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George Fikioris

Publications and source records attributed to George Fikioris.

16 recordsLinked to original sources
arXiv2026

Gibbs Phenomenon and Friedel Oscilations: Similarities, Differences, and the Educational Potential of their Comparison

We explore the similarities and differences between the Gibbs phenomenon in partial Fourier representations of discontinuous signals and Friedel oscillations in the electron density of a solid near an anomaly. Inspired by the apparent similarities of the two phenomena, we perform a detailed exploration of both from the viewpoint of an engineer being introduced to a concept from solid-state physics. Focusing on the density of an one-dimensional electronic gas confined by a square potential, we show that, despite the similarities, Friedel oscillations cannot be attributed to the inability of a partial Fourier series to describe a discontinuity. Nevertheless, the two phenomena do exhibit similarities, which can be exploited to develop intuition. By adopting an educational style, we hope to establish some common language between electrical engineers and condensed-matter physicists, hoping that this can further inspire the two communities to seek intuition and further comprehension within neighbouring but different disciplines.

cond-mat.other↗
arXiv2026

A general framework for the study of electrostatic point charges in multilayer planar structures

We develop a general framework for the electrostatic analysis of point charges in multilayer planar structures with arbitrary layer thicknesses and material parameters. Starting from a Hankel-transform analysis, we derive alternative representations of the solution and establish a Stokes-like formulation based on ``generalized reflection coefficients,'' yielding a systematic and physically transparent treatment of multilayer media. This approach extends classical image theory to parameter regimes in which the conventional image-charge series (which has an infinite number of terms) diverges. The formulation applies to arbitrary permittivity values, including negative permittivities, where overscreening effects and plasmon-resonant conditions may occur. In these regimes, we show that the boundary-value problem no longer has a unique solution because homogeneous (source-free) modes appear; and we derive Cauchy-principal-value integral representations for the particular solution. We also introduce an asymptotic ``phantom-image'' method that replaces a divergent infinite image series by a finite set of effective sources, thus providing a computationally efficient approximation in large-reflection regimes. These results furnish both practical computational tools and additional mathematical insight into the structure of electrostatic image theory in layered media.

physics.app-ph↗
arXiv2024

Fundamentals of a Null Field Method-Surface Equivalence Principle Approach for Scattering by Dielectric Cylinders

The null-field method (NFM) and the method of auxiliary sources (MAS) have been both used extensively for the numerical solution of boundary-value problems arising in diverse applications involving propagation and scattering of waves. It has been shown that, under certain conditions, the applicability of MAS may be restricted by issues concerning the divergence of the auxiliary currents, manifested by the appearance of exponentially large oscillations. In this work, we combine the NFM with the surface equivalence principle (SEP) and investigate analytically the convergence properties of the combined NFM-SEP with reference to the problem of (internal or external) line-source excitation of a dielectric cylinder. Our main purpose is to prove that (contrary to the MAS) the discrete NFM-SEP currents, when properly normalized, always converge to the corresponding continuous current densities, and thus no divergence and oscillations phenomena appear. The theoretical analysis of the NFM-SEP is accompanied by detailed comparisons with the MAS as well as with representative numerical results illustrating the conclusions.

math.NA↗
arXiv2024

Convergence, divergence, and inherent oscillations in MFS solutions of two-dimensional Laplace-Neumann problems

The method of fundamental solutions (MFS), also known as the method of auxiliary sources (MAS), is a well-known computational method for the solution of boundary-value problems. The final solution ("MAS solution") is obtained once we have found the amplitudes of $N$ auxiliary "MAS sources." Past studies have demonstrated that it is possible for the MAS solution to converge to the true solution even when the $N$ auxiliary sources diverge and oscillate. The present paper extends the past studies by demonstrating this possibility within the context of Laplace's equation with Neumann boundary conditions. One can thus obtain the correct solution from sources that, when $N$ is large, must be considered unphysical. We carefully explain the underlying reasons for the unphysical results, distinguish from other difficulties that might concurrently arise, and point to significant differences with time-dependent problems that were studied in the past.

math.NA↗
arXiv2023

An extension to "A subsemigroup of the rook monoid"

