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 EDTFiniteness of homological filling functionshttps://projecteuclid.org/euclid.involve/1522202416<strong>Joshua W. Fleming</strong>, <strong>Eduardo Martínez-Pedroza</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 4, 569--583.</p><p><strong>Abstract:</strong><br/>
Let [math] be a group. For any [math] -module [math] and any integer [math] , we define a function [math] generalizing the notion of [math] -dimensional filling function of a group. We prove that this function takes only finite values if [math] is of type [math] and [math] , and remark that the asymptotic growth class of this function is an invariant of [math] . In the particular case that [math] is a group of type [math] , our main result implies that its [math] -dimensional homological filling function takes only finite values.
</p>projecteuclid.org/euclid.involve/1522202416_20180327220024Tue, 27 Mar 2018 22:00 EDTExplicit representations of 3-dimensional Sklyanin algebras associated to a point of order 2https://projecteuclid.org/euclid.involve/1522202417<strong>Daniel J. Reich</strong>, <strong>Chelsea Walton</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 4, 585--608.</p><p><strong>Abstract:</strong><br/>
The representation theory of a 3-dimensional Sklyanin algebra [math] depends on its (noncommutative projective algebro-) geometric data: an elliptic curve [math] in [math] , and an automorphism [math] of [math] given by translation by a point. Indeed, by a result of Artin, Tate, and van den Bergh, we have that [math] is module-finite over its center if and only if [math] has finite order. In this case, all irreducible representations of [math] are finite-dimensional and of at most dimension [math] .
In this work, we provide an algorithm in Maple to directly compute all irreducible representations of [math] associated to [math] of order 2, up to equivalence. Using this algorithm, we compute and list these representations. To illustrate how the algorithm developed in this paper can be applied to other algebras, we use it to recover well-known results about irreducible representations of the skew polynomial ring [math] .
</p>projecteuclid.org/euclid.involve/1522202417_20180327220024Tue, 27 Mar 2018 22:00 EDTA classification of Klein links as torus linkshttps://projecteuclid.org/euclid.involve/1522202418<strong>Steven Beres</strong>, <strong>Vesta Coufal</strong>, <strong>Kaia Hlavacek</strong>, <strong>M. Kate Kearney</strong>, <strong>Ryan Lattanzi</strong>, <strong>Hayley Olson</strong>, <strong>Joel Pereira</strong>, <strong>Bryan Strub</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 4, 609--624.</p><p><strong>Abstract:</strong><br/>
We classify Klein links. In particular, we calculate the number and types of components in a [math] Klein link. We completely determine which Klein links are equivalent to a torus link, and which are not.
</p>projecteuclid.org/euclid.involve/1522202418_20180327220024Tue, 27 Mar 2018 22:00 EDTInterpolation on Gauss hypergeometric functions with an applicationhttps://projecteuclid.org/euclid.involve/1522202419<strong>Hina Manoj Arora</strong>, <strong>Swadesh Kumar Sahoo</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 4, 625--641.</p><p><strong>Abstract:</strong><br/>
We use some standard numerical techniques to approximate the hypergeometric function
[math]
for a range of parameter triples [math] on the interval [math] . Some of the familiar hypergeometric functional identities and asymptotic behavior of the hypergeometric function at [math] play crucial roles in deriving the formula for such approximations. We also focus on error analysis of the numerical approximations leading to monotone properties of quotients of gamma functions in parameter triples [math] . Finally, an application to continued fractions of Gauss is discussed followed by concluding remarks consisting of recent works on related problems.
</p>projecteuclid.org/euclid.involve/1522202419_20180327220024Tue, 27 Mar 2018 22:00 EDTProperties of sets of nontransitive dice with few sideshttps://projecteuclid.org/euclid.involve/1522202420<strong>Levi Angel</strong>, <strong>Matt Davis</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 4, 643--659.</p><p><strong>Abstract:</strong><br/>
We define and investigate several properties that sets of nontransitive dice might have. We prove several implications between these properties, which hold in general or for dice with few sides. We also investigate some algorithms for creating sets of 3-sided dice that realize certain tournaments.
