Artículos con la etiqueta ‘matemáticas’

Category Theory Using String Diagrams

Por • 30 ene, 2014 • Category: Ciencia y tecnología

In work of Fokkinga and Meertens a calculational approach to category theory is developed. The scheme has many merits, but sacrifices useful type information in the move to an equational style of reasoning. By contrast, traditional proofs by diagram pasting retain the vital type information, but poorly express the reasoning and development of categorical proofs. In order to combine the strengths of these two perspectives, we propose the use of string diagrams, common folklore in the category theory community, allowing us to retain the type information whilst pursuing a calculational form of proof. These graphical representations provide a topological perspective on categorical proofs, and silently handle functoriality and naturality conditions that require awkward bookkeeping in more traditional notation.

A report on Tarski’s decidability problem

Por • 29 ene, 2014 • Category: Educacion

This paper contains a list of crucial mistakes and counterexamples to some of the main statements in the paper «Elementary theory of free nonabelian groups» by O. Kharlampovich and A. Myasnikov, which was published in the Journal of Algebra in June 2006.

Benford’s Law and the Universe

Por • 27 ene, 2014 • Category: Ciencia y tecnología

Benford’s law predicts the occurrence of the n th digit of numbers in datasets originating from various sources of the world, ranging from financial data to atomic spectra. It is intriguing that although many features of Benford’s law have been proven and analysed, it is still not fully mathematically understood. In this paper we investigate the distances of galaxies and stars by comparing the first, second and third significant digit probabilities with Benford’s predictions. It is found that the distances of galaxies follow reasonably well the first digit law and the star distances agree very well with the first, second and third significant digit.

The Infinite as Method in Set Theory and Mathematics

Por • 23 ene, 2014 • Category: Opinion

El infinito como método en la teoría de conjuntos y la matemática. Este artículo da cuenta de la aparición histórica de lo infinito en la teoría de conjuntos, y de cómo lo tratamos dentro y fuera de las matemáticas. La primera sección analiza el surgimiento de lo infinito como una cuestión de método en la teoría de conjuntos. La segunda sección analiza el infinito dentro y fuera de las matemáticas, y cómo deben adoptarse.

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.

Arrow’s Theorem by Arrow Theory

Por • 21 ene, 2014 • Category: Filosofía

We give a categorical account of Arrow’s theorem, a seminal result in social choice theory.

Logical systems II: Free semantics

Por • 15 ene, 2014 • Category: Crítica

This paper is a sequel to «Logical systems I: Lambda calculi through discreteness». It provides a general 2-categorical setting for extensional calculi and shows how intensional and extensional calculi can be related in logical systems. We focus on transporting the notion of Day convolution to a 2-categorical framework, and as a complementary result we prove the convolution theorem for internal categories. We define the concept of Yoneda triangle, and show how objects in a Yoneda bitriangle get extensional semantics «for free». This includes the usual semantics for propositional calculi, Kripke semantics for intuitionistic calculi and ternary frame semantics for substructural calculi including Lambek’s lambda calculi, relevance and linear logics. We show how in this setting, one may use a model-theoretic logic to induce a structure of a proof-theoretic logic.

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.

La física teórica mejora los simuladores cuánticos

Por • 9 ene, 2014 • Category: Ciencia y tecnología

Los experimentos que se llevan a cabo con átomos fríos son muy importantes para entender la física cuántica porque permiten observar directamente los efectos cuánticos. Por eso a los átomos fríos se los conoce también como “simuladores cuánticos”. Pero la realización de estos experimentos es muy complicada. Por suerte, el trabajo teórico de un físico de la Universidad del País Vasco ha permitido establecer bajo qué condiciones los átomos bosónicos y los fermiónicos –determinantes del comportamiento cuántico de las partículas- pueden mezclarse a muy baja temperatura. El avance ayudará a comprender los efectos especiales derivados de las interacciones atómicas.

Extensionality of lambda-*

Por • 7 ene, 2014 • Category: Opinion

We prove an extensionality theorem for the «type-in-type» dependent type theory with Sigma-types. We suggest that the extensional equality type be identified with the logical equivalence relation on the free term model of type theory.