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A chapter of the book "The Princeton Companion to Mathematics," edited by Timothy Gowers, is presented. In this chapter, the author discusses the art of mathematical problem solving. The author suggests mathematics students to develop technique for problem solving with continuous practice of mathematical problems. Collections of problems using relatively elementary mathematics, are also presented.

A chapter of the book "The Princeton Companion to Mathematics," edited by Timothy Gowers, is presented. It discusses the application and the influence of mathematics in different scientific research studies including the mechanism behind the adoption of 3D structure by proteins, description of crystal structure and role of mathematics in computational chemistry.

A chapter of the book "The Princeton Companion to Mathematics," edited by Timothy Gowers, is presented. It presents brief profiles of notable ancient mathematicians including Pythagoras from Greece, Archimedes from Italy and Euclid from Alexandria, Egypt and discusses their contributions to mathematics.

A chapter of the book "The Princeton Companion to Mathematics," edited by Timothy Gowers, is presented. It discusses several mathematical theorems including Cauchy's Theorem, Dirichlet's Theorem and Ergodic Theorems. Fixed Point Theorems including Brouwer's Fixed Point Theorem and Infinite-Dimensional Fixed Point Theorem and the Fundamental Theorem of Algebra, are discussed.

A chapter of the book "The Princeton Companion to Mathematics " edited by Timothy Gowers, is presented. It discusses the origin of modern mathematics. It states that earliest mathematical documents known date back to the civilizations of the ancient Middle East, in Egypt and in Mesopotamia where a special class of people known as scribes developed. Scribes were responsible for keeping records which involved solving simple arithmetic problems.

The article discusses several topics published in the book "The Princeton Companion to Mathematics" including what is mathematics, algebra versus geometry and number theory.

A chapter of the book "The Princeton Companion to Mathematics," edited by Timothy Gowers, is presented. It discusses classical theory of algebraic numbers and discusses different branches of mathematics including Analytic Number Theory and Computational Number Theory. Current belief about the distribution of prime numbers, is also discussed.

In this paper we consider the following question: how many vertices of the discrete torus must be deleted so that no topologically nontrivial cycles remain? We look at two different edge structures for the discrete torus. For , where two vertices in are connected if their ℓ1 distance is 1, we show a nontrivial upper bound of on the number of vertices that must be deleted. For , where two vertices are connected if their ℓ∞ distance is 1, Saks, Samorodnitsky and Zosin [8] already gave a nearly tight lower bound of using arguments involving linear algebra. We give a more elementary proof which improves the bound to , which is precisely tight. [ABSTRACT FROM AUTHOR]

The systems of linear equations (homogeneous or inhomogeneous) that are partition regular, over ℕ or ℤ or ℚ, were characterized by Rado. Our aim here is to characterize those systems for which we can guarantee a nonconstant, or injective, solution. It turns out that we thereby recover an equivalence between ℕ and ℤ that is normally lost when one passes from homogeneous to inhomogeneous systems. [ABSTRACT FROM AUTHOR]