Q(sqrt(-3))-Integral Points on a Mordell Curve

We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4

Proceedings of the XIII Global Optimization Workshop: GOW'16

[Excerpt] Preface: Past Global Optimization Workshop shave been held in Sopron (1985 and 1990), Szeged (WGO, 1995), Florence (GO’99, 1999), Hanmer Springs (Let’s GO, 2001), Santorini (Frontiers in GO, 2003), San José (Go’05, 2005), Mykonos (AGO’07, 2007), Skukuza (SAGO’08, 2008), Toulouse (TOGO’10, 2010), Natal (NAGO’12, 2012) and Málaga (MAGO’14, 2014) with the aim of stimulating discussion between senior and junior researchers on the topic of Global Optimization. In 2016, the XIII Global Optimization Workshop (GOW’16) takes place in Braga and is organized by three researchers from the University of Minho. Two of them belong to the Systems Engineering and Operational Research Group from the Algoritmi Research Centre and the other to the Statistics, Applied Probability and Operational Research Group from the Centre of Mathematics. The event received more than 50 submissions from 15 countries from Europe, South America and North America. We want to express our gratitude to the invited speaker Panos Pardalos for accepting the invitation and sharing his expertise, helping us to meet the workshop objectives. GOW’16 would not have been possible without the valuable contribution from the authors and the International Scientific Committee members. We thank you all. This proceedings book intends to present an overview of the topics that will be addressed in the workshop with the goal of contributing to interesting and fruitful discussions between the authors and participants. After the event, high quality papers can be submitted to a special issue of the Journal of Global Optimization dedicated to the workshop. [...

Optimal search in discrete locations:extensions and new findings

A hidden target needs to be found by a searcher in many real-life situations, some of which involve large costs and significant consequences with failure. Therefore, efficient search methods are paramount. In our search model, the target lies in one of several discrete locations according to some hiding distribution, and the searcher's goal is to discover the target in minimum expected time by making successive searches of individual locations. In Part I of the thesis, the searcher knows the hiding distribution. Here, if there is only one way to search each location, the solution to the search problem, discovered in the 1960s, is simple; search next any location with a maximal probability per unit time of detecting the target. An equivalent solution is derived by viewing the search problem as a multi-armed bandit and following a Gittins index policy. Motivated by modern search technology, we introduce two modes---fast and slow---to search each location. The fast mode takes less time, but the slow mode is more likely to find the target. An optimal policy is difficult to obtain in general, because it requires an optimal sequence of search modes for each location, in addition to a set of sequence-dependent Gittins indices for choosing between locations. For each mode, we identify a sufficient condition for a location to use only that search mode in an optimal policy. For locations meeting neither sufficient condition, an optimal choice of search mode is extremely complicated, depending both on the hiding distribution and the search parameters of the other locations. We propose several heuristic policies motivated by our analysis, and demonstrate their near-optimal performance in an extensive numerical study. In Part II of the thesis, the searcher has only one search mode per location, but does not know the hiding distribution, which is chosen by an intelligent hider who aims to maximise the expected time until the target is discovered. Such a search game, modelled via two-person, zero-sum game theory, is relevant if the target is a bomb, intruder, or, of increasing importance due to advances in technology, a computer hacker. By Part I, if the hiding distribution is known, an optimal counter strategy for the searcher is any corresponding Gittins index policy. To develop an optimal search strategy in the search game, the searcher must account for the hider’s motivation to choose an optimal hiding distribution, and consider the set of corresponding Gittins index policies. %It follows that an optimal search strategy in the search game must be some Gittins index policy if the hiding distribution is assumed to be chosen optimally by the hider. However, the searcher must choose carefully from this set of Gittins index policies to ensure the same expected time to discover the target regardless of where it is hidden by the hider. %It follows that an optimal search strategy in the search game must be a Gittins index policy applied to a hiding distribution which is optimal from the hider's perspective. However, to avoid giving the hider any advantage, the searcher must carefully choose such a Gittins index policy among the many available. As a result, finding an optimal search strategy, or even proving one exists, is difficult. We extend several results for special cases from the literature to the fully-general search game; in particular, we show an optimal search strategy exists and may take a simple form. Using a novel test, we investigate the frequency of the optimality of a particular hiding strategy that gives the searcher no preference over any location at the beginning of the search

Computer Science for Continuous Data:Survey, Vision, Theory, and Practice of a Computer Analysis System

