50 research outputs found

    High Dimensional Expanders and Property Testing

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    We show that the high dimensional expansion property as defined by Gromov, Linial and Meshulam, for simplicial complexes is a form of testability. Namely, a simplicial complex is a high dimensional expander iff a suitable property is testable. Using this connection, we derive several testability results

    Sublinear-Time Computation in the Presence of Online Erasures

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    We initiate the study of sublinear-time algorithms that access their input via an online adversarial erasure oracle. After answering each query to the input object, such an oracle can erase tt input values. Our goal is to understand the complexity of basic computational tasks in extremely adversarial situations, where the algorithm's access to data is blocked during the execution of the algorithm in response to its actions. Specifically, we focus on property testing in the model with online erasures. We show that two fundamental properties of functions, linearity and quadraticity, can be tested for constant tt with asymptotically the same complexity as in the standard property testing model. For linearity testing, we prove tight bounds in terms of tt, showing that the query complexity is Θ(logt)\Theta(\log t). In contrast to linearity and quadraticity, some other properties, including sortedness and the Lipschitz property of sequences, cannot be tested at all, even for t=1t=1. Our investigation leads to a deeper understanding of the structure of violations of linearity and other widely studied properties. We also consider implications of our results for algorithms that are resilient to online adversarial corruptions instead of erasures

    Computação quântica : autômatos, jogos e complexidade

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    Orientador: Arnaldo Vieira MouraDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: Desde seu surgimento, Teoria da Computação tem lidado com modelos computacionais de maneira matemática e abstrata. A noção de computação eficiente foi investigada usando esses modelos sem procurar entender as capacidades e limitações inerentes ao mundo físico. A Computação Quântica representa uma ruptura com esse paradigma. Enraizada nos postulados da Mecânica Quântica, ela é capaz de atribuir um sentido físico preciso à computação segundo nosso melhor entendimento da natureza. Esses postulados dão origem a propriedades fundamentalmente diferentes, uma em especial, chamada emaranhamento, é de importância central para computação e processamento de informação. O emaranhamento captura uma noção de correlação que é única a modelos quânticos. Essas correlações quânticas podem ser mais fortes do que qualquer correlação clássica estando dessa forma no coração de algumas capacidades quânticas que vão além do clássico. Nessa dissertação, nós investigamos o emaranhamento da perspectiva da complexidade computacional quântica. Mais precisamente, nós estudamos uma classe bem conhecida, definida em termos de verificação de provas, em que um verificador tem acesso à múltiplas provas não emaranhadas (QMA(k)). Assumir que as provas não contêm correlações quânticas parece ser uma hipótese não trivial, potencialmente fazendo com que essa classe seja maior do que aquela em que há apenas uma prova. Contudo, encontrar cotas de complexidade justas para QMA(k) permanece uma questão central sem resposta por mais de uma década. Nesse contexto, nossa contribuição é tripla. Primeiramente, estudamos classes relacionadas mostrando como alguns recursos computacionais podem afetar seu poder de forma a melhorar a compreensão a respeito da própria classe QMA(k). Em seguida, estabelecemos uma relação entre Probabilistically Checkable Proofs (PCP) clássicos e QMA(k). Isso nos permite recuperar resultados conhecidos de maneira unificada e simplificada. Para finalizar essa parte, mostramos que alguns caminhos para responder essa questão em aberto estão obstruídos por dificuldades computacionais. Em um segundo momento, voltamos nossa atenção para modelos restritos de computação quântica, mais especificamente, autômatos quânticos finitos. Um modelo conhecido como Two-way Quantum Classical Finite Automaton (2QCFA) é o objeto principal de nossa pesquisa. Seu estudo tem o intuito de revelar o poder computacional provido por memória quântica de dimensão finita. Nos estendemos esse autômato com a capacidade de colocar um número finito de marcadores na fita de entrada. Para qualquer número de marcadores, mostramos que essa extensão é mais poderosa do que seus análogos clássicos determinístico e probabilístico. Além de trazer avanços em duas linhas complementares de pesquisa, essa dissertação provê uma vasta exposição a ambos os campos: complexidade computacional e autômatosAbstract: Since its inception, Theoretical Computer Science has dealt with models of computation primarily in a very abstract and mathematical way. The notion of efficient computation was investigated using these models mainly without seeking to understand the inherent capabilities and limitations of the actual physical world. In this regard, Quantum Computing represents a rupture with respect to this paradigm. Rooted on the postulates of Quantum Mechanics, it is able to attribute a precise physical notion to computation as far as our understanding of nature goes. These postulates give rise to fundamentally different properties one of which, namely entanglement, is of central importance to computation and information processing tasks. Entanglement captures a notion of correlation unique to quantum models. This quantum correlation can be stronger than any classical one, thus being at the heart of some quantum super-classical capabilities. In this thesis, we investigate entanglement from the perspective of quantum computational complexity. More precisely, we study a well known complexity class, defined in terms of proof verification, in which a verifier has access to multiple unentangled quantum proofs (QMA(k)). Assuming the proofs do not exhibit quantum correlations seems to be a non-trivial hypothesis, potentially making this class larger than the one in which only a single proof is given. Notwithstanding, finding tight complexity bounds for QMA(k) has been a central open question in quantum complexity for over a decade. In this context, our contributions are threefold. Firstly, we study closely related classes showing how computational resources may affect its power in order to shed some light on \QMA(k) itself. Secondly, we establish a relationship between classical Probabilistically Checkable Proofs and QMA(k) allowing us to recover known results in unified and simplified way, besides exposing the interplay between them. Thirdly, we show that some paths to settle this open question are obstructed by computational hardness. In a second moment, we turn our attention to restricted models of quantum computation, more specifically, quantum finite automata. A model known as Two-way Quantum Classical Finite Automaton (2QCFA) is the main object of our inquiry. Its study is intended to reveal the computational power provided by finite dimensional quantum memory. We extend this automaton with the capability of placing a finite number of markers in the input tape. For any number of markers, we show that this extension is more powerful than its classical deterministic and probabilistic analogues. Besides bringing advances to these two complementary lines of inquiry, this thesis also provides a vast exposition to both subjects: computational complexity and automata theoryMestradoCiência da ComputaçãoMestre em Ciência da Computaçã