</p>projecteuclid.org/euclid.involve/1522202420_20180327220024Tue, 27 Mar 2018 22:00 EDTNumerical studies of serendipity and tensor product elements for eigenvalue problemshttps://projecteuclid.org/euclid.involve/1522202421<strong>Andrew Gillette</strong>, <strong>Craig Gross</strong>, <strong>Ken Plackowski</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 4, 661--678.</p><p><strong>Abstract:</strong><br/>
While the use of finite element methods for the numerical approximation of eigenvalues is a well-studied problem, the use of serendipity elements for this purpose has received little attention in the literature. We show by numerical experiments that serendipity elements, which are defined on a square reference geometry, can attain the same order of accuracy as their tensor product counterparts while using dramatically fewer degrees of freedom. In some cases, the serendipity method uses only 50% as many basis functions as the tensor product method while still producing the same numerical approximation of an eigenvalue. To encourage the further use and study of serendipity elements, we provide a table of serendipity basis functions for low-order cases and a Mathematica file that can be used to generate the basis functions for higher-order cases.
</p>projecteuclid.org/euclid.involve/1522202421_20180327220024Tue, 27 Mar 2018 22:00 EDTConnectedness of two-sided group digraphs and graphshttps://projecteuclid.org/euclid.involve/1522202422<strong>Patreck Chikwanda</strong>, <strong>Cathy Kriloff</strong>, <strong>Yun Teck Lee</strong>, <strong>Taylor Sandow</strong>, <strong>Garrett Smith</strong>, <strong>Dmytro Yeroshkin</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 4, 679--699.</p><p><strong>Abstract:</strong><br/>
Two-sided group digraphs and graphs, introduced by Iradmusa and Praeger, provide a generalization of Cayley digraphs and graphs in which arcs are determined by left and right multiplying by elements of two subsets of the group. We characterize when two-sided group digraphs and graphs are weakly and strongly connected and count connected components, using both an explicit elementary perspective and group actions. Our results and examples address four open problems posed by Iradmusa and Praeger that concern connectedness and valency. We pose five new open problems.
</p>projecteuclid.org/euclid.involve/1522202422_20180327220024Tue, 27 Mar 2018 22:00 EDTNonunique factorization over quotients of PIDshttps://projecteuclid.org/euclid.involve/1522202423<strong>Nicholas R. Baeth</strong>, <strong>Brandon J. Burns</strong>, <strong>Joshua M. Covey</strong>, <strong>James R. Mixco</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 4, 701--710.</p><p><strong>Abstract:</strong><br/>
We study factorizations of elements in quotients of commutative principal ideal domains that are endowed with an alternative multiplication. This study generalizes the study of factorizations both in quotients of PIDs and in rings of single-valued matrices. We are able to completely describe the sets of factorization lengths of elements in these rings, as well as compute other finer arithmetical invariants. In addition, we provide the first example of a finite bifurcus ring.
</p>projecteuclid.org/euclid.involve/1522202423_20180327220024Tue, 27 Mar 2018 22:00 EDTLocating trinomial zeroshttps://projecteuclid.org/euclid.involve/1522202424<strong>Russell Howell</strong>, <strong>David Kyle</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 4, 711--720.</p><p><strong>Abstract:</strong><br/>
We derive formulas for the number of interior roots (i.e., zeros with modulus less than 1) and exterior roots (i.e., zeros with modulus greater than 1) for trinomials of the form [math] , where [math] . Combined with earlier work by Brilleslyper and Schaubroeck, who focus on unimodular roots (i.e., zeros that lie on the unit circle), we give a complete count of the location of zeros of these trinomials.
