By Victor J. Katz
Offers an international view of arithmetic, balancing historical, early glossy and smooth historical past. difficulties are taken from their unique resources, allowing scholars to appreciate how mathematicians in a number of occasions and locations solved mathematical difficulties. during this new version a extra worldwide point of view is taken, integrating extra non-Western insurance together with contributions from Chinese/Indian, and Islamic arithmetic and mathematicians. an extra bankruptcy covers mathematical options from different cultures. *Up so far, makes use of the result of very fresh scholarship within the historical past of arithmetic. *Provides summaries of the arguments of all vital principles within the box.
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Extra info for A History of Mathematics: An Introduction (2nd Edition)
16 Chapter Two Linear Inequalities and Theorems of the Alternative 1. Introduction It was mentioned in Chap. 1 that the presence of inequality constraints in a minimization problem constitutes the distinguishing feature between the minimization problems of the classical calculus and those of nonlinear programming. Although our main interest lies in nonlinear problems, and hence in nonlinear inequalities, linearization (that is, approximating nonlinear constraints by linear ones) will be frequently resorted to.
The type of theorem that will concern us in this chapter will involve two systems of linear inequalities and/or equalities, say systems I and II. A typical theorem of the alternative asserts that either system I has a solution, or that system II has a solution, but never both. The most famous theorem of this type is perhaps Farkas' theorem [Farkas 02, Tucker 56, Gale 60]. i Fig. 1 Geometric interpretation of Farkas' theorem: II' has solution, I' has no solution. has a solution but never both.
For 0 ^ X g 1 Hence r + A is convex. Theorem The product pY of a convex set T in Rn and the real number p is a convex set. 2 Nonlinear Programming PROOF Let zl,z* £ »T, then z1 = »xl, z2 = ^x\ where xl,x2 £ T. 0 ^ X^ 1 For Corollary // F and A are two convex sets in R", then F — A is a convex set. 2. Separation theorems for convex sets It is intuitively plausible that if we had two disjoint convex sets in Rn, then we could construct a plane such that one set would lie on one side of the plane and the other set on the other side.