arXiv subjects

Aaron Welters

Publications and source records attributed to Aaron Welters.

At least 19 recordsLinked to original sources

arXiv2026

Symmetric Bessmertny\u{i} Realizations and Field Extension Problems in Characteristic 2 - A Differential Algebra Approach

We present a short, purely algebraic proof of the Symmetric Bessmertny\u{i} Realization Theorem in the characteristic $2$ case recently proved in [EOW26]. Symmetric Bessmertny\u{i} realizations are Schur complements of affine linear symmetric matrix pencils, and they arise naturally as state-space representations in linear systems theory. In contrast with the algorithmic approach in [EOW26], we use differential algebra: by defining formal partial derivatives on multivariate rational functions over fields of positive characteristic and considering their corresponding field of constants, we obtain scalar criteria for symmetric and homogeneous symmetric realizability in characteristic $2$, effectively reducing the matrix-valued problem to its diagonal entries. As a consequence, we prove a new theorem on the field extension problem for symmetric and homogeneous symmetric Bessmertny\u{i} realizations. Finally, in the scalar case, we identify realizable rational functions with vector spaces over appropriate fields of constants and quantify the abundance of counterexamples in characteristic $2$.

math.RA↗

arXiv2026

Broadband quasistatic passive cloaking: bounds and limitations in the near-field regime

We consider here several aspects of the following challenging question: is it possible to use a passive cloak to make invisible a dielectric inclusion on a finite frequency interval in the quasistatic regime of Maxwell's equations for an observer close to the object? In this work, by considering the Dirichlet-to-Neumann (DtN) map, we not only answer negatively this question, but we go further and provide some quantitative bounds on this map that provide fundamental limits to both cloaking as well as approximate cloaking. These bounds involve the following physical parameters: the length and center of the frequency interval, the volume of the cloaking device, the volume of the obstacle, and the relative permittivity of the object. Our approach is based on two key tools: i) variational principles from the abstract theory of composites and ii) the analytic approach to deriving bounds from sum rules for passive systems. To use i), we prove a new representation theorem for the DtN map which allows us to interpret this map as an effective operator in the abstract theory of composites. One important consequence of this representation is that it allows one to incorporate the broad and deep results from the theory of composites, such as variational principles, and to apply the bounds derived from them to the DtN map. These results could be useful in other contexts other than cloaking. Next, to use ii), we show that the passivity assumption allows us to connect the DtN map (as function of the frequency) with two important classes of analytic functions, namely, Herglotz and Stieltjes functions. The sum rules for these functions, combined with the variational approach, allows us to derive new inequalities on the DtN map which impose fundamental limitations on passive cloaking, both exact and approximate, over a frequency interval. We consider both cases of lossy and lossless cloaks.

math-ph↗

arXiv2025

Effective Operators in the Theory of Composites: Hilbert Space Framework

In this chapter, the Hilbert space framework in the mathematical theory of composite materials is introduced for studying the properties of effective operators. The goal is to introduce some of the key concepts and fundamental theorems in this area while showing that they follow naturally from using only basic results in operator theory on Hilbert spaces. These concepts include the $Z$-problem as an abstraction of a constitutive equation defined in terms of a bounded linear operator on a Hilbert space with a Hodge decomposition, direct and dual $Z$-problems with the duality interpretation of the inverse of an effective operator, and the notion of an $n$-phase composite with orthogonal $Z(n)$-subspace collection. These theorems include sufficient conditions for the existence and uniqueness of both the solution of a $Z$-problem and the effective operator of a $Z$-problem, a representation formula for the effective operator as an operator Schur complement, the Dirichlet and Thomson minimization principles for the effective operator, the result on monotonicity and concavity of the effective operator map, and the Keller-Dykhne-Mendelson duality relations. Moreover, another important theorem given here (which may also be of independent interest to systems theorists) says that an effective operator of an $n$-phase composite with orthogonal $Z(n)$-subspace collection is the Schur complement of a normalized homogeneous semidefinite operator pencil (in particular, has a Bessmertny\u{\i} realization) and, up to a unitary equivalence, the converse is also true. Finally, the general theory presented here is shown to recover classical results dealing with effective conductivity but can also be applied to many other important problems involving composites in physics and engineering, e.g., in elasticity and electromagnetism.

math-ph↗

arXiv2025

Bessmertny\u{i} realizations of symmetric multivariate rational matrix functions over any field

