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Unoccupied: How a Single Word Affects Wyoming’s Ability to Regulate Tribal Hunting Through a Federal Treaty; Herrera v. Wyoming
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AN EIGENVALUE APPROACH FOR ESTIMATING THE GENERALIZED CROSS VALIDATION FUNCTION FOR CORRELATED MATRICES
No full textThis work proposes a fast estimate for the generalized cross-validation function when the design matrix of an experiment has correlated columns. The eigenvalue structure of this matrix is used to derive probability bounds satisfied by an appropriate index of proximity, which provides a simple and accurate estimate for the numerator of the generalized cross-validation function. The denominator of the function is evaluated by an analytical formula. Several simulation tests performed in statistical models having correlated design matrix with intercept confirm the reliability of the proposed probabilistic bounds and indicate the applicability of the proposed estimate for these models
Pairwise Completely Positive Matrices and Conjugate Local Diagonal Unitary Invariant Quantum States
No full textA generalization of the set of completely positive matrices called pairwise completely positive (PCP) matrices is introduced. These are pairs of matrices that share a joint decomposition so that one of them is necessarily positive semidefinite while the other one is necessarily entrywise non-negative. Basic properties of these matrix pairs are explored and several testable necessary and sufficient conditions are developed to help determine whether or not a pair is PCP. A connection with quantum entanglement is established by showing that determining whether or not a pair of matrices is pairwise completely positive is equivalent to determining whether or not a certain type of quantum state, called a conjugate local diagonal unitary invariant state, is separable. Many of the most important quantum states in entanglement theory are of this type, including isotropic states, mixed Dicke states (up to partial transposition), and maximally correlated states. As a specific application of these results, a wide family of states that have absolutely positive partial transpose are shown to in fact be separable
Tridiagonal pairs of type III with height one
No full textLet denote an algebraically closed field with characteristic . Let denote a vector space over with finite positive dimension, and let denote a tridiagonal pair on of diameter . Let denote a standard ordering of the eigenspaces of on , and let denote the corresponding eigenvalues of . It is assumed that . Let denote the dimension of . The sequence is called the {\it{shape}} of the tridiagonal pair. It is known that and there exists a unique integer such that for , for , and for . The integer is known as the {\it{height}} of the tridiagonal pair. In this paper, it is showed that the shape of a tridiagonal pair of type III with height one is either , , , , , or , , , . In each case, an interesting basis is found for and the actions of on this basis are described
Diagonal Sums of Doubly Substochastic Matrices
No full textLet denote the convex polytope of all doubly stochastic matrices, and denote the convex polytope of all doubly substochastic matrices. For a matrix , define the sub-defect of to be the smallest integer such that there exists an doubly stochastic matrix containing as a submatrix. Let denote the subset of which contains all doubly substochastic matrices with sub-defect . For a permutation of symmetric group of degree , the sequence of elements is called the diagonal of corresponding to . Let and denote the maximum and minimum diagonal sums of , respectively. In this paper, existing results of and functions are extended from to In addition, an analogue of Sylvesters law of the function on is proved
Illegitimate Succession: Vestigial Discrimination in Wyoming’s Rules of Intestate Descent
No full textResolution Of Conjectures Related To Lights Out! And Cartesian Products
No full textLights Out!\ is a game played on a grid of lights, or more generally on a graph. Pressing lights on the grid allows the player to turn off neighboring lights. The goal of the game is to start with a given initial configuration of lit lights and reach a state where all lights are out. Two conjectures posed in a recently published paper about Lights Out!\ on Cartesian products of graphs are resolved
A note on linear preservers of semipositive and minimally semipositive matrices
No full textSemipositive matrices (matrices that map at least one nonnegative vector to a positive vector) and minimally semipositive matrices (semipositive matrices whose no column-deleted submatrix is semipositive) are well studied in matrix theory. In this short note, the structure of linear maps which preserve the set of all semipositive/minimally semipositive matrices is studied. An open problem is solved, and some ambiguities in the article [J. Dorsey, T. Gannon, N. Jacobson, C.R. Johnson and M. Turnansky. Linear preservers of semi-positive matrices. {\em Linear and Multilinear Algebra}, 64:1853--1862, 2016.] are clarified
On Sign Pattern Matrices that Allow or Require Algebraic Positivity
No full textA square matrix M with real entries is algebraically positive (AP) if there exists a real polynomial p such that all entries of the matrix p(M) are positive. A square sign pattern matrix S allows algebraic positivity if there is an algebraically positive matrix M whose sign pattern is S. On the other hand, S requires algebraic positivity if matrix M, having sign pattern S, is algebraically positive. Motivated by open problems raised in a work of Kirkland, Qiao, and Zhan (2016) on AP matrices, all nonequivalent irreducible 3 by 3 sign pattern matrices are listed and classify into three groups (i) those that require AP, (ii) those that allow but not require AP, or (iii) those that do not allow AP. A necessary condition for an irreducible n by n sign pattern to allow algebraic positivity is also provided