In this thesis we present and solve selected problems arising from digital geometry and combinatorics on words. We consider digital straight lines and, equivalently, upper mechanical words with positive irrational slopes a<1 and intercept 0. We formulate a continued fraction (CF) based description of their run-hierarchical structure. Paper I gives a theoretical basis for the CF-description of digital lines. We define for each irrational positive slope less than 1 a sequence of digitization parameters which fully specifies the run-hierarchical construction. In Paper II we use the digitization parameters in order to get a description of runs using only integers. We show that the CF-elements of the slopes contain the complete information about the run-hierarchical structure of the line. The index jump function introduced by the author indicates for each positive integer k the index of the CF-element which determines the shape of the digitization runs on level k. In Paper III we present the results for upper mechanical words and compare our CF-based formula with two well-known methods, one of which was formulated by Johann III Bernoulli and proven by Markov, while the second one is known as the standard sequences method. Due to the special treatment of some CF-elements equal to 1 (essential 1's in Paper IV), our method is currently the only one which reflects the run-hierarchical structure of upper mechanical words by analogy to digital lines. In Paper IV we define two equivalence relations on the set of all digital lines with positive irrational slopes a<1. One of them groups into classes all the lines with the same run length on all digitization levels, the second one groups the lines according to the run construction in terms of long and short runs on all levels. We analyse the equivalence classes with respect to minimal and maximal elements. In Paper V we take another look at the equivalence relation defined by run construction, this time independently of the context, which makes the results more general. In Paper VI we define a run-construction encoding operator, by analogy with the well-known run-length encoding operator. We formulate and present a proof of a fixed-point theorem for Sturmian words. We show that in each equivalence class under the relation based on run length on all digitization levels (as defined in Paper IV), there exists exactly one fixed point of the run-construction encoding operator.

With the simple means of a digitized map and the wheel speed signals, it is possible to position a vehicle with an accuracy comparable to GPS. The positioning problem is a non-linear filtering problem and a particle filter has been applied to solve it. Two new approaches studied are the Auxiliary Particle Filter (APF), that aims at lowerering the variance of the error, and Rao-Blackwellization that exploits the linearities in the model. The results show that these methods require problems of higher complexity to fully utilize their advantages. Another aspect in this thesis has been to handle off-road driving scenarios, using dead reckoning. An off road detection mechanism has been developed and the results show that off-road driving can be detected accurately. The algorithm has been successfully implemented on a hand-held computer by quantizing the particle filter while keeping good filter performance.

Consider the variational inequality (VI) x* ∈C, ‹Fx*, x - x* ›≥0, x∈C (*) where C is a nonempty closed convex subset of a real Hilbert space H and F : C¡÷ H is a monotone operator form C into H. It is known that if F is strongly monotone and Lipschitzian, then VI (*) is equivalently turned into a fixed point problem of a contraction; hence Banach's contraction principle applies. However, in the case where F is inverse strongly monotone, VI (*) is equivalently transformed into a fixed point problem of a nonexpansive mapping. The purpose of this paper is to present some results which apply iterative methods for nonexpansive mappings to solve VI (*). We introduce Mann's algorithm and Halpern's algorithm and prove that the sequences generated by these algorithms converge weakly and respectively, strongly to a solution of VI (*), under appropriate conditions imposed on the parameter sequences in the algorithms.

A central component of modern computing is the idea that computation requires determinism. Contrary to this belief, the primary contribution of this work shows that useful computation can be accomplished in an error-prone fashion. Focusing on low-power computing and the increasing push toward energy conservation, the work seeks to sacrifice accuracy in exchange for energy savings. Probabilistic computing forms the basis for this error-prone computation by diverging from the requirement of determinism and allowing for randomness within computing. Implemented as probabilistic CMOS (PCMOS), the approach realizes enormous energy sav- ings in applications that require probability at an algorithmic level. Extending probabilistic computing to applications that are inherently deterministic, the biased voltage overscaling (BIVOS) technique presented here constrains the randomness introduced through PCMOS. Doing so, BIVOS is able to limit the magnitude of any resulting deviations and realizes energy savings with minimal impact to application quality. Implemented for a ripple-carry adder, array multiplier, and finite-impulse-response (FIR) filter; a BIVOS solution substantially reduces energy consumption and does so with im- proved error rates compared to an energy equivalent reduced-precision solution. When applied to H.264 video decoding, a BIVOS solution is able to achieve a 33.9% reduction in energy consumption while maintaining a peak-signal-to-noise ratio of 35.0dB (compared to 14.3dB for a comparable reduced-precision solution). While the work presented here focuses on a specific technology, the technique realized through BIVOS has far broader implications. It is the departure from the conventional mindset that useful computation requires determinism that represents the primary innovation of this work. With applicability to emerging and yet to be discovered technologies, BIVOS has the potential to contribute to computing in a variety of fashions.