    Flood propagation modelling with the Local Inertia Approximation: theoretical and numerical analysis of its physical limitations

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    Attention of the researchers has increased towards a simplification of the complete Shallow water Equations called the Local Inertia Approximation (LInA), which is obtained by neglecting the advection term in the momentum conservation equation. In the present paper it is demonstrated that a shock is always developed at moving wetting-drying frontiers, and this justifies the study of the Riemann problem on even and uneven beds. In particular, the general exact solution for the Riemann problem on horizontal frictionless bed is given, together with the exact solution of the non-breaking wave propagating on horizontal bed with friction, while some example solution is given for the Riemann problem on discontinuous bed. From this analysis, it follows that drying of the wet bed is forbidden in the LInA model, and that there are initial conditions for which the Riemann problem has no solution on smoothly varying bed. In addition, propagation of the flood on discontinuous sloping bed is impossible if the bed drops height have the same order of magnitude of the moving-frontier shock height. Finally, it is found that the conservation of the mechanical energy is violated. It is evident that all these findings pose a severe limit to the application of the model. The numerical analysis has proven that LInA numerical models may produce numerical solutions, which are unreliable because of mere algorithmic nature, also in the case that the LInA mathematical solutions do not exist. The applicability limits of the LInA model are discouragingly severe, even if the bed elevation varies continuously. More important, the non-existence of the LInA solution in the case of discontinuous topography and the non-existence of receding fronts radically question the viability of the LInA model in realistic cases. It is evident that classic SWE models should be preferred in the majority of the practical applications

    Maintaining secrecy when information leakage is unavoidable

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    Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2004.Includes bibliographical references (p. 109-115).(cont.) We apply the framework to get new results, creating (a) encryption schemes with very short keys, and (b) hash functions that leak no information about their input, yet-paradoxically-allow testing if a candidate vector is close to the input. One of the technical contributions of this research is to provide new, cryptographic uses of mathematical tools from complexity theory known as randomness extractors.Sharing and maintaining long, random keys is one of the central problems in cryptography. This thesis provides about ensuring the security of a cryptographic key when partial information about it has been, or must be, leaked to an adversary. We consider two basic approaches: 1. Extracting a new, shorter, secret key from one that has been partially compromised. Specifically, we study the use of noisy data, such as biometrics and personal information, as cryptographic keys. Such data can vary drastically from one measurement to the next. We would like to store enough information to handle these variations, without having to rely on any secure storage-in particular, without storing the key itself in the clear. We solve the problem by casting it in terms of key extraction. We give a precise definition of what "security" should mean in this setting, and design practical, general solutions with rigorous analyses. Prior to this work, no solutions were known with satisfactory provable security guarantees. 2. Ensuring that whatever is revealed is not actually useful. This is most relevant when the key itself is sensitive-for example when it is based on a person's iris scan or Social Security Number. This second approach requires the user to have some control over exactly what information is revealed, but this is often the case: for example, if the user must reveal enough information to allow another user to correct errors in a corrupted key. How can the user ensure that whatever information the adversary learns is not useful to her? We answer by developing a theoretical framework for separating leaked information from useful information. Our definition strengthens the notion of entropic security, considered before in a few different contexts.by Adam Davison Smith.Ph.D

    Supporting research and advanced development - Space programs summary, volume 3

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    Research project reports on advanced systems analysis and engineering, telecommunication, solid propellants, guidance and control, and data system

    Radiómetro por transpiração transitória : desenvolvimento de um sensor de fluxo de calor para ambientes de elevada agressividade

