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EXPONENTIAL POLYNOMIALS AND IDENTIFICATION OF POLYGONAL REGIONS FROM FOURIER SAMPLES

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arXiv 2024年

作者： Kolountzakis, Mihail N. Spyridakis, Emmanuil Department of Mathematics and Applied Mathematics University of Crete Voutes Campus Heraklion70013 Greece Institute of Computer Science Foundation of Research and Technology Hellas N. Plastira 100 Vassilika Vouton Heraklion700 13 Greece

Consider the set E(D, N) of all bivariate exponential polynomials n f(ξ, η) = X pj(ξ, η)e2πi(xjξ+yjη), j=1 where the polynomials pj ∈ C[ξ, η] have degree j, yj ∈ T = R/Z. We find a set A ⊆ Z2 that depends on N and D only and is of size O(D2N log N) such that the values of f on A determine f. Notice that the size of A is only larger by a logarithmic quantity than the number of parameters needed to write down f. We use this in order to prove some uniqueness results about polygonal regions given a small set of samples of the Fourier Transform of their indicator function. If the number of different slopes of the edges of the polygonal region is ≤ k then the region is determined from a predetermined set of Fourier samples that depends only on k and the maximum number of vertices N and is of size O(k2N log N). In the particular case where all edges are known to be parallel to the axes the polygonal region is determined from a set of O(N log N) Fourier samples that depends on N *** Codes 41A05, 41A27, 42A15, 42A16 © 2024, CC BY.

关键词： Polynomials

(f, Γ)-Divergences: interpolating between f-divergences and integral probability metrics

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The Journal of Machine Learning Research

The Journal of Machine Learning research 2022年第1期23卷 1816-1885页

作者： Jeremiah Birrell Paul Dupuis Markos A. Katsoulakis Yannis Pantazis Luc Rey-Bellet TRIPODS Institute for Theoretical Foundations of Data Science University of Massachusetts Amherst Amherst MA Division of Applied Mathematics Brown University Providence RI Department of Mathematics and Statistics University of Massachusetts Amherst Amherst MA Institute of Applied and Computational Mathematics Foundation for Research and Technology - Hellas Heraklion Greece

We develop a rigorous and general framework for constructing information-theoretic divergences that subsume both f-divergences and integral probability metrics (IPMs), such as the 1-Wasserstein distance. We prove under which assumptions these divergences, hereafter referred to as (f, Γ)-divergences, provide a notion of 'distance' between probability measures and show that they can be expressed as a two-stage mass-redistribution/mass-transport process. The (f, Γ)-divergences inherit features from IPMs, such as the ability to compare distributions which are not absolutely continuous, as well as from f-divergences, namely the strict concavity of their variational representations and the ability to control heavy-tailed distributions for particular choices of f. When combined, these features establish a divergence with improved properties for estimation, statistical learning, and uncertainty quantification applications. Using statistical learning as an example, we demonstrate their advantage in training generative adversarial networks (GANs) for heavy-tailed, not-absolutely continuous sample distributions. We also show improved performance and stability over gradient-penalized Wasserstein GAN in image generation.

关键词： f-divergences integral probability metrics Wasserstein metric variational representations GANs

ORTHOGONAL FOURIER ANALYSIS ON DOMAINS

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arXiv 2024年

作者： Kolountzakis, Mihail N. Department of Mathematics and Applied Mathematics University of Crete Voutes Campus Heraklion70013 Greece Institute of Computer Science Foundation of Research and Technology Hellas N. Plastira 100 Vassilika Vouton Heraklion700 13 Greece

In this paper we go over the history of the Fuglede or Spectral Set Conjecture as it has developed over the last 30 years or so. We do not aim to be exhaustive and we do not cover important areas of development such as the results on the problem in classes of finite groups or the version of the problem that focuses on spectral measures instead of sets. The selection of the material has been strongly influenced by personal taste, history and capabilities. We are trying to be more descriptive than detailed and we point out several open *** Codes 42B10, 43A25, 52C22 © 2024, CC BY.

关键词： Fourier analysis

Temporally Causal Discovery Tests for Discrete Time Series and Neural Spike Trains

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arXiv 2023年

作者： Theocharous, A. Gregoriou, G.G. Sapountzis, P. Kontoyiannis, I. DPMMS University of Cambridge Centre for Mathematical Sciences Wilberforce Road CambridgeCB3 0WB United Kingdom University of Crete Faculty of Medicine Foundation for Research and Technology Hellas Institute of Applied and Computational Mathematics Heraklion70013 Greece Foundation for Research and Technology Hellas Institute of Applied and Computational Mathematics Heraklion70013 Greece Statistical Laboratory DPMMS University of Cambridge Centre for Mathematical Sciences Wilberforce Road CambridgeCB3 0WB United Kingdom

