74 research outputs found

    Nonlinear Analysis and Optimization with Applications

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    Nonlinear analysis has wide and significant applications in many areas of mathematics, including functional analysis, variational analysis, nonlinear optimization, convex analysis, nonlinear ordinary and partial differential equations, dynamical system theory, mathematical economics, game theory, signal processing, control theory, data mining, and so forth. Optimization problems have been intensively investigated, and various feasible methods in analyzing convergence of algorithms have been developed over the last half century. In this Special Issue, we will focus on the connection between nonlinear analysis and optimization as well as their applications to integrate basic science into the real world

    Improving Network Reductions for Power System Analysis

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    abstract: The power system is the largest man-made physical network in the world. Performing analysis of a large bulk system is computationally complex, especially when the study involves engineering, economic and environmental considerations. For instance, running a unit-commitment (UC) over a large system involves a huge number of constraints and integer variables. One way to reduce the computational expense is to perform the analysis on a small equivalent (reduced) model instead on the original (full) model. The research reported here focuses on improving the network reduction methods so that the calculated results obtained from the reduced model better approximate the performance of the original model. An optimization-based Ward reduction (OP-Ward) and two new generator placement methods in network reduction are introduced and numerical test results on large systems provide proof of concept. In addition to dc-type reductions (ignoring reactive power, resistance elements in the network, etc.), the new methods applicable to ac domain are introduced. For conventional reduction methods (Ward-type methods, REI-type methods), eliminating external generator buses (PV buses) is a tough problem, because it is difficult to accurately approximate the external reactive support in the reduced model. Recently, the holomorphic embedding (HE) based load-flow method (HELM) was proposed, which theoretically guarantees convergence given that the power flow equations are structure in accordance with Stahl’s theory requirements. In this work, a holomorphic embedding based network reduction (HE reduction) method is proposed which takes advantage of the HELM technique. Test results shows that the HE reduction method can approximate the original system performance very accurately even when the operating condition changes.Dissertation/ThesisDoctoral Dissertation Electrical Engineering 201

    Series Representations and Approximation of some Quantile Functions appearing in Finance

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    It has long been agreed by academics that the inversion method is the method of choice for generating random variates, given the availability of a cheap but accurate approximation of the quantile function. However for several probability distributions arising in practice a satisfactory method of approximating these functions is not available. The main focus of this thesis will be to develop Taylor and asymptotic series representations for quantile functions of the following probability distributions; Variance Gamma, Generalized Inverse Gaussian, Hyperbolic, -Stable and Snedecor’s F distributions. As a secondary matter we briefly investigate the problem of approximating the entire quantile function. Indeed with the availability of these new analytic expressions a whole host of possibilities become available. We outline several algorithms and in particular provide a C++ implementation for the variance gamma case. To our knowledge this is the fastest available algorithm of its sort

    The rational SPDE approach for Gaussian random fields with general smoothness

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    A popular approach for modeling and inference in spatial statistics is to represent Gaussian random fields as solutions to stochastic partial differential equations (SPDEs) of the form Lβu=WL^{\beta}u = \mathcal{W}, where W\mathcal{W} is Gaussian white noise, LL is a second-order differential operator, and β>0\beta>0 is a parameter that determines the smoothness of uu. However, this approach has been limited to the case 2β∈N2\beta\in\mathbb{N}, which excludes several important models and makes it necessary to keep β\beta fixed during inference. We propose a new method, the rational SPDE approach, which in spatial dimension d∈Nd\in\mathbb{N} is applicable for any β>d/4\beta>d/4, and thus remedies the mentioned limitation. The presented scheme combines a finite element discretization with a rational approximation of the function x−βx^{-\beta} to approximate uu. For the resulting approximation, an explicit rate of convergence to uu in mean-square sense is derived. Furthermore, we show that our method has the same computational benefits as in the restricted case 2β∈N2\beta\in\mathbb{N}. Several numerical experiments and a statistical application are used to illustrate the accuracy of the method, and to show that it facilitates likelihood-based inference for all model parameters including β\beta.Comment: 28 pages, 4 figure

    Row Reduction Applied to Decoding of Rank Metric and Subspace Codes

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    We show that decoding of ℓ\ell-Interleaved Gabidulin codes, as well as list-ℓ\ell decoding of Mahdavifar--Vardy codes can be performed by row reducing skew polynomial matrices. Inspired by row reduction of \F[x] matrices, we develop a general and flexible approach of transforming matrices over skew polynomial rings into a certain reduced form. We apply this to solve generalised shift register problems over skew polynomial rings which occur in decoding ℓ\ell-Interleaved Gabidulin codes. We obtain an algorithm with complexity O(ℓμ2)O(\ell \mu^2) where μ\mu measures the size of the input problem and is proportional to the code length nn in the case of decoding. Further, we show how to perform the interpolation step of list-ℓ\ell-decoding Mahdavifar--Vardy codes in complexity O(ℓn2)O(\ell n^2), where nn is the number of interpolation constraints.Comment: Accepted for Designs, Codes and Cryptograph

    Phenomenology of the Higgs and Flavour Physics in the Standard Model and Beyond

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    In dieser Arbeit werden einige zukünftige Aspekte der Higgs-Messungen ein Jahrzehnt nach seiner Entdeckung untersucht, wobei der Schwerpunkt auf dem Potenzial für zukünftige Läufe des Large Hadron Collider (LHC) liegt. Insbesondere sollen anspruchsvolle Kopplungen des Higgs, wie seine Selbstkopplung und die Wechselwirkung mit leichten Quarks, untersucht werden. Der erste Teil gibt einen Überblick über die Higgs-Physik innerhalb der effektiven Feldtheorie des Standardmodells (SMEFT). Der zweite Teil befasst sich mit der Single-Higgs-Produktion, beginnend mit einer Zweischleifenberechnung der Gluonenfusionskomponente von Zh, um deren theoretische Unsicherheiten zu reduzieren. Dann wird das Potenzial für die Einschränkung der trilinearen Higgs-Selbstkopplung aus Einzel-Higgs-Raten erneut untersucht, indem ebenso schwach eingeschränkte Vier-Schwer-Quark-Operatoren einbezogen werden, die bei der nächsthöheren Ordnung in die Einzel-Higgs-Raten eingehen. Diese Operatoren korrelieren in hohem Maße mit der trilinearen Selbstkopplung, was sich auf die Anpassungen auswirkt, die für diese Kopplung anhand von Einzel-Higgs-Daten vorgenommen wurden. Der dritte Teil konzentriert sich auf die Higgs-Paarproduktion, einen wesentlichen Prozess zur Messung der Higgs-Selbstkopplung, und setzt eine multivariate Analyse ein, um ihr Potenzial zur Untersuchung der leichten Yukawa-Kopplungen zu untersuchen; dadurch wird die Empfindlichkeit der Higgs-Paarproduktion für die leichten Quark-Yukawa-Wechselwirkungen erforscht. Schließlich werden im vierten Teil einige Modelle vorgestellt, die darauf abzielen, die jüngsten Flavour-Anomalien im Lichte einer globalen SMEFT-Bayesian-Analyse zu erklären, die Flavour- und elektroschwache Präzisionsmessungen kombiniert.This thesis investigates some future aspects of Higgs measurements a decade after its discovery, focusing on the potential for future runs of the Large Hadron Collider (LHC). In particular, it aims to probe challenging couplings of the Higgs like its self-coupling and interaction with light quarks. The first part provides an overview of Higgs physics within the Standard Model Effective Field theory (SMEFT). The second part is about single-Higgs production, starting with a two-loop calculation of the gluon fusion component of Zh to reduce its theoretical uncertainties. Then, the potential for constraining the Higgs trilinear self-coupling from single Higgs rates is revisited; by including equally weaklyconstrained four-heavy-quark operators entering at the next-to-leading order in single Higgs rates. These operators highly correlate with the trilinear self-coupling, thus affecting the fits made on this coupling from single Higgs data. The third part focuses on the Higgs pair production, an essential process for measuring Higgs-self coupling, employing multivariate analysis to study its potential for probing light Yukawa couplings; thereby exploring the sensitivity of Higgs pair production for the light-quark Yukawa interactions. Finally, the fourth part showcases some models aiming to explain the recent flavour anomalies in the light of a global SMEFT Bayesian analysis combining flavour and electroweak precision measurements

    Analytic Combinatorics in Several Variables: Effective Asymptotics and Lattice Path Enumeration

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    The field of analytic combinatorics, which studies the asymptotic behaviour of sequences through analytic properties of their generating functions, has led to the development of deep and powerful tools with applications across mathematics and the natural sciences. In addition to the now classical univariate theory, recent work in the study of analytic combinatorics in several variables (ACSV) has shown how to derive asymptotics for the coefficients of certain D-finite functions represented by diagonals of multivariate rational functions. We give a pedagogical introduction to the methods of ACSV from a computer algebra viewpoint, developing rigorous algorithms and giving the first complexity results in this area under conditions which are broadly satisfied. Furthermore, we give several new applications of ACSV to the enumeration of lattice walks restricted to certain regions. In addition to proving several open conjectures on the asymptotics of such walks, a detailed study of lattice walk models with weighted steps is undertaken.Comment: PhD thesis, University of Waterloo and ENS Lyon - 259 page

    Algorithmes rapides pour les polynômes, séries formelles et matrices

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    Notes d'un cours dispensé aux Journées Nationales du Calcul Formel 2010International audienceLe calcul formel calcule des objets mathématiques exacts. Ce cours explore deux directions : la calculabilité et la complexité. La calculabilité étudie les classes d'objets mathématiques sur lesquelles des réponses peuvent être obtenues algorithmiquement. La complexité donne ensuite des outils pour comparer des algorithmes du point de vue de leur efficacité. Ce cours passe en revue l'algorithmique efficace sur les objets fondamentaux que sont les entiers, les polynômes, les matrices, les séries et les solutions d'équations différentielles ou de récurrences linéaires. On y montre que de nombreuses questions portant sur ces objets admettent une réponse en complexité (quasi-)optimale, en insistant sur les principes généraux de conception d'algorithmes efficaces. Ces notes sont dérivées du cours " Algorithmes efficaces en calcul formel " du Master Parisien de Recherche en Informatique (2004-2010), co-écrit avec Frédéric Chyzak, Marc Giusti, Romain Lebreton, Bruno Salvy et Éric Schost. Le support de cours complet est disponible à l'url https://wikimpri.dptinfo.ens-cachan.fr/doku.php?id=cours:c-2-2

    Conditional Volatility, Skewness, and Kurtosis: Existence and Persistence.

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    Recent portfolio choice asset pricing and option valuation models highlight the importance of skewness and kurtosis. Since skewness and kurtosis are related to extreme variations they are also important for Value-at-Risk measurements. Our framework builds on a GARCH model with a condi-tional generalized-t distribution for residuals. We compute the skewness and kurtosis for this model and compare the range of these moments with the maximal theoretical moments. Our model thus allows for time-varying conditional skewness and kurtosis.GARCH Stock indices Exchange rates Interest rates SNOPT VaR
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