Building on George Boole's work, Logic provides a rigorous foundation for the powerful tools in Computer Science that underlie nowadays ubiquitous processing of discrete data, such as strings or graphs. Concerning continuous data, already Alan Turing had applied "his" machines to formalize and study the processing of real numbers: an aspect of his oeuvre that we transform from theory to practice.The present essay surveys the state of the art and envisions the future of Computer Science for continuous data: natively, beyond brute-force discretization, based on and guided by and extending classical discrete Computer Science, as bridge between Pure and Applied Mathematics

Unreliable and resource-constrained decoding

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student submitted PDF version of thesis.Includes bibliographical references (p. 185-213).Traditional information theory and communication theory assume that decoders are noiseless and operate without transient or permanent faults. Decoders are also traditionally assumed to be unconstrained in physical resources like material, memory, and energy. This thesis studies how constraining reliability and resources in the decoder limits the performance of communication systems. Five communication problems are investigated. Broadly speaking these are communication using decoders that are wiring cost-limited, that are memory-limited, that are noisy, that fail catastrophically, and that simultaneously harvest information and energy. For each of these problems, fundamental trade-offs between communication system performance and reliability or resource consumption are established. For decoding repetition codes using consensus decoding circuits, the optimal tradeoff between decoding speed and quadratic wiring cost is defined and established. Designing optimal circuits is shown to be NP-complete, but is carried out for small circuit size. The natural relaxation to the integer circuit design problem is shown to be a reverse convex program. Random circuit topologies are also investigated. Uncoded transmission is investigated when a population of heterogeneous sources must be categorized due to decoder memory constraints. Quantizers that are optimal for mean Bayes risk error, a novel fidelity criterion, are designed. Human decision making in segregated populations is also studied with this framework. The ratio between the costs of false alarms and missed detections is also shown to fundamentally affect the essential nature of discrimination. The effect of noise on iterative message-passing decoders for low-density parity check (LDPC) codes is studied. Concentration of decoding performance around its average is shown to hold. Density evolution equations for noisy decoders are derived. Decoding thresholds degrade smoothly as decoder noise increases, and in certain cases, arbitrarily small final error probability is achievable despite decoder noisiness. Precise information storage capacity results for reliable memory systems constructed from unreliable components are also provided. Limits to communicating over systems that fail at random times are established. Communication with arbitrarily small probability of error is not possible, but schemes that optimize transmission volume communicated at fixed maximum message error probabilities are determined. System state feedback is shown not to improve performance. For optimal communication with decoders that simultaneously harvest information and energy, a coding theorem that establishes the fundamental trade-off between the rates at which energy and reliable information can be transmitted over a single line is proven. The capacity-power function is computed for several channels; it is non-increasing and concave.by Lav R. Varshney.Ph.D

