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Concept

Function of a real variable

In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers \mathbb{R}, or a subset of \mathbb{R} that contains an interval of positive length. Most real functions that are considered and studied are differentiable in some interval. The most widely considered such functions are the real functions, which are the real-valued functions of a real variable, that is, the functions of a real variable whose codomain is the set of real numbers. Nevertheless, the codomain of a function of a real variable may be any set. However, it is often assumed to have a structure of \mathbb{R}-vector space over the reals. That is, the codomain may be a Euclidean space, a coordinate vector, the set of matrices of real numbers of a given size, or an \mathbb{R}-algebra, such as the complex numbers or the quaternions. The structur

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Fonction (mathématiques)

vignette|Diagramme de calcul pour la fonction x \mapsto \frac{2x-1}{x+3} En mathématiques, une fonction permet de définir un résultat (le plus souvent numérique) pour chaque valeur d’un

Fonction trigonométrique

thumb|upright=1.35|Toutes les valeurs des fonctions trigonométriques d'un angle θ peuvent être représentées géométriquement. En mathématiques, les fonctions trigonométriques permettent de relier les

Dérivée

En mathématiques, la dérivée d'une fonction d'une variable réelle mesure l'ampleur du changement de la valeur de la fonction (valeur de sortie) par rapport à un petit changement de son argument (vale

MATH-123(b): Geometry

Ce cours donne une introduction à la géométrie des courbes et des surfaces.

FIN-472: Computational finance

Participants of this course will master computational techniques frequently used in mathematical finance applications. Emphasis will be put on the implementation and practical aspects.

ME-451: Advanced energetics

Methods for the rational use and conversion of energy in industrial processes : how to analyse the energy usage, calculate the heat recovery by pinch analysis, define heat exchanger network, integrate heat pumps and cogeneration units and realise exergy analysis of energy conversion systems.

Lower bounds for Pythagoras numbers of function fields

David Maximilian Grimm

We show that the transcendence degree of a real function field over an arbitrary real base field is a strict lower bound for its Pythagoras number and a weak lower bound for all its higher Pythagoras numbers.

European Mathematical Soc2015

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Proximity Operators of Discrete Information Divergences

Giovanni Chierchia, Mireille El Gheche

While phi-divergences have been extensively studied in convex analysis, their use in optimization problems often remains challenging. In this regard, one of the main shortcomings of existing methods is that the minimization of phi-divergences is usually performed with respect to one of their arguments, possibly within alternating optimization techniques. In this paper, we overcome this limitation by deriving new closed-form expressions for the proximity operator of such two-variable functions. This makes it possible to employ standard proximal methods for efficiently solving a wide range of convex optimization problems involving phi-divergences. In addition, we show that these proximity operators are useful to compute the epigraphical projection of several functions. The proposed proximal tools are numerically validated in the context of optimal query execution within database management systems, where the problem of selectivity estimation plays a central role. Experiments are carried out on small to large scale scenarios.

IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC2018

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MINLP Model and two-level Algorithm for the Simultaneous Synthesis of Heat Exchanger Networks and Utility Systems

François Maréchal, Emanuele Martelli, Alberto Mian

This work proposes a novel approach for the simultaneous synthesis of Heat Exchanger Networks (HEN) and Utility Systems. Given a set of hot and cold process streams and a set of available utility systems (e.g., gas turbine, steam cycle, boiler, etc), the method determines the optimal selection of utility systems, their arrangement and design (including steam generator), and the heat exchanger network (between process-process as well as process-utility and utility-utility streams) rigorously considering the trade-off between efficiency and capital costs. The mathematical model is formulated as a Mixed Integer NonLinear Program (MINLP) and it combines the SYNHEAT superstructure for HENs with ad hoc models/superstructures for utility systems. The challenging MINLP is solved with a two-level algorithm using at the upper level the Variable Neighbourhood Search (VNS) algorithm to optimize the integer variables, and at the lower level the SQP algorithm to optimize the real variables. The algorithm is tested on problems with up to 15 streams (corresponding to 465 binary variables).

Elsevier2015

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