</p>projecteuclid.org/euclid.involve/1522202424_20180327220024Tue, 27 Mar 2018 22:00 EDTOn the minuscule representation of type $B_n$https://projecteuclid.org/euclid.involve/1523498539<strong>William J. Cook</strong>, <strong>Noah A. Hughes</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 5, 721--733.</p><p><strong>Abstract:</strong><br/>
We study the action of the Weyl group of type [math] acting as permutations on the set of weights of the minuscule representation of type [math] (also known as the spin representation). Motivated by a previous work, we seek to determine when cycle structures alone reveal the irreducibility of these minuscule representations. After deriving formulas for the simple reflections viewed as permutations, we perform a series of computer-aided calculations in GAP. We are then able to establish that, for certain ranks, the irreducibility of the minuscule representation cannot be detected by cycle structures alone.
</p>projecteuclid.org/euclid.involve/1523498539_20180411220238Wed, 11 Apr 2018 22:02 EDTPythagorean orthogonality of compact setshttps://projecteuclid.org/euclid.involve/1523498540<strong>Pallavi Aggarwal</strong>, <strong>Steven Schlicker</strong>, <strong>Ryan Swartzentruber</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 5, 735--752.</p><p><strong>Abstract:</strong><br/>
The Hausdorff metric [math] is used to define the distance between two elements of [math] , the hyperspace of all nonempty compact subsets of [math] . The geometry this metric imposes on [math] is an interesting one — it is filled with unexpected results and fascinating connections to number theory and graph theory. Circles and lines are defined in this geometry to make it an extension of the standard Euclidean geometry. However, the behavior of lines and segments in this extended geometry is much different from that of lines and segments in Euclidean geometry. This paper presents surprising results about rays in the geometry of [math] , with a focus on attempting to find well-defined notions of angle and angle measure in [math] .
</p>projecteuclid.org/euclid.involve/1523498540_20180411220238Wed, 11 Apr 2018 22:02 EDTDifferent definitions of conic sections in hyperbolic geometryhttps://projecteuclid.org/euclid.involve/1523498541<strong>Patrick Chao</strong>, <strong>Jonathan Rosenberg</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 5, 753--768.</p><p><strong>Abstract:</strong><br/>
In classical Euclidean geometry, there are several equivalent definitions of conic sections. We show that in the hyperbolic plane, the analogues of these same definitions still make sense, but are no longer equivalent, and we discuss the relationships among them.
</p>projecteuclid.org/euclid.involve/1523498541_20180411220238Wed, 11 Apr 2018 22:02 EDTThe Fibonacci sequence under a modulus: computing all moduli that produce a given periodhttps://projecteuclid.org/euclid.involve/1523498542<strong>Alex Dishong</strong>, <strong>Marc S. Renault</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 5, 769--774.</p><p><strong>Abstract:</strong><br/>
The Fibonacci sequence [math] , when reduced modulo [math] is periodic. For example, [math] . The period of [math] is denoted by [math] , so [math] . In this paper we present an algorithm that, given a period [math] , produces all [math] such that [math] . For efficiency, the algorithm employs key ideas from a 1963 paper by John Vinson on the period of the Fibonacci sequence. We present output from the algorithm and discuss the results.
</p>projecteuclid.org/euclid.involve/1523498542_20180411220238Wed, 11 Apr 2018 22:02 EDTOn the faithfulness of the representation of $\mathrm{GL}(n)$ on the space of curvature tensorshttps://projecteuclid.org/euclid.involve/1523498543<strong>Corey Dunn</strong>, <strong>Darien Elderfield</strong>, <strong>Rory Martin-Hagemeyer</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 5, 775--785.</p><p><strong>Abstract:</strong><br/>
We prove that the standard representation of [math] on the space of algebraic curvature tensors is almost faithful by showing that the kernel of this representation contains only the identity map and its negative. We additionally show that the standard representation of [math] on the space of algebraic covariant derivative curvature tensors is faithful.
