecteuclid.org/euclid.involve/1576119628<strong>Elizabeth Anderson</strong>, <strong>Daniel Maxin</strong>, <strong>Jared Ott</strong>, <strong>Gwyneth Terrett</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 8, 1343--1355.</p><p><strong>Abstract:</strong><br/>
We analyze a logistic two-sex model with mate-finding Allee effects assuming distinct sex-related parameters. We compute the threshold of the Allee-effect strength that separates population extinction from persistence and prove that a bistability regimen appears whereby the total population either goes extinct or stabilizes at a positive level depending on the initial demographic conditions. We show that this effect is the only possible outcome as far as the population limiting behavior is concerned. In addition, we compute the optimal female-sex probability at birth that maximizes this threshold.
</p>projecteuclid.org/euclid.involve/1576119628_20191211220046Wed, 11 Dec 2019 22:00 ESTUnoriented links and the Jones polynomialhttps://projecteuclid.org/euclid.involve/1576119631<strong>Sandy Ganzell</strong>, <strong>Janet Huffman</strong>, <strong>Leslie Mavrakis</strong>, <strong>Kaitlin Tademy</strong>, <strong>Griffin Walker</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 8, 1357--1367.</p><p><strong>Abstract:</strong><br/>
The Jones polynomial is an invariant of oriented links with [math] components. When [math] , the choice of orientation does not affect the polynomial, but for [math] , changing orientations of some (but not all) components can change the polynomial. Here we define a version of the Jones polynomial that is an invariant of unoriented links; i.e., changing orientation of any sublink does not affect the polynomial. This invariant shares some, but not all, of the properties of the Jones polynomial.
The construction of this invariant also reveals new information about the original Jones polynomial. Specifically, we show that the Jones polynomial of a knot is never the product of a nontrivial monomial with another Jones polynomial.
</p>projecteuclid.org/euclid.involve/1576119631_20191211220046Wed, 11 Dec 2019 22:00 ESTNonsplit module extensions over the one-sided inverse of $k[x]$https://projecteuclid.org/euclid.involve/1576119632<strong>Zheping Lu</strong>, <strong>Linhong Wang</strong>, <strong>Xingting Wang</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 8, 1369--1377.</p><p><strong>Abstract:</strong><br/>
Let [math] be the associative [math] -algebra generated by two elements [math] and [math] with defining relation [math] . A complete description of simple modules over [math] is obtained by using the results of Irving and Gerritzen. We examine the short exact sequence [math] , where [math] and [math] are simple [math] -modules. It shows that nonsplit extension only occurs when both [math] and [math] are one-dimensional, or, under certain condition, [math] is infinite-dimensional and [math] is one-dimensional.
</p>projecteuclid.org/euclid.involve/1576119632_20191211220046Wed, 11 Dec 2019 22:00 ESTSplit Grothendieck rings of rooted trees and skew shapes via monoid representationshttps://projecteuclid.org/euclid.involve/1576119633<strong>David Beers</strong>, <strong>Matt Szczesny</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 8, 1379--1397.</p><p><strong>Abstract:</strong><br/>
We study commutative ring structures on the integral span of rooted trees and [math] -dimensional skew shapes. The multiplication in these rings arises from the smash product operation on monoid representations in pointed sets. We interpret these as Grothendieck rings of indecomposable monoid representations over [math] — the “field” of one element. We also study the base-change homomorphism from [math] -modules to [math] -modules for a field [math] containing all roots of unity, and interpret the result in terms of Jordan decompositions of adjacency matrices of certain graphs.
</p>projecteuclid.org/euclid.involve/1576119633_20191211220046Wed, 11 Dec 2019 22:00 ESTOn the classification of Specht modules with one-dimensional summandshttps://projecteuclid.org/euclid.involve/1576119634<strong>Aubrey Piper Collins</strong>, <strong>Craig J. Dodge</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 8, 1399--1413.</p><p><strong>Abstract:</strong><br/>
This paper extends a result of James to a combinatorial condition on partitions for the corresponding Specht module to have a summand isomorphic to the unique one-dimensional [math] -module over fields of characteristic 2. The work makes use of a recursively defined condition to reprove a result of Murphy and prove a new result for self-conjugate partitions. Finally we present a Python script which utilizes this work to test Specht modules for a one-dimensional summand.
