Enumerative combinatorics deals with finite sets and their cardinalities. In other words, a typical problem of enumerative combinatorics is to find the number of ways a certain pattern can be formed. In the first part of our course we will be dealing with elementary combinatorial objects and notions: permutations, combinations, compositions, Fibonacci and Catalan numbers etc. In the second part of the course we introduce the notion of generating functions and use it to study recurrence relations and partition numbers. The course is mostly self-contained. However, some acquaintance with basic linear algebra and analysis (including Taylor series expansion) may be very helpful. Do you have technical problems? Write to us: [email protected]

這是一個機率的入門課程，著重的是教授機率基本概念。課程內容和作業都使用生活化的例子，希望讓同學們快樂學習、快速培養同學們對於機率的洞察力與應用能力。

This course offers a brief introduction to the multivariate calculus required to build many common machine learning techniques. We start at the very beginning with a refresher on the “rise over run” formulation of a slope, before converting this to the formal definition of the gradient of a function. We then start to build up a set of tools for making calculus easier and faster. Next, we learn how to calculate vectors that point up hill on multidimensional surfaces and even put this into action using an interactive game. We take a look at how we can use calculus to build approximations to functions, as well as helping us to quantify how accurate we should expect those approximations to be. We also spend some time talking about where calculus comes up in the training of neural networks, before finally showing you how it is applied in linear regression models. This course is intended to offer an intuitive understanding of calculus, as well as the language necessary to look concepts up yourselves when you get stuck. Hopefully, without going into too much detail, you’ll still come away with the confidence to dive into some more focused machine learning courses in future.

Analytic Combinatorics teaches a calculus that enables precise quantitative predictions of large combinatorial structures. This course introduces the symbolic method to derive functional relations among ordinary, exponential, and multivariate generating functions, and methods in complex analysis for deriving accurate asymptotics from the GF equations. All the features of this course are available for free. It does not offer a certificate upon completion.

The lectures of this course are based on the first 11 chapters of Prof. Raymond Yeung’s textbook entitled Information Theory and Network Coding (Springer 2008). This book and its predecessor, A First Course in Information Theory (Kluwer 2002, essentially the first edition of the 2008 book), have been adopted by over 60 universities around the world as either a textbook or reference text. At the completion of this course, the student should be able to: 1) Demonstrate knowledge and understanding of the fundamentals of information theory. 2) Appreciate the notion of fundamental limits in communication systems and more generally all systems. 3) Develop deeper understanding of communication systems. 4) Apply the concepts of information theory to various disciplines in information science.

As rates of change, derivatives give us information about the shape of a graph. In this course, we will apply the derivative to find linear approximations for single-variable and multi-variable functions. This gives us a straightforward way to estimate functions that may be complicated or difficult to evaluate. We will also use the derivative to locate the maximum and minimum values of a function. These optimization techniques are important for all fields, including the natural sciences and data analysis. The topics in this course lend themselves to many real-world applications, such as machine learning, minimizing costs or maximizing profits.

Algebra is one of the definitive and oldest branches of mathematics, and design of computer algorithms is one of the youngest. Despite this generation gap, the two disciplines beautifully interweave. Firstly, modern computers would be somewhat useless if they were not able to carry out arithmetic and algebraic computations efficiently, so we need to think on dedicated, sometimes rather sophisticated algorithms for these operations. Secondly, algebraic structures and theorems can help develop algorithms for things having [at first glance] nothing to do with algebra, e.g. graph algorithms. One of the main goals of the offered course is thus providing the learners with the examples of the above mentioned situations. We believe the course to contain much material of interest to both CS and Math oriented students. The course is supported by programming assignments.

In this introductory course on Ordinary Differential Equations, we first provide basic terminologies on the theory of differential equations and then proceed to methods of solving various types of ordinary differential equations. We handle first order differential equations and then second order linear differential equations. We also discuss some related concrete mathematical modeling problems, which can be handled by the methods introduced in this course. The lecture is self contained. However, if necessary, you may consult any introductory level text on ordinary differential equations. For example, "Elementary Differential Equations and Boundary Value Problems by W. E. Boyce and R. C. DiPrima from John Wiley & Sons" is a good source for further study on the subject. The course is mainly delivered through video lectures. At the end of each module, there will be a quiz consisting of several problems related to the lecture of the week.

This course is an important part of the undergraduate stage in education for future economists. It's also useful for graduate students who would like to gain knowledge and skills in an important part of math. It gives students skills for implementation of the mathematical knowledge and expertise to the problems of economics. Its prerequisites are both the knowledge of the single variable calculus and the foundations of linear algebra including operations on matrices and the general theory of systems of simultaneous equations. Some knowledge of vector spaces would be beneficial for a student. The course covers several variable calculus, both constrained and unconstrained optimization. The course is aimed at teaching students to master comparative statics problems, optimization problems using the acquired mathematical tools. Home assignments will be provided on a weekly basis. The objective of the course is to acquire the students’ knowledge in the field of mathematics and to make them ready to analyze simulated as well as real economic situations. Students learn how to use and apply mathematics by working with concrete examples and exercises. Moreover this course is aimed at showing what constitutes a solid proof. The ability to present proofs can be trained and improved and in that respect the course is helpful. It will be shown that math is not reduced just to “cookbook recipes”. On the contrary the deep knowledge of math concepts helps to understand real life situations. Do you have technical problems? Write to us: [email protected]

Математика является базой для программиста, инженера, тестировщика. Математические модели важны для понимания того, как будет работать та или иная система, цифровая схема, программа. В нашем курсе вы познакомитесь с классической моделью дискретного устройства – конечным автоматом. Вы разберетесь, поведение каких систем можно описать этой моделью. Научитесь строить проверяющие тесты. Кроме того, мы предлагаем вам услышать мнение специалистов-практиков о роли тестирования при разработке и отладке программного обеспечения. Помимо видеолекций и традиционных тестовых заданий в курсе предусмотрен тренажер, имитирующий процесс тестирования дискретной системы. Цель курса: научить слушателя извлекать математическую модель из описания дискретной системы, строить на основе этой модели полный проверяющий тест и применять его при тестировании предъявленной реализации. Требования к знаниям слушателей: знание математики в объёме средней школы (11 классов), а также базовые знания дискретной математики и информатики. Приветствуется знание основ цифровой техники. Результаты обучения: 1. Слушатель поймет, что такое тестирование и роль формальных моделей в тестировании 2. Слушатель научится применять формальные модели для описания поведения дискретных систем 3. Слушатель научится осуществлять тестирование дискретных систем и анализировать результаты