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The Hausdorff metric measures the distance between nonempty compact sets in , the collection of which is denoted . Betweenness in can be defined in the same manner as betweenness in Euclidean geometry. But unlike betweenness in , for some elements and of there can be many elements between and at a fixed distance from . Blackburn et al. (“A missing prime configuration in the Hausdorff metric geometry”, J.Geom., 92:1–2 (2009), pp. 28–59) demonstrate that there are infinitely many positive integers such that there exist elements and having exactly different elements between and at each distance from while proving the surprising result that no such and exist for . In this vein, we prove that there do not exist elements and with exactly a countably infinite number of elements at any location between and .
The positive semidefinite minimum rank of a simple graph is defined to be the smallest possible rank over all positive semidefinite real symmetric matrices whose -th entry (for ) is nonzero whenever is an edge in and is zero otherwise. The computation of this parameter directly is difficult. However, there are a number of known bounding parameters and techniques which can be calculated and performed on a computer. We programmed an implementation of these bounds and techniques in the open-source mathematical software Sage. The program, in conjunction with the orthogonal representation method, establishes the positive semidefinite minimum rank for all graphs of order or less.
In this paper we will examine the affine algebraic curves defined on the complement of Fermat curves of degree five or higher in the affine plane. In particular we will bound the height of integral points over an affine curve outside of an exceptional set.
Building on the method used by Bhargava to prove “the fifteen theorem”, we show that every integer-valued positive definite quadratic form which represents all prime numbers must also represent 205. We further this result by proving that 205 is the smallest nontrivial composite number which must be represented by all such quadratic forms.
Given a finite field and a positive integer , we give a procedure to count the matrices with entries in with all eigenvalues in the field. We give an exact value for any field for values of up to , and prove that for fixed , as the size of the field increases, the proportion of matrices with all eigenvalues in the field approaches . As a corollary, we show that for large fields almost all matrices with all eigenvalues in the field have all eigenvalues distinct. The proofs of these results rely on the fact that any matrix with all eigenvalues in is similar to a matrix in Jordan canonical form, and so we proceed by enumerating the number of Jordan forms, and counting how many matrices are similar to each one. A key step in the calculation is to characterize the matrices that commute with a given Jordan form and count how many of them are invertible.
Lie algebras and quantum groups are not usually studied by an undergraduate. However, in the study of these structures, there are interesting questions that are easily accessible to an upper-level undergraduate. Here we look at the expansion of a nested set of brackets that appears in relations presented in a paper of Lum on toroidal algebras. We illuminate certain terms that must be in the expansion, providing a partial answer for the closed form.
We give explicit formulas for matrix coefficients of the depth-zero supercuspidal representations of over a nonarchimedean local field, highlighting the case where the test vector is a unit new vector. We also describe the partition of the set of such representations according to central character, and compute sums of matrix coefficients over all representations in a given class.
When matching socks after doing the laundry, how many unmatched socks can appear in the process of drawing one sock at a time from the basket? By connecting the problem of sock matching to the Catalan numbers, we give the probability that unmatched socks appear. We also show that, for each fixed , this probability approaches as the number of socks becomes large enough. The relation between the number of socks and the for which a given probability is first reached is also discussed, but a complete answer is open.
The method of upper and lower solutions guarantees the interval of existence of nonlinear differential equations with initial conditions. To compute the solution on this interval, we need coupled lower and upper solutions on the interval of existence. We provide both theoretical as well as numerical methods to compute coupled lower and upper solutions by using a superlinear convergence method. Further, we develop monotone sequences which converge uniformly and monotonically, and with superlinear convergence, to the unique solution of the nonlinear problem on this interval. We accelerate the superlinear convergence by means of the Gauss–Seidel method. Numerical examples are developed for the logistic equation. Our method is applicable to more general nonlinear differential equations, including Riccati type differential equations.