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Perturbation of multiple eigenvalues of Hermitian matrices
Get PDFThis paper is concerned with the perturbation of a multiple eigenvalue of the Hermitian matrix A=\mbox{diag}(\mu I,A_{22}) when it undergoes an off-diagonal perturbation whose columns have widely varying magnitudes. When some of 's columns are much smaller than the others, some copies of are much less sensitive than any existing bound suggests. We explain this phenomenon by establishing individual perturbation bounds for different copies of . They show that when is definite the th bound scales quadratically with the norm of the th column, and in the indefinite case the bound is necessarily proportional to the product of 's th column norm and 's norm. An extension to the generalized Hermitian eigenvalue problem is also presented
K-theory and the connection index
Get PDFLet G denote a split simply connected almost simple p-adic group. The classical example is the special linear group SL(n). We study the K-theory of the unramified unitary principal series of G and prove that the rank of K_0 is the connection index f(G). We relate this result to a recent refinement of the Baum-Connes conjecture, and show explicitly how generators of K_0 contribute to the K-theory of the Iwahori C*-algebra
Nonlinear pre-stress for cloaking from antiplane elastic waves
Get PDFA theory is presented showing that cloaking of objects from antiplane elastic waves can be achieved by elastic pre-stress of a neo-Hookean nonlinear elastic material. This approach would appear to eliminate the requirement of metamaterials with inhomogeneous anisotropic shear moduli and density. Waves in the pre-stressed medium are bent around the cloaked region by inducing inhomogeneous stress fields via pre-stress. The equation governing antiplane waves in the pre-stressed medium is equivalent to the antiplane equation in an unstressed medium with inhomogeneous and anisotropic shear modulus and isotropic scalar mass density. Note however that these properties are induced naturally by the pre-stress. Since the magnitude of pre-stress can be altered at will, this enables objects of varying size and shape to be cloaked by placing them inside the fluid-filled deformed cavity region
SELF-INTERSECTIONS OF IMMERSIONS AND STEENROD OPERATIONS
Get PDFWe present a formula describing the action of a gener-
alised Steenrod operation of Z2-type [14] on the cohomology class
represented by a proper self-transverse immersion f : M # X.
Our formula depends only on the Umkehr map, the characteris-
tic classes of the normal bundle, and the class represented by the
double point immersion of f. This generalises a classical result of
R. Thom [13]: If 2 Hk(X;Z2) is the ordinary cohomology class
represented by f : M # X, then Sqi() = fwi(f )
A review of some recent work on hypercyclicity
Get PDFEven linear operators on infinite-dimensional spaces can display interesting dynamical properties and yield important links among functional analysis, differential and global geometry and dynamical systems, with a wide range of applications. In particular, hypercyclicity is an essentially infinite-dimensional property, when iterations of the operator generate a dense subspace. A Frechet space admits a hypercyclic operator if and only if it
is separable and infinite-dimensional. However, by considering the semigroups generated by multiples of operators, it is possible to obtain hypercyclic behaviour on finite dimensional spaces. This article gives a brief review of some recent work on hypercyclicity of operators on Banach, Hilbert and Frechet spaces
Covariance Structure Regularization via Entropy Loss Function
Get PDFThe need to estimate structured covariance matrices arises in a variety of applications and the problem is widely studied in statistics. We propose a new method for regularizing the covariance structure of a given covariance
matrix, in which the underlying structure is usually blurred due to random noises particularly when the dimension of the covariance matrix is high. The regularization is made
by choosing an optimal structure from an available class of covariance structures in terms of minimizing the discrepancy, defined via the entropy loss function, between the given matrix and the class. A range of potential candidate structures such as tridiagonal, compound symmetry,
AR(1), and Toeplitz are considered. Simulation studies are conducted, showing that the proposed new approach is reliable in regularization of covariance structures. The approach is also applied to real data analysis, demonstrating the usefulness of the proposed approach in practice
State transitions in a model of intermittent seizure dynamics
Get PDFWe present a neural mass model of intermittent transitions into electroencephalographic (EEG) seizure rhythms in epilepsy. The route to intermittent dynamics is identied as Type 1 and state transition statistics are explored. It is demonstrated that a single framework can give rise to different distributions for seizure and non-seizure lengths in line with variability observed clinically. Further investigation of this model can give insight into the possible mechanisms underlying spontaneous seizure transitions in the epileptic brain
Triangularization of matrix polynomials
No full textFor an algebraically closed field \F,
we show that any matrix polynomial P(\l)=\sum_{j=0}^\ell \l^jA_j with A_j\in\F^{\nbym}, , can be reduced over \F[\l] to an \nbym upper triangular matrix polynomial of grade preserving the finite and infinite elementary divisors.
We also characterize the real matrix polynomials that are
triangularizable over the real numbers and show that those that are not triangularizable are quasi-triangularizable
with diagonal blocks of sizes and
A spatially adaptive iterative method for a class of nonlinear operator eigenproblems
Get PDFWe present a new algorithm for the iterative solution of nonlinear operator eigenvalue problems arising from partial differential equations. This algorithm combines automatic spatial resolution of linear operators with the infinite Arnoldi method for nonlinear matrix eigenproblems proposed in [E. JARLEBRING, W. MICHIELS, AND K. MEERBERGEN, A linear eigenvalue algorithm for the nonlinear eigenvalue problem, Numer. Math., 122 (2012)]. The iterates in this infinite Arnoldi method are functions, and each iteration requires the solution of an inhomogeneous differential equation. This formulation is independent of the spatial representation of the functions, which allows us to employ a dynamic representation with an accuracy of about the level of machine precision at each iteration, similar to what is done in the Chebfun system [Z. BATTLES AND L. N. TREFETHEN, An extension of MATLAB to continuous functions and operators, SIAM J. Sci. Comput., 25 (2004)] with its chebop functionality [T. A. DRISCOLL, F. BORNEMANN, AND L. N. TREFETHEN, The chebop system for automatic solution of differential equations, BIT, 48 (2008)], although our function representation is entirely based on coefficients instead of function values. Our approach also allows for nonlinearities in the boundary conditions of the PDE. The algorithm is illustrated with several examples, e.g., the study of eigenvalues of a vibrating string with delayed boundary feedback control