Tensor Calculus

The number of linear invariants under SO (3) as well as SO (2) of a Cartesian tensor of an arbitrary rank is studied. A linear form is defined in terms of elements of a tensor. It is established that the number of linear invariants of a tensor of rank n under SO (3) equals ...

These notes are the second part of the tensor calculus documents which started with the previous set of introductory notes (see T. Sochi, Introduction to Tensor Calculus, arXiv:1603.01660, 2016). In the present text, we continue the discussion of selected topics of the subject at a higher level expanding, when necessary, some topics and developing further concepts and techniques. Unlike the previous notes which are largely based on a Cartesian approach, the present notes are essentially based on assuming an underlying general curvilinear coordinate system.

Alfred North Whitehead, Le Principe de relativité et ses applications en physique [The Principle of Relativity. With application to Physical Science, 1922]. Traduction de Henri Vaillant relue par Sylviane Schwer, Louvain-la-Neuve, Les Éditions Chromatika, 2012. (978-2-930517-34-6 ; 147 p. ; 17 €)
Le philosophe et algébriste britannique Alfred North Whitehead (1861–1947) est une figure marquante, quoique peu connue en Europe francophone, de la pensée du XXe siècle. Le principe de relativité (1922) constitue probablement le livre le plus controversé et le moins étudié de Whitehead. Développant plus avant les conséquences épistémologiques de la « méthode de l’abstraction extensive » introduite dans ses Principes de la connaissance naturelle (1919/1925), il y propose une théorie de la relativité générale aussi — si pas plus — puissante (i.e., cohérente et applicable) que celle d’Einstein (1914) mais fondée sur une philosophie strictement processuelle qui lui permet de remédier aux tendances idéaliste et substantialiste de la relativité einsteinienne « orthodoxe ».

Henri Vaillant, ingénieur, philosophe-chercheur sur la philosophie du processus, a traduit Modes de pensée (Vrin, 2003), La Science et le monde moderne (Ontos Verlag, 2006), les Principes de la connaissance naturelle (Chromatika, 2007) et La religion en gestation (Chromatika, 2008).

Sylviane R. Schwer, ancienne élève de l'École Normale Supérieure de Fontenay-aux-Roses, agrégée de mathématiques, est professeur à l'université de Paris 13.

It is defined the torus coordinates and calculated the metric tensor of the torus surface, with all its consequences.  It is noted that the model of rotation of the torus, applied to the real universe has similar difficulties with the angular momentum as the rotation of a simple homogeneous, isotropic space.

This paper intends to introduce the Taylor series for multi-variable real functions. More than a demostration of the teorema, it shows how to expose the series in a compact notation. Generalization of the jacobean of any order of a function with multiple dependence is defined. For this we use the differential operator Nabla with multiple tensor products. Also is established a special multiplication of the derivatives with the displacement of independent variables. This multiplicaction is identified as the contraction of indexes. At the end there is a new proof of the Taylor's Theorem for vectorial and tensorial functions.  Also it is included the multi-index notation version of the series.

These notes are the second part of the tensor calculus documents which started with the previous set of introductory notes (see T. Sochi, Introduction to Tensor Calculus, arXiv:1603.01660, 2016). In the present text, we continue the discussion of selected topics of the subject at a higher level expanding, when necessary, some topics and developing further concepts and techniques. Unlike the previous notes which are largely based on a Cartesian approach, the present notes are essentially based on assuming an underlying general curvilinear coordinate system.

Esta apostila de Espaço Tensorial e Funcional Generalizado é resultado da digitação das aulas do curso de Pós-Graduação em Ciência e Engenharia de Matériais da Universidade Estadual de Ponta Grossa- UEPG, na disciplina de Mecânica do Contínuo, ministrado pelo professor Dr. Lucas Máximo Alves e de seus estudos pessoais anteriores, obtidos durante o seu doutorado no Programa de Pós-Graduação de Métodos Numéricos para a Engenharia-PPGMNE da Universidade Federal do Paraná.

I discuss Albert Einstein's 1914 review paper, "The Formal Foundation of the General Theory of Relativity" from two points of view: the main elements in the paper that appear to have led to the downfall of the Einstein-Grossman theory; and the elements that seem to have inspired Einstein during October 1915 to reformulate the 1914 Einstein-Grossmann theory in the form of the November 1915 and the 1916 General Theory of Relativity. First paper among three papers.

It has been developed a method of arbitrary degree based on Taylor series for multi-variable functions. The method is proposed for solving a system of homogeneous equations f(x)=0 in ℝᴺ. Methods of degree one and two are revised and a strategy for degree n is proposed, supported on the best estimation criterion. The multi-variable polynomials are solved sequentially from degree one of Newton-Raphson to degree n.