</p>projecteuclid.org/euclid.involve/1523498543_20180411220238Wed, 11 Apr 2018 22:02 EDTQuasipositive curvature on a biquotient of Sp$(3)$https://projecteuclid.org/euclid.involve/1523498544<strong>Jason DeVito</strong>, <strong>Wesley Martin</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 5, 787--801.</p><p><strong>Abstract:</strong><br/>
Suppose [math] denotes the unique irreducible complex [math] -dimensional representation of [math] , and consider the two subgroups [math] with [math] and [math] . We show that the biquotient [math] admits a quasipositively curved Riemannian metric.
</p>projecteuclid.org/euclid.involve/1523498544_20180411220238Wed, 11 Apr 2018 22:02 EDTSymmetric numerical ranges of four-by-four matriceshttps://projecteuclid.org/euclid.involve/1523498545<strong>Shelby L. Burnett</strong>, <strong>Ashley Chandler</strong>, <strong>Linda J. Patton</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 5, 803--826.</p><p><strong>Abstract:</strong><br/>
Numerical ranges of matrices with rotational symmetry are studied. Some cases in which symmetry of the numerical range implies symmetry of the spectrum are described. A parametrized class of [math] matrices [math] such that the numerical range [math] has fourfold symmetry about the origin but the generalized numerical range [math] does not have this symmetry is included. In 2011, Tsai and Wu showed that the numerical ranges of weighted shift matrices, which have rotational symmetry about the origin, are also symmetric about certain axes. We show that any [math] matrix whose numerical range has fourfold symmetry about the origin also has the corresponding axis symmetry. The support function used to prove these results is also used to show that the numerical range of a composition operator on Hardy space with automorphic symbol and minimal polynomial [math] is not a disk.
</p>projecteuclid.org/euclid.involve/1523498545_20180411220238Wed, 11 Apr 2018 22:02 EDTCounting eta-quotients of prime levelhttps://projecteuclid.org/euclid.involve/1523498546<strong>Allison Arnold-Roksandich</strong>, <strong>Kevin James</strong>, <strong>Rodney Keaton</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 5, 827--844.</p><p><strong>Abstract:</strong><br/>
It is known that a modular form on [math] can be expressed as a rational function in [math] , [math] and [math] . By using known theorems and calculating the order of vanishing, we can compute the eta-quotients for a given level. Using this count, knowing how many eta-quotients are linearly independent, and using the dimension formula, we can figure out a subspace spanned by the eta-quotients. In this paper, we primarily focus on the case where the level is [math] , a prime. In this case, we will show an explicit count for the number of eta-quotients of level [math] and show that they are linearly independent.
</p>projecteuclid.org/euclid.involve/1523498546_20180411220238Wed, 11 Apr 2018 22:02 EDTThe $k$-diameter component edge connectivity parameterhttps://projecteuclid.org/euclid.involve/1523498547<strong>Nathan Shank</strong>, <strong>Adam Buzzard</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 5, 845--856.</p><p><strong>Abstract:</strong><br/>
We focus on a network reliability measure based on edge failures and considering a network operational if there exists a component with diameter [math] or larger. The [math] -diameter component edge connectivity parameter of a graph is the minimum number of edge failures needed so that no component has diameter [math] or larger. This implies each resulting vertex must not have a [math] -neighbor. We give results for specific graph classes including path graphs, complete graphs, complete bipartite graphs, and a surprising result for perfect [math] -ary trees.
</p>projecteuclid.org/euclid.involve/1523498547_20180411220238Wed, 11 Apr 2018 22:02 EDTTime stopping for Tsirelson's normhttps://projecteuclid.org/euclid.involve/1523498548<strong>Kevin Beanland</strong>, <strong>Noah Duncan</strong>, <strong>Michael Holt</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 5, 857--866.</p><p><strong>Abstract:</strong><br/>
Tsirelson’s norm [math] on [math] is defined as the limit of an increasing sequence of norms [math] . For each [math] let [math] be the smallest integer satisfying [math] for all [math] with [math] . We show that [math] is [math] . This is an improvement of the upper bound of [math] given by P. Casazza and T. Shura in their 1989 monograph on Tsirelson’s space.
