| Course Code and Title
| MATH 161 Calculus I
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| Course Descriptor
| This course covers limits and continuity as well as differentiation and integration of polynomial, rational, trigonometric, logarithmic, exponential and algebraic function. The application areas include slope, velocity, extrema, area, volume and work.
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| Course LOs
| Upon the completion of this course, students are expected to be able to examine and utilize the following notions and methods.
1) Use both the limit definition and rules of differentiation to differentiate functions.
2) Sketch the graph of a function using asymptotes, critical points, the derivative test for increasing/decreasing functions, and concavity.
3) Apply differentiation to solve applied max/min problems.
4) Apply differentiation to solve related rates problems.
5) Evaluate integrals both by using Riemann sums and by using the Fundamental Theorem of Calculus.
6) Apply integration to compute arc lengths, and areas between two curves.
7) Use L’Hôpital’s rule to evaluate certain indefinite forms.
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| Course Code and Title
| MATH 251 Discrete Mathematics
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| Course Descriptor
| This course is a one semester course intended for students majoring in Mathematics or Computer Science. It introduces the students to the fundamental concepts of mathematical reasoning. The main themes of the course are logic and proof, induction and recursion, discrete structures, set theory, combinatorics, algorithms, graph theory, and their applications. The students will learn how to formulate precise mathematical statements. The course also explores several common proof techniques and exposes the students in learning how to write mathematical proofs.
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| Course LOs
| By the end of the course the student will be expected to be able to:
1) Formulate and assess logical expressions.
2) Understand the elementary set theory.
3) Construct elementary mathematical proofs, including using induction.
4) Solve simple problems related to number theory, combinatorics and graph theory.
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| Course Code and Title
| MATH 263 Calculus III
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| Course Descriptor
| This course covers analytic geometry in 3-space, partial and directional derivatives, extrema, double and triple integrals, line integrals, surface integrals, gradient, divergence, curl, multi-integral applications, as well as cylindrical and spherical coordinates.
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| Course LOs
| 1) Use new concepts to analyze situations occurring in 3-space
2) Know how to evaluate limits, derivatives and integrals of vector-valued functions
3) Know how to perform multiple integrals
4) Understand geometrical meanings of various concepts.
5) Know how to show a two-variable function has no limit
6) Know how to show a two-variable function is/is not differentiable at a given point.
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| Course Code and Title
| MATH 273 Linear Algebra with Applications
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| Course Descriptor
| This course is a one-semester course intended for mathematics, engineering and science students. It introduces students to the fundamental concepts of linear algebra. The course primarily deals with two mathematical objects: matrix and vector, and covers topics such as solutions of systems of linear equations, properties of invertible matrices, linear transformation, determinant, eigenvalues and eigenvectors, vector spaces and subspaces, linear independence, basis, coordinates, inner product, norm, orthogonal basis, similarity, and quadratic forms.
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| Course LOs
| 1) Understand systems of linear equations and how to solve them by the row reduction algorithm.
2) Formulate linear problems in the language of vectors and matrices.
3) Know the basic theory of finite dimensional real vector spaces: linear transformations, bases, dimension.
4) Know some basic matrix algebra: addition and multiplication, base change, diagonalization (by orthogonal matrices in the symmetric case), eigenvectors and eigenvalues, Gram-Schmidt process.
5) Orthogonal spaces and quadratic forms.
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| Course Code and Title
| MATH 301 Introduction to Number Theory
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| Course Descriptor
| An introductory course in the theory of numbers covering such topics as: Euclidean algorithm, Fundamental Theorem of Arithmetic, congruences, Diophantine equations, Fermat and Wilson Theorems, quadratic residues, continued fractions, Prime number theorem and applications such as cryptography.
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| Course LOs
| Upon completion of this course, students are expected to:
1) Understand divisibility and unique factorization in various rings (Z, Gaussian integers).
2) Master the use of arithmetic modulo an integer (congruences).
3) Know the basic multiplicative functions and associated theorems (Fermat, Euler, Wilson).
4) Know the elementary theory of quadratic residues, up to quadratic reciprocity.
5) Know facts related to the discrete logarithm in Z/n (primitive roots, index).
