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Asymptotic results on the condition number of FD matrices approximating semi-elliptic PDEs
No full textThis work studies the asymptotic behavior of the spectral condition number of the matrices arising from the discretization of semi-elliptic partial differential equations of the form \bdm -\left( a(x,y)u_{xx}+b(x,y)u_{yy}\right)=f(x,y), \edm on the square with Dirichlet boundary conditions, where the smooth enough variable coefficients are nonnegative functions on with zeros. In the case of coefficient functions with a single and common zero, it is discovered that apart from the minimum order of the zero also the direction that it occurs is of great importance for the characterization of the growth of the condition number of . On the contrary, when the coefficient functions have non intersecting zeros, it is proved that independently of the order their zeros, and their positions, the condition number of behaves asymptotically exactly as in the case of strictly elliptic differential equations, i.e., it grows asymptotically as . Finally, the more complicated case of coefficient functions having curves of roots is considered, and conjectures for future work are given. In conclusion, several experiments are presented that numerically confirm the developed theoretical analysis
Gershgorin type sets for eigenvalues of matrix polynomials
No full textNew localization results for polynomial eigenvalue problems are obtained, by extending the notions of the Gershgorin set, the generalized Gershgorin set, the Brauer set and the Dashnic-Zusmanovich set to the case of matrix polynomials
On Projection of a Positive Definite Matrix on a Cone of Nonnegative Definite Toeplitz Matrices
No full textWe consider approximation of a given positive definite matrix by nonnegative definite banded Toeplitz matrices. We show that the projection on linear space of Toeplitz matrices does not always preserve nonnegative definiteness. Therefore we characterize a convex cone of nonnegative definite banded Toeplitz matrices which depends on the matrix dimensions, and we show that the condition of positive definiteness given by Parter [{\em Numer. Math. 4}, 293--295, 1962] characterizes the asymptotic cone. In this paper we give methodology and numerical algorithm of the projection basing on the properties of a cone of nonnegative definite Toeplitz matrices. This problem can be applied in statistics, for example in the estimation of unknown covariance structures under the multi-level multivariate models, where positive definiteness is required. We conduct simulation studies to compare statistical properties of the estimators obtained by projection on the cone with a given matrix dimension and on the asymptotic cone
Determinantal representations of elliptic curves via Weierstrass elliptic functions
No full textHelton and Vinnikov proved that every hyperbolic ternary form admits a symmetric derminantal representation via Riemann theta functions. In the case the algebraic curve of the hyperbolic ternary form is elliptic, the determinantal representation of the ternary form is formulated by using Weierstrass -functions in place of Riemann theta functions. An example of this approach is given
On the maximal numerical range of some matrices
No full textThe maximal numerical range of a matrix is the (regular) numerical range of its compression onto the eigenspace of corresponding to its maximal eigenvalue. So, always . Conditions under which has a non-empty intersection with the boundary of are established, in particular, when . The set is also described explicitly for matrices unitarily similar to direct sums of -by- blocks, and some insight into the behavior of is provided when has codimension one
Inertia sets allowed by matrix patterns
No full textMotivated by the possible onset of instability in dynamical systems associated with a zero eigenvalue, sets of inertias \sn_n and \SN{n} for sign and zero-nonzero patterns, respectively, are introduced. For an sign pattern \mc{A} that allows inertia , a sufficient condition is given for \mc{A} and every superpattern of \mc{A} to allow \sn_n, and a family of such irreducible sign patterns for all is specified. All zero-nonzero patterns (up to equivalence) that allow \SN{3} and \SN{4} are determined, and are described by their associated digraphs
Vector Cross Product Differential and Difference Equations in R^3 and in R^7
No full textThrough a matrix approach of the -fold vector cross product in and in , some vector cross product differential and difference equations are studied. Either the classical theory or convenient Drazin inverses, of elements belonging to the class of index matrices, are applied
On the Interval Generalized Coupled Matrix Equations
No full textIn this work, the interval generalized coupled matrix equations \begin{equation*} \sum_{j=1}^{p}{{\bf{A}}_{ij}X_{j}}+\sum_{k=1}^{q}{Y_{k}{\bf{B}}_{ik}}={\bf{C}}_{i}, \qquad i=1,\ldots,p+q, \end{equation*} are studied in which , and are known real interval matrices, while and are the unknown matrices for , and . This paper discusses the so-called AE-solution sets for this system. In these types of solution sets, the elements of the involved interval matrices are quantified and all occurrences of the universal quantifier (if any) precede the occurrences of the existential quantifier . The AE-solution sets are characterized and some sufficient conditions under which these types of solution sets are bounded are given. Also some approaches are proposed which include a numerical technique and an algebraic approach for enclosing some types of the AE-solution sets
Falling Through Class
No full textThis flash essay responds lyrically to the contradictions of class identification for a citizen of the Osage Nation in northeastern Oklahoma associated with social changes resulting from the discovery and exploitation of oil and gas resources
