## Under Grad Course Learning Outcomes (CLOs)

#### Upon completion of this course, students should be able to:

1. Formulate and solve business related problems using equations and inequalities.

2. Solve system of linear equations using matrices.

3. Solve linear programing problems graphically and by the simplex method.

4. Solve financial problems involving compound interest, present and future values, and annuities.

5. Count and use descriptive statistics and basic probability concepts.

6. Recognize the Binomial and Normal distributions and their applications.

7. Apply the Binomial and Normal distributions.

#### After completion of the course, the student should be able to:

1. Compute derivative of various functions using appropriate technique.

2. Use concepts of relative minima and/or maxima, absolute minimum and/or maximum and inflection points.

3. Solve problems in optimization and exponential growth and decay.

4. Evaluate integral of some algebraic and trigonometric functions and use the Fundamental Theorem of Calculus.

5. Compute area between curves.

6. Calculate partial derivatives of a function of several variables and classify extreme values of a function of two variables and apply them to optimization problems.

7. Use basic concepts of calculus in business and economics.

#### Upon completion of the course, students should be able to:

1. Describe parametric and polar curves in plane and recognize regions and quadric surfaces in space.

2. Calculate areas, slopes, surface area, arc length for plane curves.

3. Perform vector operations in space and find equations of lines and planes in space.

4. Determine the limits and continuity of multivariable functions.

5. Calculate directional derivatives, equations of tangent planes, and gradient vectors.

6. Find extreme values of multi-variables functions.

7. Evaluate multiple integrals in rectangular, polar, cylindrical, and spherical coordinate systems.

#### Upon completion of the course, the student should be able to:

1. Solve first-order separable, exact, homogeneous, linear and Bernoulli differential equations.

2. Discuss basic theory of linear differential equations.

3. Solve real-world growth and decay, and heating and cooling problems.

4. Find general solution of homogeneous linear differential equations with constant and variable coefficients.

5. Apply undetermined coefficients and variation of parameters methods to solve linear differential equations.

6. Use series solution method to solve second order differential equations.

7. Solve systems of linear differential equations.

#### Upon successful completion of this course, a student should be able to:

1. Solve various types of ordinary differential equations.

2. Apply differential equations to solve certain real-world problems.

3. Discuss basic concepts of linear algebra.

4. Use linear algebra techniques to solve linear systems of differential equations with constant coefficients.

#### Upon completion of this course, students should be able to

1. Discuss basic concepts of elementary logic such as negation, implication, quantifiers and other logical terminology.

2. Explain elementary concepts of set theory such as intersection and union, indexed sets, relations, functions, and cardinality.

3. Discuss basic concepts in number and group theory.

4. Construct mathematical proofs of statements in elementary number theory and elementary group theory using rigorous methods such as induction and contradiction.

Math 225 Introduction to Linear Algebra
Upon completion of this course, students should be able to

1. Explain fundamental concepts such as vector spaces, subspaces, linear independence and dependence, spanning sets, bases, dimensions, and linear transformations.