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| Course Code and Title
| MATH 310 Applied Statistical Methods
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| Course Descriptor
| Descriptive statistics, measures of location and spread, graphic methods, basic probability properties, Bayes’ rule, probability distributions, expected value and variance, binomial and normal distributions, point and interval estimation, hypothesis testing, one-sample and two-sample t-tests, chi-square tests, linear regression, quality measures, and assumption verification, ANOVA and multiple comparisons.
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| Course LOs
| Upon successful completion of the course, students will have demonstrated conceptual understanding of and technical skill with the theory of statistics including:
1) Analysis of real experimental data, descriptive statistics and data displays.
2) Contingency tables, rules of probability, conditional probability, Bayes’ Theorem.
3) Normal and binomial distributions; normal approximation of discrete distributions.
4) Statistical estimation and tests of significance: one-sample and two-sample t-procedures, inference for a single population proportion, analysis of paired data.
5) Statistical principles of design, observational studies and experiments, blocking and stratification (optional), sampling concerns.
6) Linear regression models, including test of model utility, sums-of-squares and the basic regression identity, coefficient of determination, F - test and analysis of variance, multiple comparisons (Bonferroni and Tukey).
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| Course Code and Title
| MATH 321 Probability
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| Course Descriptor
| This course covers foundations of probability theory, combinatorial and counting methods, conditional probability, random variables, discrete and continuous distributions, expectation, moment generating functions, multivariate distributions, variable transformations, the Law of Large Numbers and the Central Limit Theorem.
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| Course LOs
| At the completion of this course, the students are expected to be able to:
1) Interpret and communicate the results using English
2) Understand principles of probability
3) Understand and apply independence, conditional probability, and counting techniques
4) Use continuous and discrete random variables
5) Use univariate and multivariate distributions, derive conditional and marginal distributions
6) Conduct transformations of random variables
7) Understand moments and their applications
8) Use special distributions: binomial, Poisson, normal, and others
9) Use Central Limit Theorem
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| Course Code and Title
| MATH 322 Mathematical Statistics
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| Course Descriptor
| The course starts where Probability ends and considers point estimation, method of moments, maximum likelihood estimation, unbiasedness, consistency, sampling distributions, confidence intervals, hypothesis testing, power of tests. Categorical data inference may be covered if time permits.
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| Course LOs
| Students completing the course will be able to:
1) Understand the difference between probability and likelihood functions, and find the maximum likelihood estimate for a model parameter.
2) Find confidence intervals for parameter estimates.
3) Use null hypothesis significance testing to test the significance of results and understand and compute the p-value for these tests.
4) Use specific significance tests including, z-test, t-test (one and two sample), chi-squared test.
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| Course Code and Title
| MATH 323 Actuarial Mathematics I
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| Course Descriptor
| This course covers the topics for the Financial Mathematics (FM) exam of the Society of Actuaries. They include the fundamental concepts and methods for calculating present and accumulated values for reserving, valuation, pricing, asset/liability, investment income, capital budgeting, and contingent cash flows.
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| Course LOs
| By the end of the course the student will be expected to be able to:
1) To understand basic theory of interest models,
2) To understand values of cash flows with various interest rates and payment methods
3) To be able to evaluate annuities, investment portfolios, and various cash flows
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| Course Code and Title
| MATH 351 Numerical Methods
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| Course Descriptor
| This course covers the fundamentals of numerical methods for students in science and engineering; floating-point computation, roots of equations, least-squares approximations, systems of linear equations, approximation of functions and integrals, the single nonlinear equation, and the numerical solution of ordinary differential equations; various applications in science and engineering; programming exercises in Matlab.
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| Course LOs
| By the end of the course the student will be expected to be able to:
1) Solve numerically mathematical problems (including root finding, interpolation, integration, differentiation, and ordinary differential equations)
2) Implement numerical algorithms using a programming language, i.e., Matlab.
3) Know the fundamental principles of mathematical approximations.
4) Analyze approximation errors and construct some bounds of errors.
5) Perform basic convergence analysis of an approximation method.
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| Course Code and Title
| MATH 361 Real Analysis I
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| Course Descriptor
| This course conducts a careful treatment of the theoretical aspects of the calculus of functions of a real variable. The course proceeds to cover series, absolute convergence, series of functions, and continuity, the differentiation and the integration in R, and the basic topology.
