Showing papers in "Involve, A Journal of Mathematics in 2019"
TL;DR: In this article, the authors consider a variant of the Lights Out problem in which the possible states for each vertex are indexed by the integers modulo k, and examine the space of null patterns on graphs, and use this as a way to prove theorems about Lights Out on graphs that are related to one another by two main constructions.
Abstract: The Lights Out problem on graphs, in which each vertex of the graph is in one of two states (“on” or “off”), has been investigated from a variety of perspectives over the last several decades, including parity domination, cellular automata, and harmonic functions on graphs. We consider a variant of the Lights Out problem in which the possible states for each vertex are indexed by the integers modulo k. We examine the space of “null patterns” (i.e., harmonic functions) on graphs, and use this as a way to prove theorems about Lights Out on graphs that are related to one another by two main constructions.
10 citations
TL;DR: In this paper, the authors apply results from convex geometry to determine which neural codes can be realized by arrangements of open convex sets, and restrict their attention primarily to sparse codes in low dimensions.
Abstract: 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 $\mathbb{R}^d$. Combinatorial objects known as \emph{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 $3$. Furthermore, we prove that in $\mathbb{R}^2$ and $\mathbb{R}^3$, 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 $2$.
10 citations
9 citations
TL;DR: In this article, the Erdős-Szekeres theorem for cyclic permutations of length (k−1, l−1)+2 was shown to be tight.
Abstract: We provide a cyclic permutation analogue of the Erdős–Szekeres theorem. In particular, we show that every cyclic permutation of length (k−1)(l−1)+2 has either an increasing cyclic subpermutation of length k+1 or a decreasing cyclic subpermutation of length l+1, 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 k+1 or a decreasing cyclic subpermutation of length l+1.
8 citations
8 citations
TL;DR: In this article, the game of best choice with permutation pattern-avoidance was studied under an additional assumption that the ranks of interview candidates are restricted using permutation patterns avoidance.
Abstract: 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.
8 citations
TL;DR: In this paper, the problem of pattern containment and avoidance in colored circular permutations has been explored and some interesting observations have been made, some of which are direct generalizations of previously established results.
Abstract: 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.
8 citations
TL;DR: In this paper, the authors studied the spectrum of the Kohn Laplacian on the unit spheres in ℂn and revisited Folland's classical eigenvalue computation.
Abstract: We study the spectrum of the Kohn Laplacian on the unit spheres in ℂn 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 ℂ2.
8 citations
TL;DR: In this article, the authors 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.
Abstract: 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.
7 citations
TL;DR: In this paper, the authors introduce weighted versions of the classical Cech and Vietoris-Rips complexes and show that a version of the Vietoris−Rips lemma holds for these weighted complexes.
Abstract: We introduce weighted versions of the classical Cech 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.
7 citations
TL;DR: In this paper, the authors investigated covering numbers for rings of upper triangular matrices with entries from a finite field, and they proved that if q ≥ 4, then T2(Fq) has covering number q+1, and when p is prime, Tn(Fp) has cover number p+1 for all n ≥ 2.
Abstract: A cover of a finite ring R is a collection of proper subrings {S1,…,Sm} of R such that R= ⋃i=1mSi. If such a collection exists, then R is called coverable, and the covering number of R 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 Fq be the field with q elements and let Tn(Fq) be the ring of n×n upper triangular matrices with entries from Fq. We prove that if q≠4, then T2(Fq) has covering number q+1, that T2(F4) has covering number 4, and that when p is prime, Tn(Fp) has covering number p+1 for all n≥2.
TL;DR: In this article, a complete list of prime knots with mosaic number 6 is provided, and a minimal, space-efficient knot mosaic for each of them is provided. And the tile number (or minimal mosaic tile number) of each of these knots is determined.
Abstract: In 2008, Kauffman and Lomonaco introduced the concepts of a knot mosaic and the mosaic number of a knot or link K, the smallest integer n such that K can be represented on an n-mosaic. In 2018, the authors of this paper introduced and explored space-efficient knot mosaics and the tile number of K, the smallest number of nonblank tiles necessary to depict K 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.
TL;DR: In this article, the homophonic quotient groups for French and English (the quotient of the free group generated by the French (respectively English) alphabet determined by relations representing standard pronunciation rules) were explicitly characterized.
Abstract: In 1993, the homophonic quotient groups for French and English (the quotient of the free group generated by the French (respectively English) alphabet determined by relations representing standard pronunciation rules) were explicitly characterized [5]. In this paper 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).
TL;DR: The spectrum of the Kohn Laplacian on the Rossi example was studied in this paper, and it was shown that 0 is in the essential spectrum of □ b t, which yields another proof of the global nonembeddability of the Rossi case.
Abstract: We study the spectrum of the Kohn Laplacian □ b t on the Rossi example ( S 3 , ℒ t ) . In particular we show that 0 is in the essential spectrum of □ b t , which yields another proof of the global nonembeddability of the Rossi example.
TL;DR: It is found that there is a constant probability of a large discrepancy between the numbers of quadratic residues and non-residues in the image of such subsets over uniformly random hyperelliptic curves of given degrees.
Abstract: 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 non-residues 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.
TL;DR: In this paper, it was shown that the smallest nonnegative integer for which there exists a positive integer m>k satisfying Se,p∕qk(n)=Se,p ∕qm(n) is the height of n, and that if q=2 or e=1, then k can be arbitrarily large.
