Artículos con la etiqueta ‘History and Overview (math.HO)’

DD Calculus

Por • 4 abr, 2014 • Category: Crítica

This paper introduces DD calculus and describes the basic calculus concepts of derivative and integral in a direct and non-traditional way, without limit definition: Derivative is computed from the point-slope equation of a tangent line and integral is defined as the height increment of a curve. This direct approach to calculus has three distinct features: (i) it defines derivative and (definite) integral without using limits, (ii) it defines derivative and antiderivative simultaneously via a derivative-antiderivative (DA) pair, and (iii) it posits the fundamental theorem of calculus as a natural corollary of the definitions of derivative and integral. The first D in DD calculus attributes to Descartes for his method of tangents and the second D to DA-pair.

The changing concept of matter in H. Weyl’s thought, 1918 -1930

Por • 3 abr, 2014 • Category: Educacion

During the “long decade” of transformation of mathematical physics between 1915 and 1930, H. Weyl interacted with physics in two highly productive phases and contributed to it, among others, by his widely read book on Space – Time – Matter (Raum- Zeit – Materie), (1918-1923) and on Group Theory and Quantum Mechanics (Gruppentheorie und Quantenmechanik) (1928-1931). In this time Weyl’s understanding of the constitution of matter and its mathematical description changed considerably. At the beginning of the period he started from a “dynamistic” and geometrical conception of matter, following and extending the Mie-Hilbert approch, which he gave up during the year 1920. After transitional experiments with a singularity (and in this sense topological) approach in 1921/22, he developed an open perspective of what he called an “agency theory” of matter. The idea for it was formulated already before the advent of the “new” quantum mechanics in 1925/26.

The axiomatic deduction of the quadratic Hencky strain energy by Heinrich Hencky

Por • 23 feb, 2014 • Category: Crítica

The introduction of the quadratic Hencky strain energy based on the logarithmic strain tensor log V is a milestone in the development of nonlinear elasticity theory in the first half of the 20th century. Since the original manuscripts are written in German, they are not easily accessible today. However, we believe that the deductive approach taken by Hencky deserves to be rediscovered today.

Forgotten Motives: the Varieties of Scientific Experience

Por • 13 feb, 2014 • Category: Crítica

Personal recollections about Alexandre Grothendieck and early days of his theory of motives

The straight line, the catenary, the brachistochrone, the circle, and Fermat

Por • 23 ene, 2014 • Category: Ciencia y tecnología

This paper shows that the well-known curve optimization problems which lead to the straight line, the catenary curve, the brachistochrone, and the circle, can all be handled using a unified formalism. Furthermore, from the general differential equation fulfilled by these geodesics, we can guess additional functions and the required metric. The parabola, for example, is a geodesic under a metric guessed in this way. Numerical solutions are found for the curves corresponding to geodesics in the various metrics using a ray-tracing approach based on Fermat’s principle.

Interpretations and Representations of Classical Tensors

Por • 16 ene, 2014 • Category: Ciencia y tecnología

Classical tensors, the familiar mathematical objects denoted by symbols such as t i , t ij and t ij k , are usually interpreted either as ‘coordinatizable objects’ with coordinates changing in a specific way under a change of coordinate system or as elements of tensor spaces of the form V ⊗n ⊗(V ∗ ) ⊗m . An alternative interpretation of classical tensors as linear tensor maps of the form V ⊗m →V ⊗n is presented here. In this interpretation, tensor multiplication is seen as generalized function composition. Representations of classical tensors by means of arrays are also considered.

A cognitive analysis of Cauchy’s conceptions of function, continuity, limit, and infinitesimal, with implications for teaching the calculus

Por • 12 ene, 2014 • Category: Opinion

In this paper we use theoretical frameworks from mathematics education and cognitive psychology to analyse Cauchy’s ideas of function, continuity, limit and infinitesimal expressed in his Cours D’Analyse. Our analysis focuses on the development of mathematical thinking from human perception and action into more sophisticated forms of reasoning and proof, offering different insights from those afforded by historical or mathematical analyses. It highlights the conceptual power of Cauchy’s vision and the fundamental change involved in passing from the dynamic variability of the calculus to the modern set-theoretic formulation of mathematical analysis.

Experimental library of univalent formalization of mathematics

Por • 5 ene, 2014 • Category: Ambiente

This paper contains a discussion of a library of formalized mathematics for the proof assistant Coq which the author worked on in 2011-13.

On transcendental numbers

Por • 1 ene, 2014 • Category: Opinion

Transcendental numbers play an important role in many areas of science. This paper contains a short survey on transcendental numbers and some relations among them. New inequalities for transcendental numbers are stated in Section 2 and proved in Section 4. Also, in relationship with these topics, we study the exponential function axioms related to the Yang-Baxter equation.

Love Games: A Game Theory Approach to Compatibility

Por • 26 dic, 2013 • Category: Crítica

In this note, we present a compatibility test based on John Nash’s game-theoretic notion of equilibrium strategy. The test must be taken separately by both partners, making it difficult for either partner alone to control the outcome. The mathematics behind the test including Nash’s celebrated theorem and an example from the film, “A Beautiful Mind,” are discussed as well as how to customize the test for more accurate results and how to modify the test to evaluate interpersonal relationships in other settings, not only romantic. To investigate the long-term dynamics of give and take in a relationship we introduce the “iterated dating dilemma” and apply the notion of “zero-determinant payoff strategy” introduced by Dyson and Press in 2012 for the iterated prisoner’s dilemma.