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We study the action of the Weyl group of type acting as permutations on the set of weights of the minuscule representation of type (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.
The Hausdorff metric is used to define the distance between two elements of , the hyperspace of all nonempty compact subsets of . The geometry this metric imposes on 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 , with a focus on attempting to find well-defined notions of angle and angle measure in .
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.
The Fibonacci sequence , when reduced modulo is periodic. For example, . The period of is denoted by , so . In this paper we present an algorithm that, given a period , produces all such that . 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.
We prove that the standard representation of 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 on the space of algebraic covariant derivative curvature tensors is faithful.
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 matrices such that the numerical range has fourfold symmetry about the origin but the generalized numerical range 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 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 is not a disk.
It is known that a modular form on can be expressed as a rational function in , and . 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 , a prime. In this case, we will show an explicit count for the number of eta-quotients of level and show that they are linearly independent.
We focus on a network reliability measure based on edge failures and considering a network operational if there exists a component with diameter or larger. The -diametercomponent edge connectivity parameter of a graph is the minimum number of edge failures needed so that no component has diameter or larger. This implies each resulting vertex must not have a -neighbor. We give results for specific graph classes including path graphs, complete graphs, complete bipartite graphs, and a surprising result for perfect -ary trees.
Tsirelson’s norm on is defined as the limit of an increasing sequence of norms . For each let be the smallest integer satisfying for all with . We show that is . This is an improvement of the upper bound of given by P. Casazza and T. Shura in their 1989 monograph on Tsirelson’s space.
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.
Let be a nontrivial group, and assume that for every nontrivial subgroup of . It is a simple matter to prove that or for some prime . In this note, we address the analogous (though harder) question for rings; that is, we find all nontrivial rings for which for every nontrivial subring of .
We show that the generalized -ary MacDonald codes with parameters are two-weight codes with nonzero weights , and determine the complete weight enumerator of these codes. This leads to a family of strongly regular graphs with parameters . Further, we show that the codes satisfy the Griesmer bound and are self-orthogonal for .
A poset has an interval representation if each can be assigned a real interval so that in if and only if lies completely to the left of . Such orders are called interval orders. Fishburn (1983, 1985) proved that for any positive integer , an interval order has a representation in which all interval lengths are between and if and only if the order does not contain as an induced poset. In this paper, we give a simple proof of this result using a digraph model.
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