O-Minimal Invariants for Discrete-Time Dynamical Systems

Termination analysis of linear loops plays a key rôle in several areas of computer science, including program verification and abstract interpretation. Already for the simplest variants of linear loops the question of termination relates to deep open problems in number theory, such as the decidability of the Skolem and Positivity Problems for linear recurrence sequences, or equivalently reachability questions for discrete-time linear dynamical systems. In this article, we introduce the class of o-minimal invariants , which is broader than any previously considered, and study the decidability of the existence and algorithmic synthesis of such invariants as certificates of non-termination for linear loops equipped with a large class of halting conditions. We establish two main decidability results, one of them conditional on Schanuel’s conjecture is transcendental number theory.

Trends, directions for further research, and some open problems of fractional calculus

The area of fractional calculus (FC) has been fast developing and is presently being applied in all scientific fields. Therefore, it is of key relevance to assess the present state of development and to foresee, if possible, the future evolution, or, at least, the challenges identified in the scope of advanced research works. This paper gives a vision about the directions for further research as well as some open problems of FC. A number of topics in mathematics, numerical algorithms and physics are analyzed, giving a systematic perspective for future research.

A Survey of Opponent Modeling in Adversarial Domains

Opponent modeling is the ability to use prior knowledge and observations in order to predict the behavior of an opponent. This survey presents a comprehensive overview of existing opponent modeling techniques for adversarial domains, many of which must address stochastic, continuous, or concurrent actions, and sparse, partially observable payoff structures. We discuss all the components of opponent modeling systems, including feature extraction, learning algorithms, and strategy abstractions. These discussions lead us to propose a new form of analysis for describing and predicting the evolution of game states over time. We then introduce a new framework that facilitates method comparison, analyze a representative selection of techniques using the proposed framework, and highlight common trends among recently proposed methods. Finally, we list several open problems and discuss future research directions inspired by AI research on opponent modeling and related research in other disciplines.

On the Power of Standard Information for Tractability for L2- Approximation in the Randomized Setting

We study the approximation of multivariate functions from a separable Hilbert space in the randomized setting with the error measured in the weighted L2 norm. We consider algorithms that use standard information Λstd consisting of function values or general linear information Λall consisting of arbitrary linear functionals. We investigate the equivalences of various notions of algebraic and exponential tractability in the randomized setting for Λstd and Λall for the normalized or absolute error criterion. For the normalized error criterion, we show that the power of Λstd is the same as that of Λall for all notions of exponential tractability and some notions of algebraic tractability without any condition. For the absolute error criterion, we show that the power of Λstd is the same as that of Λall for all notions of algebraic and exponential tractability without any condition. Specifically, we solve Open Problems 98, 101, 102 and almost solve Open Problem 100 as posed by E.Novak and H.Wo´zniakowski in the book: Tractability of Multivariate Problems, Volume III: Standard Information for Operators, EMS Tracts in Mathematics, Zürich, 2012.

Measurement of Energy Spectrum and Elemental Composition of PeV Cosmic Rays: Open Problems and Prospects

Cosmic rays represent one of the most important energy transformation processes of the universe. They bring information about the surrounding universe, our galaxy, and very probably also the extragalactic space, at least at the highest observed energies. More than one century after their discovery, we have no definitive models yet about the origin, acceleration and propagation processes of the radiation. The main reason is that there are still significant discrepancies among the results obtained by different experiments located at ground level, probably due to unknown systematic uncertainties affecting the measurements. In this document, we will focus on the detection of galactic cosmic rays from ground with air shower arrays up to 1018 eV. The aim of this paper is to discuss the conflicting results in the 1015 eV energy range and the perspectives to clarify the origin of the so-called `knee’ in the all-particle energy spectrum, crucial to give a solid basis for models up to the end of the cosmic ray spectrum. We will provide elements useful to understand the basic techniques used in reconstructing primary particle characteristics (energy, mass, and arrival direction) from the ground, and to show why indirect measurements are difficult and results are still conflicting.