In this paper, we prove the following. First, every square matrix whose entries are multivariable rational functions over a field $\mathbb{F}$ has a Bessmertny\u{i} realization, i.e., is the Schur complement of an affine linear square matrix pencil with coefficients in $\mathbb{F}$. Second, if the matrix is also symmetric and the characteristic of the field $\mathbb{F}$ is not two then it has a symmetric Bessmertny\u{i} realization (i.e., the pencil can be chosen to consist of symmetric matrices) and counterexamples are given to prove this statement is false in general for fields of characteristic two. Third, for fields of characteristic two (e.g., binary or Boolean field), we completely characterize those functions that have a symmetric Bessmertny\u{i} realization. Finally, analogous results hold when restricted to the class of homogeneous degree-one rational functions. To solve these realization problems, i.e., finding such structured Bessmertny\u{i} realizations for a given multivariate rational function, we use state-space methods from systems theory to produce realizations for algebraic operations on Schur complements such as sums, products, inverses, and symmetrization, which become the elementary building blocks of our constructions. Further complications arise over fields of characteristic two, so a large part of the paper is devoted to developing additional methods to decide if the symmetric realization problem can be solved and, if so, to construction the symmetric realization for a given symmetric rational matrix function. Our motivations are discussed in the context of multidimensional linear systems theory on generalizing state-space representations for rational functions including the Givone-Roesser and Fornasini-Marchesini realizations.

math.RA↗

arXiv2025

Kharitonov's Theorem with Degree Drop: a Wronskian Approach

In this paper, we present a simplified proof of Kharitonov's Theorem, an important result on determining the Hurwitz stability of interval polynomials. Our new approach to the proof, which is based on the Wronskian of a pair of polynomials, is not only more elementary in comparison to known methods, but is able to handle the degree drop case with ease.

math.OC↗

arXiv2024

Complete characterization of symmetric Kubo-Ando operator means satisfying Moln\'ar's weak associativity

We provide a complete characterization of a subclass of weakly associative means of positive operators in the class of symmetric Kubo-Ando means. This class, which includes the geometric mean, was first introduced and studied in L. Moln\'ar, ``Characterizations of certain means of positive operators," Linear Algebra Appl. 567 (2019) 143-166, where he gives a characterization of this subclass (which we call the Moln\'ar class of means) in terms of the properties of their representing operator monotone functions. Moln\'ar's paper leaves open the problem of determining if the geometric mean is the only such mean in that subclass. Here we give a negative answer to this question by constructing an order-preserving bijection between this class and a class of real measurable odd periodic functions bounded in absolute value by $1/2$. Each member of the latter class defines a Molnar mean by an explicit exponential-integral representation. From this we are able to understand the order structure of the Moln\'ar class and construct several infinite families of explicit examples of Moln\'ar means that are not the geometric mean. Our analysis also shows how to modify Moln\'ar's original characterization so that the geometric mean is the only one satisfying the requisite set of properties.

math.FA↗

arXiv2023

Matrix monotonicity and concavity of the principal pivot transform

We prove the (generalized) principal pivot transform is matrix monotone, in the sense of the Löwner ordering, under minimal hypotheses. This improves on the recent results of J. E. Pascoe and R. Tully-Doyle, Monotonicity of the principal pivot transform, Linear Algebra Appl. 662 (2022) in two ways. First, we use the ``generalized" principal pivot transform, where matrix inverses in the classical definition of the principal pivot transform are replaced with Moore-Penrose pseudoinverses. Second, the hypotheses on matrices for which monotonicity holds is relaxed and, in particular, we find the weakest hypotheses possible for which it can be true. We also prove the principal pivot transform is a matrix convex function on positive semi-definite matrices that have the same kernel (and, in particular, on positive definite matrices). Our proof is a corollary of a minimization variational principle for the principal pivot transform.

math.FA↗

arXiv2022

On the Spectral Theory of Linear Differential-Algebraic Equations with Periodic Coefficients

In this paper, we consider the spectral theory of linear differential-algebraic equations (DAEs) for periodic DAEs in canonical form, i.e., \begin{equation*} J \frac{df}{dt}+Hf=λWf, \end{equation*} where $J$ is a constant skew-Hermitian $n\times n$ matrix that is not invertible, both $H=H(t)$ and $W=W(t)$ are $d$-periodic Hermitian $n\times n$-matrices with Lebesgue measurable functions as entries, and $W(t)$ is positive semidefinite and invertible for a.e. $t\in \mathbb{R}$ (i.e., Lebesgue almost everywhere). Under some additional hypotheses on $H$ and $W$, called the local index-1 hypotheses, we study the maximal and the minimal operators $L$ and $L_0'$, respectively, associated with the differential-algebraic operator $\mathcal{L}=W^{-1}(J\frac{d}{dt}+H)$, both treated as an unbounded operators in a Hilbert space $L^2(\mathbb{R};W)$ of weighted square-integrable vector-valued functions. We prove the following: (i) the minimal operator $L_0'$ is a densely defined and closable operator; (ii) the maximal operator $L$ is the closure of $L_0'$; (iii) $L$ is a self-adjoint operator on $L^2(\mathbb{R};W)$ with no eigenvalues of finite multiplicity, but may have eigenvalues of infinite multiplicity. As an important application, we show that for 1D photonic crystals with passive lossless media, Maxwell's equations for the electromagnetic fields become, under separation of variables, periodic DAEs in canonical form satisfying our hypotheses so that our spectral theory applies to them (a primary motivation for this paper).

