110 research outputs found

    Cyclic proof systems for modal fixpoint logics

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    This thesis is about cyclic and ill-founded proof systems for modal fixpoint logics, with and without explicit fixpoint quantifiers.Cyclic and ill-founded proof-theory allow proofs with infinite branches or paths, as long as they satisfy some correctness conditions ensuring the validity of the conclusion. In this dissertation we design a few cyclic and ill-founded systems: a cyclic one for the weak Grzegorczyk modal logic K4Grz, based on our explanation of the phenomenon of cyclic companionship; and ill-founded and cyclic ones for the full computation tree logic CTL* and the intuitionistic linear-time temporal logic iLTL. All systems are cut-free, and the cyclic ones for K4Grz and iLTL have fully finitary correctness conditions.Lastly, we use a cyclic system for the modal mu-calculus to obtain a proof of the uniform interpolation property for the logic which differs from the original, automata-based one

    Integration on the Surreals

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    Conway's real closed field No of surreal numbers is a sweeping generalization of the real numbers and the ordinals to which a number of elementary functions such as log and exponentiation have been shown to extend. The problems of identifying significant classes of functions that can be so extended and of defining integration for them have proven to be formidable. In this paper, we address this and related unresolved issues by showing that extensions to No, and thereby integrals, exist for most functions arising in practical applications. In particular, we show they exist for a large subclass of the resurgent functions, a subclass that contains the functions that at infinity are semi-algebraic, semi-analytic, analytic, meromorphic, and Borel summable as well as generic solutions to linear and nonlinear systems of ODEs possibly having irregular singularities. We further establish a sufficient condition for the theory to carry over to ordered exponential subfields of No more generally and illustrate the result with structures familiar from the surreal literature. We work in NBG less the Axiom of Choice (for both sets and proper classes), with the result that the extensions of functions and integrals that concern us here have a "constructive" nature in this sense. In the Appendix it is shown that the existence of such constructive extensions and integrals of substantially more general types of functions (e.g. smooth functions) is obstructed by considerations from the foundations of mathematics.Comment: This paper supersedes the positive portion of O. Costin, P. Ehrlich and H. Friedman, "Integration on the surreals: a conjecture of Conway, Kruskal and Norton", arXiv:1505.02478v3, 24 Aug 2015. A separate paper superseding the negative portion of the earlier arXiv preprint is in preparation by H. Friedman and O. Costi

    Spin-glass in diverse geometrie (da sottili film a sistemi cubici): simulazioni incontrano esperimenti

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    Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Físicas, Departamento de Física Teórica I, leída el 19-07-2021During my Ph.D., we made small progress in the secular question on the nature of the spin-glass system. However, we demonstrated the unique and powerful combination of experiments, theory, and simulations addressing complex dynamics. On one hand, the experimental progress on the sample preparation and the increased precision on the measurements of key physical observables have opened new prospects in the spinglass investigation. On the other hand, the Janus II special-purpose supercomputer, in combination with theory, is suficient to extend simulation time and length scales tovalues explored experimentally. The text is organized into four parts. In the following paragraphs, we introduce briefly each of them...Esta tesis presenta nuestra modesta contribución a la comprensión y modelización de los vidrios de espín. En particular, hemos mostrado que la combinación resultados teóricos, numéricos y experimentales permite revolucionar la comprensión de la dinámica de estos sistemas complejos. Desde el punto de vista experimental, los progresos en la preparación de muestras se alían con la alta precisión de las medidas para abrir nuevas perspectivas previamente insospechadas. Por otro lado, el uso combinado del superordenador Janus II y el análisis teórico ha permitido obtener (e interpretar)simulaciones en una escala temporal comparable a la de los experimentos. Para explicar de manera coherente estos avances, hemos dividido el texto en cuatro partes cuyo contenido describimos a continuación...Fac. de Ciencias FísicasTRUEunpu

    Magnetic Hybrid-Materials

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    Externally tunable properties allow for new applications of suspensions of micro- and nanoparticles in sensors and actuators in technical and medical applications. By means of easy to generate and control magnetic fields, fluids inside of matrices are studied. This monnograph delivers the latest insigths into multi-scale modelling, manufacturing and application of those magnetic hybrid materials

    The Universality Problem

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    The theme of this thesis is to explore the universality problem in set theory in connection to model theory, to present some methods for finding universality results, to analyse how these methods were applied, to mention some results and to emphasise some philosophical interrogations that these aspects entail. A fundamental aspect of the universality problem is to find what determines the existence of universal objects. That means that we have to take into consideration and examine the methods that we use in proving their existence or nonexistence, the role of cardinal arithmetic, combinatorics etc. The proof methods used in the mathematical part will be mostly set-theoretic, but some methods from model theory and category theory will also be present. A graph might be the simplest, but it is also one of the most useful notions in mathematics. We show that there is a faithful functor F from the category L of linear orders to the category G of graphs that preserves model theoretic-related universality results (classes of objects having universal models in exactly the same cardinals, and also having the same universality spectrum). Trees constitute combinatorial objects and have a central role in set theory. The universality of trees is connected to the universality of linear orders, but it also seems to present more challenges, which we survey and present some results. We show that there is no embedding between an ℵ2-Souslin tree and a non-special wide ℵ2 tree T with no cofinal branches. Furthermore, using the notion of ascent path, we prove that the class of non-special ℵ2-Souslin tree with an ω-ascent path a has maximal complexity number, 2ℵ2 = ℵ3. Within the general framework of the universality problem in set theory and model theory, while emphasising their approaches and their connections with regard to this topic, we examine the possibility of drawing some philosophical conclusions connected to, among others, the notions of mathematical knowledge, mathematical object and proof