We consider the problem of detecting causal relationships between discrete time series, in the presence of potential confounders. A hypothesis test is introduced for identifying the temporally causal influence of (xn) on (yn), causally conditioned on a possibly confounding third time series (zn). Under natural Markovian modeling assumptions, it is shown that the null hypothesis, corresponding to the absence of temporally causal influence, is equivalent to the underlying ‘causal conditional directed information rate’ being equal to zero. The plug-in estimator for this functional is identified with the log-likelihood ratio test statistic for the desired test. This statistic is shown to be asymptotically normal under the alternative hypothesis and asymptotically χ2 distributed under the null, facilitating the computation of p-values when used on empirical data. The effectiveness of the resulting hypothesis test is illustrated on simulated data, validating the underlying theory. The test is also employed in the analysis of spike train data recorded from neurons in the V4 and FEF brain regions of behaving animals during a visual attention task. There, the test results are seen to identify interesting and biologically relevant information. © 2023, CC BY-NC-ND.

关键词： Time series

Dynamics of self-accelerating electron beams in a homogeneous magnetic field

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Physical Review A 2021年第1期103卷 013519-013519页

作者： Michael Goutsoulas Nikolaos K. Efremidis Department of Mathematics and Applied Mathematics University of Crete 70013 Heraklion Crete Greece Institute of Applied and Computational Mathematics Foundation for Research and Technology-Hellas (FORTH) 70013 Heraklion Crete Greece

We examine the dynamics of electron beams that, in free space, are self-accelerating, in the presence of an additional magnetic field. We focus our attention on the case of Airy beams that follow parabolic trajectories and in generalized classes of beams associated with power-law trajectories. We study the interplay between beam self-acceleration and the circular motion caused by the magnetic field. In the case of Airy beams, using an integral representation, we find closed-form solutions for the electron wave function. We also derive asymptotic formulas for the beam trajectories, both for Airy beams and for self-accelerating power-law beams. A ray optics description is rather useful for the interpretation of the beam dynamics. Our results are in excellent agreement with direct numerical simulations.

关键词： Beam control Cold atoms & matter waves

CURVES IN THE FOURIER ZEROS OF POLYTOPAL REGIONS AND THE POMPEIU PROBLEM

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arXiv 2024年

We prove that any finite union P of interior-disjoint polytopes in Rd has the Pompeiu property, a result first proved by Williams [Wil76]. This means that if a continuous function f on Rd integrates to 0 on any congruent copy of P then f is identically 0. By a fundamental result of Brown, Schreiber and Taylor [BST73] this is equivalent to showing that the Fourier–Laplace transform of the indicator function of P does not vanish identically on any 0-centered complex sphere in Cd. Our proof initially follows the recent one of Machado and Robins [MR23] who are using the Brion–Barvinok formula for the Fourier–Laplace transform of a polytope. But we simplify this method considerably by removing the use of properties of Bessel function zeros. Instead we use some elementary arguments on the growth of linear combinations of exponentials with rational functions as coefficients. Our approach allows us to prove the non-existence of complex spheres of any center in the zero-set of the Fourier–Laplace transform. The planar case is even simpler in that we do not even need the Brion–Barvinok formula. We then go further in the question of which sets can be contained in the null set of the Fourier–Laplace transform of a polytope by extending results of Engel [Eng23] who showed that rationally parametrized hypersurfaces, under some mild conditions, cannot be contained in this null-set. We show that a rationally parametrized curve which is not contained in an affine hyperplane in Cd cannot be contained in this null-set. Results about curves parametrized by meromorphic functions are also *** Codes 52B11, 42B37, 42B99 © 2024, CC BY.

关键词： Rational functions

Multipurpose composites with switched surface functionality and morphology based on oligoperoxide-modified electrospun nanofiber

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applied Nanoscience (Switzerland) 2023年第11期13卷 7155-7170页

作者： Datsyuk, Vitaliy Mitina, Nataliya Zaichenko, Alexander Trotsenko, Svitlana Balaban, Oksana Harhay, Khrystyna Teucher, Markus Izhyk, Oleh Grygorchak, Ivan Sygellou, Labrini Trotsenko, Oleksandr Stoika, Rostyslav Klyuchivska, Olga Fachbereich Physik Institut Für Experimentalphysik Freie Universität Berlin Arnimallee 14 Berlin14195 Germany Department of Organic Chemistry Institute of Chemistry and Chemical Technologies and Department of Applied Physics and Nanomaterials Science Institute of Applied Mathematics and Fundamental Sciences Lviv Polytechnic National University Lviv79013 Ukraine Institute of Chemical Engineering and High-Temperature Processes Foundation of Research and Technology Hellas Laboratory of Surface Characterization Patras26504 Greece Department of Textiles Merchandising and Interiors University of Georgia Nanostructure Materials Lab AthensGA30602 United States Institute of Cell Biology NAS of Ukraine Lviv79005 Ukraine