<p>With the simple means of a digitized map and the wheel speed signals, it is possible to position a vehicle with an accuracy comparable to GPS. The positioning problem is a non-linear filtering problem and a particle filter has been applied to solve it. Two new approaches studied are the Auxiliary Particle Filter (APF), that aims at lowerering the variance of the error, and Rao-Blackwellization that exploits the linearities in the model. The results show that these methods require problems of higher complexity to fully utilize their advantages.</p><p>Another aspect in this thesis has been to handle off-road driving scenarios, using dead reckoning. An off road detection mechanism has been developed and the results show that off-road driving can be detected accurately. The algorithm has been successfully implemented on a hand-held computer by quantizing the particle filter while keeping good filter performance.</p>

The increasing complexity of heterogeneous cellular networks (HetNets) due to the irregular deployment of small cells demands significant rethinking in the way cellular networks are perceived, modeled and analyzed. In addition to threatening the relevance of classical models, this new network paradigm also raises questions regarding the feasibility of state-of-the-art simulation-based approach for system design. This dissertation proposes a fundamentally new approach based on random spatial models that is not only tractable but also captures current deployment trends fairly accurately. First, this dissertation presents a general baseline model for HetNets consisting of K different types of base stations (BSs) that may differ in terms of transmit power, deployment density and target rate. Modeling the locations of each class of BSs as an independent Poisson Point Process (PPP) allows the derivation of surprisingly simple expressions for coverage probability and average rate. One interpretation of these results is that adding more BSs or tiers does not necessarily change the coverage probability, which indicates that fears of "interference overload" in HetNets are probably overblown. Second, a flexible notion of BS load is incorporated by introducing a new idea of conditionally thinning the interference field. For this generalized model, the coverage probability is shown to increase when lightly loaded small cells are added to the existing macrocellular networks. This is due to the fact that owing to the smaller loads, small cells typically transmit less often than macrocells, thus contributing less to the interference power. The same idea of conditional thinning is also shown to be useful in modeling the non-uniform user distributions, especially when the users lie closer to the BSs. Third, the baseline model is extended to study multi-antenna HetNets, where BSs across tiers may additionally differ in terms of the number of transmit antennas, number of users served and the multi-antenna transmission strategy. Using novel tools from stochastic orders, a tractable framework is developed to compare the performance of various multi-antenna transmission strategies for a fairly general spatial model, where the BSs may follow any general stationary distribution. The analysis shows that for a given total number of transmit antennas in the network, it is preferable to spread them across many single-antenna BSs vs. fewer multi-antenna BSs. Fourth, accounting for the load on the serving BS, downlink rate distribution is derived for a generalized cell selection model, where shadowing, following any general distribution, impacts cell selection while fading does not. This generalizes the baseline model and all its extensions, which either ignore the impact of channel randomness on cell selection or lumps all the sources of randomness into a single random variable. As an application of these results, it is shown that in certain regimes, shadowing naturally balances load across various tiers and hence reduces the need for artificial cell selection bias. Fifth and last, a slightly futuristic scenario of self-powered HetNets is considered, where each BS is powered solely by a self-contained energy harvesting module that may differ across tiers in terms of the energy harvesting rate and energy storage capacity. Since a BS may not always have sufficient energy, it may not always be available to serve users. This leads to a notion of availability region, which characterizes the fraction of time each type of BS can be made available under variety of strategies. One interpretation of this result is that the self-powered BSs do not suffer performance degradation due to the unreliability associated with energy harvesting if the availability vector corresponding to the optimal system performance lies in the availability region. / text

