Applying pure mathematics: mathematics in the natural sciences

The book of nature is written in the language of mathematics. This quotation, attributed to Galileo, seemed to hold to an unreasonable1 extent in the era of quantum mechanics.

Edexcel IGCSE Further Pure Mathematics: Student Book.

Este libro desarrolla los conocimientos básicos de matemáticas para aprobar los exámenes de Edexcel IGCSE nivel de enseñanza secundaria. Los temas del libro son : índices y funciones logarítmicas (reglas de índices, escribir una expresión como logaritmo, leyes de los logaritmos), la función al cuadrado ( resolviendo ecuaciones al cuadrado), igualdades y desigualdades (dividiendo un polinomio), gráficos y funciones (usando gráficos de funciones para resolver ecuaciones, dibujando las curvas de las funciones) series (sumando los términos de una secuencia aritmética), usando vectores para describir la posición de un punto, cálculo, trigonometría (usando las reglas del seno y el coseno).

[Lecture notes on pure mathematics. Convex sets]

Includes bibliographies.

Pure mathematics for engineers,

Mode of access: Internet.

Encyclopaedia of pure mathematics.

Articles reprinted from Encyclopaedia metropolitana.

A complete course of pure mathematics /

Mode of access: Internet.

Questions in pure mathematics proposed at the B.A. and B.Sc. pass and honours examinations of the University of London. With complete solutions.

Mode of access: Internet.

Partial regularity of weak solutions of the liquid crystal equilibrium system

For a parameter, we consider the modified relaxed energy of the liquid crystal system. Each minimizer of the modified relaxed energy is a weak solution to the liquid crystal equilibrium system. We prove the partial regularity of minimizers of the modified relaxed energy. We also prove the existence of infinitely many weak solutions for the special boundary value x.

Partial regularity of minimizers of a functional involving forms and maps

We discuss the partial regularity of minimizers of energy functionals such as (1)/(p)integral(Omega)[sigma(u)dA(p) + (1)/(2)delu(2p)]dx, where u is a map from a domain Omega is an element of R-n into the m-dimensional unit sphere of Rm+1 and A is a differential one-form in Omega.

Extremal circular permutations with concave functions

Let {a(1), a(2), ..., a(n)} be a set of n distinct real numbers and let alpha(1), alpha(2), ..., alpha(n) an be a permutation of the numbers. We construct the permutation to maximise L-f = Sigma(i=1)(n) f(\alpha(i+1) - alpha(i)\), for any increasing concave function f, where we denote alpha(n+1) equivalent to alpha(1). The optimal permutation depends on the particular numbers {a(1), a(2), ..., a(n)} and the function f, contrary to a postulate by Chao and Liang (European J. Combin. 13 (1992) 325). (C) 2004 Elsevier Ltd. All rights reserved.

Existence of solutions to some equilibrium problems

The concept of a monotone family of functions, which need not be countable, and the solution of an equilibrium problem associated with the family are introduced. A fixed-point theorem is applied to prove the existence of solutions to the problem.

Regularity and relaxed problems of minimizing biharmonic maps into spheres

For n >= 5 and k >= 4, we show that any minimizing biharmonic map from Omega subset of R-n to S-k is smooth off a closed set whose Hausdorff dimension is at most n - 5. When n = 5 and k = 4, for a parameter lambda is an element of [0, 1] we introduce lambda-relaxed energy H-lambda of the Hessian energy for maps in W-2,W-2 (Omega; S-4) so that each minimizer u(lambda) of H-lambda is also a biharmonic map. We also establish the existence and partial regularity of a minimizer of H-lambda for lambda is an element of [0, 1).