Isomonodromic deformation of resonant rational connections

We analyze isomonodromic deformations of rational connections on the Riemann sphere with Fuchsian and irregular singularities. The Fuchsian singularities are allowed to be of arbitrary resonant index; the irregular singularities are also allowed to be resonant in the sense that the leading coefficient matrix at each singularity may have arbitrary Jordan canonical form, with a genericity condition on the Lidskii submatrix of the subleading term. We also give the relevant notion of isomonodromic tau function extending the one of non-resonant deformations introduced by Miwa-Jimbo-Ueno. The tau function is expressed purely in terms of spectral invariants of the matrix of the connection.Comment: 48 page

Exceptional Points of Degeneracy Induced by Linear Time-Periodic Variation

We present a general theory of exceptional points of degeneracy (EPD) in periodically time-variant systems. We show that even a single resonator with a time-periodic component is able to develop EPDs, contrary to parity-time- (PT) symmetric systems that require two coupled resonators. An EPD is a special point in a system parameter space at which two or more eigenmodes coalesce in both their eigenvalues and eigenvectors into a single degenerate eigenmode. We demonstrate the conditions for EPDs to exist when they are directly induced by time-periodic variation of a system without loss and gain elements. We also show that a single resonator system with zero time-average loss-gain exhibits EPDs with purely real resonance frequencies, yet the resonator energy grows algebraically in time since energy is injected into the system from the time-variation mechanism. Although the introduced concept and formalism are general for any time-periodic system, here, we focus on the occurrence of EPDs in a single LC resonator with time-periodic modulation. These findings have significant importance in various electromagnetic and photonic systems and pave the way for many applications, such as sensors, amplifiers, and modulators. We show a potential application of this time-varying EPD as a highly sensitive sensor

Higher order derivatives of matrix functions

We present theory for general partial derivatives of matrix functions on the form $f(A(x))$ where $A(x)$ is a matrix path of several variables ($x=(x_1,\dots,x_j)$). Building on results by Mathias [SIAM J. Matrix Anal. Appl., 17 (1996), pp. 610-620] for the first order derivative, we develop a block upper triangular form for higher order partial derivatives. This block form is used to derive conditions for existence and a generalized Dalecki\u{i}-Kre\u{i}n formula for higher order derivatives. We show that certain specializations of this formula lead to classical formulas of quantum perturbation theory. We show how our results are related to earlier results for higher order Fr\'echet derivatives. Block forms of complex step approximations are introduced and we show how those are related to evaluation of derivatives through the upper triangular form. These relations are illustrated with numerical examples.Comment: 24 pages, 2 figure

On the design of robust deadbeat regulators

This paper considers the synthesis of state feedback gains which provide robustness against perturbation in deadbeat regulation. It is formulated as an unconstrained optimization problem. Through a posteriori perturbation analysis of the closed-loop eigenvalues, the justification of the use of a new objective function to measure the robustness of deadbeat systems is established. The objective function does not require the computation of eigenvectors and has simple analytical gradient and Hessian. A numerical example is employed to illustrate the effectiveness of the proposed method.published_or_final_versio

