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Applicable from fall 2013 and thereafter.
|Year.||Course Title.||Field.||1st semester.||2nd semester.|
|Freshman||Introduction to Computer Science||Computing and implementations||3||3|
|SophomoreJunior||*Vector Calculus||Analysis||3|| |
|SophomoreJunior||Ordinary Differential Equation||Analysis||3|| |
|SophomoreJunior||Introduction to Partial Differential Equations||Analysis|| ||3|
|SophomoreJunior||Introduction to Practice of Mathematics Software||Computing and implementations||3|| |
|SophomoreJunior||Computational Mathematics||Computing and implementations|| ||3|
|SophomoreJunior||Discrete Mathematics||Others||3|| |
|SophomoreJunior||Introduction to Analysis||Analysis||4||4|
|Junior||Complex Analysis||Analysis|| ||3|
| ||Total|| || ||66 Credits|
For old version see Downloads.
Courses Description (click the title for more information)
|This course offers the general concepts and principles in physics. The main objectives of this course are to motivate students to explore the beauty of physics in various disciplines, and to train students the physics problem-solving ability....more|
|This course provides a first introduction into the theory of differentiation and integration. The course mainly serves as a bridge between high school mathematics and university mathematics. Its main goal is to make students acquainted with rigorous mathematical thinking. This is done via learning basic concepts such as limits, continuity, differentiability, etc. on the one hand and fundamental theorems such as the intermediate value theorem, the extreme value theorem, the mean value theorem, etc. on the other hand.Moreover, the course is intended to train students problem solving skills as well as writing and oral skills. Finally, the course equips students with the basic tools needed in the more applied sciences and is the entrance door to more advanced courses on mathematics....more|
|Introduction to Computer Science|
|The goal of this course is twofold. On the one hand, the course examines the basic ideas of the science of computing. Lectures cover a wide variety of topics such as computer organization, different number systems and its conversions, operating systems.
On the other hand, the course introduces the theory and practice of computer programming. The emphasis is on techniques of program development using the C++ language. It aims to help students feel confident of their ability to write programs to solve problems in mathematics or engineering....more
|This course provides a first introduction into the origin, theory and application of Linear Algebra. ...more|
|The main goal of the course Vector Calculus is to explore the mathematical operations which one encounters in carrying out concrete calculations in multivariable Calculus, as well as the fundamental concepts of vector analysis.
Indeed, the scope of the course is to let students become familiar with the ideas of taking integrals of scalar - valued (or vector – valued) functions over curves or surfaces, as well as the applications of the theorems of Green, Stokes and Gauss.
In addition, the course will also introduce students to the basic geometric concepts of curves and surfaces in either Euclidean spaces of dimension 2 or 3.
The prerequisites of this course are Calculus and linear algebra.
|Differential Equation is an immediate-connected course of Calculus and Linear Algebra, as its content consists of various applications of these basic mathematics. Differential Equation is originated from Newtonian mechanics and celestial mechanics, and still plays an important role in modern physics including quantum mechanics. On the other hand, mathematical modeling has prevailed in neural science, infectious diseases, gene networks, circuit theory, financial engineering, and traffic flows, etc. Although nonlinear differential equations can not be solved into solutions of exact forms in general, it is still feasible to apply various mathematical ideas and concepts such as analysis and geometry in understanding at least a certain part of the dynamics. Moreover, combining numerical computation with geometric delineation of or phase portraits enhances the investigation of differential equations....more|
|Introduction to Partial Differential Equation|
|The main driving force which completely changed the outlook of the scientific-technological aspect of human civilization was the scientific revolution initiated by Newton. The scientific revolution of Newton was the climax of human scientific thought which began immediately after a series of previous great artistic and scientific achievements by the giants since the time of Renaissance. In general, Newton’s scientific revolution, through his invention of Newtonian mechanics, is generally regarded as one of the triumphs of human’s rational thinking in recent human history.
Basically speaking, the main principle of Newtonian mechanics can be realized mathematically as a second-order ordinary differential equation. To a certain extent, the past developments of mathematical tools, which are capable of addressing mathematical problems arising from this second-order O.D.E. in Newtonian mechanics, constituted the whole historical development of Calculus.
To put it in simple words, we can say that Calculus was invented for the sole purpose of solving differential equations. Indeed, being witnesses to the tremendous success of Newtonian mechanics, mathematicians and physicists since the time of Newton began to apply the main ideas of Newtonian Mechanics to all the other branches of Natural Science. In particular, in the studies of the physical phenomena of a vibrating string with fixed end-points, the first partial differential equation, which is known as the wave equation (or the d’Alembert’s equation), was derived for the very first time in the early days of the historical development of P.D.E.’s theory. A partial differential equation is, by definition, an equation satisfied by some of the partial derivatives of the unknown multi-variable function.
