Some instructive mathematical errors

We describe various errors in the mathematical literature, and consider how some of them might have been avoided, or at least detected at an earlier stage, using tools such as Maple or Sage. Our examples are drawn from three broad categories of errors. First, we consider some significant errors made by highly-regarded mathematicians. In some cases these errors were not detected until many years after their publication. Second, we consider in some detail an error that was recently detected by the author. This error in a refereed journal led to further errors by at least one author who relied on the (incorrect) result. Finally, we mention some instructive errors that have been detected in the author's own published papers.Comment: 25 pages, 77 references. References updated in v

an introduction to the history of European computer science and technology

(essays and documents

Concentration Inequalities in the Wild: Case Studies in Blockchain & Reinforcement Learning

Concentration inequalities (CIs) are a powerful tool that provide probability bounds on how a random variable deviates from its expectation. In this dissertation, first I describe a blockchain protocol that I have developed, called Graphene, which uses CIs to provide probabilistic guarantees on performance. Second, I analyze the extent to which CIs are robust when the assumptions they require are violated, using Reinforcement Learning (RL) as the domain. Graphene is a method for interactive set reconciliation among peers in blockchains and related distributed systems. Through the novel combination of a Bloom filter and an Invertible Bloom Lookup Table, Graphene uses a fraction of the network bandwidth used by deployed work for one- and two-way synchronization. It is a fast and implementation-independent algorithm that uses CIs for parameterizing an IBLT so that it is optimal in size for a given desired decode rate. I characterize performance improvements through analysis, detailed simulation, and deployment results for Bitcoin Cash, a prominent cryptocurrency. Implementations of Graphene, IBLTs, and the IBLT optimization algorithm are all open-source code. Second, I analyze the extent to which existing methods rely on accurate training data for a specific class of RL algorithms, known as Safe and Seldonian RL. Several Seldonian RL algorithms have a component called the safety test, which uses CIs to lower bound the performance of a new policy with training data collected from another policy. I introduce a new measure of security to quantify the susceptibility to corruptions in training data, and show that a couple of Seldonian RL methods are extremely sensitive to even a few data corruptions, completely breaking the probability bounds guaranteed by CIs. I then introduce a new algorithm, called Panacea, that is more robust against data corruptions, and demonstrate its usage in practice on some RL problems, including a grid-world and diabetes treatment simulation

On the Mechanics Behind Academic Progress

In wissensintensiven Gesellschaften wird akademischem Fortschritt eine immer größere Bedeutung beigemessen. Sowohl die Generierung neuen Wissens als auch die Ausbildung hochqualifizierter Arbeitskräfte spielen hierbei eine Schlüsselrolle. Jüngste Studien legen allerdings nahe, dass es zunehmend schwieriger wird, neue Ideen zu finden und dass längere Bildungswege das individuelle Innovationspotenzial erheblich reduzieren. Vor diesem Hintergrund widmet sich die vorliegende Dissertation den Mechanismen, die akademischem Fortschritt zugrunde liegen. Im ersten Teil der Arbeit liegt der Fokus auf Spillover-Effekten in der Wissensproduktion. Der Ursprung dieser Effekte wird primär bei herausragenden Forscherinnen und Forschern vermutet, zu deren Ermittlung Metadaten zu 15,6 Millionen Publikationen zusammengestellt wurden. Innerhalb der Forschungselite erweist sich eine Subgruppe von 162 Personen als zentral für die Identifikation kausaler Effekte. Konstituierendes Merkmal dieser Personengruppe ist ihr unerwarteter Tod und ein damit verbundenes vorzeitiges Karriereende. Auf diese letalen Schocks folgend zeigt sich ein signifikanter Rückgang im Publikationsoutput ihres direkten Kollaborationsnetzwerks. Spillover-Effekte werden zwar aggregiert deutlich, treten über das Fächerspektrum allerdings in sehr heterogenen Formen auf. Der zweite Teil der Arbeit geht auf die institutionelle Ebene über und untersucht, wie effizient europäische Universitäten in Forschung und Lehre agieren. Effizienz wird hierbei über die Relation von Inputs und Outputs modelliert, die neben bibliometrischen Daten auch amtliche Statistiken zu Finanz- und Studierendenzahlen miteinbeziehen. Vergleichbar mit dem Resultat des ersten Teils wird offenbar, dass universitäre Effizienz je nach fachlichem Schwerpunkt von einer Reihe unterschiedlicher Faktoren bedingt wird. Abschließend erfolgt im dritten Teil der Arbeit die Betrachtung eines groß angelegten Programms zur Förderung der Hochschullehre in Deutschland. Die kompetitive Mittelvergabe des Qualitätspakts Lehre lässt darauf schließen, dass die Drittmittelakquise deutlich positive Pfadabhängigkeiten aufweist und somit das potenzielle Risiko birgt, finanzielle Ungleichheiten zu intensivieren