Structured perturbation theory for eigenvalues of symplectic matrices

The problem of computing eigenvalues, eigenvectors, and invariant subspaces of symplectic matrices plays a major role in many applications, in particular in control theory when the focus is on discrete systems. If standard numerical methods for the solution of the symplectic eigenproblem are applied that do not take into account the special symmetry structure of the problem, then not only the existing symmetry in the spectrum of symplectic matrices may be lost in finite precision arithmetic, but more importantly other relevant intrinsic features or invariants may be ignored although they have a major influence in the corresponding computed eigenvalues. The importance of structure-preservation has been acknowledged in the Numerical Linear Algebra community since several decades, and consequently many algorithms have been developed for the symplectic eigenvalue problem that preserve the given structure at each iteration step. The error analysis for such algorithms requires a corresponding stucture-preserving perturbation theory. This is the general framework in which this dissertation can be placed. In this work, a first order perturbation theory for eigenvalues of real or complex Jsymplectic matrices under structure-preserving perturbations is developed. Since the class of symplectic matrices has an underlying multiplicative structure, Lidskii’s classical formulas for small additive perturbations of the form b A = A+εB cannot be applied directly, so a new multiplicative perturbation theory is first developed: given an arbitrary square matrix A, we obtain the leading terms of the asymptotic expansions in the small, real parameter ε of multiplicative perturbations b A(ε) = (I +εB +· · · )A(I +εC +· · · ) of A for arbitrary matrices B and C. The analysis is separated in two complementary cases, depending on whether the unperturbed eigenvalue is zero or not. It is shown that in either case the leading exponents are obtained from the partial multiplicities of the eigenvalue of interest, and the leading coefficients generically involve only appropriately normalized left and right eigenvectors of A associated with that eigenvalue, with no need of generalized eigenvectors. It should be noted that, although initially motivated by the needs for the symplectic case, this multiplicative (unstructured) perturbation theory is of independent interest and stands on its own. After showing that any small structured perturbation bS of a symplectic matrix S can be written as bS = bS(ε) = (I + εB + · · · ) S with Hamiltonian first-order coefficient B, we apply the previously obtained Lidskii-like formulas for multiplicative perturbations to the symplectic case by exploiting the particular connections that symplectic structure induces in the Jordan form between normalized left and right eigenvectors. Special attention is given to eigenvalues on the unit circle, particularly to the exceptional eigenvalues ±1, whose behavior under structure-preserving perturbations is known to differ significantly from the behavior under arbitrary ones. Also, several numerical examples are generated Although the approach described above via multiplicative expansions works in most situations, there is a very specific one, the one we call the nongeneric case, which requires a separate, completely different analysis. It corresponds to the case in which, in the absence of structure, the rank of the perturbation would break an odd number of oddsized Jordan blocks corresponding to the eigenvalue either 1 or −1. Since this is not allowed by symplecticity, one among that odd number of Jordan blocks does not break, but increases its size by one becoming an even-sized block. This very special behavior lies outside of the theory developed for what we might call the generic cases, and requires a completely different perturbation analysis, based on Newton diagram techniques like the one performed to obtain the multiplicative expansions. The main difference with the previous expansions is that in this nongeneric case the leading coefficients depend not only on eigenvectors, but also on first generalized Jordan vectors.El problema de calcular autovalores, autovectores y subespacios invariantes de matrices simplécticas juega un papel crucial en muchas aplicaciones, en particular en la Teoría de Control cuando ésta se centra en sistemas discretos. Si para resolver el problema simpléctico de autovalores se emplean métodos numéricos estándar que no tienen en cuenta la simetría especial del problema, entonces no solo se perderá en aritmética finita la simetría natural del espectro de las matrices simplécticas, sino que, aún más importante, podemos estar ignorando otras características o invariantes intrínsecas que tienen una influencia crucial en los correspondientes autovalores calulados. La importancia de preservar la estructura ha sido reconocida por la comunidad del Álgebra Lineal Numérica desde hace varias décadas y, en consecuencia, se han desarrollado diversos algoritmos para el problema simpléctico de autovalores que mantienen la estructura simpléctica en cada paso del proceso iterativo. El análisis de errores de tales algoritmos demanda una teoría de perturbación asociada que también preserve la estructura. Este es el marco general en el que se puede inscribir esta tesis doctoral. En este trabajo se desarrolla una teoría de perturbación de autovalores de matrices J-simplécticas frente a perturbaciones que preservan la simplecticidad de la matriz. Dado que la clase de matrices simplécticas tiene una estructura multiplicativa subyacente, las fórmulas clásicas de Lidskii para perturbaciones aditivas pequeñas de la forma b A = A + εB no se pueden aplicar de manera directa, de modo que desarrollamos una nueva teoría de perturbación multiplicativa: dada cualquier matriz cuadrada A, obtenemos el término director del desarrollo asintótico en el parámetro real (y pequeño) ε de autovalores de perturbaciones multiplicativas b A(ε) = (I + εB + · · · )A(I + εC + · · · ) de A para matrices arbitrarias B y C. El análisis se separa en dos casos complementarios, dependiendo de que el autovalor a perturbar sea nulo o no. Se demuestra que en ambos casos los exponentes directores se obtienen a partir de las multiplicidades parciales del autvalor bajo estudio, y que los coeficientes directores solo involucran genéricamente autovectores derechos e izquierdos adecuadamente normalizados, sin necesidad de autovalor generalizado alguno. Debe señalarse que, aunque inicialmente motivados por la necesidad para el caso simpléctico, esta teoría (no estructurada) de perturbación multiplicativa reviste interés per se independientemente de su aplicación al caso simpléctico. Tras mostrar que cualquier perturbación estructurada peque na bS de una matriz simpléctica S puede escribirse como bS = bS(ε) = (I + εB + · · · ) S con coeficiente de primer orden B Hamiltoniano, aplicamos las fórmulas tipo Lidskii obtenidas para perturbaciones multiplicativas al caso simpléctico, explotando la particular conexión que la estructura simpléctica induce entre los autovectores derechos e izquierdos normalizados por la forma de Jordan. Especial atención se le dedica a los autovalores sobre el círculo unidad, particularmente a los autovalores excepcionales ±1, cuyo comportamiento frente a perturbaciones estructuradas es sabido que difiere muy significativemente del comportamiento frente a perturbaciones arbitrarias. Además, presentamos varios ejemplos numéricos que ilustran (y confirman) los desarrollos asintóticos obtenidos. Aunque el enfoque que acabamos de describir via desarrollos multiplicativos funciona en la mayor parte de las situaciones, hay una muy específica, la que llamamos el caso no-genérico, que requiere de un análisis por separado, completamete distinto del anterior. Corresponde al caso en que, en ausencia de estructura, el rango de la perturbación rompería un número impar de bloques de Jordan de tamaño impar asociados a uno de los autovalores 1 ó −1. Como esto es incompatible con la simplecticidad, uno de entre los bloques de tamaño impar no se rompe, sino que incrementa en uno su dimensión, conviertiéndose en un bloque de Jordn de tamaño par. Este comportamiento tan especial no está explicado por la teoría de lo que podríamos llamar los casos ‘genéricos’ , y requiere de un análisis de perturbación completamente distinto, basado en técnicas del Diagrama de Newton, como el llevado a cabo para obtener los desarrollos multiplicativos. La diferencia principal con los desarrollos anteriores es que en el caso no genérico los coeficientes directores dependen no solo de autovectores, sino también de vectores primeros generalizados de Jordan.Este trabajo ha sido desarrollado en el Departamento de Matemáticas de la Universidad Carlos III de Madrid (UC3M) bajo la dirección del profesor Julio Moro Carreño. Se contó con una beca de la UC3M de ayuda al estudio de máster y posteriormente con un contrato predoctoral de la UC3M. Adicionalmente se recibió ayuda parcial del proyecto de investigación: “Matemáticas e Información Cuántica: de las Álgebras de Operadores al Muestreo Cuántico” (Ministerio de Economía y Competitividad de España, Número de proyecto: MTM2014-54692-P).Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de MadridPresidente: Luis Alberto Ibort Latre.- Secretario: Rafael Cantó Colomina.- Vocal: Francisco Enrique Velasco Angul

