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Anders Karlsson

Publications and source records attributed to Anders Karlsson.

At least 37 records · Page 2Linked to original sources
arXiv2022

An efficient full wave solver for eddy currents

An integral equation reformulation of the Maxwell transmission problem is presented. The reformulation uses techniques such as tuning of free parameters and augmentation of close-to-rank-deficient operators. It is designed for the eddy current regime and works both for surfaces of genus $0$ and $1$. Well-conditioned systems and field representations are obtained despite the Maxwell transmission problem being ill-conditioned for genus $1$ surfaces due to the presence of Neumann eigenfields. Furthermore, it is shown that these eigenfields, for ordinary conductors in the eddy current regime, are different from the classical Neumann eigenfields for superconductors. Numerical examples, based on the reformulation, give an unprecedented $13$-digit accuracy both for transmitted and scattered fields.

math.AP↗
arXiv2022

Volumes of spheres and special values of zeta functions of $\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}$

The volume of the unit sphere in every dimension is given a new interpretation as a product of special values of the zeta function of $\mathbb{Z}$, akin to volume formulas of Minkowski and Siegel in the theory of arithmetic groups. A product formula is found for this zeta function that specializes to Catalan numbers. Moreover, certain closed-form expressions for various other zeta values are deduced, in particular leading to an alternative perspective on Euler's values of the Riemann zeta function.

math.NT↗
arXiv2021

Comparison of integral equations for the Maxwell transmission problem with general permittivities

Two recently derived integral equations for the Maxwell transmission problem are compared through numerical tests on simply connected axially symmetric domains for non-magnetic materials. The winning integral equation turns out to be entirely free from false eigenwavenumbers for any passive materials, also for purely negative permittivity ratios and in the static limit, as well as free from false essential spectrum on non-smooth surfaces. It also appears to be numerically competitive to all other available integral equation reformulations of the Maxwell transmission problem, despite using eight scalar surface densities.

physics.comp-ph↗
arXiv2021

Comments on the cosmic convergence of nonexpansive maps

This note discusses some aspects of the asymptotic behaviour of nonexpansive maps. Using metric functionals, we make a connection to the invariant subspace problem and prove a new result for nonexpansive maps of $\ell^{1}$. We also point out some inaccurate assertions appearing in the literature on this topic.

math.FA↗
arXiv2021

Deep limits and cut-off phenomena for neural networks

We consider dynamical and geometrical aspects of deep learning. For many standard choices of layer maps we display semi-invariant metrics which quantify differences between data or decision functions. This allows us, when considering random layer maps and using non-commutative ergodic theorems, to deduce that certain limits exist when letting the number of layers tend to infinity. We also examine the random initialization of standard networks where we observe a surprising cut-off phenomenon in terms of the number of layers, the depth of the network. This could be a relevant parameter when choosing an appropriate number of layers for a given learning task, or for selecting a good initialization procedure. More generally, we hope that the notions and results in this paper can provide a framework, in particular a geometric one, for a part of the theoretical understanding of deep neural networks.

cs.LG↗
arXiv2020

An extended charge-current formulation of the electromagnetic transmission problem

A boundary integral equation formulation is presented for the electromagnetic transmission problem where an incident electromagnetic wave is scattered from a bounded dielectric object. The formulation provides unique solutions for all combinations of wavenumbers in the closed upper half-plane for which Maxwell's equations have a unique solution. This includes the challenging combination of a real positive wavenumber in the outer region and an imaginary wavenumber inside the object. The formulation, or variants thereof, is particularly suitable for numerical field evaluations as confirmed by examples involving both smooth and non-smooth objects.

physics.comp-ph↗
arXiv2020

Hahn-Banach for metric functionals and horofunctions

It is observed that a natural analog of the Hahn-Banach theorem is valid for metric functionals but fails for horofunctions. Several statements of the existence of invariant metric functionals for individual isometries and 1-Lipschitz maps are proved. Various other definitions, examples and facts are pointed out related to this topic. In particular it is shown that the metric (horofunction) boundary of every infinite Cayley graphs contains at least two points.

math.MG↗
arXiv2019

Spectral zeta functions

This paper discusses the simplest examples of spectral zeta functions, especially those associated with graphs, a subject which has not been much studied. The analogy and the similar structure of these functions, such as their parallel definition in terms of the heat kernel and their functional equations, are emphasized. Another theme is to point out various contexts in which these non-classical zeta functions appear. This includes Eisenstein series, the Langlands program, Verlinde formulas, Riemann hypotheses, Catalan numbers, Dedekind sums, and hypergeometric functions. Several open-ended problems are suggested with the hope of stimulating further research.

math.NT↗
arXiv2019

Elements of a metric spectral theory

This paper discusses a general method for spectral type theorems using metric spaces instead of vector spaces. Advantages of this approach are that it applies to genuinely non-linear situations and also to random versions. Metric analogs of operator norm, spectral radius, eigenvalue, linear functional, and weak convergence are suggested. Applications explained include generalizations of the mean ergodic theorem, the Wolff-Denjoy theorem and Thurston's spectral theorem for surface homeomorphisms.

