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Papers by José Carlos Santos

Documenta Mathematica

Chapter 1. Generalities Definitions of group, isomorphism, representation, vector space and algeb... more Chapter 1. Generalities Definitions of group, isomorphism, representation, vector space and algebra. Biographical notes on Galois, Abel and Jacobi are given. Chapter 2. Lie groups and Lie algebras Lie Groups, infinitesimal generators, structure constants, Cartan's metric tensor, simple and semisimple groups and algebras, compact and non-compact groups. Biographical notes on Euler, Lie and Cartan are given. Chapter 3. Rotations: SO(3) and SU(2) Rotations and reflections, connectivity, center, universal covering group. Chapter 4. Representations of SU(2) Irreducible representations, Casimir operators, addition of angular momenta, Clebsch-Gordan coefficients, the Wigner-Eckart theorem, multiplicity. Biographical notes on Casimir, Weyl, Clebsch, Gordan and Wigner are given. Chapter 5. The so(n) algebra and Clifford numbers Spin(n), spinors and semispinors, Schur's lemma. Biographical notes on Clifford and Schur are given. Chapter 6. Reality properties of spinors Conjugate, orthogonal and symplectic representations. Chapter 7. Clebsch-Gordan series for spinors Antisymmetrie tensors, duality. Chapter 8. The center and outer automorphisms of Spin(n) Inversion, 12, 14 and 12 x 12 centers. A biographical note on Dynkin is given.

The American Mathematical Monthly

The Mathematical Gazette, 2007

To understand what people do when they do mathematics and write programs emulating that process i... more To understand what people do when they do mathematics and write programs emulating that process is a continuous research topic in artificial intelligence, automated reasoning and symbolic computation. We present the first release of an web application for athematics education that is being developed within AGILMAT (Automatic Generation of Interactive Drills for Mathematics Learning, POSI/CHS/48565/2002) project. AGILMAT aims at developing a system to automatically create and solve mathematics exercises that is flexible enough to be easily customizable to different curricula and users. Its major guiding principles are: the abstraction and formal representation of the problems that may be actually solved by algebraic algorithms covered by the curricula; the customization of these models by adding further constraints; and designing flexible solvers that emulate the steps students usually take to solve the generated drills.

rbhm.org.br

Einstein published his first article about the Theory of Relativity in 1905 but, at first, few ph... more Einstein published his first article about the Theory of Relativity in 1905 but, at first, few physicists displayed interest in this subject. Relativity began to be known by a wider audience only in 1908, when Hermann Minkowski gave his lecture “Space and Time”. ...

Extracta mathematicae, 2007

Let G be a topological group which acts in a continuous and transitive way on a topological space... more Let G be a topological group which acts in a continuous and transitive way on a topological space M. Sufficient conditions are given that assure that, for every m∈ M, the map from G onto M defined by g↦→ g· m is an open map. Some consequences of the ...

Mathematics Magazine, 2004

A classical exercise for college students is to ask them to prove that the sine func-tion is not ... more A classical exercise for college students is to ask them to prove that the sine func-tion is not a polynomial or, more generally, a rational function. This follows from the fact that the sine function has an infinite number of zeros, but this cannot hap-pen to a rational function unless it is ...

Documenta Mathematica

Chapter 1. Generalities Definitions of group, isomorphism, representation, vector space and algeb... more Chapter 1. Generalities Definitions of group, isomorphism, representation, vector space and algebra. Biographical notes on Galois, Abel and Jacobi are given. Chapter 2. Lie groups and Lie algebras Lie Groups, infinitesimal generators, structure constants, Cartan's metric tensor, simple and semisimple groups and algebras, compact and non-compact groups. Biographical notes on Euler, Lie and Cartan are given. Chapter 3. Rotations: SO(3) and SU(2) Rotations and reflections, connectivity, center, universal covering group. Chapter 4. Representations of SU(2) Irreducible representations, Casimir operators, addition of angular momenta, Clebsch-Gordan coefficients, the Wigner-Eckart theorem, multiplicity. Biographical notes on Casimir, Weyl, Clebsch, Gordan and Wigner are given. Chapter 5. The so(n) algebra and Clifford numbers Spin(n), spinors and semispinors, Schur's lemma. Biographical notes on Clifford and Schur are given. Chapter 6. Reality properties of spinors Conjugate, orthogonal and symplectic representations. Chapter 7. Clebsch-Gordan series for spinors Antisymmetrie tensors, duality. Chapter 8. The center and outer automorphisms of Spin(n) Inversion, 12, 14 and 12 x 12 centers. A biographical note on Dynkin is given.

The American Mathematical Monthly

The Mathematical Gazette, 2007

To understand what people do when they do mathematics and write programs emulating that process i... more To understand what people do when they do mathematics and write programs emulating that process is a continuous research topic in artificial intelligence, automated reasoning and symbolic computation. We present the first release of an web application for athematics education that is being developed within AGILMAT (Automatic Generation of Interactive Drills for Mathematics Learning, POSI/CHS/48565/2002) project. AGILMAT aims at developing a system to automatically create and solve mathematics exercises that is flexible enough to be easily customizable to different curricula and users. Its major guiding principles are: the abstraction and formal representation of the problems that may be actually solved by algebraic algorithms covered by the curricula; the customization of these models by adding further constraints; and designing flexible solvers that emulate the steps students usually take to solve the generated drills.

rbhm.org.br

Einstein published his first article about the Theory of Relativity in 1905 but, at first, few ph... more Einstein published his first article about the Theory of Relativity in 1905 but, at first, few physicists displayed interest in this subject. Relativity began to be known by a wider audience only in 1908, when Hermann Minkowski gave his lecture “Space and Time”. ...

Extracta mathematicae, 2007

Let G be a topological group which acts in a continuous and transitive way on a topological space... more Let G be a topological group which acts in a continuous and transitive way on a topological space M. Sufficient conditions are given that assure that, for every m∈ M, the map from G onto M defined by g↦→ g· m is an open map. Some consequences of the ...

Mathematics Magazine, 2004

A classical exercise for college students is to ask them to prove that the sine func-tion is not ... more A classical exercise for college students is to ask them to prove that the sine func-tion is not a polynomial or, more generally, a rational function. This follows from the fact that the sine function has an infinite number of zeros, but this cannot hap-pen to a rational function unless it is ...