Abstract: Let n be a positive integer and S2(n) be the sum of the squares of its decimal digits. When there exists a positive integer k such that the k-th iterate of S2 on n is 1, i.e., S2k(n)=1, then n 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 Se,p∕q(n) be the sum of the e-th powers of the digits of n base pq. Let k be the smallest nonnegative integer for which there exists a positive integer m>k satisfying Se,p∕qk(n)=Se,p∕qm(n). We prove that such a k, called the height of n, exists for all n, and that, if q=2 or e=1, then k can be arbitrarily large.
TL;DR: In this article, the length of all closed sub-Riemannian geodesics on the three-sphere was determined, and the lengths of all the closed subriemannians on the 3D space were determined.
Abstract: We determine the lengths of all closed sub-Riemannian geodesics on the three-sphere. Our methods are elementary and allow us to avoid using explicit formulas for the sub-Riemannian geodesics.
TL;DR: In this article, the authors classify cyclic Leibniz algebras over an arbitrary field and give examples of leitneralgebra with maximal Lie quotients that exhaust all 2-dimensional possibilities.
Abstract: 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.
TL;DR: In this paper, the authors consider optimal transportation with constraint, as did Korman and McCann (2013), and provide simplifications and generalizations of their examples and results, and provide some new examples.
Abstract: 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.
TL;DR: In this article, a complete description of simple modules over $R$ was obtained by using the results of Irving and Gerritzen, and it was shown that nonsplit extension only occurs when both $U$ and $V$ are one-dimensional.
Abstract: Let $R$ be the associative $k$-algebra generated by two elements $x$ and $y$ with defining relation $yx=1$. A complete description of simple modules over $R$ is obtained by using the results of Irving and Gerritzen. We examine the short exact sequence $0\rightarrow U\rightarrow E \rightarrow V\rightarrow 0$, where $U$ and $V$ are simple $R$-modules. It shows that nonsplit extension only occurs when both $U$ and $V$ are one-dimensional, or, under certain condition, $U$ is infinite-dimensional and $V$ is one-dimensional.
TL;DR: In this article, the minimum mean-squared error for 2-means clustering with two spheres is investigated, where the outcomes of the vector-valued random variable to be clustered are on two spheres, and the underlying probability distribution is the normalized surface measure.
Abstract: We study the minimum mean-squared error for 2-means clustering when the outcomes of the vector-valued random variable to be clustered are on two spheres, that is, the surface of two touching balls of unit radius in n-dimensional Euclidean space, and the underlying probability distribution is the normalized surface measure. For simplicity, we only consider the asymptotics of large sample sizes and replace empirical samples by the probability measure. The concrete question addressed here is whether a minimizer for the mean-squared error identifies the two individual spheres as clusters. Indeed, in dimensions n≥3, the minimum of the mean-squared error is achieved by a partition obtained from a separating hyperplane tangent to both spheres at the point where they touch. In dimension n=2, however, the minimizer fails to identify the individual spheres; an optimal partition is associated with a hyperplane that does not contain the intersection of the two spheres.
TL;DR: In this article, a graph theoretic analogue of an orbifold called an orbigraph is defined, and the relationship between the singular vertices of a Riemannian manifold and the spectrum of its adjacency matrix is analyzed.
Abstract: A Riemannian orbifold is a mildly singular generalization of a Riemannian manifold that is locally modeled on $R^n$ 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 analogue 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.
TL;DR: In this paper, it was shown that every doubled regular n-gon admits a 1∕(2n)-geodesic, where n is the length of the geodesic.
Abstract: We study 1∕k-geodesics, those closed geodesics that minimize on any subinterval of length L∕k, where L is the length of the geodesic. We investigate the existence and behavior of these curves on doubled polygons and show that every doubled regular n-gon admits a 1∕(2n)-geodesic. For the doubled regular p-gons, with p an odd prime, we conjecture that k=2p is the minimum value for k such that the space admits a 1∕k-geodesic.
TL;DR: In this article, the edge Folkman number Fe(G1, G2; k) is defined as the smallest order of any Kk-free graph F such that any red-blue coloring of the edges of F contains either a red G1 or a blue G2.
Abstract: Edge Folkman numbers Fe(G1, G2; k) can be viewed as a generalization of more commonly studied Ramsey numbers. Fe(G1, G2; k) is defined as the smallest order of any Kk-free graph F such that any red-blue coloring of the edges of F contains either a red G1 or a blue G2. In this note, first we discuss edge Folkman numbers involving graphs Js = Ks− e, including the results Fe(J3,Kn;n+ 1) = 2n−1, Fe(J3, Jn;n) = 2n− 1, and Fe(J3, Jn;n+ 1) = 2n− 3. Our modification of computational methods used previously in the study of classical Folkman numbers is applied to obtain upper bounds on Fe(J4, J4; k) for all k > 4.
TL;DR: In this paper, the authors examined the combinatorics arising from the arithmetic of these representations, with a particular emphasis on understanding the Zeckendorf tree that encodes them.
Abstract: 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.
TL;DR: In this article, 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.
Abstract: 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.
TL;DR: In this article, the authors explore the number of fixed points of the discrete exponentiation map by a statistical analysis of experimental data and show that the number can in many cases be modeled by a binomial distribution.
Abstract: The map x ↦ x x modulo p 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.