The problem of determining the tensors and the vector invariants that describe the vector field of the order accelerations is generally avoided in rigid body kinematics. This paper extends the discussion from velocities and accelerations to order accelerations. Using the tensor calculus and the dual numbers algebra, a computing method for studying the order acceleration field properties is proposed for the case of the general motion of the rigid body. This approach uses the isomorphism between the Lie group of the rigid displacements and the Lie group of the orthogonal dual tensors.

These are general notes on tensor calculus which can be used as a reference for an introductory course on tensor algebra and calculus. A basic knowledge of calculus and linear algebra with some commonly used mathematical terminology is presumed.

Ronny Desmet & Michel Weber (edited by), Whitehead. The Algebra of Metaphysics. Applied Process Metaphysics Summer Institute Memorandum, Louvain-la-Neuve, Les Éditions Chromatika, 2010. (978-2-930517-08-7 ; 378 p. ; 40 €)
Drawing upon the major Harvard works —Science and the Modern World (1925), Process and Reality (1929) and Adventures of Ideas (1933)—, the essays gathered here on the occasion of the creation of the Applied Process Metaphysics Summer Institute, seek, first, to introduce into Whitehead’s thought by clarifying what is at stake in his philosophy and by providing a synoptic vision of his key categories in light of their historical development and, second, to foster a creative dialogue among all participants. These essays give the opportunity to travel through most aspects of Whitehead’s legacy: anthropology, ecology, education, epistemology, metaphysics, psychology, political theory and relativity physics.

Ronny Desmet holds an M.A. in mathematics (1983), an M.A. and a Ph.D. in philosophy (2005 and 2010). He is currently research assistant at the Centre of Logic and Philosophy of Science, Free University of Brussels. His research and publications have focused on Whitehead’s philosophy of mathematics and relativity.

Michel Weber holds an M.A. and a Ph.D. in philosophy (1991 and 1997). He is the director of the Centre for Philosophical Practice “Chromatiques whiteheadiennes.” He has published more than 30 books, e.g., Whitehead’s Pancreativism. The Basics (2006) and the Handbook of Whiteheadian Process Thought (2008).

The Semi-Algebraic framework for the approximate Canonical Polyadic (CP) decomposition via SImultaneaous matrix diagonalization (SECSI) is an efficient tool for the computation of the CP decomposition. The SECSI framework reformulates the CP decomposition into a set of joint eigenvalue decomposition (JEVD) problems. Solving all JEVDs, we obtain multiple estimates of the factor matrices and the best estimate is chosen in a subsequent step by using an exhaustive search or some heuristic strategy that reduces the computational complexity. Moreover, the SECSI framework retains the option of choosing the number of JEVDs to be solved, thus providing an adjustable complexity-accuracy trade-off. In this work, we provide an analytical performance analysis of the SECSI framework for the computation of the approximate CP decomposition of a noise corrupted low-rank tensor, where we derive closed-form expressions of the relative mean square error for each of the estimated factor matrices. These expressions are obtained using a first-order perturbation analysis and are formulated in terms of the second-order moments of the noise, such that apart from a zero mean, no assumptions on the noise statistics are required. Simulation results exhibit an excellent match between the obtained closed-form expressions and the empirical results. Moreover, we propose a new Performance Analysis based Selection (PAS) scheme to choose the final factor matrix estimate. The results show that the proposed PAS scheme outperforms the existing heuristics, especially in the high SNR regime.

Before developing his 1915 General Theory of Relativity, Einstein held the "Entwurf" theory. Tullio Levi-Civita from Padua, one of the founders of tensor calculus, objected to a major problematic element in this theory, which reflected its global problem: its field equations were restricted to an adapted coordinate system. Einstein proved that his gravitational tensor was a covariant tensor for adapted coordinate systems. In an exchange of letters and postcards that began in March 1915 and ended in May 1915, Levi-Civita presented his objections to Einstein's above proof. Einstein tried to find ways to save his proof, and found it hard to give it up. Finally Levi-Civita convinced Einstein about a fault in his arguments. However, only in spring 1916, long after Einstein had abandoned the 1914 theory, did he finally understand the main problem with his 1914 gravitational tensor. In autumn 1915 the G\"ottingen brilliant mathematician David Hilbert found the central flaw in Einstein's 1914 derivation. On March 30, 1916, Einstein sent to Hilbert a letter admitting, "The error you found in my paper of 1914 has now become completely clear to me".