</p>projecteuclid.org/euclid.involve/1576119634_20191211220046Wed, 11 Dec 2019 22:00 ESTThe monochromatic column problem with a prime number of colorshttps://projecteuclid.org/euclid.involve/1576119635<strong>Loran Crowell</strong>, <strong>Steve Szabo</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 8, 1415--1422.</p><p><strong>Abstract:</strong><br/>
Let [math] be a sequence of [math] pairwise coprime positive integers, [math] , and [math] be a sequence of [math] different colors. Let [math] be an [math] matrix of colors in which row [math] consists of blocks of [math] consecutive entries of the same color with colors 0 through [math] repeated cyclically. The monochromatic column problem is to determine the number of columns of [math] in which every entry is the same color. The solution for a prime number of colors is provided.
</p>projecteuclid.org/euclid.involve/1576119635_20191211220046Wed, 11 Dec 2019 22:00 ESTTotal Roman domination edge-critical graphshttps://projecteuclid.org/euclid.involve/1576119636<strong>Chloe Lampman</strong>, <strong>Kieka (C. M.) Mynhardt</strong>, <strong>Shannon Ogden</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 8, 1423--1439.</p><p><strong>Abstract:</strong><br/>
A total Roman dominating function on a graph [math] is a function [math] such that every vertex [math] with [math] is adjacent to some vertex [math] with [math] , and the subgraph of [math] induced by the set of all vertices [math] such that [math] has no isolated vertices. The weight of [math] is [math] . The total Roman domination number [math] is the minimum weight of a total Roman dominating function on [math] . A graph [math] is [math] - [math] -edge-critical if [math] for every edge [math] , and [math] - [math] -edge-supercritical if it is [math] - [math] -edge-critical and [math] for every edge [math] . We present some basic results on [math] -edge-critical graphs and characterize certain classes of [math] -edge-critical graphs. In addition, we show that, when [math] is small, there is a connection between [math] - [math] -edge-critical graphs and graphs which are critical with respect to the domination and total domination numbers.
</p>projecteuclid.org/euclid.involve/1576119636_20191211220046Wed, 11 Dec 2019 22:00 ESTStructured sequences and matrix rankshttps://projecteuclid.org/euclid.involve/1584669663<strong>Charles Johnson</strong>, <strong>Yaoxian Qu</strong>, <strong>Duo Wang</strong>, <strong>John Wilkes</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 13, Number 1, 1--8.</p><p><strong>Abstract:</strong><br/>
We consider infinite sequences from a field and all matrices whose rows consist of distinct consecutive subsequences. We show that these matrices have bounded rank if and only if the sequence is a homogeneous linear recurrence; moreover, it is a [math] -term linear recurrence if and only if the maximum rank is [math] . This means, in particular, that the ranks of matrices from the sequence of primes are unbounded. Though not all matrices from the primes have full rank, because of the Green–Tao theorem, we conjecture that square matrices whose entries are a consecutive sequence of primes do have full rank.
</p>projecteuclid.org/euclid.involve/1584669663_20200319220108Thu, 19 Mar 2020 22:01 EDTAnalysis of steady states for classes of reaction-diffusion equations with hump-shaped density-dependent dispersal on the boundaryhttps://projecteuclid.org/euclid.involve/1584669664<strong>Quinn Morris</strong>, <strong>Jessica Nash</strong>, <strong>Catherine Payne</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 13, Number 1, 9--19.</p><p><strong>Abstract:</strong><br/>
We study a two-point boundary-value problem describing steady states of a population dynamics model with diffusion, logistic growth, and nonlinear density-dependent dispersal on the boundary. In particular, we focus on a model in which the population exhibits hump-shaped density-dependent dispersal on the boundary, and explore its effects on existence, uniqueness and multiplicity of steady states.
</p>projecteuclid.org/euclid.involve/1584669664_20200319220108Thu, 19 Mar 2020 22:01 EDTThe $L$-move and Markov theorems for trivalent braidshttps://projecteuclid.org/euclid.involve/1584669665<strong>Carmen Caprau</strong>, <strong>Gabriel Coloma</strong>, <strong>Marguerite Davis</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 13, Number 1, 21--50.</p><p><strong>Abstract:</strong><br/>
The [math] -move for classical braids extends naturally to trivalent braids. We follow the [math] -move approach to the Markov theorem to prove a one-move Markov-type theorem for trivalent braids. We also reformulate this [math] -move Markov theorem and prove a more algebraic Markov-type theorem for trivalent braids. Along the way, we provide a proof of the Alexander theorem analogue for spatial trivalent graphs and trivalent braids.