A new approach toward obtaining polymer nanofibers via electrospinning the blend of polybenzimidazole (PBI) and surface-active oligoperoxide for further creation of smart polymer composites with tailored surface functionality via polymerization initiated from the large-area PBI nanofiber sheets is presented. The peroxide fragments localized on the nanofiber surface due to poor immiscibility of the polymers in the blend provide "fiber-initiated" radical polymerization aiming variation of the functionality, height, and packing density of grafted polymer brushes according to the application requirements. PBI-based nanofiber 2D sheets of the variable packing density of grafted poly(hydroxyethyl methacrylate) (poly(HEMA)) or poly(dimethylaminoethyl methacrylate) (poly(DMAEMA)) brushes were synthesized. The structure, morphology, and special unique properties of new nanofiber-based composites studied using combined techniques demonstrate the ability of our advanced technology to produce smart functional materials for enhanced multi-purpose performance. The nanofiber sheets functionalized with grafted poly(HEMA) brushes were studied as biocompatible scaffolds for the immobilization and growth of mammalian cells and tissues. Poly(DMAEMA)-modified nanofiber composites possess enhanced dielectric permittivity providing their potential application in nanoelectronics. © 2023, King Abdulaziz City for Science and technology.

关键词： Polymerization

Relaxation Dynamics of a Liquid in the Vicinity of an Attractive Surface: The Process of Escaping from the Surface

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arXiv 2023年

作者： Behbahani, Alireza F. Harmandaris, Vagelis Institut für Physik Johannes Gutenberg-Universität Mainz Staudingerweg 7 MainzD-55099 Germany Institute of Applied and Computational Mathematics Foundation for Research and Technology - Hellas HeraklionGR-71110 Greece Computation-based Science and Technology Research Center The Cyprus Institute Nicosia2121 Cyprus Department of Mathematics and Applied Mathematics University of Crete HeraklionGR-71110 Greece

We analyze the displacements of the particles of a glass-forming molecular liquid perpendicular to a confining solid surface, using extensive molecular dynamics simulations with atomistic models. In the vicinity of an attractive surface, the liquid molecules are trapped. Transient localization of liquid molecules near the surface introduces a relaxation process, related to the escape of molecules from the surface, into the dynamics of the interfacial liquid layer. To describe this process, we analyze several dynamical observables of the confined liquid. The self-intermediate scattering function and the mean-squared displacement of the particles located in the interfacial layer are dominated by the process of escaping from the surface. This relaxation process is also associated with a strong heterogeneity in the mobility of the interfacial particles. The studied model liquid is hydrogenated methyl methacrylate. For the confining wall, we consider different models, namely a periodic single layer of graphene and a frozen amorphous configuration of the bulk liquid (frozen wall). Near graphene, where the liquid molecules form a layered structure and adopt parallel-to-surface orientation, a clear separation between small-scale movements of the molecules near the surface and the process of escaping from the surface is observed. This is reflected in the three-step relaxation of the interfacial layer. However, near the frozen wall, where the liquid molecules do not have a preferential alignment, a clear three-step relaxation is not seen, even though the dynamical quantities are controlled by the process of escaping from the surfacea Copyright © 2023, The Authors. All rights reserved.

关键词： Molecular dynamics

SPECTRALITY OF A MEASURE CONSISTING OF TWO LINE SEGMENTS

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arXiv 2025年

作者： Kolountzakis, Mihail N. Wu, Sha Department of Mathematics and Applied Mathematics University of Crete Voutes Campus Heraklion70013 Greece Institute of Computer Science Foundation of Research and Technology Hellas N. Plastira 100 Vassilika Vouton Heraklion700 13 Greece School of Mathematics Hunan University Changsha410082 China

MSC Codes 42C15, 42C30Take an interval [t, t+ 1] on the x-axis together with the same interval on the y-axis and let ρ be the normalized one-dimensional Lebesgue measure on this set of two segments. Continuing the work done by Lai, Liu and Prince [LLP21] as well as Ai, Lu and Zhou [ALZ23] we examine the spectrality of this measure for all different values of t (being spectral means that there is an orthonormal basis for L2(ρ) consisting of exponentials e2πi(λ1x+λ2y)). We almost complete the study showing that for (Formula presenetd) and for all t ∉ Q the measure ρ is not spectral. The only remaining undecided case is the case (Formula presenetd) (plus space). We also observe that in all known cases of spectral instances of this measure the spectrum is contained in a line and we give an easy necessary and sufficient condition for such measures to have a line spectrum. © 2025, CC BY.

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