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Concept# Algebra

Summary

Algebra () is the study of variables and the rules for manipulating these variables in formulas; it is a unifying thread of almost all of mathematics.
Elementary algebra deals with the manipulation of variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in education, to the study of algebraic structures such as groups, rings, and fields. Linear algebra, which deals with linear equations and linear mappings, is used for modern presentations of geometry, and has many practical applications (in weather forecasting, for example). There are many areas of mathematics that belong to algebra, some having "algebra" in their name, such as commutative algebra, and some not, such as Galois theory.
The word algebra is not only used for naming an area of mathematics and some subareas; it is also used for naming some sorts of algebraic structures, such as an algebra over

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This work contains the study of the algebra called al-Badī‘ fī al-ḥisāb (literally : "the Wonderful on calculation"), written by the Persian mathematician Abu Bakr Muḥammad ibn al-Ḥusain al-Karaǧi (previously known as al-Karẖī, native from Karaǧ, Persia. Written c. 1010 in Bagdad, this work takes an important place in history of mathematics in general. Of particular interest are the first known appearance of a theory on root extracting of algebraic polynomials, and the beginning of a tendency to get rid of illustrating formulas and the resolutions of equations with help of geometric figures, which makes it a pure algebraic text. This work of high level adresses to a public with advanced mathematic knowledge. This algebra is, by will of the author, written in three main parts (books), containing part of Euclid's Elements (book I), a theory on root extracting of algebraic polynomials (book II), and a collection of problems on indeterminate analysis (book III). Some chapters are written hastily, while others go into the details. We provide a complete translation of the Badī‘, based on the transcription of the manuscript 36,1 of the Vatican library Barberini Orientale by Adel Anbouba (edited in Beyrouth in 1964), as well as a glossary. This translation comes with a mathematical commentary, and includes a list of significant words used by the author. We will also relate this algebra with other prior and later works containing the same problems.

In this thesis, we explore possible stabilisation methods for the reduce basis approximation of advection-diffusion problems, for which the advection term is dominating. The options we consider are mainly inspired by the Variational Multiscale method (VMS), which decomposes the solution of a variational problem into its coarse scale component, from a coarse scale space, and a fine scale component, from a fine scale space. Our stabilisation proposals are divided into three classes. The first one groups methods that rely on a stabilisation parameter. The second class uses VMS at the algebraic level to attempt stabilisation. Finally the third class is also inspired by VMS at the algebraic level, but with the additional constraint that the fine scale space is orthogonal to the coarse scale space. Numericals tests reported in this thesis show that the methods of the first class is not viable options as the best stabilisation parameter among those tested is the stabilisation parameter that is used at the high fidelity level. Although the stabilisation methods of the second class give accurate results when applied to stable problems, they were also dismissed by the numerical tests, as they did not improve the accuracy of the already stabilised problem. The third class also performs well when applied to stable problems. It has been shown in [7] one of those methods can improve accuracy. However in the current implementation, this result was not achieved here.

2016Motivated by the work of Kupershmidt (J. Nonlin. Math. Phys. 6 (1998), 222 -245) we discuss the occurrence of left symmetry in a generalized Virasoro algebra. The multiplication rule is defined, which is necessary and sufficient for this algebra to be quasi-associative. Its link to geometry and nonlinear systems of hydrodynamic type is also recalled. Further, the criteria of skew-symmetry, derivation and Jacobi identity making this algebra into a Lie algebra are derived. The coboundary operators are defined and discussed. We deduce the hereditary operator and its generalization to the corresponding 3 ary bracket. Further, we derive the so-called rho compatibility equation and perform a phase-space extension. Finally, concrete relevant particular cases are investigated.