A recent paper studied an inverse submonoid $M_n$ of the rook monoid, by representing the nonzero elements of $M_n$ via certain triplets belonging to $\mathbb{Z}^3$. In this short note, we allow the triplets to belong to $\mathbb{R}^3$. We thus study a new inverse monoid $\overline{M}_n$, which is a supermonoid of $M_n$. We point out similarities and find essential differences. We show that $\overline{M}_n$ is a noncommutative, periodic, combinatorial, fundamental, completely semisimple, and strongly $E^*$-unitary inverse monoid.

math.CO↗
arXiv2023

The Uniqueness Theorem for Nonlocal Hydrodynamic Media

We investigate a fundamental electromagnetic theorem, namely the uniqueness theorem, in the context of nonlocal electromagnetics, as simulated by a popular semiclassical model, the Hydrodynamic Drude Model (HDM) and extensions thereof such as the Generalized Nonlocal Optical Response (GNOR). The derivations and proofs presented here give a theoretical foundation to the use of the Additional Boundary Conditions (ABCs), whose necessity is recognized and underlined in virtually all implementations and applications of HDM. Our proofs follow a mathematically relaxed style, borrowing from the literature of established electromagnetics textbooks that study the matter from an engineering perspective. Through this simpler route we deduce clear and intuitive material-response requirements for uniqueness to hold, while using a familiar parlance in a topic that is mostly studied through a physics perspective. Two numerical examples that examine the problem from either a semianalytical or a purely numerical viewpoint support our findings.

physics.app-ph↗
arXiv2022

A subsemigroup of the rook monoid

We define a subsemigroup $S_n$ of the rook monoid $R_n$ and investigate its properties. To do this, we represent the nonzero elements of $S_n$ (which are $n\times n$ matrices) via certain triplets of integers, and develop a closed-form expression representing the product of two elements; these tools facilitate straightforward deductions of a great number of properties. For example, we show that $S_n$ consists solely of idempotents and nilpotents, find the numbers of idempotents and nilpotents, and compute nilpotency indexes. Furthermore, we give a necessary and sufficient condition for the $j$th root of a nonzero element to exist in $S_n$, show that existence implies uniqueness, and compute the said root explicitly. We also point to several combinatorial aspects; describe a number of subsemigroups of $S_n$; and, using rook $n$-diagrams, graphically interpret many of our results.

math.CO↗
arXiv2021

Eigenvalue contour lines of Kac-Murdock-Szego matrices with a complex parameter

A previous paper studied the so-called borderline curves of the Kac--Murdock--Szegő matrix $K_{n}(ρ)=\left[ρ^{|j-k|}\right]_{j,k=1}^{n}$, where $ρ\in\mathbb{C}$. These are the level curves (contour lines) in the complex-$ρ$ plane on which $K_n(ρ)$ has a type-1 or type-2 eigenvalue of magnitude $n$, where $n$ is the matrix dimension. Those curves have cusps at all critical points $ρ=ρ_c$ at which multiple (double) eigenvalues occur. The present paper determines corresponding curves pertaining to eigenvalues of magnitude $N\ne n$. We find that these curves no longer present cusps; and that, when $N<n$, the cusps have in a sense transformed into loops. We discuss the meaning of the winding numbers of our curves. Finally, we point out possible extensions to more general matrices.

math.SP↗
arXiv2020

Eigenvalue bifurcations in Kac-Murdock-Szego matrices with a complex parameter

For complex $ρ$, the spectral properties of the Toeplitz matrix $K_{n}(ρ)=\left[ρ^{|j-k|}\right]_{j,k=1}^{n}$, often called the Kac-Murdock-Szegο matrix, have been examined in detail in two recent papers. The second paper, in particular, introduced the concept of borderline curves. These are two closed curves in the complex-$ρ$ plane that consist of all the $ρ$ for which $K_n(ρ)$ possesses some eigenvalue whose magnitude equals the matrix dimension $n$. The purpose of the present paper is to examine eigenvalue bifurcations in both a qualitative and a quantitative manner, and to discuss connections between bifurcations and the borderline curves.

math.SP↗
arXiv2020

On the Application of Numerical Methods to Hallen's Equation: The Case of a Lossy Medium

