On Functions of quasi Toeplitz matrices

Let $a(z)=\sum_{i\in\mathbb Z}a_iz^i$ be a complex valued continuous function, defined for $|z|=1$, such that $\sum_{i=-\infty}^{+\infty}|ia_i|<\infty$. Consider the semi-infinite Toeplitz matrix $T(a)=(t_{i,j})_{i,j\in\mathbb Z^+}$ associated with the symbol $a(z)$ such that $t_{i,j}=a_{j-i}$. A quasi-Toeplitz matrix associated with the continuous symbol $a(z)$ is a matrix of the form $A=T(a)+E$ where $E=(e_{i,j})$, $\sum_{i,j\in\mathbb Z^+}|e_{i,j}|<\infty$, and is called a CQT-matrix. Given a function $f(x)$ and a CQT matrix $M$, we provide conditions under which $f(M)$ is well defined and is a CQT matrix. Moreover, we introduce a parametrization of CQT matrices and algorithms for the computation of $f(M)$. We treat the case where $f(x)$ is assigned in terms of power series and the case where $f(x)$ is defined in terms of a Cauchy integral. This analysis is applied also to finite matrices which can be written as the sum of a Toeplitz matrix and of a low rank correction

Integrable hierarchies and the mirror model of local CP1

We study structural aspects of the Ablowitz-Ladik (AL) hierarchy in the light of its realization as a two-component reduction of the two-dimensional Toda hierarchy, and establish new results on its connection to the Gromov-Witten theory of local CP1. We first of all elaborate on the relation to the Toeplitz lattice and obtain a neat description of the Lax formulation of the AL system. We then study the dispersionless limit and rephrase it in terms of a conformal semisimple Frobenius manifold with non-constant unit, whose properties we thoroughly analyze. We build on this connection along two main strands. First of all, we exhibit a manifestly local bi-Hamiltonian structure of the Ablowitz-Ladik system in the zero-dispersion limit. Secondarily, we make precise the relation between this canonical Frobenius structure and the one that underlies the Gromov-Witten theory of the resolved conifold in the equivariantly Calabi-Yau case; a key role is played by Dubrovin's notion of "almost duality" of Frobenius manifolds. As a consequence, we obtain a derivation of genus zero mirror symmetry for local CP1 in terms of a dual logarithmic Landau-Ginzburg model.Comment: 27 pages, 1 figur

Quasi-block Toeplitz matrix in matlab

In this paper we try to approximate any properties of quasi-block Toeplitz matrix (QBT), by means of a finite number of parameters. A quasi-block Toeplitz (QBT) matrix is a semi-infinite block matrix of the kind F = T(F) + E where T(F) = (Fj−k)j,k∈Z, that Fk are m × m matrices such that Ʃi∈Z|Fi| has bounded entries, and E = (ei,j )i,j∈Z+ is a compact correction. Also, we should say the norms ‖ F ‖w= Ʃi∈Z ‖ Fi ‖ and ‖ E ‖2 are finite. QBT-matrices are done with any given precision. The norm ‖ F ‖QBT = α ‖ F ‖w + ‖ E ‖2, is for α = (1 + √5)/2. These matrices are a Banach algebra with the standard arithmetic operations. We try to analysis some structures and computational properties for arithmetic operations of QBT matrices with some MATLAB commands.Publisher's Versio

Efficient resonance computations for Helmholtz problems based on a Dirichlet-to-Neumann map

We present an efficient procedure for computing resonances and resonant modes of Helmholtz problems posed in exterior domains. The problem is formulated as a nonlinear eigenvalue problem (NEP), where the nonlinearity arises from the use of a Dirichlet-to-Neumann map, which accounts for modeling unbounded domains. We consider a variational formulation and show that the spectrum consists of isolated eigenvalues of finite multiplicity that only can accumulate at infinity. The proposed method is based on a high order finite element discretization combined with a specialization of the Tensor Infinite Arnoldi method. Using Toeplitz matrices, we show how to specialize this method to our specific structure. In particular we introduce a pole cancellation technique in order to increase the radius of convergence for computation of eigenvalues that lie close to the poles of the matrix-valued function. The solution scheme can be applied to multiple resonators with a varying refractive index that is not necessarily piecewise constant. We present two test cases to show stability, performance and numerical accuracy of the method. In particular the use of a high order finite element discretization together with TIAR results in an efficient and reliable method to compute resonances

The Dirichlet Markov Ensemble

We equip the polytope of $n\times n$ Markov matrices with the normalized trace of the Lebesgue measure of $\mathbb{R}^{n^2}$. This probability space provides random Markov matrices, with i.i.d. rows following the Dirichlet distribution of mean $(1/n,...,1/n)$. We show that if \bM is such a random matrix, then the empirical distribution built from the singular values of\sqrt{n} \bM tends as $n\to\infty$ to a Wigner quarter--circle distribution. Some computer simulations reveal striking asymptotic spectral properties of such random matrices, still waiting for a rigorous mathematical analysis. In particular, we believe that with probability one, the empirical distribution of the complex spectrum of \sqrt{n} \bM tends as $n\to\infty$ to the uniform distribution on the unit disc of the complex plane, and that moreover, the spectral gap of \bM is of order $1-1/\sqrt{n}$ when $n$ is large.Comment: Improved version. Accepted for publication in JMV