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| Course LOs
| 1) Understand the real and complex filed, state their properties, and prove some of them.
2) Use properties of compact, finite, countable and connected sets to prove theorems
3) Know, prove, and apply the lemmas and theorems about convergence sequences and series
4) Understand and reproduce definitions and theorems about functions and their proofs.
5) Understand the differences between sequences of numbers and functions
6) Define the derivative of a function and know the related theorems and proofs.
7) Understand the construction of the Riemann integral, integrable functions and inequalities involving integrals.
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| Course Code and Title
| MATH 371 Introduction to Mathematical Biology
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| Course Descriptor
| This course provides an introduction to the use of discrete and differential equations in the biological sciences. Biological models will include single species and interacting population dynamics, modeling infectious and dynamic diseases, regulation of cell function, molecular interactions, neural and biological oscillators. Mathematical tools such as phase portraits, bifurcation diagrams, perturbation theory, and parameter estimation techniques that are introduced to analyze and interpret biological models will also be covered.
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| Course LOs
| At the completion of this course, the students are expected to be able to:
1) Formulate and solve mathematical models of evolution.
2) Use techniques from difference and differential equations to describe spread of disease and other biological material.
3) Explain how these techniques are applied in scientific studies and applied in ecology and epidemiology.
4) Find the equilibria of a single-population model and their stability.
5) Use techniques from nonlinear differential equations to explain the oscillation in population dynamics and chemical systems.
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| Course Code and Title
| MATH 407 Graph Theory
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| Course Descriptor
| Examines basic concepts and applications of graph theory, where graph refers to a set of vertices and edges that join some pairs of vertices; topics include subgraphs, connectivity, trees, cycles, vertex and edge coloring, planar graphs and their colorings. Draws applications from computer science, operations research, chemistry, the social sciences, and other branches of mathematics, but emphasis is placed on theoretical aspects of graphs.
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| Course LOs
| The course participants
1) become familiar with graphs both as mathematical objects and data structures;
2) learn a variety of graph properties and their interrelatedness;
3) apply skills acquired in discrete mathematics, linear algebra and/or computation to the analysis and calculation of graph properties;
4) understand graphs as a unifying abstraction of natural and computing systems;
5) are enabled and motivated to begin independent project research work in discrete mathematics and computation.
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| Course Code and Title
| MATH 411 Linear Programming
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| Course Descriptor
| This course introduces the concepts and theoretical aspects of linear optimization. The course starts with formulation of real problem into a linear program (LP), geometric interpretation of linear optimization, integer programming and LP relaxation, duality theory, simplex method, sensitivity analysis, robust optimization, ellipsoid and interior point methods, and semidefinite optimization, applications (e.g., game theory and network optimization).
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| Course LOs
| 1) Be able to construct a linear program of various simple real-life problems
2) Understand simplex method and several variations of the simplex method
3) Know the fundamental theorem of linear programming
4) Be able to solve a small LP problem using simplex method
5) Understand the behavior of the simplex method
6) Be able to transform a non-standard LP problem into standard ones
7) Understand the concept of duality
8) Know an example of interior point methods and understand its constructions.
9) Understand the role of Lagrange multipliers in linear programming
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| Course Code and Title
| MATH 412 Nonlinear Optimization
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| Course Descriptor
| The course covers an introduction into convex sets and convex functions, unconstrained and constrained optimization, existence of solutions (Weierstrass’ theorem), unconstrained optimization: first and second order conditions, equality constraints: Lagrange’s optimality theorem, second order conditions, inequality constraints: Karush-Kuhn-Tucker theorem, convex and non-convex optimization, some computational methods: line search, steepest descent, Newton, quasi Newton, trust region, feasible points, penalty methods, and augmented Lagrangian methods.
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| Course LOs
| At the end of the class , the students will grasp the knowledge of
1) Convexity
2) Fenchel-Rockafeller Duality
3) Convex Optimization
4) Lagrange Duality
5) Dynamic Programming Principle
6) Pontryagin Maximum Principle
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| Course Code and Title
| MATH 417 Cryptography
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| Course Descriptor
| The course is an introduction to modern cryptographic algorithms and to their cryptanalysis, with an emphasis on the fundamental principles of information security. Topics include: classical cryptosystems, modern block and stream ciphers, public key ciphers, digital signatures, hash functions, key distribution and agreement.
