Learn the mathematics behind the Fibonacci numbers, the golden ratio, and how they are related. These topics are not usually taught in a typical math curriculum, yet contain many fascinating results that are still accessible to an advanced high school student. The course culminates in an explanation of why the Fibonacci numbers appear unexpectedly in nature, such as the number of spirals in the head of a sunflower. Download the lecture notes: https://www.math.ust.hk/~machas/fibonacci.pdf Watch the promotional video: https://youtu.be/VWXeDFyB1hc

O curso "Explorando os recursos educacionais da Khan Academy" tem como objetivo ajudar todos aqueles que desejam utilizar a plataforma como um recurso pedagógico. No decorrer do curso, você terá a oportunidade de conhecer a plataforma do ponto de vista do estudante e do ponto de vista do tutor, sabendo exatamente como utilizá-la em favor da aprendizagem. Ao longo do curso, você terá todas as orientações de que precisa para utilizar a plataforma com seus alunos. A plataforma Khan Academy foi concebida para oferecer aprendizado personalizado para todas as idades, com exercícios, vídeos e um painel de acompanhamento que habilita os estudantes a aprender no seu próprio ritmo. Para o professor, a Khan Academy é uma poderosa ferramenta pedagógica que possibilita personalizar o percurso de aprendizagem dos alunos, dinamizar a experiência educacional com recursos de gamificação e monitorar o progresso de todos e de cada um, com relatórios simples e objetivos. Este curso é autoinstrucional, ou seja, a aprendizagem acontece na interação com o conteúdo, com outros participantes e por meio da realização das atividades propostas. Assim, o cursista tem mais flexibilidade para estudar no seu próprio ritmo, de acordo com a sua disponibilidade de tempo. Como material didático, disponibilizamos textos, links, vídeo tutoriais, quizzes e uma proposta de atividade final, que envolve a sistematização da experiência de uso da plataforma como recurso pedagógico. Para ser certificado, o cursista deve obter 100% de aproveitamento nas avaliações ofertadas ao longo do curso, que são apresentadas em formato de teste, com possibilidade de mais de uma tentativa. Instituições envolvidas no projeto Khan Academy no Brasil: A Fundação Lemann é parceira internacional da Khan Academy no Brasil e tem experiência na implementação da plataforma em diferentes contextos educacionais, incluindo ONGs, instituições de ensino públicas e privadas. Como instituições parceiras do projeto no Brasil, temos o Instituto Península, o Instituto Natura e o Ismart. Ainda, agradecemos o apoio do Instituto Singularidades no desenvolvimento deste programa de formação. Participe de outros cursos da Fundação Lemann no Coursera - https://www.coursera.org/lemann

A very beautiful classical theory on field extensions of a certain type (Galois extensions) initiated by Galois in the 19th century. Explains, in particular, why it is not possible to solve an equation of degree 5 or more in the same way as we solve quadratic or cubic equations. You will learn to compute Galois groups and (before that) study the properties of various field extensions. We first shall survey the basic notions and properties of field extensions: algebraic, transcendental, finite field extensions, degree of an extension, algebraic closure, decomposition field of a polynomial. Then we shall do a bit of commutative algebra (finite algebras over a field, base change via tensor product) and apply this to study the notion of separability in some detail. After that we shall discuss Galois extensions and Galois correspondence and give many examples (cyclotomic extensions, finite fields, Kummer extensions, Artin-Schreier extensions, etc.). We shall address the question of solvability of equations by radicals (Abel theorem). We shall also try to explain the relation to representations and to topological coverings. Finally, we shall briefly discuss extensions of rings (integral elemets, norms, traces, etc.) and explain how to use the reduction modulo primes to compute Galois groups. PREREQUISITES A first course in general algebra — groups, rings, fields, modules, ideals. Some knowledge of commutative algebra (prime and maximal ideals — first few pages of any book in commutative algebra) is welcome. For exercises we also shall need some elementary facts about groups and their actions on sets, groups of permutations and, marginally, the statement of Sylow's theorems. ASSESSMENTS A weekly test and two more serious exams in the middle and in the end of the course. For the final result, tests count approximately 30%, first (shorter) exam 30%, final exam 40%. There will be two non-graded exercise lists (in replacement of the non-existent exercise classes...) Do you have technical problems? Write to us: [email protected]

In this course, you will learn the science behind how digital images and video are made, altered, stored, and used. We will look at the vast world of digital imaging, from how computers and digital cameras form images to how digital special effects are used in Hollywood movies to how the Mars Rover was able to send photographs across millions of miles of space. The course starts by looking at how the human visual system works and then teaches you about the engineering, mathematics, and computer science that makes digital images work. You will learn the basic algorithms used for adjusting images, explore JPEG and MPEG standards for encoding and compressing video images, and go on to learn about image segmentation, noise removal and filtering. Finally, we will end with image processing techniques used in medicine. This course consists of 7 basic modules and 2 bonus (non-graded) modules. There are optional MATLAB exercises; learners will have access to MATLAB Online for the course duration. Each module is independent, so you can follow your interests.

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