</p>projecteuclid.org/euclid.involve/1584669665_20200319220108Thu, 19 Mar 2020 22:01 EDTLow stages of the Taylor tower for $r$-immersionshttps://projecteuclid.org/euclid.involve/1584669666<strong>Bridget Schreiner</strong>, <strong>Franjo Šarčević</strong>, <strong>Ismar Volić</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 13, Number 1, 51--75.</p><p><strong>Abstract:</strong><br/>
We study the beginning of the Taylor tower, supplied by manifold calculus of functors, for the space of [math] -immersions, which are immersions without [math] -fold self-intersections. We describe the first [math] layers of the tower and discuss the connectivities of the associated maps. We also prove several results about [math] -immersions that are of independent interest.
</p>projecteuclid.org/euclid.involve/1584669666_20200319220108Thu, 19 Mar 2020 22:01 EDTA new go-to sampler for Bayesian probit regressionhttps://projecteuclid.org/euclid.involve/1584669667<strong>Scott Simmons</strong>, <strong>Elizabeth J. McGuffey</strong>, <strong>Douglas VanDerwerken</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 13, Number 1, 77--89.</p><p><strong>Abstract:</strong><br/>
This paper introduces an independent-proposal Metropolis–Hastings sampler for Bayesian probit regression. The Gibbs sampler of Albert and Chib has been the default sampler since its introduction in 1993. We conduct a simulation study comparing the two methods under various combinations of predictor variables and sample sizes. The proposed sampler is shown to outperform that of Albert and Chib in terms of efficiency measured through autocorrelation, effective sample size, and computation time. We then demonstrate performance of the samplers on real data applications with analogous results.
</p>projecteuclid.org/euclid.involve/1584669667_20200319220108Thu, 19 Mar 2020 22:01 EDTCharacterizing optimal point sets determining one distinct trianglehttps://projecteuclid.org/euclid.involve/1584669668<strong>Hazel N. Brenner</strong>, <strong>James S. Depret-Guillaume</strong>, <strong>Eyvindur A. Palsson</strong>, <strong>Robert Stuckey</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 13, Number 1, 91--98.</p><p><strong>Abstract:</strong><br/>
We determine the maximum number of points in [math] which form exactly [math] distinct triangles, where we restrict ourselves to the case of [math] . We denote this quantity by [math] . It is known from the work of Epstein et al. ( Integers 18 (2018), art. id. A16) that [math] . Here we show somewhat surprisingly that [math] and [math] , whenever [math] , and characterize the optimal point configurations. This is an extension of a variant of the distinct distance problem put forward by Erdős and Fishburn ( Discrete Math. 160 :1-3 (1996), 115–125).
</p>projecteuclid.org/euclid.involve/1584669668_20200319220108Thu, 19 Mar 2020 22:01 EDTSolutions of periodic boundary value problemshttps://projecteuclid.org/euclid.involve/1584669669<strong>R. Aadith</strong>, <strong>Paras Gupta</strong>, <strong>Jagan Mohan Jonnalagadda</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 13, Number 1, 99--107.</p><p><strong>Abstract:</strong><br/>
We deal with a resonant boundary value problem involving a second-order differential equation with periodic boundary conditions. First, we modify the problem at resonance and consider an equivalent nonresonant boundary value problem. Next, we obtain sufficient conditions for the existence of solutions of the modified boundary value problem, using fixed-point theory. Consequently, these conditions suffice for the existence of solutions of the original boundary value problem. We demonstrate the applicability of established results through examples.
</p>projecteuclid.org/euclid.involve/1584669669_20200319220108Thu, 19 Mar 2020 22:01 EDTA few more trees the chromatic symmetric function can distinguishhttps://projecteuclid.org/euclid.involve/1584669670<strong>Jake Huryn</strong>, <strong>Sergei Chmutov</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 13, Number 1, 109--116.</p><p><strong>Abstract:</strong><br/>
A well-known open problem in graph theory asks whether Stanley’s chromatic symmetric function , a generalization of the chromatic polynomial of a graph, distinguishes between any two nonisomorphic trees. Previous work has proven the conjecture for a class of trees called spiders . This paper generalizes the class of spiders to [math] -spiders , where normal spiders correspond to [math] , and verifies the conjecture for [math] .
