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BOUNDS ON THE SUM OF MINIMUM SEMIDEFINITE RANK OF A GRAPH AND ITS COMPLEMENT
No full textThe minimum semi-definite rank (msr) of a graph is the minimum rank among all positive semi-definite matrices associated to the graph. The graph complement conjecture gives an upper bound for the sum of the msr of a graph and the msr of its complement. It is shown that when the msr of a graph is equal to its independence number, the graph complement conjecture holds with a better upper bound. Several sufficient conditions are provided for the msr of different classes of graphs to equal to its independence number
Positive and Z-operators on Closed Convex Cones
No full textLet be a closed convex cone with dual \dual{K} in a finite-dimensional real Hilbert space. A \emph{positive operator} on is a linear operator such that L\of{K} \subseteq K. Positive operators generalize the nonnegative matrices and are essential to the Perron-Frobenius theory. It is said that is a \emph{\textbf{Z}-operator} on if % \begin{equation*} \ip{L\of{x}}{s} \le 0 \;\text{ for all } \pair{x}{s} \in \cartprod{K}{\dual{K}} \text{ such that } \ip{x}{s} = 0. \end{equation*} % The \textbf{Z}-operators are generalizations of \textbf{Z}-matrices (whose off-diagonal elements are nonpositive) and they arise in dynamical systems, economics, game theory, and elsewhere. In this paper, the positive and \textbf{Z}-operators are connected. This extends the work of Schneider, Vidyasagar, and Tam on proper cones, and reveals some interesting similarities between the two families
Estimators Comparison of Separable Covariance Structure with One Component as Compound Symmetry Matrix
No full textThe maximum likelihood estimation (MLE) of separable covariance structure with one component as compound symmetry matrix has been widely studied in the literature. Nevertheless, the proposed estimates are not given in explicit form and can be determined only numerically. In this paper we give an alternative form of MLE and we show that this new algorithm is much quicker than the algorithms given in the literature.\\ Another estimator of covariance structure can be found by minimizing the entropy loss function. In this paper we give three methods of finding the best approximation of separable covariance structure with one component as compound symmetry matrix and we compare the quickness of proposed algorithms.\\ We conduct simulation studies to compare statistical properties of MLEs and entropy loss estimators (ELEs), such us biasedness, variability and loss. Another estimator of covariance structure can be found by minimizing the entropy loss function. In this paper we give three methods of finding the best approximation of separable covariance structure with one component as compound symmetry matrix and we compare the quickness of proposed algorithms. We conduct simulation studies to compare statistical properties of MLEs and entropy loss estimators (ELEs), such us biasedness and variability