math.SP↗

arXiv2022

Effective operators and their variational principles for discrete electrical network problems

Using a Hilbert space framework inspired by the methods of orthogonal projections and Hodge decompositions, we study a general class of problems (called Z-problems) that arise in effective media theory, especially within the theory of composites, for defining the effective operator. A new and unified approach is developed, based on block operator methods, for obtaining solutions of the Z-problem, formulas for the effective operator in terms of the Schur complement, and associated variational principles (e.g., the Dirichlet and Thomson minimization principles) that lead to upper and lower bounds on the effective operator. In the case of finite-dimensional Hilbert spaces, this allows for a relaxation of the standard hypotheses on positivity and invertibility for the classes of operators usually considered in such problems, by replacing inverses with the Moore-Penrose pseudoinverse. As we develop the theory, we show how it applies to the classical example from the theory of composites on the effective conductivity in the periodic conductivity problem in the continuum (2d and 3d) under the standard hypotheses. After that, we consider the following three important and diverse examples of discrete electrical network problems in which our theory applies under the relaxed hypotheses. First, an operator-theoretic reformulation of the discrete Dirichlet-to-Neumann (DtN) map for an electrical network on a finite linear graph is given and used to relate the DtN map to the effective operator of an associated Z-problem.\ Second, we show how the classical effective conductivity of an electrical network on a finite linear graph is essentially the effective operator of an associated Z-problem. Finally, we consider electrical networks on periodic linear graphs and develop a discrete analog to classical example of the periodic conductivity equation and effective conductivity in the continuum.

math-ph↗

arXiv2021

Continuity of the roots of a nonmonic polynomial and applications in multivariate stability theory

We study continuity of the roots of nonmonic polynomials as a function of their coefficients using only the most elementary results from an introductory course in real analysis and the theory of single variable polynomials. Our approach gives both qualitative and quantitative results in the case that the degree of the unperturbed polynomial can change under a perturbation of its coefficients, a case that naturally occurs, for instance, in stability theory of polynomials, singular perturbation theory, or in the perturbation theory for generalized eigenvalue problems. An application of our results in multivariate stability theory is provided which is important in, for example, the study of hyperbolic polynomials or realizability and synthesis problems in passive electrical network theory, and will be of general interest to mathematicians as well as physicists and engineers.

math.CA↗

arXiv2021

Extension of the Bessmertnyi Realization Theorem for Rational Functions of Several Complex Variables

We prove a realization theorem for rational functions of several complex variables which extends the main theorem of M. Bessmertnyi, "On realizations of rational matrix functions of several complex variables," in Vol. 134 of Oper. Theory Adv. Appl., pp. 157-185, Birkhäuser Verlag, Basel, 2002. In contrast to Bessmertnyi's approach of solving large systems of linear equations, we use an operator theoretical approach based on the theory of Schur complements. This leads to a simpler and more "natural" construction to solving the realization problem as we need only apply elementary algebraic operations to Schur complements such as sums, products, inverses, and compositions. A novelty of our approach is the use of Kronecker product as opposed to the matrix product in the realization problem. As such our synthetic approach leads to a solution of the realization problem that has potential for further extensions and applications within multidimensional systems theory especially for those linear models associated with electric circuits, networks, and composites.

math.CV↗

arXiv2021

A Short Proof of the Symmetric Determinantal Representation of Polynomials

We provide a short proof of the theorem that every real multivariate polynomial has a symmetric determinantal representation, which was first proved in J. W. Helton, S. A. McCullough, and V. Vinnikov, Noncommutative convexity arises from linear matrix inequalities, J. Funct. Anal. 240 (2006), 105-191. We then provide an example using our approach and extend our results from the real field $\mathbb{R}$ to an arbitrary field $\mathbb{F}$ different from characteristic $2$. The new approach we take is only based on elementary results from the theory of determinants, the theory of Schur complements, and basic properties of polynomials.

math.CV↗

arXiv2016

Analyticity of the Dirichlet-to-Neumann map for the time-harmonic Maxwell's equations

In this chapter of the book entitled, "Extending the Theory of Composites to Other Areas of Science" [edited by Graeme W. Milton, 2016] we derive the analyticity properties of the electromagnetic Dirichlet-to-Neumann map for the time-harmonic Maxwell's equations for passive linear multicomponent media. Moreover, we discuss the connection of this map to Herglotz functions for isotropic and anisotropic multicomponent composites.

math.AP↗

arXiv2016

A rigorous approach to the field recursion method for two-component composites with isotropic phases

