Spectral graph theory with applications to quantum adiabatic optimization

In this dissertation I draw a connection between quantum adiabatic optimization, spectral graph theory, heat-diffusion, and sub-stochastic processes through the operators that govern these processes and their associated spectra. In particular, we study Hamiltonians which have recently become known as ``stoquastic'' or, equivalently, the generators of sub-stochastic processes. The operators corresponding to these Hamiltonians are of interest in all of the settings mentioned above. I predominantly explore the connection between the spectral gap of an operator, or the difference between the two lowest energies of that operator, and certain equilibrium behavior. In the context of adiabatic optimization, this corresponds to the likelihood of solving the optimization problem of interest. I will provide an instance of an optimization problem that is easy to solve classically, but leaves open the possibility to being difficult adiabatically. Aside from this concrete example, the work in this dissertation is predominantly mathematical and we focus on bounding the spectral gap. Our primary tool for doing this is spectral graph theory, which provides the most natural approach to this task by simply considering Dirichlet eigenvalues of subgraphs of host graphs. I will derive tight bounds for the gap of one-dimensional, hypercube, and general convex subgraphs. The techniques used will also adapt methods recently used by Andrews and Clutterbuck to prove the long-standing ``Fundamental Gap Conjecture''.

Undersampled Critical Branching Processes on Small-World and Random Networks Fail to Reproduce the Statistics of Spike Avalanches

The power-law size distributions obtained experimentally for neuronal avalanches are an important evidence of criticality in the brain. This evidence is supported by the fact that a critical branching process exhibits the same exponent t~3=2. Models at criticality have been employed to mimic avalanche propagation and explain the statistics observed experimentally. However, a crucial aspect of neuronal recordings has been almost completely neglected in the models: undersampling. While in a typical multielectrode array hundreds of neurons are recorded, in the same area of neuronal tissue tens of thousands of neurons can be found. Here we investigate the consequences of undersampling in models with three different topologies (two-dimensional, small-world and random network) and three different dynamical regimes (subcritical, critical and supercritical). We found that undersampling modifies avalanche size distributions, extinguishing the power laws observed in critical systems. Distributions from subcritical systems are also modified, but the shape of the undersampled distributions is more similar to that of a fully sampled system. Undersampled supercritical systems can recover the general characteristics of the fully sampled version, provided that enough neurons are measured. Undersampling in two-dimensional and small-world networks leads to similar effects, while the random network is insensitive to sampling density due to the lack of a well-defined neighborhood. We conjecture that neuronal avalanches recorded from local field potentials avoid undersampling effects due to the nature of this signal, but the same does not hold for spike avalanches. We conclude that undersampled branching-process-like models in these topologies fail to reproduce the statistics of spike avalanches.

A Recommender System based on Idiotypic Artificial Immune Networks

The immune system is a complex biological system with a highly distributed, adaptive and self-organising nature. This paper presents an Artificial Immune System (AIS) that exploits some of these characteristics and is applied to the task of film recommendation by Collaborative Filtering (CF). Natural evolution and in particular the immune system have not been designed for classical optimisation. However, for this problem, we are not interested in finding a single optimum. Rather we intend to identify a sub-set of good matches on which recommendations can be based. It is our hypothesis that an AIS built on two central aspects of the biological immune system will be an ideal candidate to achieve this: Antigen-antibody interaction for matching and idiotypic antibody-antibody interaction for diversity. Computational results are presented in support of this conjecture and compared to those found by other CF techniques.

Problems in number theory and hyperbolic geometry

In the first part of this thesis we generalize a theorem of Kiming and Olsson concerning the existence of Ramanujan-type congruences for a class of eta quotients. Specifically, we consider a class of generating functions analogous to the generating function of the partition function and establish a bound on the primes ℓ for which their coefficients c(n) obey congruences of the form c(ℓn + a) ≡ 0 (mod ℓ). We use this last result to answer a question of H.C. Chan. In the second part of this thesis [S2] we explore a natural analog of D. Calegari’s result that there are no hyperbolic once-punctured torus bundles over S^1 with trace field having a real place. We prove a contrasting theorem showing the existence of several infinite families of pairs (−χ, p) such that there exist hyperbolic surface bundles over S^1 with trace field of having a real place and with fiber having p punctures and Euler characteristic χ. This supports our conjecture that with finitely many known exceptions there exist such examples for each pair ( −χ, p).

A Recommender System based on the Immune Network

Abstract-The immune system is a complex biological system with a highly distributed, adaptive and self-organising nature. This paper presents an artificial immune system (AIS) that exploits some of these characteristics and is applied to the task of film recommendation by collaborative filtering (CF). Natural evolution and in particular the immune system have not been designed for classical optimisation. However, for this problem, we are not interested in finding a single optimum. Rather we intend to identify a sub-set of good matches on which recommendations can be based. It is our hypothesis that an AIS built on two central aspects of the biological immune system will be an ideal candidate to achieve this: Antigen - antibody interaction for matching and antibody - antibody interaction for diversity. Computational results are presented in support of this conjecture and compared to those found by other CF techniques.

