On the resolution of the generalized nonlinear complementarity problem

Minimization of a differentiable function subject to box constraints is proposed as a strategy to solve the generalized nonlinear complementarity problem ( GNCP) defined on a polyhedral cone. It is not necessary to calculate projections that complicate and sometimes even disable the implementation of algorithms for solving these kinds of problems. Theoretical results that relate stationary points of the function that is minimized to the solutions of the GNCP are presented. Perturbations of the GNCP are also considered, and results are obtained related to the resolution of GNCPs with very general assumptions on the data. These theoretical results show that local methods for box-constrained optimization applied to the associated problem are efficient tools for solving the GNCP. Numerical experiments are presented that encourage the use of this approach.Minimization of a differentiable function subject to box constraints is proposed as a strategy to solve the generalized nonlinear complementarity problem ( GNCP) defined on a polyhedral cone. It is not necessary to calculate projections that complicate an122303321CNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICOFAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOsem informaçãosem informaçã

Second order strategies for complementarity problems

Orientadores: Sandra Augusta Santos, Roberto AndreaniTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação CientificaResumo: Neste trabalho reformulamos o problema de complementaridade não linear generalizado (GNCP) em cones poliedrais como um sistema não linear com restrição de não negatividade em algumas variáveis, e trabalhamos na resolução de tal reformulação por meio de estratégias de pontos interiores. Em particular, definimos dois algoritmos e provamos a convergência local de tais algoritmos sob hipóteses usuais. O primeiro algoritmo é baseado no método de Newton, e o segundo, no método tensorial de Chebyshev. O algoritmo baseado no método de Chebyshev pode ser visto como um método do tipo preditor-corretor. Tal algoritmo, quando aplicado a problemas em que as funções envolvidas são afins, e com escolhas adequadas dos parâmetros, torna-se o bem conhecido algoritmo preditor-corretor de Mehrotra. Também apresentamos resultados numéricos que ilustram a competitividade de ambas as propostas.Abstract: In this work we reformulate the generalized nonlinear complementarity problem (GNCP) in polyhedral cones as a nonlinear system with nonnegativity in some variables and propose the resolution of such reformulation through interior-point methods. In particular we define two algorithms and prove the local convergence of these algorithms under standard assumptions. The first algorithm is based on Newton's method and the second, on the Chebyshev's tensorial method. The algorithm based on Chebyshev's method may be considered a predictor-corrector one. Such algorithm, when applied to problems for which the functions are affine, and the parameters are properly chosen, turns into the well-known Mehrotra's predictor corrector algorithm. We also present numerical results that illustrate the competitiveness of both proposals.DoutoradoOtimizaçãoDoutor em Matemática Aplicad

Generalisations of holographic hydrodynamics

In der vorliegenden Dissertation werden Eigenschaften stark gekoppelter hydrodynamischer Theorien untersucht, die mittels einer dualen Beschreibung als höherdimensionale gravitative Systeme aufgefasst werden können. Besonderes Augenmerk liegt hierbei auf der Berechnung physikalischer Größen wie Viskositäten oder Diffusionskonstanten. Diese werden hinsichtlich der Frage betrachtet, ob sie allgemeingültigen, universellen Gesetzmäßigkeiten folgen, die man aus der Beschreibung mittels einer Gravitationstheorie ableiten kann. Die theoretische Grundlage bildet hierbei die Dualität konformer Quantenfeldtheorien im Minkowski Raum und höherdimensionaler Stringtheorien im Anti-de Sitter Raum, die AdS/CFT Korrespondenz. Einen besonders interessanten Grenzfall stellt der Limes starker Kopplung und hoher Anzahl von Freiheitsgraden der konformen Feldtheorie dar, in dem sich die duale Beschreibung zu klassischer Gravitationstheorie im AdS Raum vereinfacht. Mittels störungstheoretischer Betrachtung der Fluktuationen von Schwarzen Loch Lösungen der Gravitationstheorie lassen sich universelle hydrodynamische Eigenschaften der stark gekoppelten Feldtheorie beschreiben. Eines der Hauptergebnisse dieses Forschungsgebietes ist der Nachweis, dass Fluide, die durch eine einfache duale Gravitationstheorie mit ungebrochener Rotationsinvarianz beschrieben werden können, ein universelles Verhältnis aus Scherviskosität und Entropiedichte besitzen. Erstaunlicherweise stimmt dieses Verhältnis parametrisch mit dem gemessenen Wert des stark gekoppelten Quark-Gluonen-Plasmas überein, ohne dass eine direkte Beschreibung dieser QCD Phase momentan möglich ist. In der vorliegenden Arbeit wird die Konstruktion eines ähnlichen, universellen Zusammenhangs beschrieben. In der hydrodynamischen Beschreibung supersymmetrischen Feldtheorien existiert eine Diffusionskonstante, die, ähnlich der Scherviskosität, den spurfreien Teil der Konstitutivgleichung des Supersymmetriestroms beschreibt. Wir berechnen diese Konstante in supersymmetrischen Theorien allgemeiner Dimension mittels verschiedener unabhängiger Rechnungen. Dazu betrachten wir als duale Gravitationstheorie eine generische Supergravitationstheorie. Die Bewegungsgleichung des zum Supersymmetriestrom dualen Gravitinos in Schwarzen Loch Hintergründen wird gelöst und erlaubt die Berechnung der retardierten Greenschen Funktion des Supersymmetriestroms der Feldtheorie. Diese besitzt einen Pol, der die charakteristische Schalldispersionsrelation des Phoninos beschreibt, des Goldstonefermions spontan gebrochener Supersymmetrie aufgrund endlicher Temperatur. In dieser Dispersionsrelation findet sich die besagte Diffusionskonstante, die sich auch mittels einer neuartigen Kubo-Formel direkt aus der Greenschen Funktion berechnen lässt. Das Hauptergebnis der Arbeit bildet hierbei die Etablierung eines Zusammenhangs dieser Diffusionskonstante und eines universell gültigen Absorptionsquerschnitts auf der dualen Seite der Gravitationstheorie, der die Absorption von Spinoren von einem Schwarzen Loch Hintergrund beschreibt. Eine weitere bedeutende Entwicklung besteht in der Entdeckung eines neuartigen Transportkoeffizienten, der einen beobachtbaren induzierten Strom aufgrund der Vortizität eines Fluids beschreibt. Dieser stellt die klassische Manifestation eines quantenmechanischen Effektes dar, der entsteht, wenn die zugrunde liegende mikroskopische Theorie eine quantenmechanische chirale Anomalie aufweist. Wir untersuchen diesen Effekt mithilfe eines theoretischen Ansatzes, der verschiedene Zugänge zum Verhältnis von Hydrodynamik und Gravitation miteinander vereint. Dazu werden rotierende D3-Branen effektiv als asymptotisch flache Verallgemeinerungen von fünf-dimensionalen AdS Reissner-Nordström Schwarzen Löchern beschrieben. Die Fluktuationen dieses Hintergrundes beschreiben nun eine effektive hydrodynamische Theorie auf einer Fläche in festem Abstand zur Singularität des Schwarzen Lochs, auf der die Fluktuationen Dirichlet Randbedingungen annehmen. Diese Herangehensweise erlaubt es uns den erwähnten Quanteneffekt nicht nur am Rand des AdS Raums zu betrachten, sondern auch am Horizont des Schwarzen Lochs, auf jeder Fläche mit konstantem Radius dazwischen oder sogar im asymptotisch flachen Raum.In the present thesis we study properties of strongly coupled hydrodynamic theories which may be described in terms of a dual higher dimensional gravitational system. Particular attention is given to the computation of physical quantities like the theories' viscosities and diffusion constants. These are analysed with regard to the question of whether they follow generally applicable, universal laws which may be derived from the description in terms of a gravitational theory. The theoretical foundation for this is laid by the duality between conformal quantum field theories in Minkowski space and higher-dimensional string theories on Anti-de Sitter space, the AdS/CFT correspondence. A particularly interesting simplification is given by the limit of strong coupling and large number of degrees of freedom of the conformal field theory in which the dual description reduces to a classical theory of gravity on AdS space. By using a perturbative treatment of fluctuations of the gravitational theory's black hole solutions one may describe universal hydrodynamic properties of the strongly coupled field theory. One of the main results within this area of research is the proof that fluids which may be described by a simple dual gravitational theory with unbroken rotational invariance possess a universal ratio of shear viscosity and entropy density. Astonishingly, this ratio parametrically agrees with the value measured for the strongly-coupled quark gluon plasma, although a direct treatment of this QCD phase is at present not available. In the following work we describe the construction of a similar, universal relation. In the hydrodynamic description of supersymmetric field theories there exists a further diffusion constant which, similarly to the shear viscosity, appears in the traceless part of the constitutive relation of the supersymmetry current. We compute this constant in supersymmetric theories of arbitrary dimension via different independent calculations. For doing so we look at a generic supergravity theory as the gravitational dual. The equation of motion of a gravitino, which is the dual field to the supersymmetry current, is solved in a black hole background and allows for the computation of retarded Green's functions of the field theory's supersymmetry current. This has a pole which describes the characteristic sound dispersion relation of the phonino, the Goldstone fermion of spontaneously broken supersymmetry due to finite temperature. In this dispersion relation we find the aforementioned diffusion constant which we also obtain directly from the correlator via a new Kubo formula. The main result of this project is the establishment of a relation of the supersound diffusion constant and a universally applicable absorption cross section on the dual gravitational side which describes the absorption of spinors by a black hole. A further important development is the discovery of a new transport coefficient which describes the observable current that is induced by the vorticity of a fluid. This illustrates the classical manifestation of a quantum mechanical effect which appears when the underlying microscopic theory possesses a quantum mechanical chiral anomaly. We investigate this effect within a theoretical framework which unifies several different approaches at the interplay of hydrodynamics and gravitational physics. We effectively describe rotating D3-branes as asymptotically flat generalisations of five-dimensional AdS Reissner-Nordström black holes. The fluctuations of this background describe an effective hydrodynamical theory on a surface at a finite distance from the black hole's singularity, on which the fluctuations satisfy Dirichlet boundary conditions. This approach allows us to study the mentioned quantum effect not only at the boundary of AdS space, but also at the black hole's horizon, at a surface in between at finite radius, or even in asymptotically flat space

Classical Systems in Quantum Mechanics

If we admit that quantum mechanics (QM) is universal theory, then QM should contain also some description of classical mechanical systems. The presented text contains description of two different ways how the mathematical description of kinematics and dynamics of classical systems emerges from the mathematical formalism of QM. The first of these ways is to obtain an equivalent description of QM (with finite number of degrees of freedom) as a classical Hamiltonian field theory and afterwards restrict it in dependence of specific classical system to obtain the classical Hamiltonian mechanics of that finite system. The second way is transition to QM of systems with infinite number of degrees of freedom - i.e. of macroscopic systems - and extract from it classical mechanics (with finite number of degrees of freedom) of macroscopic variables of this quantum system. The last chapter contains some considerations concerning the "measurement problem in QM", in which a measured quantum "microsystem" has to be dynamically connected with changes of classical states of a macroscopic quantum (sub-)system - the measuring device. Several models of this process are presented.Comment: book format, 230 pages, extensive bibliography and index, hypertext; alternative versions appear on the author`s home page: davinci.fmph.uniba.sk/~bona1/CSinQM/CSinQM-June7.pd

Foundations of Quantum Theory: From Classical Concepts to Operator Algebras

Quantum physics; Mathematical physics; Matrix theory; Algebr

Cosmological consequences of Quantum Gravity proposals

In this thesis, we study the implications of Quantum Gravity models for the dynamics of spacetime and the ensuing departures from classical General Relativity. The main focus is on cosmological applications, particularly the impact of quantum gravitational effects on the dynamics of a homogenous and isotropic cosmological background. Our interest lies in the consequences for the evolution of the early universe and singularity resolution, as well as in the possibility of providing an alternative explanation for dark matter and dark energy in the late universe. The thesis is divided into two main parts, dedicated to alternative (and complementary) ways of tackling the problem of Quantum Gravity. The first part is concerned with cosmological applications of background independent approaches to Quantum Gravity, both in the context of loop quantisation and in quantum geometrodynamics. Particularly relevant in this work is the Group Field Theory approach, which we use to study the effective dynamics of the emergent universe from a full theory of Quantum Gravity (i.e. without symmetry reduction). In the second part, modified gravity theories are introduced as tools to provide an effective description of quantum gravitational effects, e.g. by introducing new degrees of freedom and symmetries. Particularly relevant in this respect is local conformal invariance, which finds a natural realisation in the framework of Weyl geometry. We build a modified theory of gravity based on such symmetry principle, and argue that new fields in the extended gravitational sector may play the role of dark matter. New degrees of freedom are also natural in models with varying fundamental `constants', which we examine critically. Finally, we discuss prospects for future work and point at directions for the derivation of realistic cosmological models from Quantum Gravity candidates.Comment: PhD thesis, King's College London (supervisor: Mairi Sakellariadou), 282 pages, 20 figures; submitted in September 201

