Quick Index on Probability and Statistics

Quick Index on Probability and Statistics

Calculus (Latin, calculus, a small stone used for counting) is a branch in mathematics focused on limits, functions, derivatives, integrals, and infinite series. Leibniz and Newton are usually both credited with the invention of calculus. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. In the framework of calculus foundations: function, limit, continuous function, theorems about continuous functions which are based on properties of real numbers, bounded above must have least upper bound(有上界必有上确界limitado superiormente,supremo/infimo可以导致闭区间连续必有界). derivatives: definition, differentiability and continuation, theorems about the rules of derivatives, maximum, monotonicity, convexity/convavity, mean value theorem, L'Hôpital's rule. integrals: definitions and theorems about lower/upper sum and integrability(soma superior包括闭区间上连续必可积), fundamental theorem of calculus, rigorous definition of trigonometric logarithm and exponential functions, pi is irrational, theorems about the tricks of integrals, formulae of integral for elementary functions.
 * Cálculo integral e diferencial

Comments: Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. Generally, the definition of natural numbers starts from Peano's fudamental axioms of analysis for the definition of natural numbers and its properties. Through the definitions of field, ordered field and complete ordered field, the fundamental axiom of analysis is introduced, which is usually introduced indirectly in calculus in a not so rigorous way.

Applications: physics(density, velocity, Maxwell equation), chemistry(reaction rate, radioactive decay), biology(population dyamics), mathematics(linear algebra->best fit, analytic geometry->slope concavity, statistic->probability distribution), economics(marginal cost, marginal revenue)

theorems concerning Taylor polynomial, e is irrational as well as transcendental, theorems concerning infinite sequences(如单调有界必收敛,有界数列必有收敛子列não crescente/decrescente) theorem concerning convergence of series, theorem concerning uniform convergence(包含处处连续但处处不可导).
 * Seqüências e séries numéricas

Mário's book Ch.1-2
 * Probabilidade

Magalhães's book
 * Inferência estatística

Mário's book Ch.5
 * Cadeias de Markov

Mário's book Ch.6
 * Equação Mestra

Mário's book Ch.7 (c.f. Ch.4 metodo de Langevin), Binder's book
 * Simulação de Monte Carlo

Su's book
 * Introdução à mecânica estatística

Kerson Huang, Ch., Landau's theory on the second order phase transition, Symmetry spontaneous breaking, Monte Carlo approach/Glauber model
 * Modelo de Ising

The scientific method in practice book
 * Método científico

The Mathematically Biology book, (c.f. oscillation in classical mechanics & reação quimica)
 * Equações de Lotka-Volterra