Η έννοια των δικτύων εμφανίζεται πολύ συχνά με διάφορες μορφές. Δίκτυο μπορούμε να θεωρήσουμε ένα σύνολο υπολογιστών που συνδέονται μεταξύ τους υπακούοντας σε κάποιο πρωτόκολλο επικοινωνίας αλλά και μια ομάδα ανθρώ- πων που συνδέονται μέσω κάποιας κοινωνικής σχέσης, ενός εργασιακού χώρου αλλά και ως χρήστες ενός forum ή μίας πλατφόρμας κοινωνικής δικτύωσης. Σε οποιαδήποτε περίπτωση, ένα δίκτυο μπορεί να αναπαρασταθεί με τη μορφή ενός γράφηματος, όπου οι κόμβοι αναπαριστούν τα άτομα/υπολογιστές και οι ακμές τη μεταξύ τους σχέση ανάλογα με το πρόβλημα. Στα πλαίσια ενός τέτοιου δικτύου μας ενδιαφέρει η συμπεριφορά των κόμβων στην περίπτωση που συμβεί ένα γεγονός που αλλάζει την κατάστασή τους. Στην περίπτωση που αναφερόμαστε σε μία κοινωνική ομάδα ή μία πόλη, αυτό το φαι- νόμενο μπορεί να είναι το ξέσπασμα μίας επιδημίας που εξαπλώνεται στο δίκτυο αλλά και μίας είδησης/φήμης, όπου ενημερώνεται το δίκτυο. Στην πρώτη περί- πτωση, μας ενδιαφέρει να περιορίσουμε την επιδημία, αλλάζοντας τοπολογικά το δίκτυο ενώ στη δεύτερη περίπτωση, είναι επιθυμητό να διευκολύνουμε την εξά- πλωση της είδησης, έτσι ώστε να ενημερωθούν όσο το δυνατόν, περισσότεροι κόμβοι(χρήστες). Η συμπεριφορά του δικτύου σε ένα τέτοιο φαινόμενο, μπορεί να προσομοιωθεί από ένα δυναμικό σύστημα. Με τον όρο δυναμικό σύστημα αναφερόμαστε σε ένα σύστημα που έχει ένα σύνολο καταστάσεων, όπου κάθε κατάσταση, προκύπτει σε συνάρτηση με την προηγούμενη. Παραδείγματα εφαρμογής ενός δυναμικού συστήματος σε δίκτυο, εμφανίζονται σε διάφορους τομείς όπως στην οικολογία, τη διάχυση πληροφορίας, το viral marketing, την επιδημιολογία. Τα δυναμικά συστήματα που προσομοιώνουν τη συμπεριφορά του δικτύου σε τέτοια φαινόμενα, χρησιμοποιούν επιδημιολογικά μοντέλα για να περιγρά- ψουν τις δυνατές καταστάσεις στις οποίες μπορεί να περιέλθει ένας κόμβος. Στη συγκεκριμένη εργασία, χρησιμοποιήσαμε το μοντέλο SIS (Susceptible-Infected-Susceptible) [8].Το μοντέλο SIS δηλώνει ότι ένας κόμβος μπορεί να είναι είτε επιρρεπής στο να ασθενήσει (susceptible) είτε ασθενής (infected). Αυτό σημαί- νει πως ένας κόμβος δεν θεραπεύεται ποτέ πλήρως αλλά υπάρχει πιθανότητα να ασθενήσει πάλι. Με βάση τη βιβλιογραφία, σε ένα τέτοιο δυναμικό σύστημα, αναζητούμε κά- ποια σημεία (fixed points) στα οποία το σύστημα θα ισορροπεί. Υπάρχουν σημεία τα οποία είναι σημεία ισορροπίας αλλά δεν είναι σταθερά. Σε αυτά τα σημεία, το σύστημα μπορεί στιγμιαία να ισορροπήσει αλλά ξεφεύγει πολύ εύκολα από αυτό. Αναζητούμε συνεπώς, σταθερά σημεία ισορροπίας, τα λεγόμενα stable fixed points. Έχει αποδειχτεί [6] ότι μπορούμε σε αυτά στα σημεία να καθορίσουμε τις απαραίτητες συνθήκες για να είναι σταθερά, περιορίζοντας τη μέγιστη ιδιοτιμή του μητρώου γειτνίασης που περιγράφει το δίκτυο. Ορίζονται δηλαδή κατώφλια (thresholds) κατά περίπτωση, που περιορίζουν την μέγιστη ιδιοτιμή του δικτύου με τέτοιο τρόπο ώστε το σύστημα, να βρίσκεται σε κατάσταση ισορροπίας. Στην πε- ρίπτωση που αναφερόμαστε στο φαινόμενο της επιδημίας, στόχος είναι στο αντί- στοιχο σημείο ισορροπίας η μέγιστη ιδιοτιμή να είναι κάτω του κατωφλίου, έτσι ώστε να εξασφαλίσουμε τον περιορισμό εξάπλωσης της επιδημίας στο δίκτυο. Στην περίπτωση που αναφερόμαστε σε μία είδηση ή ένα ανταγωνιστικό προϊόν, η μέγιστη ιδιοτιμή θέλουμε να είναι άνω του αντίστοιχου κατωφλίου έτσι ώστε να έχουμε εξάπλωση στο δίκτυο. Επομένως ανάλογα με την περίπτωση, αντιμε- τωπίζουμε διαφορετικά τα κατώφλια που υπολογίζονται για το αντίστοιχο σημείο ισορροπίας. Στα πλαίσια της μεταπτυχιακής εργασίας, χρησιμοποιήσαμε το μοντέλο SIS για να περιγράψουμε το φαινόμενο όπου ένας ιός εξαπλώνεται σε ένα δίκτυο όπου οι κόμβοι του δικτύου, έχουν διαφορετική ευαισθησία απέναντί του. Πραγματο- ποιήσαμε μαθηματική περιγραφή του μοντέλου, ορίζοντας τα απαραίτητα κατώ- φλια έτσι ώστε το σύστημα να ισορροπεί ανάλογα με το σημείο ισορροπίας αλλά και το είδος του γραφήματος. Επίσης, πραγματοποιήσαμε προσομοίωση του μο- ντέλου σε συνθετικά γραφήματα (κλίκα, αυθαίρετο γράφημα κ.α), επαληθεύοντας τη συμπεριφορά που υποδεικνύει το μαθηματικό μοντέλο. / Which is the appropriate answer about the definition of a network? One could answer that a group of people who share a relationship (colleagues, students etc) could be referred to, as a network. Another possible definition, is a computer network. Consequently, it is obvious that the idea of a network can be found in various ways in our daily life. In the same terms, suppose we have one competing idea/product or a virus that propagates over a multiple profile social (or other) network. Can we predict what proportion of the network will actually get ”infected” (e.g., spread the idea or buy the competing product), when the nodes of the network appear to have different sensitivity based on their profile? For example, if there are two profiles A, B in a network and the nodes of profile A and profile B are susceptible to a highly spreading virus with probabilities βA and βB respectively, what percentage of both profiles will actually get infected from the virus in the end? The behavior of such a network, can be simulated using dynamical systems theory. We consider a dynamical system as a system with a set of possible states where each future state, is computed based on the previous state. Dynamical System Applications, can be found in many fields such as viral marketing, ecology, information diffusion and virus propagation. In order to simulate the rumor or virus which is spreading across the network, one has to use virus propagation models. The selection of the appropriate model, depends on the special attributes and characteristics of the spreading rumor/virus and it should cover all the possible states in which a node in the network can be (sick, healthy,susceptible, informed, not informed etc). According to Dynamical Systems Theory, we are looking for possible fixed points where the system is in equilibrium. In particular, we would require each fixed point to be a stable attractor and not lead the system far away from the equilibrium point due to opposing forces (stable fixed points). It has been proven that limiting the leading eigenvalue of the adjacency matrix of the graph, is the only condition required, in order for the system to be in equilibrium state, in the corresponding fixed point. In this paper, we assume an SIS propagation model [8] which is applied on a heterogeneous network. That is, we assume that there is no fair game using the terminology of [14, 3]. In the SIS model, each node can be either in a susceptible (S) state or in the infected state (I) and as result there is no permanent immunity and every node can get infected multiple times.Since this is the first theoretical treatment of heterogeneous environments for virus propagation, we choose to work in the simple model of SIS and not in other models. Suppose that we are given a social network and a rumor that spreads over it, where the nodes of the network represent people with high/low sensitivity to the rumor and the links represent the association of the nodes, how will the rumor propagate over the network? That is, can we determine whether all members of the network will reproduce the rumor to their neighbors and ”infect” them or the rumor will spread in a small group in the network and die out quickly? Similarly, which is the tipping point where such a rumor or infectious virus will take off? It would be very helpful if we could find the specific point when the ”virus” spreads all over the network and an epidemic occurs. Finally, what is the case when the nodes have different endurance/sensitivity to the ”virus” and have temporary or permanent immunity? Our basic assumption and innovation when compared to all previous approaches is that there is no fair-play and nodes have different profile against the virus. That is, the network is heterogeneous with respect to the virus, which means that nodes have different sensitivity to it. This is one of our main contributions in comparison with previous results where all nodes appear to have the same behavior towards the virus and the same model parameters. The propagation model which is followed, resembles the SIS (no immunity like flu) model where nodes are either susceptible or infected but with modifications. All nodes can get infected from one another, despite the difference of their profiles. We prove and present the tipping point where the virus is about to spread all over the network or the rumor ”infect” every member of the network and result in a ”viral” phenomenon. Our main contribution, is that we provide answers for the questions above, for special topologies such as the clique as well as arbitrary graphs of high or low connectivity. In particular, to the best of our knowledge, we are the first to provide theoretical and experimental findings on the propagation of a virus over a heterogeneous network. We prove that in the case of two profiles, if one profile has high sensitivity to the virus and the other one has low sensitivity, actually nodes from both profiles will get infected proportionally in the case where the network is a clique. For arbitrary networks, we prove necessary conditions for the virus to die out allowing for multiple profiles (not just two), while at the same time we give directions to prove other interesting cases. The problem has many applications in the field of viral marketing, medicine, ecology and other.