Partial differential equations arise naturally in the study of geometry, and in almost all physical laws of Natural Sciences or practical engineering problems. We can even say that, without the developments of the theory of partial differential equations, there could be no development of Science at all in recent human history. Indeed, some of the most famous partial differential equations which appeared in the history of Science are:Laplace equation, Wave equation, Heat equation, Maxwell equation, Schrodinger equation, Navier-Stokes equation and Euler equation, Boltzmann equation, KdV equation.
Basically, one can say that the studies of these equations encompass almost all the mathematical problems arising from Physics and different areas of engineering.
In order to solve these partial differential equations, mathematicians, physicists, and engineers had developed various kinds of mathematical methods, among these the most famous one is the Fourier analysis. As a matter of fact, the subject of Fourier analysis constitutes almost all of the modern mathematical analysis. However, as the development of the theory of partial differential equations became more mature and sophisticated, researchers began to realize the importance of acquiring a better understanding about various kinds of nonlinear phenomena. This naturally motivated the developments of the theory of dynamical systems in recent and modern times, including the recent studies of the theory of Chaos.
Besides standard tools borrowed from mathematical analysis, numerical analysis is another important methodology which is currently used in the studies of solutions to partial differential equations. Through carrying out numerical simulations with the aid of high-speed computers, the use of “Numerical Analysis” in the studies of P.D.E.’s can be regarded as carrying out mathematical experiments. These mathematical experiments through numerical simulations not only help us to gain a deeper understanding about the equation being studied, but even may provide us with clues which can lead to rigorous proofs of mathematical assertions related to the equation being studied.
Finally, but not the least, we should never forget that the sole purpose of a differential equation is to describe natural phenomena. Even priori to our efforts devoted in solving a specific differential equation, the equation itself has already revealed some of the secrets of the Nature to us. In other words, Differential equations themselves naturally speak the language of the Nature. So, it is absolutely true that knowing the basics of Physics will be extremely helpful for the studies of partial differential equations....more
|Computational mathematics is the study of algorithms use to solve mathematical problems. In this course, we introduce numerical methods for solving systems of linear equations and nonlinear equations. Approximation theory including polynomial interpolation, numerical differentiation and numerical integration will also be covered. Numerical errors arisen from various approximations in some of the above mentioned topics will be analyzed. Moreover, round-off errors from computations under floating point arithmetic will be discussed. Finally, students will learn methods for solving ordinary differential equations numerically and understand the stability/consistency of the numerical methods they use....more|
|This course is intedned as an elementary introduction to the theory of probability for students in mathematics. It attempts to present not only the mathematics of probability theory but also, through numerous examples, the many diverse possible applications of this subject. The prerequisite for this course is very fundamental, including the basic counting in high school and a one-year course in calculus....more|
|The origin of statistics can be traced back to the urge of governments to gather information about economic development and population. However, modern statistics is not only restricted to these two areas anymore, but also studies problems from psychology, medicin, anthropology, epigenetics, agriculture etc. Below, a more detailed description of the course is given....more|
|Discrete Mathematics, or called Combinatorics, is an important branch of Mathematics, and its influence continues to expand. Part of the reason for tremendous growth of Combinatorics is the development of computer, which needs combinatorial thinking to perform and analyze its correct and efficient functioning. Another reason for the recent growth of Combinatorics is its ideas and techniques applicable to many areas, for example, the physical sciences, the social sciences, the biological sciences, information theory and so on. This course prepares students the ability concerned with the existence, enumeration, analysis and optimization of discrete structure....more|
|Introduction to Analysis|
|The purpose of this course is to lay down the rigorous foundation for the study of real number system, real sequences, real series and real-valued functions. The course provides a bridge from “freshman” calculus to graduate courses that use analytic ideas, e.g. , real and complex analysis, partial and ordinary differential equations , numerical analysis, fluid mechanics , and differential geometry....more|
|Abstract algebra is the subject area in mathematics that studies algebraic structures of mathematical objects, such as integers, complex numbers, matrices, roots of polynomials, symmetries of polyhedrons, and et al. Contemporary mathematices make extensive use of abstract algebra.
For example, algebraic number theory and algebraic geometry apply algebraic methods to problems in number theory and geometry, respectively. In recent years, abstract algebra also figures prominently in coding theory and cryptography, which are essential in telecommunication nowadays. In this one-year course, we will study three kinds of algebraic structures that appear most often in mathematics, namely, groups, rings, and fields, culminating in the proof of the theorem that a general quintic equation has no solutions in radicals....more
|Complex Analysis is an important area with many application in Physics, Number Theory, Combinatorics, etc. The main goal of complex analysis is to study analytic functions. In this undergraduate course, student will learn the basics of the theory. Topics covered include analytic functions, line integrals, singularities, the residue theorem, conformal mappings, analytic continuation, etc....more|