A previous paper analyzed in detail the difficulties associated with the application of numerical methods to Hallen's integral equation with the approximate kernel for the case of a lossless surrounding medium. The present paper extends to the case where the medium is conducting and points out similarities and differences between the two cases. Our main device is an analytical/asymptotic study of the antenna of infinite length.

physics.comp-ph↗
arXiv2019

An Effective-Current Approach for Hallén's Equation in Center-Fed Dipole Antennas with Finite Conductivity

We propose a remedy for the unphysical oscillations arising in the current distribution of carbon nanotube and imperfectly conducting antennas center-driven by a delta-function generator when the approximate kernel is used. We do so by formulating an effective current, which was studied in detail in a 2011 and a 2013 paper for a perfectly conducting linear cylindrical antenna of infinite length, with application to the finite-length antenna. We discuss our results in connection with the perfectly conducting antenna, providing perturbative corrections to the current distribution for a large conductance, as well as presenting a delta-sequence and the field of a Hertzian dipole for the effective current in the limit of vanishing conductance. To that end, we employ both analytical tools and numerical methods to compare with experimental results.

physics.comp-ph↗
arXiv2018

Double, borderline, and extraordinary eigenvalues of Kac-Murdock-Szeg\"o matrices with a complex parameter

For all sufficiently large complex $\rho$, and for arbitrary matrix dimension $n$, it is shown that the Kac--Murdock--Szeg\H{o} matrix $K_n(\rho)=\left[\rho^{|j-k|}\right]_{j,k=1}^{n}$ possesses exactly two eigenvalues whose magnitude is larger than $n$. We discuss a number of properties of the two "extraordinary" eigenvalues. Conditions are developed that, given $n$, allow us-without actually computing eigenvalues-to find all values $\rho$ that give rise to eigenvalues of magnitude $n$, termed "borderline" eigenvalues. The aforementioned values of $\rho$ form two closed curves in the complex-$\rho$ plane. We describe these curves, which are $n$-dependent, in detail. An interesting borderline case arises when an eigenvalue of $K_n(\rho)$ equals $-n$: apart from certain exceptional cases, this occurs if and only if the eigenvalue is a double one; and if and only if the point $\rho$ is a cusp-like singularity of one of the two closed curves.

math.NA↗
arXiv2018

Spectral properties of Kac-Murdock-Szeg\"o matrices with a complex parameter

When $0\lt \rho \lt 1$, the Kac-Murdock-Szeg\"o matrix $K_n(\rho)=\left[\rho^{\lvert j-k \rvert}\right]_{j,k=1}^n$ is a Toeplitz correlation matrix with many applications and very well known spectral properties. We study the eigenvalues and eigenvectors of $K_n(\rho)$ for the general case where $\rho$ is complex, pointing out similarities and differences to the case $0\lt \rho \lt 1$. We then specialize our results to real $\rho$ with $\rho \gt 1$, emphasizing the continuity of the eigenvalues as functions of $\rho$. For $\rho \gt 1$, we develop simple approximate formulas for the eigenvalues and pinpoint all eigenvalues' locations. Our study starts from a certain polynomial whose zeros are connected to the eigenvalues by elementary formulas. We discuss relations of our results to earlier results of W. F. Trench.

math.NA↗
arXiv2017

Suboscillations with arbitrary shape

We report a method for constructing bandpass functions that approximate a given analytic function with arbitrary accuracy over a finite interval. A corollary is that bandpass functions can be obtained that oscillate arbitrarily slower than their minimum frequency component, a counter-intuitive phenomenon known as suboscillations.

math-ph↗
arXiv2015

Superoscillations with arbitrary polynomial shape

We present a method for constructing superoscillatory functions the superoscillatory part of which approximates a given polynomial with arbitrarily small error in a fixed interval. These functions are obtained as the product of the polynomial with a sufficiently flat, bandlimited envelope function whose Fourier transform has at least N-1 continuous derivatives and an N-th derivative of bounded variation, N being the order of the polynomial. Polynomials of arbitrarily high order can be approximated if the Fourier transform of the envelope is smooth, i.e. a bump function.

math-ph↗