Made available in DSpace on 2014-06-11T19:26:55Z (GMT). No. of bitstreams: 0 Previous issue date: 2009-02-27Bitstream added on 2014-06-13T18:30:54Z : No. of bitstreams: 1 galves_apt_me_sjrp.pdf: 719114 bytes, checksum: 3cc285d329c0d629cf2c6ff65c13a201 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Em Topologia, mais especificamente em Topologia Algébrica, temos alguns resultados clássicos que de alguma forma estão relacionados. No desenvolvimento deste trabalho, estudamos alguns desses resultados, a saber: Teorema de Lefschetz-Hopf, Teorema do Ponto Fixo de Lefschetz, Teorema do Ponto Fixo de Brouwer, Teorema da Curva de Jordan e o Teorema Clássico de Borsuk-Ulam. Além disso, tivemos como objetivo principal mostrar relações existentes entre esses teoremas a partir do Teorema de Lefschetz-Hopf. / In Topology, more specifically in Algebraic Topology, we have some classical results that are in some way related. In developing this work, we studied some of these results, namely the Lefschetz-Hopf Theorem, the Lefschetz Fixed Point Theorem, the Brouwer Fixed Point Theorem, the Jordan Curve Theorem and the Classic Borsuk-Ulam Theorem. Moreover, our main objective was to show relationships among those theorems by using Lefschetz-Hopf Theorem.

