Situations where resources are shared among users appear in a wide variety of domains, from lines at stores and toll booths to queues in telecommunication networks. The management of these shared resources can have direct consequences on users, whether it be waiting times or blocking probabilities. In this course, you'll learn how to describe a queuing system statistically, how to model the random evolution of queue lengths over time and calculate key performance indicators, such as an average delay or a loss probability. This course is aimed at engineers, students and teachers interested in network planning. Practical coursework will be carried out using ipython notebooks on a Jupyterhub server which you will be given access to. Student testimonial "Great MOOC ! The videos, which are relatively short, provide a good recap on Markov chains and how they apply to queues. The quizzes work well to check if you've understood." Loïc, beta-tester "The best MOOC on edX! I'm finishing week 2 and I've never seen that much care put in a course lab! And I love these little gotchas you put into quizzes here and there! Thank you!" rka444, learner from Session 1, February - March 2018

Linear algebra is at the core of all of modern mathematics, and is used everywhere from statistics and data science, to economics, physics and electrical engineering. However, learning the subject is not principally about acquiring computational ability, but is more a matter of fluency in its language and theory. In this course, we will start with systems of linear equations, and connect them to vectors and vector spaces, matrices, and linear transformations. We will be emphasizing the vocabulary throughout, so that students become comfortable working with the different aspects. We will then introduce matrix and vector operations such as matrix multiplication and inverses, paying particular attention to their underlying purposes. Students will learn not just how to calculate them, but also why they work the way that they do. We willdiscuss the key concepts of basis and dimension, which form the foundation for many of the more advanced concepts of linear algebra. The last chapter concerns inner products, which allow us to use linear algebra for approximating solutions; we will see how this allows for applications ranging from statistics and linear regression to digital audio.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29 are all prime numbers and they hold special significance. Mathematicians from ancient times to the 21st century have been working on prime numbers, as they're one of the most mysterious and important subjects in mathematics. In this course, I will present several attractive topics on prime numbers. You will learn basic concepts of prime numbers from the beginning. They obey mysterious laws. Some laws are easily verified by hand, some laws were discovered 100 years ago, and some laws are yet to be discovered. Surprisingly, prime numbers are also applied to cryptography today. You will also learn how to construct practical cryptosystems using prime numbers. The original course " Fun with Prime Numbers " was first offered in 2015 and attracted many students. This course will be offered as its refined and upgraded version. All the lecture videos will be renewed, and a new topic on cryptography will be added so as to enliven and satisfy even the students who took the previous course. No previous knowledge of prime numbers is required in this course. Calculating with a pen and paper, you will explore the mysterious world of prime numbers. The course is designed to encourage you to attack unsolved problems, and hopefully, discover new laws of your own in the future!

How do populations grow? How do viruses spread? What is the trajectory of a glider? Many real-life problems can be described and solved by mathematical models. This course will introduce you to the modelling cycle which includes: analyzing a problem, formulating it as a mathematical model, calculating solutions and validating your results. All models are (systems of) ordinary differential equations, and you will learn more about those by watching videos and reading short texts, and more importantly, by completing well-crafted exercises. You will learn how to implement Euler's method in a (Python) program, and finally, you will learn how to write about your findings in a scientific way (with LaTeX). In the verified track of this course you will additionally: Consolidate the new theoretical skills with graded problem sets about five real-life applications. Work on your own modelling project (individually or in a team). Because mathematical modelling is only learned by doing it yourself, you complete your own modelling project on a self-defined real-life problem. You will be guided through the project by completing a list of smaller tasks. This course is aimed at Bachelor students from Mathematics, Engineering and Science disciplines. The course is for anyone who would to use mathematical modelling for solving real world problems, including business owners, researchers and students.

Introduction to unconstrained nonlinear optimization, Newton’s algorithms and descent methods.

Introduction to linear optimization, duality and the simplex algorithm.

