The history of degenerate (bipartite) extremal graph problems

This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version of our survey presented in Erdos 100. In this version 2 only a citation was complete

Some Applications of Hyperbolic Geometry in String Perturbation Theory

In this thesis, we explore some applications of recent developments in the hyperbolic geometry of Riemann surfaces and moduli spaces thereof in string theory. First we show how a proper decomposition of the moduli space of hyperbolic surfaces can be achieved using the hyperbolic parameters. The decomposition is appropriate to define off-shell amplitudes in bosonic-string, heterotic-string and type-II superstring theories. Since the off-shell amplitudes in bosonic-string theory are dependent on the choice of local coordinates around the punctures, we associate local coordinates around the punctures in various regions of the moduli space. The next ingredient to define the off-shell amplitudes is to provide a method to integrate the off-shell string measure over the moduli space of hyperbolic surfaces. We next show how the integrals appearing in the definition of bosonic-string, heterotic-string and type-II superstring amplitudes can be computed by lifting them to appropriate covering spaces of the moduli space. In heterotic-string and typeII superstring theories, we also need to provide a proper distribution of picture-changing operators. We provide such a distribution. Finally, we illustrate the whole construction in few examples. We then describe the construction of a consistent string field theory using the tools from hyperbolic geometry

Path planning for robotic truss assembly

A new Potential Fields approach to the robotic path planning problem is proposed and implemented. Our approach, which is based on one originally proposed by Munger, computes an incremental joint vector based upon attraction to a goal and repulsion from obstacles. By repetitively adding and computing these 'steps', it is hoped (but not guaranteed) that the robot will reach its goal. An attractive force exerted by the goal is found by solving for the the minimum norm solution to the linear Jacobian equation. A repulsive force between obstacles and the robot's links is used to avoid collisions. Its magnitude is inversely proportional to the distance. Together, these forces make the goal the global minimum potential point, but local minima can stop the robot from ever reaching that point. Our approach improves on a basic, potential field paradigm developed by Munger by using an active, adaptive field - what we will call a 'flexible' potential field. Active fields are stronger when objects move towards one another and weaker when they move apart. An adaptive field's strength is individually tailored to be just strong enough to avoid any collision. In addition to the local planner, a global planning algorithm helps the planner to avoid local field minima by providing subgoals. These subgoals are based on the obstacles which caused the local planner to fail. A best-first search algorithm A* is used for graph search

Evolutionary dynamics of populations with genotype-phenotype map

In this thesis we develop a multi-scale model of the evolutionary dynamics of a population of cells, which accounts for the mapping between genotype and phenotype as determined by a model of the gene regulatory network. We study topological properties of genotype-phenotype networks obtained from the multi-scale model. Moreover, we study the problem of evolutionary escape and survival taking into account a genotype-phenotype map. An outstanding feature of populations with genotype-phenotype map is that selective pressures are determined by the phenotype, rather than genotypes. Our multi-scale model generates the evolution of a genotype-phenotype network represented by a pseudo-bipartite graph, that allows formulate a topological definition of the concepts of robustness and evolvability. We further study the problem of evolutionary escape for cell populations with genotype-phenotype map, based on a multi-type branching process. We present a comparative analysis between genotype-phenotype networks obtained from the multi-scale model and networks constructed assuming that the genotype space is a regular hypercube. We compare the effects on the probability of escape and the escape rate associated to the evolutionary dynamics between both classes of graphs. We further the study of evolutionary escape by analysing the long term survival conditioned to escape. Traditional approaches to the study of escape assume that the reproduction number of the escape genotype approaches infinity, and, therefore, survival is a surrogate of escape. Here, we analyse the process of survival upon escape by taking advantage of the fact that the natural setting of the escape problem endows the system with a separation of time scales: an initial, fast-decaying regime where escape actually occurs, is followed by a much slower dynamics within the (neutral network of) the escape phenotype. The probability of survival is analysed in terms of topological features of the neutral network of the escape phenotype.En aquesta tesi es desenvolupa un model multi-escala de la dinàmica evolutiva d'una població de cèl·lules, tenint en compte la correspondència entre el genotip i el fenotip determinat per un model de la xarxa de regulació genètica. Estudiem les propietats topològiques de les xarxes genotip-fenotip obtingudes a partir del model multi-escala. D'altra banda, s'estudia el problema de la fugida evolutiva i la supervivència, tenint en compte una aplicació entre genotip i fenotip. Una característica destacable de les poblacions amb aplicació genotip-fenotip és que les pressions selectives actuen sobre els fenotips, en lloc dels genotips. El nostre model multi-escala genera l'evolució d'una xarxa genotip-fenotip representada per un graf pseudo-bipartit, el qual permet formular una definició topològica dels conceptes de robustesa y capacitat evolutiva. A més a més, estudiem el problema de fugida evolutiva de poblacions de cèl¿lules amb una aplicació genotip-fenotip, basat en en un procés de ramificació multi-tipus. Presentem un anàlisi comparatiu entre les xarxes de genotip-fenotip obtingudes a partir del model multi-escala i les xarxes construïdes assumint un espai de genotips de tipus hipercub regular. Comparem els efectes de la probabilitat de fugida i la freqüència d'escapament associades a la dinàmica evolutiva entre ambdues classes de grafs. Anem més enllà de l'estudi de fugida evolutiva mitjançant l'anàlisi de la supervivència a llarg plaç condicionat a fugir. Els enfocaments tradicionals per a l'estudi de la fugida o escapament suposen una taxa de reproducció en el genotip de fugida propera a infinit. Per tant, la supervivència és equivalent a la fugida. Aquí analitzem el procés de supervivència suposant fugida aprofitant el fet que l'entorn natural del problema de fugida dota al sistema amb una separació d'escales de temps: un règim inicial, de temps ràpid, on la fugida realment es produeix; seguit d'una dinàmica molt més lenta dins de la (xarxa neutra del) fenotip de fugida. La probabilitat de supervivència s'analitza en termes de les característiques topològiques de la xarxa neutra del fenotip de fugidaPostprint (published version

Distributed cooperation of multiple robots under operational constraints via lean communication

Η αυτόνομη λειτουργία των ρομπότ εντός περίπλοκων χώρων εργασίας αποτελεί ένα επίκαιρο θέμα έρευνας και η αυτόνομη πλοήγηση είναι αναμφισβήτητα ένα θεμελιώδες κομμάτι αυτής. Επιπλέον, καθώς οι εργασίες που τα ρομπότ καλούνται να εκπληρώσουν αυξάνονται σε πολυπλοκότητα μέρα με τη μέρα, η χρήση πολύ-ρομποτικών συστημάτων, τα οποία εμφανίζουν γενικά υψηλότερη ευρωστία και ευελιξία, αυξάνεται προοδευτικά. Ως εκ τούτου, τα προβλήματα αυτόνομης πλοήγησης που πρέπει να επιλυθούν γίνονται όλο και πιο απαιτητικά, αυξάνοντας την ανάγκη για πιο αποτελεσματικά και σθεναρά σχήματα σχεδιασμού πορείας και κίνησης

Data compression and harmonic analysis

In this paper we review some recent interactions between harmonic analysis and data compression. The story goes back of course to Shannon’

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum