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Project Euclid (Hosted at Cornell University Library) (203.669 recursos)

Involve. A Journal of Mathematics

1. A simple proof characterizing interval orders with interval lengths between 1 and $k$

Boyadzhiyska, Simona; Isaak, Garth; Trenk, Ann N.
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.

2. On generalized MacDonald codes

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] .

3. Rings isomorphic to their nontrivial subrings

Lojewski, Jacob; Oman, Greg
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] .

4. Enumeration of stacks of spheres

Endicott, Lauren; May, Russell; Shacklette, Sienna
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.

5. Time stopping for Tsirelson's norm

Beanland, Kevin; Duncan, Noah; Holt, Michael
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.

6. The $k$-diameter component edge connectivity parameter

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.

7. Counting eta-quotients of prime level

Arnold-Roksandich, Allison; James, Kevin; Keaton, Rodney
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 ...

8. Symmetric numerical ranges of four-by-four matrices

Burnett, Shelby L.; Chandler, Ashley; Patton, Linda J.
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...

9. Quasipositive curvature on a biquotient of Sp$(3)$

DeVito, Jason; Martin, Wesley
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.

10. On the faithfulness of the representation of $\mathrm{GL}(n)$ on the space of curvature tensors

Dunn, Corey; Elderfield, Darien; Martin-Hagemeyer, Rory
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.

11. The Fibonacci sequence under a modulus: computing all moduli that produce a given period

Dishong, Alex; Renault, Marc S.
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.

12. Different definitions of conic sections in hyperbolic geometry

Chao, Patrick; Rosenberg, Jonathan
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.

13. Pythagorean orthogonality of compact sets

Aggarwal, Pallavi; Schlicker, Steven; Swartzentruber, Ryan
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...

14. On the minuscule representation of type $B_n$

Cook, William J.; Hughes, Noah A.
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.

15. Locating trinomial zeros

Howell, Russell; Kyle, David
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.

16. Nonunique factorization over quotients of PIDs

Baeth, Nicholas R.; Burns, Brandon J.; Covey, Joshua M.; Mixco, James R.
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.

17. Connectedness of two-sided group digraphs and graphs

Chikwanda, Patreck; Kriloff, Cathy; Lee, Yun Teck; Sandow, Taylor; Smith, Garrett; Yeroshkin, Dmytro
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.

18. Numerical studies of serendipity and tensor product elements for eigenvalue problems

Gillette, Andrew; Gross, Craig; Plackowski, Ken
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...

19. Properties of sets of nontransitive dice with few sides

Angel, Levi; Davis, Matt
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.

20. Interpolation on Gauss hypergeometric functions with an application

Arora, Hina Manoj; Sahoo, Swadesh Kumar
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.

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