An Artificial Immune System Based Recommender

The immune system is a complex biological system with a highly distributed, adaptive and self-organising nature. This paper presents an artificial immune system (AIS) that exploits some of these characteristics and is applied to the task of film recommendation by collaborative filtering (CF). Natural evolution and in particular the immune system have not been designed for classical optimisation. However, for this problem, we are not interested in finding a single optimum. Rather we intend to identify a sub-set of good matches on which recommendations can be based. It is our hypothesis that an AIS built on two central aspects of the biological immune system will be an ideal candidate to achieve this: Antigen - antibody interaction for matching and antibody - antibody interaction for diversity. Computational results are presented in support of this conjecture and compared to those found by other CF techniques. Notes: Uwe Aickelin, University of the West of England, Coldharbour Lane, Bristol, BS16 1QY, UK

A Recommender System based on Idiotypic Artificial Immune Networks

The immune system is a complex biological system with a highly distributed, adaptive and self-organising nature. This paper presents an Artificial Immune System (AIS) that exploits some of these characteristics and is applied to the task of film recommendation by Collaborative Filtering (CF). Natural evolution and in particular the immune system have not been designed for classical optimisation. However, for this problem, we are not interested in finding a single optimum. Rather we intend to identify a sub-set of good matches on which recommendations can be based. It is our hypothesis that an AIS built on two central aspects of the biological immune system will be an ideal candidate to achieve this: Antigen-antibody interaction for matching and idiotypic antibody-antibody interaction for diversity. Computational results are presented in support of this conjecture and compared to those found by other CF techniques.

Complex functions with Geogebra

Complex functions, generally feature some interesting peculiarities, seen as extensions real functions, complementing the study of real analysis. However, the visualization of some complex functions properties requires the simultaneous visualization of two-dimensional spaces. The multiple Windows of GeoGebra, combined with its ability of algebraic computation with complex numbers, allow the study of the functions defined from ℂ to ℂ through traditional techniques and by the use of Domain Colouring. Here, we will show how we can use GeoGebra for the study of complex functions, using several representations and creating tools which complement the tools already provided by the software. Our proposals designed for students of the first year of engineering and science courses can and should be used as an educational tool in collaborative learning environments. The main advantage in its use in individual terms is the promotion of the deductive reasoning (conjecture / proof). In performed the literature review few references were found involving this educational topic and by the use of a single software.

Two-State Thermodynamics of Supercooled Water

Water has been called the “most studied and least understood” of all liquids, and upon supercooling its behavior becomes even more anomalous. One particularly fruitful hypothesis posits a liquid-liquid critical point terminating a line of liquid-liquid phase transitions that lies just beyond the reach of experiment. Underlying this hypothesis is the conjecture that there is a competition between two distinct hydrogen-bonding structures of liquid water, one associated with high density and entropy and the other with low density and entropy. The competition between these structures is hypothesized to lead at very low temperatures to a phase transition between a phase rich in the high-density structure and one rich in the low-density structure. Equations of state based on this conjecture have given an excellent account of the thermodynamic properties of supercooled water. In this thesis, I extend that line of research. I treat supercooled aqueous solutions and anomalous behavior of the thermal conductivity of supercooled water. I also address supercooled water at negative pressures, leading to a framework for a coherent understanding of the thermodynamics of water at low temperatures. I supplement analysis of experimental results with data from the TIP4P/2005 model of water, and include an extensive analysis of the thermodynamics of this model.

Unpolarized transverse momentum dependent parton distribution and fragmentation functions at next-to-next-to-leading order

The transverse momentum dependent parton distribution/fragmentation functions (TMDs) are essential in the factorization of a number of processes like Drell-Yan scattering, vector boson production, semi-inclusive deep inelastic scattering, etc. We provide a comprehensive study of unpolarized TMDs at next-to-next-to-leading order, which includes an explicit calculation of these TMDs and an extraction of their matching coefficients onto their integrated analogues, for all flavor combinations. The obtained matching coefficients are important for any kind of phenomenology involving TMDs. In the present study each individual TMD is calculated without any reference to a specific process. We recover the known results for parton distribution functions and provide new results for the fragmentation functions. The results for the gluon transverse momentum dependent fragmentation functions are presented for the first time at one and two loops. We also discuss the structure of singularities of TMD operators and TMD matrix elements, crossing relations between TMD parton distribution functions and TMD fragmentation functions, and renormalization group equations. In addition, we consider the behavior of the matching coefficients at threshold and make a conjecture on their structure to all orders in perturbation theory.

Corpos tatuados : experiências sensíveis em realidade aumentada móvel

Tese (doutorado)—Universidade de Brasília, Instituto de Artes, Programa de Pós-Graduação em Artes, 2015.