Query processing in complex modern traffic networks

The transport sector generates about one quarter of all greenhouse gas emissions worldwide. In the European Union (EU), passenger cars and light-duty trucks make up for over half of these traffic-related emissions. It is evident that everyday traffic is a serious environmental threat. At the same time, transport is a key factor for the ambitious EU climate goals; among them, for instance, the reduction of greenhouse gas emissions by 85 to 90 percent in the next 35 years. This thesis investigates complex traffic networks and their requirements from a computer science perspective. Modeling of and query processing in modern traffic networks are pivotal topics. Challenging theoretical problems are examined from different perspectives, novel algorithmic solutions are provided. Practical problems are investigated and solved, for instance, employing qualitative crowdsourced information and sensor data of various sources. Modern traffic networks are often modeled as graphs, i.e., defined by sets of nodes and edges. In conventional graphs, the edges are assigned numerical weights, for instance, reflecting cost criteria like distance or travel time. In multicriteria networks, the edges reflect multiple, possibly dynamically changing cost criteria. While these networks allow for diverse queries and meaningful insight, query processing usually is significantly more complex. Novel means for computation are required to keep query processing efficient. The crucial task of computing optimal paths is particularly expensive under multiple criteria. The most established set of optimal paths in multicriteria networks is referred to as path skyline (or set of pareto-optimal paths). Until now, computing the path skyline either required extensive precomputation or networks of minor size or complexity. Neither of these demands can be made on modern traffic networks. This thesis presents a novel method which makes on-the-fly computation of path skylines possible, even in dynamic networks with three or more cost criteria. Another problem examined is the exponentially growth of path skylines. The number of elements in a path skyline is potentially exponential in the number of cost criteria and the number of edges between start and target. This often produces less meaningful results, sometimes hindering usability. These drawbacks emphasize the importance of the linear path skyline which is investigated in this thesis. The linear path skyline is based on a different notion of optimality. By the notion of optimality, the linear path skyline is a subset of the conventional path skyline but in general contains less and more diverse elements. Thus, the linear path skyline facilitates interpretation while in general reducing computational effort. This topic is first studied in networks with two cost criteria and subsequently extended to more cost criteria. These cost criteria are not limited to purely quantitative measures like distance and travel time. This thesis examines the integration of qualitative information into abstractly modeled road networks. It is proposed to mine crowdsourced data for qualitative information and use this information to enrich road network graphs. These enriched networks may in turn be used to produce routing suggestions which reflect an opinion of the crowd. From data processing to knowledge extracting, network enrichment and route computation, the possibilities and challenges of crowdsourced data as a source for information are surveyed. Additionally, this thesis substantiates the practicability of network enrichment in real-world experiments. The description of a demonstration framework which applies some of the presented methods to the use case of tourist route recommendation serves as an example. The methods may also be applied to a novel graph-based routing problem proposed in this thesis. The problem extends the family of Orienteering Problems which find frequent application in tourist routing and other tasks. An approximate solution to this NP-hard problem is presented and evaluated on a large scale, real-world, time-dependent road network. Another central aspect of modern traffic networks is the integration of sensor data, often referred to as telematics. Nowadays, manifold sensors provide a plethora of data. Using this data to optimize traffic is and will continue to be a challenging task for research and industry. Some of the applications which qualify for the integration of modern telematics are surveyed in this thesis. For instance, the abstract problem of consumable and reoccurring resources in road networks is studied. An application of this problem is the search for a vacant parking space. Taking statistical and real-time sensor information into account, a stochastic routing algorithm which maximizes the probability of finding a vacant space is proposed. Furthermore, the thesis presents means for the extraction of driving preferences, helping to better understand user behavior in traffic. The theoretical concepts partially find application in a demonstration framework described in this thesis. This framework provides features which were developed for a real-world pilot project on the topics of electric and shared mobility. Actual sensor car data collected in the project, gives insight to the challenges of managing a fleet of electric vehicles.Verkehrsmittel erzeugen rund ein Viertel aller Treibhausgas-Emissionen weltweit. Für über die Hälfte der verkehrsbedingten Emissionen in der Europäischen Union (EU) zeichnen PKW und Kleinlaster verantwortlich. Die Tragweite ökologischer Konsequenzen durch alltäglichen Verkehr ist enorm. Zugleich ist ein Umdenken im Bezug auf Verkehr entscheidend, um die ehrgeizigen