Interactions in Higher-Spin Gravity: a Holographic Perspective

This thesis is an elaboration of recent results on the holographic re-construction of metric-like interactions in higher-spin gauge theories on anti-de Sitter space (AdS), employing their conjectured holographic duality with free conformal field theories (CFTs). After reviewing the general approach and establishing the necessary intermediate results, we extract explicit expressions for the complete cubic action on AdS$_{d+1}$ and the quartic self-interaction of the scalar on AdS$_4$ for the type A minimal bosonic higher-spin theory from the three- and four- point correlation functions of single-trace operators in the free scalar $O\left(N\right)$ vector model. For this purpose tools were developed to evaluate tree-level three-point Witten diagrams involving fields of arbitrary integer spin and the conformal partial wave expansions of tree-level four-point Witten diagrams, which are underpinned by the ambient space formulation of AdS space and CFT. We also discuss the implications of the holographic duality on the locality properties of interactions in higher-spin gauge theories.Comment: Ph.D. thesis, 246 pages, 14 figures. Advisor: Prof. Dr. Johanna Erdmenger. v2: Minor edits, refs added and typos fixed. Published as a topical review in Journal of Physics

Model counting for reactive systems

Model counting is the problem of computing the number of solutions for a logical formula. In the last few years, it has been primarily studied for propositional logic, and has been shown to be useful in many applications. In planning, for example, propositional model counting has been used to compute the robustness of a plan in an incomplete domain. In information-flow control, model counting has been applied to measure the amount of information leaked by a security-critical system. In this thesis, we introduce the model counting problem for linear-time properties, and show its applications in formal verification. In the same way propositional model counting generalizes the satisfiability problem for propositional logic, counting models for linear-time properties generalizes the emptiness problem for languages over infinite words to one that asks for the number of words in a language. The model counting problem, thus, provides a foundation for quantitative extensions of model checking, where not only the existence of computations that violate the specification is determined, but also the number of such violations. We solve the model counting problem for the prominent class of omega-regular properties. We present algorithms for solving the problem for different classes of properties, and show the advantages of our algorithms in comparison to indirect approaches based on encodings into propositional logic. We further show how model counting can be used for solving a variety of quantitative problems in formal verification, including probabilistic model checking, quantitative information-flow in security-critical systems, and the synthesis of approximate implementations for reactive systems.Das Modellzählproblem fragt nach der Anzahl der Lösungen einer logischen Formel, und wurde in den letzten Jahren hauptsächlich für Aussagenlogik untersucht. Das Zählen von Modellen aussagenlogischer Formeln hat sich in vielen Anwendungen als nützlich erwiesen. Im Bereich der künstlichen Intelligenz wurde das Zählen von Modellen beispielsweise verwendet, um die Robustheit eines Plans in einem unvollständigen Weltmodell zu bewerten. Das Zählen von Modellen kann auch verwendet werden, um in sicherheitskritischen Systemen die Menge an enthüllten vertraulichen Daten zu messen. Diese Dissertation stellt das Modellzählproblem für Linearzeiteigenschaften vor, und untersucht dessen Rolle in der Welt der formalen Verifikation. Das Zählen von Modellen für Linearzeiteigenschaften führt zu neuen quantitativen Erweiterungen klassischer Verifikationsprobleme, bei denen nicht nur die Existenz eines Fehlers in einem System zu überprüfen ist, sondern auch die Anzahl solcher Fehler. Wir präsentieren Algorithmen zur Lösung des Modellzählproblems für verschiedene Klassen von Linearzeiteigenschaften und zeigen die Vorteile unserer Algorithmen im Vergleich zu indirekten Ansätzen, die auf Kodierungen der untersuchten Probleme in Aussagenlogik basieren. Darüberhinaus zeigen wir wie das Zählen von Modellen zur Lösung einer Vielzahl quantitativer Probleme in der formalen Verifikation verwendet werden kann. Dies beinhaltet unter anderem die Analyse probabilistischer Modelle, die Kontrolle quantitativen Informationsflusses in sicherheitskritischen Systemen, und die Synthese von approximativen Implementierungen für reaktive Systeme

Mathematical Economics

This book is devoted to the application of fractional calculus in economics to describe processes with memory and non-locality. Fractional calculus is a branch of mathematics that studies the properties of differential and integral operators that are characterized by real or complex orders. Fractional calculus methods are powerful tools for describing the processes and systems with memory and nonlocality. Recently, fractional integro-differential equations have been used to describe a wide class of economical processes with power law memory and spatial nonlocality. Generalizations of basic economic concepts and notions the economic processes with memory were proposed. New mathematical models with continuous time are proposed to describe economic dynamics with long memory. This book is a collection of articles reflecting the latest mathematical and conceptual developments in mathematical economics with memory and non-locality based on applications of fractional calculus

Computing With Distributed Information

The age of computing with massive data sets is highlighting new computational challenges. Nowadays, a typical server may not be able to store an entire data set, and thus data is often partitioned and stored on multiple servers in a distributed manner. A natural way of computing with such distributed data is to use distributed algorithms: these are algorithms where the participating parties (i.e., the servers holding portions of the data) collaboratively compute a function over the entire data set by sending (preferably small-size) messages to each other, where the computation performed at each participating party only relies on the data possessed by it and the messages received by it. We study distributed algorithms focused on two key themes: convergence time and data summarization. Convergence time measures how quickly a distributed algorithm settles on a globally stable solution, and data summarization is the approach of creating a compact summary of the input data while retaining key information. The latter often leads to more efficient computation and communication. The main focus of this dissertation is on design and analysis of distributed algorithms for important problems in diverse application domains centering on the themes of convergence time and data summarization. Some of the problems we study include convergence time of double oral auction and interdomain routing, summarizing graphs for large-scale matching problems, and summarizing data for query processing