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| Course LOs
| 1) Understand classical and modern approaches to encryption.
2) Understand the difference between absolute and computational security.
3) Exposure to basic cryptographic concepts such as: block cipher, stream cipher, hash functions, public-key encryption, RSA, elliptic curves, digital signatures.
4) Perform some simple cryptanalysis under various assumptions.
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| Course Code and Title
| MATH 425 Stochastic Processes
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| Course Descriptor
| There first part of this course deals with discrete time Markov chains, including Markov property, transition probabilities, classification of states, visits to a fixed state, notion of limiting behavior, reducibility, recurrence. The course continues to Poisson processes, continuous-time chains, birth-and-death processes, renewal processes, branching processes, and queuing theory with applications. There is also a brief introduction to martingales and Brownian motion.
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| Course LOs
| At the end of the class, the students will grasp the knowledge of
1) Random Walks
2) Conditional Expectation
3) Markov Chains
4) Poisson Processes
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| Course Code and Title
| MATH 440 Regression Analysis
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| Course Descriptor
| The course starts with simple linear regression, diagnostic tests and plots, quality measures, matrix description of regression model. It continues with the multiple regression, predictor subset selection, interactions, variable transformations, use of categorical predictors, model validation, remedial measures. Some other topics that are considered are autocorrelation and logistic regression.
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| Course LOs
| When given observations of two or more variables, the student will be able to:
1) Select appropriate set of predictors,
2) Model numerical response using a single or multiple explanatory variables to investigate relationships between variables,
3) Examine the appropriateness of a regression model and use remedial measures when the model is not appropriate,
4) Interpret modeling results correctly, effectively, and in context without relying on statistical jargon,
5) Prepare reports and presentations with reproducible code.
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| Course Code and Title
| MATH 441 Design of Experiments
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| Course Descriptor
| The course begins with one-way and two-way ANOVA for fixed effects, diagnostics and remedial measures, multiple comparisons. We proceed to covering studies with unequal sample sizes, multifactor ANOVA, random effects, randomized block designs, nested designs, repeated measure designs, Latin Square and related designs.
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| Course LOs
| By the end of the course students should be able to:
1) Understand the benefits and limitations of common experimental designs;
2) Plan a sound experimental design, including the use of orthogonality, randomization, blocking, and replication (through a sample size analysis);
3) Plan a practical experimental design, including the use of parsimonious methods when experimental resources are limited;
4) Analyze factorial experiments using exploratory data analysis, informal and formal inferential methods; and
5) Communicate experimental designs to technical and non-technical audiences.
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| Course Code and Title
| MATH 444 Nonparametric Methods
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| Course Descriptor
| A large part of this course is dedicated to considering nonparametric analogs of parametric tests. These analogs include the sign test, Wilcoxon signed-rank test, permutation test, Wilcoxon rank-sum test, Kruskal-Wallis test, Friedman test and various chi-square tests. The course also considers goodness-of-fit tests, various permutation procedures, bootstrapping methods, procedures for censored data.
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| Course LOs
| On completion of the course, the student should be able to:
1) give an account of the most common non-parametric methods for the analysis of data;
2) apply the most common non-parametric methods in practice;
3) decide when a non-parametric method is more suitable than a parametric method;
4) use statistical software for performing permutation tests and bootstrap on data sets.
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| Course Code and Title
| MATH 460 Topology
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| Course Descriptor
| A first course in topology, the first part of the course will deal with metric spaces, including limits of sequences, cluster points of sets, the two definitions of compactness, continuity and convergence. The later part of the course will cover topological spaces and their properties. Elementary properties of homotopy theory will be covered at the end.
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| Course LOs
| At the end of the course, students will be familiar with:
1) Topological spaces, bases and subbases of a topology. Different ways to define a topology.
2) Product spaces, compact spaces, Tychonoff’s theorem.
3) The different separation axioms.
4) Continuous functions, homeomorphic topological spaces.
5) Metrizable spaces. Examples and counterexamples.
