Integration over curves and surfaces defined by the closest point mapping

We propose a new formulation for integrating over smooth curves and surfaces that are described by their closest point mappings. Our method is designed for curves and surfaces that are not defined by any explicit parameterization and is intended to be used in combination with level set techniques. However, contrary to the common practice with level set methods, the volume integrals derived from our formulation coincide exactly with the surface or line integrals that one wish to compute. We study various aspects of this formulation and provide a geometric interpretation of this formulation in terms of the singular values of the Jacobian matrix of the closest point mapping. Additionally, we extend the formulation - initially derived to integrate over manifolds of codimension one - to include integration along curves in three dimensions. Some numerical examples using very simple discretizations are presented to demonstrate the efficacy of the formulation.Comment: Revised the pape

Copyright and Machine Learning: from human-created input to computer-generated output

This thesis examines the legality of unauthorized reproduction of in-copyright works for the purpose of being used in Machine Learning processes. It focuses primarily on US and EU copyright systems as environments for Artificial Intelligence technological developments. Machine Learning uses of creative works differ from traditional ones: they do not involve human readers, they do not display protected expression to the public and they analyse works to extract information not protected by copyright. Hence they raise a question of whether they actually should fall within a reach of exclusive rights of copyright holders. In respect of the US copyright system, this study addresses the fair use doctrine that under certain conditions allows unauthorized reproduction of works. The research makes an attempt to apply the doctrine to Machine Learning uses by drawing parallels with recent case law on other technological uses of copyrighted works. As regards the EU copyright realities, this research discusses Machine Learning uses within the scope of newly proposed copyright exception for Text and Data Mining. It firstly analyses whether exempting these uses from a copyright reach would meet the three-step test requirements. After that, it critically assesses the scopes of the exception proposed and negotiated on the EU policymaking level. Additionally, this study discusses relations between AI-generated works and original human-created works used during the training process. It touches upon a question of possible reproduction of protected expression from original works in secondary ones and copyright-related consequences of that

Generalizations of Goursat\u27s Theorem for Groups

Petrillo\u27s recent article in the College Mathematics Journal explained a theorem of Goursat on the subgroups of a direct product of two groups. In this note, we extend this theorem to commutative rings, and to modules over commutative rings and fields

Coarsening in High Order, Discrete, Ill-Posed Diffusion Equations

We study the discrete version of a family of ill-posed, nonlinear diffusion equations of order 2n. The fourth order (n=2) version of these equations constitutes our main motivation, as it appears prominently in image processing and computer vision literature. It was proposed by You and Kaveh as a model for denoising images while maintaining sharp object boundaries (edges). The second order equation (n=1) corresponds to another famous model from image processing, namely Perona and Malik\u27s anisotropic diffusion, and was studied in earlier papers. The equations studied in this paper are high order analogues of the Perona-Malik equation, and like the second order model, their continuum versions violate parabolicity and hence lack well-posedness theory. We follow a recent technique from Kohn and Otto, and prove a weak upper bound on the coarsening rate of the discrete in space version of these high order equations in any space dimension, for a large class of diffusivities. Numerical experiments indicate that the bounds are close to being optimal, and are typically observed

Aktywności przyśrodkowej kory czołowej szczura podczas treningu dyskryminacji węchowej - rejestracja pojedynczych komórek

Po sesji doświadczalnej podczas której szczur uczy się kierować właściwym zapachem by odnaleźć nagrodę pokarmową, w przyśrodkowej korze czołowej (medial frontal cortex, mFCx) następuje istotny wzrost ekspresji c-fos [6]. W celu poznania dynamiki aktywacji mFCx podczas wykonywania tego zadania konieczne było zastosowanie metod elektrofizjologicznych. Dzięki chronicznej implantacji elektrod osadzonych na mikromanipulatorach rejestrowaliśmy aktywność pojedynczych komórek mFCx podczas kolejnych faz sesji treningowej. Około 35% badanych neuronów zaangażowanych było w samo wykonanie zadania - ich aktywność wzrastała w czasie trwania kolejnych prób równolegle ze skróceniem latencji poprawnych odpowiedzi. W przypadku 40% komórek zaobserwowaliśmy toniczny wzrost częstości wyładowań w 2- 3minutowych okresach międzypróbowych. Sądzimy, że populacja aktywna pomiędzy próbami może być częścią systemu pamięci przechowującą świeżo zdobyte informacje do wykorzystania w bliższej i dalszej przyszłości.Zadanie pt. „Digitalizacja i udostępnienie w Cyfrowym Repozytorium Uniwersytetu Łódzkiego kolekcji czasopism naukowych wydawanych przez Uniwersytet Łódzki” nr 885/P-DUN/2014 dofinansowane zostało ze środków MNiSW w ramach działalności upowszechniającej naukę

Lyapunov Functionals that Lead to Exponential Stability and Instability in Finite Delay Volterra Difference Equations

We use Lyapunov functionals to obtain sufficient conditions that guarantee exponential stability of the zero solution of the finite delay Volterra difference equation. Also, by displaying a slightly different Lyapunov functional, we obtain conditions that guarantee the instability of the zero solution. The highlight of the paper is the relaxing of the condition |a(t)| \u3c 1. Moreover, we provide examples in which we show that our theorems provide an improvement of some recent results

When Numerical Analysis Crosses Paths with Catalan and Generalized Motzkin Numbers

We study a linear doubly indexed sequence that contains the Catalan numbers and relates to a class of generalized Motzkin numbers. We obtain a closed form formula, a generating function and a nonlinear recursion relation for this sequence. We show that a finite difference scheme with compact stencil applied to a nonlinear differential operator acting on the Euclidean distance function is exact, and exploit this exactness to produce the nonlinear recursion relation. In particular, the nonlinear recurrence relation is obtained by using standard error analysis techniques from numerical analysis. This work shows a connection between numerical analysis and number theory, and illustrates an interesting occurrence of the Catalan and generalized Motzkin numbers in a context a priori void of combinatorial objects

NATO wobec zagrożeń środowiska informacyjnego

Abstract: The article concerns the presentation and characterization of the most common threats to the information environment. The authoress will refer to the activities of NATO in this area and the involvement in initiatives that meet contemporary threats to the information environment, including disinformation, propaganda, and in a broader sense - the operation of the impact implemented through the information environment. The authoress also indicates efforts to identify and counteract information attacks.Streszczenie: Artykuł dotyczy pokazania i scharakteryzowania najpowszechniejszych zagrożeń środowiska informacyjnego. Autorka odniesie się do działalności NATO w tym obszarze oraz zaangażowania w inicjatywy, które wychodzą naprzeciw współczesnym zagrożeniom dla środowiska informacyjnego, w tym dezinformacji, propagandy, a w ujęciu szerszym – operacji wpływu realizowanej poprzez środowisko informacyjne. Wskaże także wysiłki w rozpoznanie i przeciwdziałanie atakom informacyjnym