</p>projecteuclid.org/euclid.involve/1523498548_20180411220238Wed, 11 Apr 2018 22:02 EDTEnumeration of stacks of sphereshttps://projecteuclid.org/euclid.involve/1523498549<strong>Lauren Endicott</strong>, <strong>Russell May</strong>, <strong>Sienna Shacklette</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 5, 867--875.</p><p><strong>Abstract:</strong><br/>
As a three-dimensional generalization of fountains of coins, we analyze stacks of spheres and enumerate two particular classes, so-called “pyramidal” stacks and “Dominican” stacks. Using the machinery of generating functions, we obtain exact formulas for these types of stacks in terms of the sizes of their bases.
</p>projecteuclid.org/euclid.involve/1523498549_20180411220238Wed, 11 Apr 2018 22:02 EDTRings isomorphic to their nontrivial subringshttps://projecteuclid.org/euclid.involve/1523498550<strong>Jacob Lojewski</strong>, <strong>Greg Oman</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 5, 877--883.</p><p><strong>Abstract:</strong><br/>
Let [math] be a nontrivial group, and assume that [math] for every nontrivial subgroup [math] of [math] . It is a simple matter to prove that [math] or [math] for some prime [math] . In this note, we address the analogous (though harder) question for rings; that is, we find all nontrivial rings [math] for which [math] for every nontrivial subring [math] of [math] .
</p>projecteuclid.org/euclid.involve/1523498550_20180411220238Wed, 11 Apr 2018 22:02 EDTOn generalized MacDonald codeshttps://projecteuclid.org/euclid.involve/1523498551<strong>Padmapani Seneviratne</strong>, <strong>Lauren Melcher</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 5, 885--892.</p><p><strong>Abstract:</strong><br/>
We show that the generalized [math] -ary MacDonald codes [math] with parameters [math] are two-weight codes with nonzero weights [math] , [math] and determine the complete weight enumerator of these codes. This leads to a family of strongly regular graphs with parameters [math] . Further, we show that the codes [math] satisfy the Griesmer bound and are self-orthogonal for [math] .
</p>projecteuclid.org/euclid.involve/1523498551_20180411220238Wed, 11 Apr 2018 22:02 EDTA simple proof characterizing interval orders with interval lengths between 1 and $k$https://projecteuclid.org/euclid.involve/1523498552<strong>Simona Boyadzhiyska</strong>, <strong>Garth Isaak</strong>, <strong>Ann N. Trenk</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 5, 893--900.</p><p><strong>Abstract:</strong><br/>
A poset [math] has an interval representation if each [math] can be assigned a real interval [math] so that [math] in [math] if and only if [math] lies completely to the left of [math] . Such orders are called interval orders . Fishburn (1983, 1985) proved that for any positive integer [math] , an interval order has a representation in which all interval lengths are between [math] and [math] if and only if the order does not contain [math] as an induced poset. In this paper, we give a simple proof of this result using a digraph model.
</p>projecteuclid.org/euclid.involve/1523498552_20180411220238Wed, 11 Apr 2018 22:02 EDTLights Out for graphs related to one another by constructionshttps://projecteuclid.org/euclid.involve/1540432906<strong>Laura E. Ballard</strong>, <strong>Erica L. Budge</strong>, <strong>Darin R. Stephenson</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 2, 181--201.</p><p><strong>Abstract:</strong><br/>
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 [math] . 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.