Alfred North Whitehead, Les Principes de la connaissance naturelle [An Enquiry Concerning the Principles of Natural Knowledge, 1919/1925]. Traduction de Henri Vaillant, relue par Sylviane Schwer, Louvain-la-Neuve, Les Éditions Chromatika, 2007. (978-2-930517-00-1 ; 296 p. ; 30 € ; http://www.i6doc.com/fr/livre/?GCOI=28001100104050)

Le philosophe et algébriste britannique Alfred North Whitehead (1861–1947) est une figure marquante, quoique peu connue en Europe francophone, de la pensée du XXe siècle. Son parcours intellectuel est exceptionnel : à Cambridge (1880–1910), il enseigne les mathématiques appliquées et publie un monumental traité d’algèbre (A Treatise on Universal Algebra, with Applications, 1898) avant de se lancer, avec son ancien élève Bertrand Russell (1872–1970), dans le projet des Principia Mathematica (1910, 1912, 1913) ; il poursuit son enseignement à Londres (1910–1924) tout en se concentrant à présent sur les implications épistémologiques de la relativité einsteinienne ; enfin, à Harvard (1924–1947), il enseigne la philosophie et conçoit le projet d’une nouvelle synthèse métaphysique qui culmine avec la publication de Process and Reality (1929).

Quelle que soit la manière dont on comprenne les différents aspects, nuances et implications du développement de la pensée de Whitehead, ses Principes de la connaissance naturelle (1919/1925) demeurent un texte-charnière qui voit la mise en pratique d’outils logiques (méréo-topologiques) dans un contexte épistémologique relativiste à l’horizon duquel se profile une réforme ontologique majeure : c’est bien dans les Principes qu’il introduit le concept d’ « avancée créatrice ».

Henri Vaillant, Ingénieur, Philosophe-chercheur sur la philosophie du processus, est le fondateur, avec Jean-Marie Breuvart, du Groupe de Recherche sur l’Actualité et la Créativité. H.Vaillant a traduit Modes de pensée chez Vrin (2003), La science et le monde moderne chez ontos verlag (2006) et La religion en gestation chez Chromatika (2008). Sa traduction du Principe de relativité est sous presse (également chez Chromatika).

Sylviane R. Schwer, ancienne élève de l'École Normale Supérieure de Fontenay-aux-Roses, agrégée de mathématiques, est professeur à l'université de Paris 13.

The space fabric deforms logarithmically and this is consistent with black holes and red holes.
The preconditions for the warping of space are given.
The surprising result is that Newton's Law of Action and Reaction is violated or at least set in a new context. Curious? Well it's just a short paper. Take a look :)

Because undervalued in classical Relativity, following Ferrarese's approach, I simply expose the necessity of dealing with this topic of surface geometry as basis for extension of Einstein's theory into Tailherer's Vortex theory of gravity by the introduction of the metric deformation speed to account for gravitational radiation.

The problem of determining the tensors and the vector invariants that describe the vector field of the nth order accelerations is generally avoided in rigid body kinematics. This paper extends the discussion from velocities and accelerations to nth order accelerations. Using the tensor calculus and the dual numbers algebra, a computing method for studying the nth order acceleration field properties is proposed for the case of the general motion of the rigid body. This approach uses the isomorphism between the Lie group of the rigid displacements \( S{\mathbb{E}}_{3} \) and the Lie group of the orthogonal dual tensors \( \underline{{S{\mathbb{O}}}}_{3} \).

In this paper an extension of the classical attitude Wahba's problem to a new problem that involves both position and attitude is presented. This new problem is illustrated in the dual algebra of orthogonal dual tensors. Our approach makes use of the isomorphism between the special euclidean group SE 3 and the orthogonal dual tensors group SO 3 , which allows the development of a new technique for solving the proposed problem. This new Wahba problem is directly related to different applications such as point-to-point registration and sensor calibration.

The relative orbital motion is a six d.o.f motion, generated by the coupling of the relative translational motion with the rotational one. This paper is focused on developing a new relative orbital motion parametrization method using dual rigid bases. Our studies showed that, in the dual tensors free module, the dual rigid bases can completely characterize the relative orbital motion from the Euclidean three dimensional space. The combination between a dual rigid basis and its reciprocal provides a natural computational instrument that can be used to solve many problems in the kinematics, dynamics and control of relative orbital motion setups.

Projecting the square 1 through the diagonal of its hypotenuse we can build a new prime square 1 with an irrational symmetry. Combining the rational and irrational symmetries we can get new prime squares which roots will be irrational. The zero points displaced in this way through the infinite diagonal should be coincident with the Riemann function zeros. But it seems it also could be necessary to consider other different lines related to the mirror symmetry of the Riemann's critical line.

The properties of the n-order accelerations distribution of a rigid body under general motion are systematically studied based on tensor algebra. A tensor represents an R -linear mapping of the R -linear space of free vectors V into V . It may be represented by a matrix only after a basis in V was selected. Preferring an orthonormal basis simplifies the calculus, but this particular choice may conceal some essential properties. This paper uses the notion of rigid basis associated to a spherical rigid motion in order to point out some properties of the n-order accelerations tensor.