A evolução técnica e orgânica das Unidades de Manobra de Cavalaria no pós-Guerra Fria (1993-2015)

A Guerra e os Exércitos que nela tomam parte, nunca se mantiveram como imutáveis ao longo de séculos de História Militar. Analisando os últimos 22 anos, verifica-se que existiram alterações vividas no contexto Político-Militar Internacional que, por sua vez, repercutiram-se e moldaram a conjetura nacional. São diversas as modificações reproduzindo-se em diversos níveis das estruturas Militares e, consequentemente, afetando a Arma de Cavalaria, também ela marcadamente alvo de alterações. Com o intuito de estudar parte dessas mesmas alterações, surge o presente relatório subordinando-se ao tema A evolução Técnica e Orgânica das Unidades de Manobra de Cavalaria no pós-Guerra Fria (1993-2015), enquadrado no Mestrado em Cavalaria ministrado pela Academia Militar. Com a elaboração do presente relatório, pretende-se concorrer para identificar uma parte relevante das alterações sofridas pelas Unidades de Manobra de Cavalaria, aos níveis técnico e orgânico, desde o fim da Guerra Fria até à atualidade. Com vista a alcançar este objetivo, numa abordagem dedutiva estudam-se as Unidades de Manobra da Cavalaria do Escalão Grupo até ao Escalão Pelotão e compara-se a sua evolução nos últimos 22 anos, mediante duas variáveis distintas. A primeira ao nível Orgânico, onde são abordados itens como a Composição Esquemática e o número de Efetivos explanados nos Quadros Orgânicos das respetivas Unidades. A segunda variável ao nível dos Principais Sistemas de Armas que equiparam e equipam as Unidades em estudo. Nesta variável são estudados itens como a Mobilidade e o Poder de Fogo desses mesmos Sistemas de Armas. No sentido de operacionalizar a comparação acima referida foi empregue o Método de Procedimento Comparativo, numa abordagem Mista, onde são utilizados dados recolhidos através de Análise Documental e de Pesquisa Bibliográfica. O Relatório inicia-se com um breve Quadro Conceptual, apresentando-se de seguida uma síntese relativa ao contexto Político-Militar Internacional vivido na Guerra Fria e Pós-Guerra Fria. Posteriormente, são apresentadas as Unidades de Manobra de Cavalaria e todos os resultados referentes à pesquisa. Fruto da análise feita, são várias as conclusões que foram sendo determinadas. Com respeito à Orgânica, conclui-se que existiram alterações significativas aos Quadros Orgânicos nos anos de 1993, 2006, 2009 e 2015. Neste âmbito, destacam-se ao Escalão Grupo a extinção do Grupo de Auto Metralhadoras e a criação do Grupo de Reconhecimento. Salientam-se ao Escalão Esquadrão mudanças na Composição Esquemática das Unidades, denotando-se ainda uma tendência para a diminuição dos meios que equipam as Unidades de Reconhecimento aprontadas pelo Regimento de Cavalaria 6 e Quartel de Cavalaria. O decréscimo dos Efetivos também é uma alteração patente nos 23 anos estudados. No que concerne aos Sistemas de Armas, salienta-se o Grupo de Carros de Combate como a Unidade com a maior evolução sofrida devido à transição por três Carros de Combate distintos durante o período em estudo. Verifica-se ainda uma disparidade entre os Sistemas de Armas que atualmente equipam as Unidades de Reconhecimento sediadas no Regimento de Cavalaria 6 e o Quartel de Cavalaria.

Shear-free perfect fluids with a γ-law equation of state

 We investigate all shear-free perfect fluid solutions of the Einstein field equations where the pressure and energy density satisfy a (Formula presented.)-law equation of state with (Formula presented.). We prove that such a fluid is either non rotating or non expanding. As a consequence, it follows by combining our result with those of Collins and Wainwright that any such shear-free perfect fluid which models either an expand universe or a collapsing star must in fact be a Friedmann–Robertson–Walker spacetime.

TensorPack : a Maple-based software package for the manipulation of algebraic expressions of tensors in general relativity

In this paper we: (1) introduce TensorPack, a software package for the algebraic manipulation of tensors in covariant index format in Maple; (2) briefly demonstrate the use of the package with an orthonormal tensor proof of the shearfree conjecture for dust. TensorPack is based on the Riemann and Canon tensor software packages and uses their functions to express tensors in an indexed covariant format. TensorPack uses a string representation as input and provides functions for output in index form. It extends the functionality to basic algebra of tensors, substitution, covariant differentiation, contraction, raising/lowering indices, symmetry functions and other accessory functions. The output can be merged with text in the Maple environment to create a full working document with embedded dynamic functionality. The package offers potential for manipulation of indexed algebraic tensor expressions in a flexible software environment.

Ill-defined versus precise pre-play communication in the traveler's dilemma

This article reports results from an experiment comparing the effects of vague versus precise pre-play communication in a highly competitive two-player game with conflicting interests. In the classic Traveler's Dilemma, non-binding precise messages about intent of play are pure cheap talk. We conjecture that a form of imprecise pre-play communication whereby subjects can submit ill-defined messages may help foster cooperation because of their vagueness. Comparing behavior both across modes of communication and to a baseline case without communication, we find that cooperation is highest when players can communicate using precise numerical messages. When communication with ill-defined messages is allowed, then conditional on receiving a message, subjects act more cooperatively than when no message is received. However, overall, the ability to exchange ill-defined messages does not substantially improve cooperation.