Generalisations of holographic hydrodynamics

In der vorliegenden Dissertation werden Eigenschaften stark gekoppelter hydrodynamischer Theorien untersucht, die mittels einer dualen Beschreibung als höherdimensionale gravitative Systeme aufgefasst werden können. Besonderes Augenmerk liegt hierbei auf der Berechnung physikalischer Größen wie Viskositäten oder Diffusionskonstanten. Diese werden hinsichtlich der Frage betrachtet, ob sie allgemeingültigen, universellen Gesetzmäßigkeiten folgen, die man aus der Beschreibung mittels einer Gravitationstheorie ableiten kann. Die theoretische Grundlage bildet hierbei die Dualität konformer Quantenfeldtheorien im Minkowski Raum und höherdimensionaler Stringtheorien im Anti-de Sitter Raum, die AdS/CFT Korrespondenz. Einen besonders interessanten Grenzfall stellt der Limes starker Kopplung und hoher Anzahl von Freiheitsgraden der konformen Feldtheorie dar, in dem sich die duale Beschreibung zu klassischer Gravitationstheorie im AdS Raum vereinfacht. Mittels störungstheoretischer Betrachtung der Fluktuationen von Schwarzen Loch Lösungen der Gravitationstheorie lassen sich universelle hydrodynamische Eigenschaften der stark gekoppelten Feldtheorie beschreiben. Eines der Hauptergebnisse dieses Forschungsgebietes ist der Nachweis, dass Fluide, die durch eine einfache duale Gravitationstheorie mit ungebrochener Rotationsinvarianz beschrieben werden können, ein universelles Verhältnis aus Scherviskosität und Entropiedichte besitzen. Erstaunlicherweise stimmt dieses Verhältnis parametrisch mit dem gemessenen Wert des stark gekoppelten Quark-Gluonen-Plasmas überein, ohne dass eine direkte Beschreibung dieser QCD Phase momentan möglich ist. In der vorliegenden Arbeit wird die Konstruktion eines ähnlichen, universellen Zusammenhangs beschrieben. In der hydrodynamischen Beschreibung supersymmetrischen Feldtheorien existiert eine Diffusionskonstante, die, ähnlich der Scherviskosität, den spurfreien Teil der Konstitutivgleichung des Supersymmetriestroms beschreibt. Wir berechnen diese Konstante in supersymmetrischen Theorien allgemeiner Dimension mittels verschiedener unabhängiger Rechnungen. Dazu betrachten wir als duale Gravitationstheorie eine generische Supergravitationstheorie. Die Bewegungsgleichung des zum Supersymmetriestrom dualen Gravitinos in Schwarzen Loch Hintergründen wird gelöst und erlaubt die Berechnung der retardierten Greenschen Funktion des Supersymmetriestroms der Feldtheorie. Diese besitzt einen Pol, der die charakteristische Schalldispersionsrelation des Phoninos beschreibt, des Goldstonefermions spontan gebrochener Supersymmetrie aufgrund endlicher Temperatur. In dieser Dispersionsrelation findet sich die besagte Diffusionskonstante, die sich auch mittels einer neuartigen Kubo-Formel direkt aus der Greenschen Funktion berechnen lässt. Das Hauptergebnis der Arbeit bildet hierbei die Etablierung eines Zusammenhangs dieser Diffusionskonstante und eines universell gültigen Absorptionsquerschnitts auf der dualen Seite der Gravitationstheorie, der die Absorption von Spinoren von einem Schwarzen Loch Hintergrund beschreibt. Eine weitere bedeutende Entwicklung besteht in der Entdeckung eines neuartigen Transportkoeffizienten, der einen beobachtbaren induzierten Strom aufgrund der Vortizität eines Fluids beschreibt. Dieser stellt die klassische Manifestation eines quantenmechanischen Effektes dar, der entsteht, wenn die zugrunde liegende mikroskopische Theorie eine quantenmechanische chirale Anomalie aufweist. Wir untersuchen diesen Effekt mithilfe eines theoretischen Ansatzes, der verschiedene Zugänge zum Verhältnis von Hydrodynamik und Gravitation miteinander vereint. Dazu werden rotierende D3-Branen effektiv als asymptotisch flache Verallgemeinerungen von fünf-dimensionalen AdS Reissner-Nordström Schwarzen Löchern beschrieben. Die Fluktuationen dieses Hintergrundes beschreiben nun eine effektive hydrodynamische Theorie auf einer Fläche in festem Abstand zur Singularität des Schwarzen Lochs, auf der die Fluktuationen Dirichlet Randbedingungen annehmen. Diese Herangehensweise erlaubt es uns den erwähnten Quanteneffekt nicht nur am Rand des AdS Raums zu betrachten, sondern auch am Horizont des Schwarzen Lochs, auf jeder Fläche mit konstantem Radius dazwischen oder sogar im asymptotisch flachen Raum.In the present thesis we study properties of strongly coupled hydrodynamic theories which may be described in terms of a dual higher dimensional gravitational system. Particular attention is given to the computation of physical quantities like the theories' viscosities and diffusion constants. These are analysed with regard to the question of whether they follow generally applicable, universal laws which may be derived from the description in terms of a gravitational theory. The theoretical foundation for this is laid by the duality between conformal quantum field theories in Minkowski space and higher-dimensional string theories on Anti-de Sitter space, the AdS/CFT