Pure Σ2\Sigma_2-Elementarity beyond the Core

We display the entire structure ${\cal R}_2$ coding $\Sigma_1$- and $\Sigma_2$-elementarity on the ordinals. This will enable the analysis of pure $\Sigma_3$-elementary substructures.Comment: Extensive rewrite of the introduction. Mathematical content of sections 2 and 3 unchanged, extended introduction to section 2. Removed section 4. Theorem 4.3 to appear elsewhere with corrected proo

Pure patterns of order 2

We provide mutual elementary recursive order isomorphisms between classical ordinal notations, based on Skolem hulling, and notations from pure elementary patterns of resemblance of order $2$, showing that the latter characterize the proof-theoretic ordinal of the fragment $\Pi^1_1$-$\mathrm{CA}_0$ of second order number theory, or equivalently the set theory $\mathrm{KPl}_0$. As a corollary, we prove that Carlson's result on the well-quasi orderedness of respecting forests of order $2$ implies transfinite induction up to the ordinal of $\mathrm{KPl}_0$. We expect that our approach will facilitate analysis of more powerful systems of patterns.Comment: corrected Theorem 4.2 with according changes in section 3 (mainly Definition 3.3), results unchanged. The manuscript was edited, aligned with reference [14] (moving former Lemma 3.5 there), and argumentation was revised, with minor corrections in (the proof of) Theorem 4.2; results unchanged. Updated revised preprint; to appear in the APAL (2017

Tracking chains revisited

The structure ${\cal C}_2:=(1^\infty,\le,\le_1,\le_2)$, introduced and first analyzed in Carlson and Wilken 2012 (APAL), is shown to be elementary recursive. Here, $1^\infty$ denotes the proof-theoretic ordinal of the fragment $\Pi^1_1$-$\mathrm{CA}_0$ of second order number theory, or equivalently the set theory $\mathrm{KPl}_0$, which axiomatizes limits of models of Kripke-Platek set theory with infinity. The partial orderings $\le_1$ and $\le_2$ denote the relations of $\Sigma_1$- and $\Sigma_2$-elementary substructure, respectively. In a subsequent article we will show that the structure ${\cal C}_2$ comprises the core of the structure ${\cal R}_2$ of pure elementary patterns of resemblance of order $2$. In Carlson and Wilken 2012 (APAL) the stage has been set by showing that the least ordinal containing a cover of each pure pattern of order $2$ is $1^\infty$. However, it is not obvious from Carlson and Wilken 2012 (APAL) that ${\cal C}_2$ is an elementary recursive structure. This is shown here through a considerable disentanglement in the description of connectivity components of $\le_1$ and $\le_2$. The key to and starting point of our analysis is the apparatus of ordinal arithmetic developed in Wilken 2007 (APAL) and in Section 5 of Carlson and Wilken 2012 (JSL), which was enhanced in Carlson and Wilken 2012 (APAL) specifically for the analysis of ${\cal C}_2$.Comment: The text was edited and aligned with reference [10], Lemma 5.11 was included (moved from [10]), results unchanged. Corrected Def. 5.2 and Section 5.3 on greatest immediate $\le_1$-successors. Updated publication information. arXiv admin note: text overlap with arXiv:1608.0842

Generically Computable Linear Orderings

We study notions of generic and coarse computability in the context of computable structure theory. Our notions are stratified by the $\Sigma_\beta$ hierarchy. We focus on linear orderings. We show that at the $\Sigma_1$ level all linear orderings have both generically and coarsely computable copies. This behavior changes abruptly at higher levels; we show that at the $\Sigma_{\alpha+2}$ level for any $\alpha\in\omega_1^{ck}$ the set of linear orderings with generically or coarsely computable copies is $\mathbf{\Sigma}_1^1$-complete and therefore maximally complicated. This development is new even in the general analysis of generic and coarse computability of countable structures. In the process of proving these results we introduce new tools for understanding generically and coarsely computable structures. We are able to give a purely structural statement that is equivalent to having a generically computable copy and show that every relational structure with only finitely many relations has coarsely and generically computable copies at the lowest level of the hierarchy.Comment: 35 page