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    Doutoramento em Engenharia MecânicaThe development of a new instrument for the measurement of convective and radiative is proposed, based on the transient operation of a transpiration radiometer. Current transpiration radiometers rely on steady state temperature measurements in a porous element crossed by a know gas mass flow. As a consequence of the porous sensing element’s intrinsically high thermal inertia, the instrument’s time constant is in the order of several seconds. The proposed instrument preserves established advantages of transpiration radiometers while incorporating additional features that broaden its applicability range. The most important developments are a significant reduction of the instrument’s response time and the possibility of separating and measuring the convective and radiative components of the heat flux. These objectives are achieved through the analysis of the instrument’s transient response, a pulsed gas flow being used to induce the transient behavior.Propõe-se o desenvolvimento de um novo instrumento para medição de fluxos de calor convectivos e radiativos, baseado na operação de um radiómetro de transpiração em regime transitório. Os radiómetros de transpiração atuais baseiam-se em medições de temperatura em regime estacionário num elemento poroso atravessado por um caudal mássico gasoso conhecido. Como consequência da inércia térmica intrinsecamente elevada do elemento sensível poroso, a constante de tempo do instrumento é da ordem dos segundos. O instrumento proposto preservará as vantagens estabelecidas dos radiómetros de transpiração incorporando características adicionais que alargarão a gama de aplicabilidade. As novas características mais importantes serão uma redução significativa do tempo de resposta do instrumento e a possibilidade de medir separadamente as componentes radiativa e convectiva do fluxo de calor. Estes objetivos serão conseguidos através da análise da resposta transitória do instrumento, utilizando-se um caudal pulsado de gás para induzir o comportamento transitório

    High-order renormalization of scalar quantum fields

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    Thema dieser Dissertation ist die Renormierung von perturbativer skalarer Quantenfeldtheorie bei großer Schleifenzahl. Der Hauptteil der Arbeit ist dem Einfluss von Renormierungsbedingungen auf renormierte Greenfunktionen gewidmet. Zunächst studieren wir Dyson-Schwinger-Gleichungen und die Renormierungsgruppe, inklusive der Gegenterme in dimensionaler Regularisierung. Anhand zahlreicher Beispiele illustrieren wir die verschiedenen Größen. Alsdann diskutieren wir, welche Freiheitsgrade ein Renormierungsschema hat und wie diese mit den Gegentermen und den renormierten Greenfunktionen zusammenhängen. Für ungekoppelte Dyson-Schwinger-Gleichungen stellen wir fest, dass alle Renormierungsschemata bis auf eine Verschiebung des Renormierungspunktes äquivalent sind. Die Verschiebung zwischen kinematischer Renormierung und Minimaler Subtraktion ist eine Funktion der Kopplung und des Regularisierungsparameters. Wir leiten eine neuartige Formel für den Fall einer linearen Dyson-Schwinger Gleichung vom Propagatortyp her, um die Verschiebung direkt aus der Mellintransformation des Integrationskerns zu berechnen. Schließlich berechnen wir obige Verschiebung störungstheoretisch für drei beispielhafte nichtlineare Dyson-Schwinger-Gleichungen und untersuchen das asymptotische Verhalten der Reihenkoeffizienten. Ein zweites Thema der vorliegenden Arbeit sind Diffeomorphismen der Feldvariable in einer Quantenfeldtheorie. Wir präsentieren eine Störungstheorie des Diffeomorphismusfeldes im Impulsraum und verifizieren, dass der Diffeomorphismus keinen Einfluss auf messbare Größen hat. Weiterhin untersuchen wir die Divergenzen des Diffeomorphismusfeldes und stellen fest, dass die Divergenzen Wardidentitäten erfüllen, die die Abwesenheit dieser Terme von der S-Matrix ausdrücken. Trotz der Wardidentitäten bleiben unendlich viele Divergenzen unbestimmt. Den Abschluss bildet ein Kommentar über die numerische Quadratur von Periodenintegralen.This thesis concerns the renormalization of perturbative quantum field theory. More precisely, we examine scalar quantum fields at high loop order. The bulk of the thesis is devoted to the influence of renormalization conditions on the renormalized Green functions. Firstly, we perform a detailed review of Dyson-Schwinger equations and the renormalization group, including the counterterms in dimensional regularization. Using numerous examples, we illustrate how the various quantities are computable in a concrete case and which relations they satisfy. Secondly, we discuss which degrees of freedom are present in a renormalization scheme, and how they are related to counterterms and renormalized Green functions. We establish that, in the case of an un-coupled Dyson-Schwinger equation, all renormalization schemes are equivalent up to a shift in the renormalization point. The shift between kinematic renormalization and Minimal Subtraction is a function of the coupling and the regularization parameter. We derive a novel formula for the case of a linear propagator-type Dyson-Schwinger equation to compute the shift directly from the Mellin transform of the kernel. Thirdly, we compute the shift perturbatively for three examples of non-linear Dyson-Schwinger equations and examine the asymptotic growth of series coefficients. A second, smaller topic of the present thesis are diffeomorphisms of the field variable in a quantum field theory. We present the perturbation theory of the diffeomorphism field in momentum space and find that the diffeomorphism has no influence on measurable quantities. Moreover, we study the divergences in the diffeomorphism field and establish that they satisfy Ward identities, which ensure their absence from the S-matrix. Nevertheless, the Ward identities leave infinitely many divergences unspecified and the diffeomorphism theory is perturbatively unrenormalizable. Finally, we remark on a third topic, the numerical quadrature of Feynman periods
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