Subgraph densities in a surface

Given a fixed graph H that embeds in a surface $\Sigma$ , what is the maximum number of copies of H in an n-vertex graph G that embeds in $\Sigma$ ? We show that the answer is $\Theta(n^{f(H)})$ , where f(H) is a graph invariant called the ‘flap-number’ of H, which is independent of $\Sigma$ . This simultaneously answers two open problems posed by Eppstein ((1993) J. Graph Theory17(3) 409–416.). The same proof also answers the question for minor-closed classes. That is, if H is a $K_{3,t}$ minor-free graph, then the maximum number of copies of H in an n-vertex $K_{3,t}$ minor-free graph G is $\Theta(n^{f'(H)})$ , where f′(H) is a graph invariant closely related to the flap-number of H. Finally, when H is a complete graph we give more precise answers.

Computability-theoretic complexity of effective Banach spaces

<p><b>We investigate the geometry of effective Banach spaces, namely a sequenceof approximation properties that lies in between a Banach space having a basis and the approximation property.</b></p> <p>We establish some upper bounds on suchproperties, as well as proving some arithmetical lower bounds. Unfortunately,the upper bounds obtained in some cases are far away from the lower bound.</p> <p>However, we will show that much tighter bounds will require genuinely newconstructions, and resolve long-standing open problems in Banach space theory.</p> <p>We also investigate the effectivisations of certain classical theorems in Banachspaces.</p>

Computability-theoretic complexity of effective Banach spaces

<p><b>We investigate the geometry of effective Banach spaces, namely a sequenceof approximation properties that lies in between a Banach space having a basis and the approximation property.</b></p> <p>We establish some upper bounds on suchproperties, as well as proving some arithmetical lower bounds. Unfortunately,the upper bounds obtained in some cases are far away from the lower bound.</p> <p>However, we will show that much tighter bounds will require genuinely newconstructions, and resolve long-standing open problems in Banach space theory.</p> <p>We also investigate the effectivisations of certain classical theorems in Banachspaces.</p>

Federated Learning in Edge Computing: A Systematic Survey

Edge Computing (EC) is a new architecture that extends Cloud Computing (CC) services closer to data sources. EC combined with Deep Learning (DL) is a promising technology and is widely used in several applications. However, in conventional DL architectures with EC enabled, data producers must frequently send and share data with third parties, edge or cloud servers, to train their models. This architecture is often impractical due to the high bandwidth requirements, legalization, and privacy vulnerabilities. The Federated Learning (FL) concept has recently emerged as a promising solution for mitigating the problems of unwanted bandwidth loss, data privacy, and legalization. FL can co-train models across distributed clients, such as mobile phones, automobiles, hospitals, and more, through a centralized server, while maintaining data localization. FL can therefore be viewed as a stimulating factor in the EC paradigm as it enables collaborative learning and model optimization. Although the existing surveys have taken into account applications of FL in EC environments, there has not been any systematic survey discussing FL implementation and challenges in the EC paradigm. This paper aims to provide a systematic survey of the literature on the implementation of FL in EC environments with a taxonomy to identify advanced solutions and other open problems. In this survey, we review the fundamentals of EC and FL, then we review the existing related works in FL in EC. Furthermore, we describe the protocols, architecture, framework, and hardware requirements for FL implementation in the EC environment. Moreover, we discuss the applications, challenges, and related existing solutions in the edge FL. Finally, we detail two relevant case studies of applying FL in EC, and we identify open issues and potential directions for future research. We believe this survey will help researchers better understand the connection between FL and EC enabling technologies and concepts.

Extrachromosomal Circular DNA (eccDNA): From Chaos to Function

Extrachromosomal circular DNA (eccDNA) is a type of double-stranded circular DNA that is derived and free from chromosomes. It has a strong heterogeneity in sequence, length, and origin and has been identified in both normal and cancer cells. Although many studies suggested its potential roles in various physiological and pathological procedures including aging, telomere and rDNA maintenance, drug resistance, and tumorigenesis, the functional relevance of eccDNA remains to be elucidated. Recently, due to technological advancements, accumulated evidence highlighted that eccDNA plays an important role in cancers by regulating the expression of oncogenes, chromosome accessibility, genome replication, immune response, and cellular communications. Here, we review the features, biogenesis, physiological functions, potential functions in cancer, and research methods of eccDNAs with a focus on some open problems in the field and provide a perspective on how eccDNAs evolve specific functions out of the chaos in cells.