In this chapter of the book entitled, "Extending the Theory of Composites to Other Areas of Science" [edited by Graeme W. Milton, 2016] we give a rigorous derivation of the field equation recursion method in the abstract theory of composites to two-component composites with isotropic phases. This method is of great interest since it has proven to be a powerful tool in developing sharp bounds for the effective tensor of a composite material. The reason is that the effective tensor $\bf L_*$ can be interpreted in the general framework of the abstract theory of composites as the $Z$-operator on a certain orthogonal $Z(2)$ subspace collection. The base case of the recursion starts with an orthogonal $Z(2)$ subspace collection on a Hilbert space $\cal H$, the $Z$-problem, and the associated $Y$-problem. We provide some new conditions for the solvability of both the $Z$-problem and the associated $Y$-problem. We also give explicit representations of the associated $Z$-operator and $Y$-operator and study their analytical properties. An iteration method is then developed from a hierarchy of subspace collections and their associated operators which leads to a continued fraction representation of the initial effective tensor $\bf L_*$.

math-ph↗

arXiv2015

On overdamping phenomena in gyroscopic systems composed of high-loss and lossless components

Using a Lagrangian framework, we study overdamping phenomena in gyroscopic systems composed of two components, one of which is highly lossy and the other is lossless. The losses are accounted by a Rayleigh dissipative function. As we have shown previously, for such a composite system the modes split into two distinct classes, high-loss and low-loss, according to their dissipative behavior. A principal result of this paper is that for any such system a rather universal phenomenon of selective overdamping occurs. Namely, first of all the high-loss modes are all overdamped, i.e., non-oscillatory, as are an equal number of low-loss modes. Second of all, the rest of the low-loss modes remain oscillatory (i.e., the underdamped modes) each with an extremely high quality factor (Q-factor) that actually increases as the loss of the lossy component increases. We prove that selective overdamping is a generic phenomenon in Lagrangian systems with gyroscopic forces and give an analysis of the overdamping phenomena in such systems. Moreover, using perturbation theory, we derive explicit formulas for upper bound estimates on the amount of loss required in the lossy component of the composite system for the selective overdamping to occur in the generic case, and give Q-factor estimates for the underdamped modes. Central to the analysis is the introduction of the notion of a "dual" Lagrangian system and this yields significant improvements on some results on modal dichotomy and overdamping. The effectiveness of the theory developed here is demonstrated by applying it to an electric circuit with a gyrator element and a high-loss resistor.

math-ph↗

arXiv2014

Lagrangian Framework for Systems Composed of High-Loss and Lossless Components

Using a Lagrangian mechanics approach, we construct a framework to study the dissipative properties of systems composed of two components one of which is highly lossy and the other is lossless. We have shown in our previous work that for such a composite system the modes split into two distinct classes, high-loss and low-loss, according to their dissipative behavior. A principal result of this paper is that for any such dissipative Lagrangian system, with losses accounted by a Rayleigh dissipative function, a rather universal phenomenon occurs, namely, selective overdamping: The high-loss modes are all overdamped, i.e., non-oscillatory, as are an equal number of low-loss modes, but the rest of the low-loss modes remain oscillatory each with an extremely high quality factor that actually increases as the loss of the lossy component increases. We prove this result using a new time dynamical characterization of overdamping in terms of a virial theorem for dissipative systems and the breaking of an equipartition of energy.

math-ph↗

arXiv2014

Speed-of-light limitations in passive linear media

We prove that well-known speed of light restrictions on electromagnetic energy velocity can be extended to a new level of generality, encompassing even nonlocal chiral media in periodic geometries, while at the same time weakening the underlying assumptions to only passivity and linearity of the medium (either with a transparency window or with dissipation). As was also shown by other authors under more limiting assumptions, passivity alone is sufficient to guarantee causality and positivity of the energy density (with no thermodynamic assumptions). Our proof is general enough to include a very broad range of material properties, including anisotropy, bianisotropy (chirality), nonlocality, dispersion, periodicity, and even delta functions or similar generalized functions. We also show that the "dynamical energy density" used by some previous authors in dissipative media reduces to the standard Brillouin formula for dispersive energy density in a transparency window. The results in this paper are proved by exploiting deep results from linear-response theory, harmonic analysis, and functional analysis that had previously not been brought together in the context of electrodynamics.

physics.optics↗

arXiv2012

Dissipative Properties of Systems Composed of High-Loss and Lossless Components

We study here dissipative properties of systems composed of two components one of which is highly lossy and the other is lossless. A principal result of our studies is that all the eigenmodes of such a system split into two distinct classes characterized as high-loss and low-loss. Interestingly, this splitting is more pronounced the higher the loss of the lossy component. In addition, the real frequencies of the high-loss eigenmodes can become very small and even can vanish entirely, which is the case of overdamping.

math-ph↗