correspondence. A particularly interesting simplification is given by the limit of strong coupling and large number of degrees of freedom of the conformal field theory in which the dual description reduces to a classical theory of gravity on AdS space. By using a perturbative treatment of fluctuations of the gravitational theory's black hole solutions one may describe universal hydrodynamic properties of the strongly coupled field theory. One of the main results within this area of research is the proof that fluids which may be described by a simple dual gravitational theory with unbroken rotational invariance possess a universal ratio of shear viscosity and entropy density. Astonishingly, this ratio parametrically agrees with the value measured for the strongly-coupled quark gluon plasma, although a direct treatment of this QCD phase is at present not available. In the following work we describe the construction of a similar, universal relation. In the hydrodynamic description of supersymmetric field theories there exists a further diffusion constant which, similarly to the shear viscosity, appears in the traceless part of the constitutive relation of the supersymmetry current. We compute this constant in supersymmetric theories of arbitrary dimension via different independent calculations. For doing so we look at a generic supergravity theory as the gravitational dual. The equation of motion of a gravitino, which is the dual field to the supersymmetry current, is solved in a black hole background and allows for the computation of retarded Green's functions of the field theory's supersymmetry current. This has a pole which describes the characteristic sound dispersion relation of the phonino, the Goldstone fermion of spontaneously broken supersymmetry due to finite temperature. In this dispersion relation we find the aforementioned diffusion constant which we also obtain directly from the correlator via a new Kubo formula. The main result of this project is the establishment of a relation of the supersound diffusion constant and a universally applicable absorption cross section on the dual gravitational side which describes the absorption of spinors by a black hole. A further important development is the discovery of a new transport coefficient which describes the observable current that is induced by the vorticity of a fluid. This illustrates the classical manifestation of a quantum mechanical effect which appears when the underlying microscopic theory possesses a quantum mechanical chiral anomaly. We investigate this effect within a theoretical framework which unifies several different approaches at the interplay of hydrodynamics and gravitational physics. We effectively describe rotating D3-branes as asymptotically flat generalisations of five-dimensional AdS Reissner-Nordström black holes. The fluctuations of this background describe an effective hydrodynamical theory on a surface at a finite distance from the black hole's singularity, on which the fluctuations satisfy Dirichlet boundary conditions. This approach allows us to study the mentioned quantum effect not only at the boundary of AdS space, but also at the black hole's horizon, at a surface in between at finite radius, or even in asymptotically flat space

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

Progress in Group Field Theory and Related Quantum Gravity Formalisms

Following the fundamental insights from quantum mechanics and general relativity, geometry itself should have a quantum description; the search for a complete understanding of this description is what drives the field of quantum gravity. Group field theory is an ambitious framework in which theories of quantum geometry are formulated, incorporating successful ideas from the fields of matrix models, ten-sor models, spin foam models and loop quantum gravity, as well as from the broader areas of quantum field theory and mathematical physics. This special issue collects recent work in group field theory and these related approaches, as well as other neighbouring fields (e.g., cosmology, quantum information and quantum foundations, statistical physics) to the extent that these are directly relevant to quantum gravity research

Reformulation of variational inequalities on a simplex and compactification of complementarity problems

Many variational inequality problems (VIPs) can be reduced, by a compactification procedure, to a VIP on the canonical simplex. Reformulations of this problem are studied, including smooth reformulations with simple constraints and unconstrained reformulations based on the penalized Fischer-Burmeister function. It is proved that bounded level set results hold for these reformulations under quite general assumptions on the operator. Therefore, it can be guaranteed that minimization algorithms generate bounded sequences and, under monotonicity conditions, these algorithms necessarily nd solutions of the original problem. Some numerical experiments are presented