klimapolitischen Ziele der EU zu erfüllen. Dazu gehört unter anderem, Treibhausgas-Emissionen bis 2050 um 85 bis 90 Prozent zu verringern. Die vorliegende Arbeit widmet sich den komplexen Anforderungen an Verkehr und Verkehrsnetzwerke aus der Sicht der Informatik. Dabei spielen sowohl die Modellierung von als auch die Anfragebearbeitung in modernen Verkehrsnetzwerken eine entscheidende Rolle. Theoretische Fragestellungen werden aus unterschiedlichen Persepektiven beleuchtet, neue Algorithmen werden vorgestellt. Ebenso werden praktische Fragestellungen untersucht und gelöst, etwa durch die Einbindung nutzergenerierten Inhalts oder die Verwendung von Sensordaten aus unterschiedlichen Quellen. Moderne Verkehrsnetzwerke werden häufig als Graphen modelliert, d.h., durch Knoten und Kanten dargestellt. Man unterscheidet zwischen konventionellen Graphen und sogenannten Multiattributs-Graphen. Während die Kanten konventioneller Graphen numerische Gewichte tragen, die statische Kostenkriterien wie Distanz oder Reisezeit modellieren, beschreiben die Kantengewichte in Multiattributs-Graphen mehrere, möglicherweise dynamisch veränderliche Kostenkriterien. Das erlaubt einerseits vielseitige Anfragen und aussagekräftige Erkenntnisse, macht die Anfragebearbeitung jedoch ungleich komplexer und verlangt deshalb nach neuen Berechnungsmethoden. Eine besonders aufwendige Anfrage ist die Berechnung optimaler Pfade, zugleich eine der zentralsten Fragestellungen. Die gängigste Menge optimaler Pfade wird als Pfad-Skyline (auch: Menge der pareto-optimalen Pfade) bezeichnet. Die effiziente Berechnung der Pfad-Skyline setzte bisher überschaubare Netzwerke oder beträchtliche Vorberechnungen voraus. Keine der beiden Bedingung kann in modernen Verkehrsnetzwerken erfüllt werden. Diese Arbeit stellt deshalb eine Methode vor, die die Berechnung der Pfad-Skyline erheblich beschleunigt, selbst in dynamischen Netzwerken mit drei oder mehr Kostenkriterien. Außerdem wird das Problem des exponentiellen Wachstums der Pfad-Skyline betrachtet. Die Anzahl der Elemente der Pfad-Skyline wächst im schlechtesten Fall exponentiell in der Anzahl der Kostenkriterien sowie in der Entfernung zwischen Start und Ziel. Dies kann zu unübersichtlichen und wenig aussagekräftigen Resultatmengen führen. Diese Nachteile unterstreichen die Bedeutung der linearen Pfad-Skyline, die auch im Rahmen diese Arbeit untersucht wird. Die lineare Pfad-Skyline folgt einer anderen Definition von Optimalität. Stets ist die lineare Pfad-Skyline eine Teilmenge der konventionellen Pfad-Skyline, meist enthält sie deutlich weniger, unterschiedlichere Resultate. Dadurch lässt sich die lineare Pfad-Skyline im Allgemeinen schneller berechnen und erleichtert die Interpretation der Resultate. Die Berechnung der linearen Pfad-Skyline wird erst für Netzwerke mit zwei Kostenkriterien, anschließend für Netzwerke mit beliebig vielen Kostenkriterien untersucht. Kostenkriterien sind nicht notwendigerweise auf rein quantitative Maße wie Distanz oder Reisezeit beschränkt. Diese Arbeit widmet sich auch der Integration qualitativer Informationen, mit dem Ziel, intuitivere und greifbarere Routingergebnisse zu erzeugen. Dazu wird die Möglichkeit untersucht, abstrakte Straßennetzwerke mit qualitativen Informationen anzureichern, wobei die Informationen aus nutzergenerierten Daten geschöpft werden. Solche sogenannten Enriched Networks ermöglichen die Berechnung von Pfaden, die in gewisser Weise das Wissen der Nutzer reflektieren. Von der Datenverarbeitung, über die Extraktion von Wissen, bis hin zum Network-Enrichment und der Pfadberechnung, gibt diese Arbeit einen überblick zum Thema. Weiterhin wird die Praktikabilität dieses Vorgehens mit Experimenten auf Realdaten untermauert. Die Beschreibung eines Demonstrationstools für den Anwendungsfall der Navigation von Touristen dient als anschauliches Beispiel. Die vorgestellten Methoden sind darüber hinaus auch anwendbar auf ein neues, graphentheoretisches Routingproblem, das in dieser Arbeit vorgestellt wird. Es handelt sich dabei um eine zeitabängige Erweiterung der Familie der Orienteering Probleme, die häufig Anwendung finden, etwa auch im der Bereich der Touristennavigation. Das vorgestellte Problem ist NP-schwer lässt sich jedoch dank eines hier vorgestellten Algorithmus effizient approximieren. Die Evaluation untermauert die Effizienz des vorgestellten Lösungsansatzes und ist zugleich die erste Auswertung eines zeitabhängigen Orienteering Problems auf einem großformatigen Netzwerk. Ein weiterer zentraler Aspekt moderner Verkehrsnetzwerke ist die Integration von Sensordaten, oft unter dem Begriff Telematik zusammengefasst. Heutzutage generiert eine Vielzahl von Sensoren Unmengen an Daten. Diese Daten zur Verkehrsoptimierung einzusetzen ist und bleibt eine wichtige Aufgabe für Wissenschaft und Industrie. Einige der Anwendungen, die sich für den Einsatz von Telematik anbieten, werden in dieser Arbeit untersucht. So wird etwa das abstrakte Problem konsumierbarer und wiederkehrender Ressourcen im Straßennetzwerk untersucht. Ein alltägliches Beispiel für dieses Problem ist die Parkplatzsuche. Der vorgeschlagene Algorithmus, der die Wahrscheinlichkeit maximiert, einen freien Parkplatz zu finden, baut auf die Verwendung statistischer sowie aktueller Sensordaten. Weiterhin werden Methoden zur Ableitung von Fahrerpräferenzen entwickelt. Die theoretischen Fundamente finden zum Teil in einem hier beschriebenen Demonstrationstool Anwendung. Das Tool veranschaulicht Features, die für ein Pilotprojekt zu den Themen Elektromobilität und Fahrzeugflotten entwickelt wurden. Im Rahmen eines Pilotversuchs wurden Sensordaten von Elektrofahrzeugen erhoben, die Einblick in die Herausforderungen beim Management von Elektrofahrzeugflotten geben