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| Course Code and Title
| MATH 471 Nonlinear Differential Equations
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| Course Descriptor
| The course studies nonlinear differential equations and the time-behavior of their solutions. The course begins with a quick review on first order (logistic) differential equations, and continues with classification into (non) autonomous and (non)linear equations, interpretation of the solution and its general and near equilibrium behavior, the phase-plane, Poincaré map, critical points, periodic solutions, Poincaré-Bendixson theorem, the concept of stability (Lyapunov), stability analysis by linearization, perturbation theory, Poincaré-Lindstedt method, averaging, bifurcation, one dimensional chaos, fractal sets, and Hamiltonian systems.
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| Course LOs
| At the completion of this course, the students are expected to be able to:
1) Draw and interpret phase diagrams.
2) Distinguish between stable and unstable equilibrium points.
3) Define and compute the index of an equilibrium point.
4) Use linear approximation methods for analyzing dynamical systems.
5) Find limit cycles and describe their stability.
6) Apply some perturbation methods to nonlinear systems.
7) Understand and apply the concepts of Poincaré and Liapunov stability.
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| Course Code and Title
| MATH 476 Numerical Methods for Partial Differential Equations
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| Course Descriptor
| The course is a blend of conceptual and practical aspects of numerical solutions of partial differential equations, and aims at students of senior level. Various numerical methods are discussed: finite difference, finite volume, finite element, and spectral methods, and explicit/implicit methods for time integration. Conceptual aspects include convergence, consistency, stability, Courant-Friedrichs-Lewy criteria, Lax equivalence theorem, error analysis, Fourier-von Neumann stability analysis. Discussions are tailored towards numerical solutions of model problems: Poisson equation, heat equation, advection equation, wave equation, advection-diffusion equation, and Stokes equation. Students are required to implement some methods in Matlab for 1D/2D Poisson, advection, and advection-diffusion equations.
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| Course LOs
| 1) Understand various numerical methods for solving boundary value problems for main types of partial differential equations.
2) Be able to perform several discretization procedure for reducing a boundary value problem into a system of algebraic equations for elliptic problems and solving such systems of equations numerically.
3) Be able to apply several spatial discretization methods and time integrators for reducing an initial/ boundary value problem into a system of differential equations and solving such systems numerically.
4) Understand various applications of the boundary value and initial/boundary value problems in science and engineering and why the numerical methods are necessary for solving these problems in practice.
5) Be able to discretize some transport equations.
6) Be able to develop and implement some numerical procedures for solutions of a project problem approved by the instructor.
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| Course Code and Title
| MATH 477 Applied Finite Element Methods
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| Course Descriptor
| Piecewise polynomial interpolations, basis shape functions in natural coordinates, local and global shape functions in spatial one and two dimensions, initial/boundary value problems in science and engineering with industrial applications, Galerkin method, Rayleigh-Ritz method, local and global finite element matrices, connectivity and nodal degrees of freedom, numerical integration, numerical linear algebra, Newton’s method, conjugate gradient method, industrial applications of finite elements for heat transfer, structural, and fluid flows, industrial models and software application, validation and presentation of simulation results.
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| Course LOs
| 1) Understand a range of initial/boundary value problems in science, engineering, and industries that can be modeled and solved numerically by finite elements
2) Understand the Rayleigh-Ritz and Galerkin procedures in finite element analysis
3) Construct a variety of finite element shape functions in 1-d and 2d
4) Express piecewise finite element interpolation in compact mathematical forms
5) Understand the book-keeping and assembling techniques to form the finite element matrix equations
6) Be able to use standard time discretization procedures for initial/boundary value problems
7) Be familiar with basic operations of at least one finite element software
8) Be able to write report and to present numerical simulation results for industrial models
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| Course Code and Title
| MATH 481 Partial Differential Equations
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| Course Descriptor
| An introductory course on partial differential equations, it focuses on classical linear and quasi-linear partial differential equations. The course begins with the classification of partial differential equations (hyperbolic, parabolic, and elliptic) and derivation of some model problems (heat, wave, and Laplace equation) that fall under this classification. Solution methods for model problems are discussed, which include separation of variables, Fourier series and transforms, method of characteristics, Sturm-Liouville eigenvalue problem, and Green’s function. Existence and uniqueness of the solutions for the model problem will also be discussed.