</p>projecteuclid.org/euclid.involve/1540432906_20181024220224Wed, 24 Oct 2018 22:02 EDTA characterization of the sets of periods within shifts of finite typehttps://projecteuclid.org/euclid.involve/1540432907<strong>Madeline Doering</strong>, <strong>Ronnie Pavlov</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 2, 203--220.</p><p><strong>Abstract:</strong><br/>
We characterize precisely the possible sets of periods and least periods for the periodic points of a shift of finite type (SFT). We prove that a set is the set of least periods of some mixing SFT if and only if it is either [math] or cofinite, and the set of periods of some mixing SFT if and only if it is cofinite and closed under multiplication by arbitrary natural numbers. We then use these results to derive similar characterizations for the class of irreducible SFTs and the class of all SFTs. Specifically, a set is the set of (least) periods for some irreducible SFT if and only if it can be written as a natural number times the set of (least) periods for some mixing SFT, and a set is the set of (least) periods for an SFT if and only if it can be written as the finite union of the sets of (least) periods for some irreducible SFTs. Finally, we prove that the possible sets of (least) periods of mixing sofic shifts are exactly the same as for mixing SFTs, and that the same is not true for the class of nonmixing sofic shifts.
</p>projecteuclid.org/euclid.involve/1540432907_20181024220224Wed, 24 Oct 2018 22:02 EDTNumerical secondary terms in a Cohen–Lenstra conjecture on real quadratic fieldshttps://projecteuclid.org/euclid.involve/1540432910<strong>Codie Lewis</strong>, <strong>Cassandra Williams</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 2, 221--233.</p><p><strong>Abstract:</strong><br/>
In 1984, Cohen and Lenstra made a number of conjectures regarding the class groups of quadratic fields. In particular, they predicted the proportion of real quadratic fields with class number divisible by an odd prime. We numerically investigate the difference between reality and these predictions. Using 4 million data points, we perform a curve fitting of the difference with a monomial term and demonstrate that there is reason to believe the term can be effectively approximated within the scope of our data set for odd primes less than 30. We use cross-validation to show that including our monomial term as a secondary term to the original conjecture reduces the overall error.
</p>projecteuclid.org/euclid.involve/1540432910_20181024220224Wed, 24 Oct 2018 22:02 EDTCurves of constant curvature and torsion in the 3-spherehttps://projecteuclid.org/euclid.involve/1540432914<strong>Debraj Chakrabarti</strong>, <strong>Rahul Sahay</strong>, <strong>Jared Williams</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 2, 235--255.</p><p><strong>Abstract:</strong><br/>
We describe the curves of constant (geodesic) curvature and torsion in the three-dimensional round sphere. These curves are the trajectory of a point whose motion is the superposition of two circular motions in orthogonal planes. The global behavior may be periodic or the curve may be dense in a Clifford torus embedded in the 3-sphere. This behavior is very different from that of helices in three-dimensional Euclidean space, which also have constant curvature and torsion.
</p>projecteuclid.org/euclid.involve/1540432914_20181024220224Wed, 24 Oct 2018 22:02 EDTProperties of RNA secondary structure matching modelshttps://projecteuclid.org/euclid.involve/1540432915<strong>Nicole Anderson</strong>, <strong>Michael Breunig</strong>, <strong>Kyle Goryl</strong>, <strong>Hannah Lewis</strong>, <strong>Manda Riehl</strong>, <strong>McKenzie Scanlan</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 2, 257--280.</p><p><strong>Abstract:</strong><br/>
RNA secondary structures have been modeled using a variety of matching families. We first explore the intersections of different matching families which are models for likely RNA secondary structures. We then introduce their respective enumeration sequences and prove our proposed equations for enumeration. Next, we prove a formula for the number of matchings with a given crossing number for a variety of matching families. Then we develop a new statistic called the pseudoknot number and find the maximum pseudoknot number on a given set of matchings. We end by providing a comparison between the crossing number, nesting number, and pseudoknot number for three matching families on nine edges.