</p>projecteuclid.org/euclid.involve/1584669670_20200319220108Thu, 19 Mar 2020 22:01 EDTOne-point hyperbolic-type metricshttps://projecteuclid.org/euclid.involve/1584669671<strong>Marina Borovikova</strong>, <strong>Zair Ibragimov</strong>, <strong>Miguel Jimenez Bravo</strong>, <strong>Alexandro Luna</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 13, Number 1, 117--136.</p><p><strong>Abstract:</strong><br/>
We study basic properties of one-parametric families of the [math] -metric, the scale-invariant Cassinian metric and the half-Apollonian metric on locally compact, noncomplete metric spaces. We first establish basic properties of these metrics on once-punctured general metric spaces and obtain sharp estimates between these metrics, and then we show that all these properties, except for [math] -hyperbolicity, extend to the settings of locally compact noncomplete metric spaces. We also show that these metrics are [math] -hyperbolic only if the underlying space is a once-punctured metric space.
</p>projecteuclid.org/euclid.involve/1584669671_20200319220108Thu, 19 Mar 2020 22:01 EDTSome generalizations of the ASR search algorithm for quasitwisted codeshttps://projecteuclid.org/euclid.involve/1584669672<strong>Nuh Aydin</strong>, <strong>Thomas H. Guidotti</strong>, <strong>Peihan Liu</strong>, <strong>Armiya S. Shaikh</strong>, <strong>Robert O. VandenBerg</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 13, Number 1, 137--148.</p><p><strong>Abstract:</strong><br/>
One of the most important and challenging problems in coding theory is explicit construction of linear codes with the best possible parameters. It is well known that the class of quasitwisted (QT) codes is asymptotically good and contains many linear codes with best known parameters (BKLCs). A search algorithm (ASR) on QT codes has been particularly effective to construct such codes. Recently, the ASR algorithm was generalized based on the notion of code equivalence. In this work, we introduce a new generalization of the ASR algorithm to include a broader scope of QT codes. As a result of implementing this algorithm, we have found eight new linear codes over the field [math] . Furthermore, we have found seven additional new codes from the standard constructions of puncturing, shortening or Construction X. We also introduce a new search algorithm that can be viewed as a further generalization of ASR into the class multitwisted (MT) codes. Using this method, we have found many codes with best known parameters with more direct and desirable constructions than what is currently available in the database of BKLCs.
</p>projecteuclid.org/euclid.involve/1584669672_20200319220108Thu, 19 Mar 2020 22:01 EDTContinuous factorization of the identity matrixhttps://projecteuclid.org/euclid.involve/1584669673<strong>Yuying Dai</strong>, <strong>Ankush Hore</strong>, <strong>Siqi Jiao</strong>, <strong>Tianxu Lan</strong>, <strong>Pavlos Motakis</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 13, Number 1, 149--164.</p><p><strong>Abstract:</strong><br/>
We investigate conditions under which the identity matrix [math] can be continuously factorized through a continuous [math] matrix function [math] with domain in [math] . We study the relationship of the dimension [math] , the diagonal entries of [math] , and the norm of [math] to the dimension [math] and the norms of the matrices that witness the factorization of [math] through [math] .
</p>projecteuclid.org/euclid.involve/1584669673_20200319220108Thu, 19 Mar 2020 22:01 EDTAlmost excellent unique factorization domainshttps://projecteuclid.org/euclid.involve/1584669674<strong>Sarah M. Fleming</strong>, <strong>Susan Loepp</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 13, Number 1, 165--180.</p><p><strong>Abstract:</strong><br/>
Let [math] be a complete local (Noetherian) domain such that [math] . In addition, suppose [math] contains the rationals, [math] , and the set of all principal height-1 prime ideals of [math] has the same cardinality as [math] . We construct a universally catenary local unique factorization domain [math] such that the completion of [math] is [math] and such that there exist uncountably many height-1 prime ideals [math] of [math] such that [math] is a field. Furthermore, in the case where [math] is a normal domain, we can make [math] “close” to excellent in the following sense: the formal fiber at every prime ideal of [math] of height not equal to 1 is geometrically regular, and uncountably many height-1 prime ideals of [math] have geometrically regular formal fibers.
</p>projecteuclid.org/euclid.involve/1584669674_20200319220108Thu, 19 Mar 2020 22:01 EDT