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| Course LOs
| By the end of the course the student will be expected to be able to:
1) Solve first- and second-order linear equations using the method of characteristics;
2) Solve first- and second-order linear equations using separation of variables;
3) Solve first- and second-order linear equations using Fourier analysis;
4) Solve elliptic equations using Green’s functions;
5) Prove basic properties of linear equations;
6) Understand the basic theory of weak derivatives and weak solutions.
7) Solve first-order nonlinear equations.
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| In the first two years of the program, students complete most of their NU common core courses and introductory math courses for core courses at the upper level. These introductory math courses are designed to provide them with basic mathematical “cookbook” skills. They learn techniques applicable in many scientific disciplines, broadly related to the notion of function and of linear equations. They also learn how to formulate their solution with simple arguments and sentences, using the proper connectives for a logically sound argument (taught in MATH 251). This prepares them for the more challenging “proof-based” courses such as MATH 361 Real Analysis I and MATH 302 Abstract Algebra I.
The third year is the stage where students start to learn advanced mathematics and their applications. Each semester in the third year, students are expected to take four math courses (two core course and two electives at 300 or 400 level). Also, students have an opportunity to learn how to perform research in mathematics via MATH 350 Research Methods in the third year.
In the fourth, students continue to take more electives in their interest. The math department provides various elective courses at 400 level, which are classified as pure math, applied math, and statistics and financial math. Students are required to take at least four elective courses at 400 level so that they can be equipped with deep background for deeper materials and research in graduate schools, and real-world applications in industry. Also, all students pass two research courses among Capstone Project 1, Capstone Project 2, Honor’s Capstone Project 1 and Honor’s Capstone Project 2. A student will gain experiences doing research independently or by collaboration through these course. Qualified students who meet the criteria set by the department are eligible to take Honor’s Capstone Project. A project may be one-semester-long or two-semester-long, and a student who takes a capstone project course will write an interim report or a final report at the end of semester and give an oral presentation to the supervisor and another faculty at NU who will give a final grade for the semester.
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| Mathematics majors must earn at least 240 ECTS credits. Requirements for the completion of BS in Math consist of five parts:
1. University Common Core Framework (UCCF) requirements
2. Mathematics Core
3. Mathematics electives
4. Technical electives.
5. General electives.
1. University Common Core Framework (UCCF) requirements (78 ECTS)
UCCF consists of 8 categories. Students majoring in math must take MATH 161 in the category of MATH, which is the prerequisite for MATH 162.
2. Mathematics Core (78 ECTS)
Students must pass MATH 162, 251, 263, 273, 274, 302, 321, 322, 351, 361, and two research courses
3. Mathematics Electives (42 ECTS)
3.1 Students must earn 42 ECTS from 300- or 400-level courses among which 24 ECTS must be
from 400-level courses.
3.2 Some special math electives :
Only eligible students are allowed to take the following courses, and students must receive the consent of the faculty member who will supervise this course.
(1) MATH 399 Internship: MATH 399 Internship is an elective 6 ECTS workload, which provides an opportunity for a student to gain practical hands-on work experience in a related field of interest.
· Eligibility: The student must have 50 ECTS in math courses (MATH) to register for MATH 399, and not be under academic or non-academic probation. The student must neither be conditional nor be suspended from any academic activity.
· The student works directly with a faculty at NU or a recognized company on anything related to mathematics, statistics or their applications. The job performed at internships must involve with mathematics or statistics and must be pre-approved by the mathematics department by an official offer letter from the hosting organization or company.
· If the internship is performed off campus, a student must provide the faculty who supervise the internship with official information from the host institution that details the expected work, outcome, and workload.
(2) MATH 497/498 Directed Study I, II: MATH 497/498 are a 6 ECTS course intended for 3rd- and 4th-year students who are interested in building their knowledge of the material not included in the courses offered in the undergraduate program of the math department. To enroll in MATH 497/498, students must have a Math GPA 3.2.
4. Technical Electives (18 ECTS)
For math majors, a technical elective course is a course taken in the following: BIOL, ECON, CHEM, PHYS, SEDS, SMG. (MATH courses cannot be a technical electives for math majors.) At least one course at the 300-level must be taken.
5. General electives
In addition to the previous requirements, students can take any courses offered at NU to reach the total 240 ECTS requirement.
※ Grade Requirements:All courses taken to satisfy the requirements for Mathematics Core, Mathematics Electives and MATH 161 in UCCF must be passed with minimum C-.
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