</p>projecteuclid.org/euclid.involve/1540432915_20181024220224Wed, 24 Oct 2018 22:02 EDTInfinite sums in totally ordered abelian groupshttps://projecteuclid.org/euclid.involve/1540432918<strong>Greg Oman</strong>, <strong>Caitlin Randall</strong>, <strong>Logan Robinson</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 2, 281--300.</p><p><strong>Abstract:</strong><br/>
The notion of convergence is absolutely fundamental in the study of calculus. In particular, it enables one to define the sum of certain infinite sets of real numbers as the limit of a sequence of partial sums, thus obtaining so-called convergent series . Convergent series, of course, play an integral role in real analysis (and, more generally, functional analysis) and the theory of differential equations. An interesting textbook problem is to show that there is no canonical way to “sum” uncountably many positive real numbers to obtain a finite (i.e., real) value. Plenty of solutions to this problem, which make strong use of the completeness property of the real line, can be found both online and in textbooks. In this note, we show that there is a more general reason for the nonfiniteness of uncountable sums. In particular, we present a canonical definition of “convergent series”, valid in any totally ordered abelian group, which extends the usual definition encountered in elementary analysis. We prove that there are convergent real series of positive numbers indexed by an arbitrary countable well-ordered set and, moreover, that any convergent series in a totally ordered abelian group indexed by an arbitrary well-ordered set has but countably many nonzero terms.
</p>projecteuclid.org/euclid.involve/1540432918_20181024220224Wed, 24 Oct 2018 22:02 EDTOn the minimum of the mean-squared error in 2-means clusteringhttps://projecteuclid.org/euclid.involve/1540432919<strong>Bernhard G. Bodmann</strong>, <strong>Craig J. George</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 2, 301--319.</p><p><strong>Abstract:</strong><br/>
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 [math] -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 [math] , 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 [math] , 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.
</p>projecteuclid.org/euclid.involve/1540432919_20181024220224Wed, 24 Oct 2018 22:02 EDTFailure of strong approximation on an affine conehttps://projecteuclid.org/euclid.involve/1540432920<strong>Martin Bright</strong>, <strong>Ivo Kok</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 2, 321--327.</p><p><strong>Abstract:</strong><br/>
We use the Brauer–Manin obstruction to strong approximation on a punctured affine cone to explain why some mod [math] solutions to a homogeneous Diophantine equation of degree [math] cannot be lifted to coprime integer solutions.
</p>projecteuclid.org/euclid.involve/1540432920_20181024220224Wed, 24 Oct 2018 22:02 EDTQuantum metrics from traces on full matrix algebrashttps://projecteuclid.org/euclid.involve/1540432923<strong>Konrad Aguilar</strong>, <strong>Samantha Brooker</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 2, 329--342.</p><p><strong>Abstract:</strong><br/>
We prove that, in the sense of the Gromov–Hausdorff propinquity, certain natural quantum metrics on the algebras of [math] -matrices are separated by a positive distance when [math] is not prime.
</p>projecteuclid.org/euclid.involve/1540432923_20181024220224Wed, 24 Oct 2018 22:02 EDTSolving Scramble Squares puzzles with repetitionshttps://projecteuclid.org/euclid.involve/1540432924<strong>Jason Callahan</strong>, <strong>Maria Mota</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 12, Number 2, 343--349.</p><p><strong>Abstract:</strong><br/>
A Scramble Squares puzzle consists of nine square pieces with half of an image on each side. To solve the puzzle the pieces are arranged in a [math] grid so that sides of adjacent pieces form complete images. A repetition is a half-image that appears more than once on a piece. Previous research uses a graph-theoretical approach to establish necessary and sufficient conditions for solutions without repetitions to [math] Scramble Squares puzzles. We use a similar approach to establish necessary and sufficient conditions for solutions with repetitions to [math] Scramble Squares puzzles.
</p>projecteuclid.org/euclid.involve/1540432924_20181024220224Wed, 24 Oct 2018 22:02 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 EST