Made available in DSpace on 2014-06-11T19:27:10Z (GMT). No. of bitstreams: 0 Previous issue date: 2010-12-09Bitstream added on 2014-06-13T20:47:43Z : No. of bitstreams: 1 cobra_tt_me_rcla.pdf: 485593 bytes, checksum: 107d36859b5a9c932411b3a54094c4ac (MD5) / O principal objetivo deste trabalho é apresentar conceitos básicos sobre coincidências e pontos fixos de aplicações contínuas usando como ferramentas os Lemas Combinatórios de Sperner e grau de aplicações. Apresentamos também um cálculo do número de Lefschetz de f; g : T2 ¡! T3, onde Th denota uma superfície de genus h, através da fórmula dada por Gonçalves e Oliveira em [3] / The main goal of this work is present basic concepts on coincidences and fixed points of continuous maps with Sperner’s Combinatorial Lemmas, and degree maps approaches. We also present a calculation of the Lefschetz number of f; g : T2 ¡! T3, where Th denotes surface of genus h, by using the formula given by Gonçalves and Oliveira in [3]

In this work we will demonstrate that the Cauchy problem associated with the Korteweg-de Vries equation, denoted by KdV, and Korteweg-de Vries modified equation, denoted by mKdV, with initial data in the space of Sobolev Hs(|R), is locally well-posed on Hs(|R), with s>3/4 for KdV and s&#8805;1/4 for mKdV, where the notion of well-posedness includes existence, uniqueness, persistence property of solution and continuous dependence of solution with respect to the initial data. This result is based on the works of Kenig, Ponce and Vega. The technique used to obtain these results is based on fixed point Banach theorem combined with the regularizantes effects of the group associated with the linear part. / Fundação de Amparo a Pesquisa do Estado de Alagoas / Neste trabalho demonstraremos que o problema de Cauchy associado as equações de Korteweg-de Vries, denotada por KdV, e de Korteweg-de Vries modificada, denotada por mKdV, com dado inicial no espaço de Sobolev Hs(|R), é bem posto localmente em Hs(|R), com s>3/4 para a KdV e s&#8805;1/4 para a mKdV, onde a noção de boa postura inclui a existência, unicidade, a propriedade de persistência da solução e dependência contínua da solução com relação ao dado inicial. Este resultado é baseado nos trabalhos de Kenig, Ponce e Vega. A técnica utilizada para obter tais resultados se baseia no Teorema do Ponto Fixo de Banach combinada com os efeitos regularizantes do grupo associado com a parte linear.