Marches al\'eatoires et \'el\'ements contractants sur des espaces CAT(0)

This thesis is dedicated to random walks on spaces with non-positive curvature. In particular, we study the case of group actions on CAT(0) spaces that admit contracting elements, that is, whose properties mimic those of loxodromic isometries in Gromov-hyperbolic spaces. In this context, we prove several limit laws, among which the almost sure convergence to the boundary without moment assumption, positivity of the drift and a central limit theorem. In a second part, we study boundary maps and stationary measures on affine buildings of type $\tilde{A}_2$, and we show that there always exists a hyperbolic isometry for a non-elementary action by isometries on such a space. Our approach involves the use of hyperbolic models for CAT(0) spaces, which were constructed by H.~Petyt, D.~Spriano and A.~Zalloum, and measured boundary theory, whose principles come from H.~Furstenberg.Comment: 194 pages. Introduction and presentation of the context in Frenc

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 23rd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2020, which took place in Dublin, Ireland, in April 2020, and was held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2020. The 31 regular papers presented in this volume were carefully reviewed and selected from 98 submissions. The papers cover topics such as categorical models and logics; language theory, automata, and games; modal, spatial, and temporal logics; type theory and proof theory; concurrency theory and process calculi; rewriting theory; semantics of programming languages; program analysis, correctness, transformation, and verification; logics of programming; software specification and refinement; models of concurrent, reactive, stochastic, distributed, hybrid, and mobile systems; emerging models of computation; logical aspects of computational complexity; models of software security; and logical foundations of data bases.

Introduction to Particle and Astroparticle Physics : Multimessenger Astronomy and its Particle Physics Foundations -2/E

This book introduces particle physics, astrophysics and cosmology starting from experiment. It provides a unified view of these fields, which is needed to answer our questions to the Universe–a unified view that has been lost somehow in recent years due to increasing specialization. This is the second edition of a book we published only three years ago, a book which had a success beyond our expectations. We felt that the recent progress on gravitational waves, gamma ray and neutrino astrophysics deserved a new edition including all these new developments: multimessenger astronomy is now a reality. In addition, the properties of the Higgs particle are much better known now than three years ago. Thanks to this second edition we had the opportunity to fix some bugs, to extend the material related to exercises, and to change in a more logical form the order of some items. Last but not least, our editor encouraged us a lot to write a second edition. Particle physics has recently seen the incredible success of the so-called standard model. A 50-year long search for the missing ingredient of the model, the Higgs particle, has been concluded successfully, and some scientists claim that we are close to the limit of the physics humans may know

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 23rd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2020, which took place in Dublin, Ireland, in April 2020, and was held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2020. The 31 regular papers presented in this volume were carefully reviewed and selected from 98 submissions. The papers cover topics such as categorical models and logics; language theory, automata, and games; modal, spatial, and temporal logics; type theory and proof theory; concurrency theory and process calculi; rewriting theory; semantics of programming languages; program analysis, correctness, transformation, and verification; logics of programming; software specification and refinement; models of concurrent, reactive, stochastic, distributed, hybrid, and mobile systems; emerging models of computation; logical aspects of computational complexity; models of software security; and logical foundations of data bases.

Notes in Pure Mathematics & Mathematical Structures in Physics

These Notes deal with various areas of mathematics, and seek reciprocal combinations, explore mutual relations, ranging from abstract objects to problems in physics.Comment: Small improvements and addition

Tracking chains of Σ2-elementarity

AbstractWe apply the ordinal arithmetical tools that were developed in Wilken (2007) [10] and Carlson and Wilken (in press) [4] in order to introduce tracking chains as the crucial means in the arithmetical analysis of (pure) elementary patterns of resemblance of order 2; see Carlson (2001) [2], Carlson (2009) [3], and Carlson and Wilken (in preparation) [5]. Although underlying heuristics for an analysis of Σ2-elementarity within the structure R2 is given in [5], this article is independent of [5] and provides a complete arithmetical analysis of the structure R2 below the least ordinal α such that any pure pattern of order 2 has a covering below α. α is shown to be the proof-theoretic ordinal of KPℓ0