Exceptional points of degeneracy and branch points for coupled transmission lines - Linear-algebra and bifurcation theory perspectives

We provide a new angle to investigate exceptional points of degeneracy (EPD) relating the current linear-algebra point of view to bifurcation theory. We apply these concepts to EPDs related to propagation in waveguides supporting two modes (in each direction), described as a coupled transmission line. We show that EPDs are singular points of the dispersion function associated with the fold bifurcation connecting multiple branches of dispersion spectra. This provides an important connection between various modal interaction phenomena known in guided-wave structures with recent interesting effects observed in quantum mechanics, photonics, and metamaterials systems described in terms of the algebraic EPD formalism. Since bifurcation theory involves only eigenvalues, we also establish the connection to the linear-algebra point of view by casting the system eigenvectors in terms of eigenvalues, analytically showing that the coalescence of two eigenvalues results automatically in the coalescence of the two respective eigenvectors. Therefore, for the studied two-coupled transmission-line problem, the eigenvalue degeneracy explicitly implies an EPD. Furthermore, we discuss in some detail the fact that EPDs define branch points in the complex frequency plane, we provide simple formulas for these points, and we show that parity-time (PT) symmetry leads to real-valued EPDs occurring on the real-frequency axis

First order structure-preserving perturbation theory for eigenvalues of symplectic matrices

A first order perturbation theory for eigenvalues of real or complex J-symplectic matrices under structure- preserving perturbations is developed. As main tools structured canonical forms and Lidskii-like formulas for eigenvalues of multiplicative perturbations are used. Explicit formulas, depending only on appropriately normalized left and right eigenvectors, are obtained for the leading terms of asymptotic expansions describing the perturbed eigenvalues. Special attention is given to eigenvalues on the unit circle, especially to the exceptional eigenvalues ±1, whose behavior under structure-preserving perturbations is known to differ significantly from the behavior under general perturbations. Several numerical examples are used to illustrate the asymptotic expansions