Analysis: Theory and Practice

There are several sections. The goal is to link complex analysis and imbeddings of graphs so that graph imbeddings can be performed systematically using identification spaces based on group actions in the complex plane. We give a short primer of basic calculus and then explain some of the finer aspects that obfuscate a complete solution to the Riemann Hypothesis. This is not central to our main goal. However, the exercise is meant to help the reader familiarize themselves with integration in the complex plane. The goal was to juxtapose various topics in the fields of combinatorics and analysis. There is one main theorem that explains the treatment of functions on adjacency matrices. Section 1. Basic Calculus. Subsets of power sets form a calculus on a space if the operations union and intersection based in the original subset form a σ-algebra over the entire space. The development of limits is crucial to the understanding of basic calculus and analysis. The Bolzano-Weierstrass result is that every infinite sequence in a closed and bounded set has an accumulation point. A space U is compact if and only if every open cover has a finite subcover. A closed and bounded set is compact, but the coverse is not necessarily true. If a set is compact and closed, then it is also complete. Any Cauchy sequence has a limit, where a Cauchy sequence is one such that for all > 0 there is N such that if m,n > N the distance between xn and xm, has |xn − xm| < . If every Cauchy sequence in a space U is also convergent, then U is sequentially compact. Every compact set is sequentially compact. The space U is a complete space if and only if every sequence {xn} ⊂ U which is Cauchy has xn → x ∈ U. That is, complete spaces contain all the limit points of all the sequences in U. A complete space is closed; likewise, any closed subset of a complete space is also complete. Let f, g be continuous on a finite closed interval and have derivatives on the corresponding interior K of the interval. Theorem 1 (IVT). Then there exists c ∈ K such that f ′(c)[g(b)−g(a)] = g′(c)[f(b)− f(a)]. Theorem 2 (Power Rule). The derivative of f(x) = x has f ′(x) = nxn−1. If a derivative of a function is strictly positive on an interval then the function is strictly increasing on that same interval. If f is defined on a closed interval, then the maxima of f occurs when f ′(0) or at one or both of the endpoints of the interval. Theorem 3 (Chain Rule). If f ◦ g(x) is differentiated relative to the variable x, then [f ◦ g(x)]′ = [f ′ ◦ g(x)][g′(x)]. 1