As modern life science research becomes ever more quantitative, the need for mathematical modeling becomes ever more important. A deeper and mechanistic understanding of complicated biological processes can only come from the understanding of complex interactions at many different scales, for instance, the molecular, the cellular, individual organisms and population levels. In this course, through case studies, we will examine some simplified and idealized mathematical models and their underlying mathematical framework so that we learn how to construct simplified representations of complex biological processes and phenomena. We will learn how to analyze these models both qualitatively and quantitatively and interpret the results in a biological fashion by providing predictions and hypotheses that experimentalists may verify. 当现代生命科学研究变得更加量化，建立数学模型的需求变得越来越重要。对复杂生物现象的深入理解最终是建立在了解发生于多时空间尺度的复杂生物学相互作用上，例如，分子尺度，细胞尺度，个体和群体尺度上。通过研究一些案例，我们将建立一些简化的数学模型以及其背后的基本数学框架。同时，我们将学习如何建立基本生物学过程的简单表征，以及如何定量和定性和定量地的分析这些模型，并将它们的结果以生物学的方式进行解释，以期提供实验学家进行检验的假说和预测。

In this course, we go beyond the calculus textbook, working with practitioners in social, life and physical sciences to understand how calculus and mathematical models play a role in their work. Through a series of case studies, you’ll learn: How standardized test makers use functions to analyze the difficulty of test questions; How economists model interaction of price and demand using rates of change, in a historical case of subway ridership; How an x-ray is different from a CT-scan, and what this has to do with integrals; How biologists use differential equation models to predict when populations will experience dramatic changes, such as extinction or outbreaks; How the Lotka-Volterra predator-prey model was created to answer a biological puzzle; How statisticians use functions to model data, like income distributions, and how integrals measure chance; How Einstein’s Energy Equation, E=mc2 is an approximation to a more complicated equation. With real practitioners as your guide, you’ll explore these situations in a hands-on way: looking at data and graphs, writing equations, doing calculus computations, and making educated guesses and predictions. This course provides a unique supplement to a course in single-variable calculus. Key topics include application of derivatives, integrals and differential equations, mathematical models and parameters. This course is for anyone who has completed or is currently taking a single-variable calculus course (differential and integral), at the high school (AP or IB) or college/university level. You will need to be familiar with the basics of derivatives, integrals, and differential equations, as well as functions involving polynomials, exponentials, and logarithms. This is a course to learn applications of calculus to other fields, and NOT a course to learn the basics of calculus. Whether you’re a student who has just finished an introductory Calculus course or a teacher looking for more authentic examples for your classroom, there is something for you to learn here, and we hope you’ll join us!

Matrix Algebra underlies many of the current tools for experimental design and the analysis of high-dimensional data. In this introductory online course in data analysis, we will use matrix algebra to represent the linear models that commonly used to model differences between experimental units. We perform statistical inference on these differences. Throughout the course we will use the R programming language to perform matrix operations. Given the diversity in educational background of our students we have divided the series into seven parts. You can take the entire series or individual courses that interest you. If you are a statistician you should consider skipping the first two or three courses, similarly, if you are biologists you should consider skipping some of the introductory biology lectures. Note that the statistics and programming aspects of the class ramp up in difficulty relatively quickly across the first three courses. You will need to know some basic stats for this course. By the third course will be teaching advanced statistical concepts such as hierarchical models and by the fourth advanced software engineering skills, such as parallel computing and reproducible research concepts. These courses make up two Professional Certificates and are self-paced: Data Analysis for Life Sciences: PH525.1x: Statistics and R for the Life Sciences PH525.2x: Introduction to Linear Models and Matrix Algebra PH525.3x: Statistical Inference and Modeling for High-throughput Experiments PH525.4x: High-Dimensional Data Analysis Genomics Data Analysis: PH525.5x: Introduction to Bioconductor PH525.6x: Case Studies in Functional Genomics PH525.7x: Advanced Bioconductor This class was supported in part by NIH grant R25GM114818.

Introduction to the mathematical concept of networks, and to two important optimization problems on networks: the transshipment problem and the shortest path problem. Short introduction to the modeling power of discrete optimization, with reference to classical problems. Introduction to the branch and bound algorithm, and the concept of cuts.