math.MG↗
arXiv2019

Physical-density integral equation methods for scattering from multi-dielectric cylinders

An integral equation-based numerical method for scattering from multi-dielectric cylinders is presented. Electromagnetic fields are represented via layer potentials in terms of surface densities with physical interpretations. The existence of null-field representations then adds superior flexibility to the modeling. Local representations are used for fast field evaluation at points away from their sources. Partially global representations, constructed as to reduce the strength of kernel singularities, are used for near-evaluations. A mix of local- and partially global representations is also used to derive the system of integral equations from which the physical densities are solved. Unique solvability is proven for the special case of scattering from a homogeneous cylinder under rather general conditions. High achievable accuracy is demonstrated for several examples found in the literature.

physics.comp-ph↗
arXiv2018

On a Helmholtz transmission problem in planar domains with corners

A particular mix of integral equations and discretization techniques is suggested for the solution of a planar Helmholtz transmission problem with relevance to the study of surface plasmon waves. The transmission problem describes the scattering of a time harmonic transverse magnetic wave from an infinite dielectric cylinder with complex permittivity and sharp edges. Numerical examples illustrate that the resulting scheme is capable of obtaining total magnetic and electric fields to very high accuracy in the entire computational domain.

physics.comp-ph↗
arXiv2018

Resonances in axially symmetric dielectric objects

A high-order convergent and robust numerical solver is constructed and used to find complex eigenwavenumbers and electromagnetic eigenfields of dielectric objects with axial symmetry. The solver is based on Fourier--Nyström discretization of combined integral equations for the transmission problem and can be applied to demanding resonance problems at microwave, terahertz, and optical wavelengths. High achievable accuracy, even at very high wavenumbers, makes the solver ideal for benchmarking and for assessing the performance of general purpose commercial software.

physics.comp-ph↗
arXiv2016

An explicit kernel-split panel-based Nyström scheme for integral equations on axially symmetric surfaces

A high-order accurate, explicit kernel-split, panel-based, Fourier-Nyström discretization scheme is developed for integral equations associated with the Helmholtz equation in axially symmetric domains. Extensive incorporation of analytic information about singular integral kernels and on-the-fly computation of nearly singular quadrature rules allow for very high achievable accuracy, also in the evaluation of fields close to the boundary of the computational domain.

math.NA↗
arXiv2015

Spectral zeta functions of graphs and the Riemann zeta function in the critical strip

We initiate the study of spectral zeta functions $ζ_{X}$ for finite and infinite graphs $X$, instead of the Ihara zeta function, with a perspective towards zeta functions from number theory and connections to hypergeometric functions. The Riemann hypothesis is shown to be equivalent to an approximate functional equation of graph zeta functions. The latter holds at all points where Riemann's zeta function $ζ(s)$ is non-zero. This connection arises via a detailed study of the asymptotics of the spectral zeta functions of finite torus graphs in the critcal strip and estimates on the real part of the logarithmic derivative of $ζ(s)$. We relate $ζ_{\mathbb{Z}}$ to Euler's beta integral and show how to complete it giving the functional equation $ξ_{\mathbb{Z}}(1-s)=ξ_{\mathbb{Z}}(s)$. This function appears in the theory of Eisenstein series although presumably with this spectral intepretation unrecognized. In higher dimensions $d$ we provide a meromorphic continuation of $ζ_{\mathbb{Z}^{d}}(s)$ to the whole plane and identify the poles. From our aymptotics several known special values of $ζ(s)$ are derived as well as its non-vanishing on the line $Re(s)=1$. We determine the spectral zeta functions of regular trees and show it to be equal to a specialization of Appell's hypergeometric function $F_{1}$ via an Euler-type integral formula due to Picard.

math.NT↗
arXiv2015

Determination of normalized electric eigenfields in microwave cavities with sharp edges

The magnetic field integral equation for axially symmetric cavities with perfectly conducting piecewise smooth surfaces is discretized according to a high-order convergent Fourier--Nyström scheme. The resulting solver is used to accurately determine eigenwavenumbers and normalized electric eigenfields in the entire computational domain.

physics.comp-ph↗
arXiv2015

Subadditive and Multiplicative Ergodic Theorems

A result for subadditive ergodic cocycles is proved that provides more delicate information than Kingman's subadditive ergodic theorem. As an application we deduce a multiplicative ergodic theorem generalizing an earlier result of Karlsson-Ledrappier, showing that the growth of a random product of semi-contractions is always directed by some horofunction. We discuss applications of this result to ergodic cocycles of bounded linear operators, holomorphic maps and topical operators, as well as a random mean ergodic theorem.

math.DS↗
arXiv2014

Determination of normalized magnetic eigenfields in microwave cavities

The magnetic field integral equation for axially symmetric cavities with perfectly conducting surfaces is discretized according to a high-order convergent Fourier--Nyström scheme. The resulting solver is used to determine eigenwavenumbers and normalized magnetic eigenfields to very high accuracy in the entire computational domain.

physics.comp-ph↗