Open Math Notes

This is a much enhanced set of lecture notes presented as part of the mathematics programme in the Mathematics Department of King's College, London. The aim here is to offer a clear exposition of many of the essential results from calculus. The student is expected to be both curious and willing to persevere. This is not intended to be a cookbook of recipes but rather to present not just the what but also the why.

Ivan Wilde · retired - Maths Dept, King's College, London · Date posted: October 23, 2023

Send feedback to the author(s)

This book is on the presentation of the fundamental concepts in the number theory:
divisibility, greatest common divisor, Euclidean algorithm, congruences, diophantine equations. The three classical theorems: Wilson’s theorem, Fermat’s Little theorem and Euler’s
theorem together with its applications are also presented.

Len Meas · Royal University of Phnom Penh · Date posted: October 9, 2023

Send feedback to the author(s)

This is a course on the theory of statistics at the advanced undergraduate (or beginning graduate) level. The goal is to prepare students to use probability to understand statistical inference. Some more advanced topics include decision theory and its relation to Bayes procedures. There is also material on order statistics and on the bootstrap.

William Faris · University of Arizona · Date posted: October 2, 2023

Send feedback to the author(s)

Introductory course: Includes elementary solution via separation of variables and change of variables, general theory (Peano existence theorem, uniqueness of solutions, maximal solutions, continuity in initial conditions), linear ODE (existence, uniqueness, vector space of solutions, fundamental matrix solution, variation of constants, higher-order, Wronskian, constant coefficients), qualitative theory (stability at fixed points, phase portraits, Lyapunov's theorem, linearization, limit sets).

Peter Philip · LMU Munich · Date posted: August 9, 2023

Send feedback to the author(s)

Part I is an intermediate level probability course given at the Independent University of Moscow. The unusual feature is an emphasis on discrete-parameter martingales. The development using examples based on gambling schemes makes no mention of conditional expectation.

Part II is a more systematic review of basic theory, leading up to the common definition of martingales in terms of conditional expectation.

William Faris · University of Arizona · Date posted: August 8, 2023

Send feedback to the author(s)