    Deepening the knowledge of Spin Glasses: Metastate, Off-equilibrium phenomena and Temperature Chaos

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    Esta tesis se centra en el estudio de los vidrios de espín desde el punto de vista numérico, abordando diversos problemas del campo con datos muy precisos generados por el supercomputador Janus II.La tesis cuenta con un capítulo introductorio que pretende recoger los aspectos teóricos y experimentales más relevantes del campo así como motivar el estudio de los vidrios de espín desde un punto de vista experimental. Posteriormente se describen los resultados obtenidos durante la tesis en tres bloques diferentes.El primer bloque está centrado en la primera construcción numérica del metaestado. Utilizando simulaciones y análisis de alta calidad, realizamos un estudio numérico sobre un sistema que solo había sido abordado desde el punto de vista teórico y con la que podemos discriminar, parcialmente y dentro de nuestra precisión, diferentes propuestas teóricas que explican la naturaleza de los vidrios de espín desde un punto de vista fundamental.El segundo bloque está dedicado a los sistemas fuera del equilibrio. En los vidrios de espín, este régimen es de especial relevancia, pues es donde se enmarcan la práctica totalidad de los experimentos. En estos trabajos, resolvemos una discrepancia entre experimentos y simulaciones numéricas y también estudiamos por primera vez en vidrios de espín el efecto Mpemba, un efecto conocido y recientemente estudiado en otros campos de la física estadística.El último de los bloques está centrado en el estudio del caos en temperatura de los vidrios de espín. Este bloque cuenta con tres capítulos. El primero de ellos está destinado a introducir el concepto de caos en temperatura y a presentar los trabajos relevantes que se han realizado en el campo. El segundo se centra en el estudio del caos en temperatura en sistemas equilibrados. En este capítulo, los propios métodos numéricos de simulación se relacionan con el fenómeno del caos en temperatura a través de un novedoso procedimiento presentado en trabajos previos. Este procedimiento es mejorado mediante la introducción de un método variacional. Finalmente, el tercer capítulo de este bloque está dedicado al primer estudio numérico del efecto del caos en temperatura en sistemas fuera del equilibrio. Este capítulo se sirve de ideas previas que relacionan los efectos de estática y dinámica en los vidrios de espín así como del análisis de eventos raros para estudiar el caos en este régimen, que como se ha mencionado anteriormente, tiene especial relevancia debido a las circunstancias en las que de desarrollan los experimentos.Por último, la tesis cuenta con varios apéndices en los que se dan detalles técnicos, algunos fundamentales para la reproducibilidad de los trabajos desarrollados.<br /

    Друга міжнародна конференція зі сталого майбутнього: екологічні, технологічні, соціальні та економічні питання (ICSF 2021). Кривий Ріг, Україна, 19-21 травня 2021 року

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    Second International Conference on Sustainable Futures: Environmental, Technological, Social and Economic Matters (ICSF 2021). Kryvyi Rih, Ukraine, May 19-21, 2021.Друга міжнародна конференція зі сталого майбутнього: екологічні, технологічні, соціальні та економічні питання (ICSF 2021). Кривий Ріг, Україна, 19-21 травня 2021 року

    Making up Numbers

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    "Making up Numbers: A History of Invention in Mathematics offers a detailed but accessible account of a wide range of mathematical ideas. Starting with elementary concepts, it leads the reader towards aspects of current mathematical research. The book explains how conceptual hurdles in the development of numbers and number systems were overcome in the course of history, from Babylon to Classical Greece, from the Middle Ages to the Renaissance, and so to the nineteenth and twentieth centuries. The narrative moves from the Pythagorean insistence on positive multiples to the gradual acceptance of negative numbers, irrationals and complex numbers as essential tools in quantitative analysis. Within this chronological framework, chapters are organised thematically, covering a variety of topics and contexts: writing and solving equations, geometric construction, coordinates and complex numbers, perceptions of ‘infinity’ and its permissible uses in mathematics, number systems, and evolving views of the role of axioms. Through this approach, the author demonstrates that changes in our understanding of numbers have often relied on the breaking of long-held conventions to make way for new inventions at once providing greater clarity and widening mathematical horizons. Viewed from this historical perspective, mathematical abstraction emerges as neither mysterious nor immutable, but as a contingent, developing human activity. Making up Numbers will be of great interest to undergraduate and A-level students of mathematics, as well as secondary school teachers of the subject. In virtue of its detailed treatment of mathematical ideas, it will be of value to anyone seeking to learn more about the development of the subject.

    Modes of Truth

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    The aim of this volume is to open up new perspectives and to raise new research questions about a unified approach to truth, modalities, and propositional attitudes. The volume’s essays are grouped thematically around different research questions. The first theme concerns the tension between the theoretical role of the truth predicate in semantics and its expressive function in language. The second theme of the volume concerns the interaction of truth with modal and doxastic notions. The third theme covers higher-order solutions to the semantic and modal paradoxes, providing an alternative to first-order solutions embraced in the first two themes. This book will be of interest to researchers working in epistemology, logic, philosophy of logic, philosophy of language, philosophy of mathematics, and semantics
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