2. Discuss inner product spaces and orthonormal bases.

3. Solve linear systems and compute determinants and matrix inverses.

4. Apply the Gram-Schmidt process to construct orthonormal bases.

5. Determine matrix representations of linear transformations.

6. Compute eigenvalues and eigenvectors and use them in diagonalization and in classifying real quadratic forms.

#### Upon completing this course, students should be able to:

1. Define vector space, subspace, basis, dimension  and spanning set of a vector space.

2. Compute eigenvalues, eigenvectors, inverse and rank of  matrices.

3. Construct an orthogonal matrix using eigenvectors of a symmetric matrix.

4. Compute different types of integrals using Green's, Stokes' and Divergence theorems

5. Explain geometry of a complex plane and state properties of analytic functions.

6. Calculate the Taylor and Laurent series of a function of complex variable about a given point.

7. Compute integrals using Cauchy-Goursat theorem, Cauchy's integral formula and Residue theorem.

#### Upon completion of this course, students should be able to:

1. Express definition of a mathematical language, mathematical formulae and nature of mathematical axioms with proofs.

2. Explain well-orders, ordinal numbers, transfinite recursion and induction, cardinal numbers, and cardinality.

3. Describe Godel's completeness theorem and compactness theorem for first-order logic.

4. Explain the system ZFC as a formalization of set theory.

5. Discuss nonstandard models of arithmetic.

6. Use the axiom of choice in discussing cardinality.

7. Compute limit of Goodstein sequences.

#### Upon completion of this course, students should be able to:

1. Recall the history of numeration.

2. Discuss acquisition of basic knowledge of arithmetic.

3. Recall the beginning development of fractions.

4. Describe the beginning of algebra.

5. Explain coding theories and geometry.

6. Recognize the beginning of trigonometry.

7. Employ real-world applications for algebra, geometry and trigonometry.

#### Math 323 Modern Algebra IUpon completion of this course, students should be able to:

1. Define normal subgroups, factor groups and homomorphisms.

2. Discuss the fundamental theorem of finite Abelian groups.

3. Explain integral domains and fields.

4. Define ideals, factor rings and ring homomorphisms.

5. Explain factorization of polynomials and factor rings of polynomials over a field.

6. Define irreducible elements and unique factorization.

7. Discuss principal ideal domains.

#### Math 325 Linear AlgebraUpon completion of this course, students should be able to:

1. Recognize vector spaces, bases and dimension of a vector space, and linear transformations.

2. Discuss inner product spaces and orthogonality.

3. Explain dual spaces and bilinear forms.

4. Explain Hermitian and unitary operators.

5. Discuss Polynomials of matrices.

6. Discuss Triangulation of matrices and Cayley-Hamilton Theorem.

7. Demonstrate the computational skills required to manipulate such concepts.

#### Upon completion of this course, students will be able to:

1. Calculate line integral along plane or space curves and  surface integral over surfaces in 3-space.

2. Compute different types of integrals using Green's, Stokes' and Divergence theorems

3. Evaluate Laplace transform, inverse Laplace transform, and Fourier integral of a function.

4. Find Fourier series, Fourier cosine/sine series, Bessel and Legendre series of a function.

5. Evaluate eigenvalues and eigenfunctions for a Sturm-Liouville boundary-value problem.

6. Solve boundary-value problems for wave, heat, and Laplace equations in various coordinate systems by variable separable method.

7. Use Laplace, inverse Laplace, Fourier, and inverse Fourier transforms to solve linear initial and boundary-value problems.

#### Math 341 Advanced Calculus IUpon completion of this course, students should be able to:

1. Identify different classes of real numbers.

2. Apply  concepts of limit and continuity.

3. Distinguish between the concepts of continuity and uniform continuity.

4. Apply properties of differentiation of functions of one variable.

5. Compute Riemann sums and apply them to evaluate integrals.

6. Interpret and apply the fundamental theorem of calculus.

#### Upon completion of this course, students should be able to

1. Explain basic ideas of spherical and hyperbolic geometry.

2. Explain theorems, facts and lemmas in a formal mathematical system.

3. Explain main contributions of Saccheri, Bolyai, Lobachevskii, Gauss, Poincaré, and others in the development of hyperbolic geometry.

4. Explain relation between matrices and geometric transformations.

#### After completion of the course, students should be able to:

1. Use Taylor series to approximate functions, errors, and their upper bounds.

2. Devise algorithms to approximate roots of equations.

3. Analyze data using least squares method.

4. Use polynomials to interpolate collected data or approximate function.

5. Program algorithms to compute derivatives and integral of a function, estimate error and upper bound.

6. Approximate numerical solutions of linear systems of equations and IVPs in ODEs.

7. Apply numerical and computer programming tools to solve engineering problems.

#### Math 399 Summer TrainingUpon successfully completion of summer training, the students should be able to:

1. Develop self-learning capabilities and recognize their importance for career development.

2. Recognize ethical responsibility (work ethics) and practice professional integrity.

3. Communicate effectively in oral, written, and graphical format.

4. Link theory to practice in the real-life workplace.

#### Upon successful completion of this course, a student should be able to:

1. Describe linear algebra and statistics fundamental to many machine learning algorithms.

2. Apply linear algebra concepts to probability and statistics.

3. Apply linear algebra to optimization problems.

4. Use linear algebra and statistics in selected machine learning algorithms.

#### Math 423 Modern Algebra IIUpon completion of this course, students should be able to:

1. Explain finitely generated Abelian groups, solvable & nilpotent groups and Sylow theorems.

2. Discuss factorization in integral domains and, specifically, in important classes of unique factorization domains.

3. Explain field extensions, finite fields, and Galois groups.

4. Prove statements and construct examples of  some classes of groups, rings and fields.

#### MATH 424 Applied AlgebraUpon completion of the course, the student should be able to:

1. Describe basic notions of semigroups, monoids, and groups.

2. Discuss basic concepts of Boolean algebra.

3. Discuss relationship between mathematics, technology, and the societal implications in science.

4. Explain automata theory and error correction codes.

5. Apply basic concepts of Algebra in natural and computer science.

#### Math 427 Number TheoryUpon completion of this course, students should be able to

1. Solve questions about divisibility and primes both theoretically and computationally.

2. Apply the theorems of Fermat, Euler, and Wilson in computing and/or proving some statements in Number Theory.

3. Solve polynomial congruences and systems of linear congruences in one variable.

4. Find the order of integers and primitive roots modulo primes.

5. Use quadratic reciprocity law in computing and proving some statements in number theory.

6. Solve problems involving arithmetic functions.

7. Solve some types of Diophantine equations and problems on selected applications of Number theory.

#### Math 432 Applied Matrix TheoryUpon completion of this course, students should be able to

1. Apply matrix theory to solve systems of linear ODEs.

2. Recognize different classes of matrices and their properties.

3. Use Rayleigh Principle  to minimize/maximize quotients of quadratic functions.

4. Perform the Gram-Schmidt othogonalization process.

5. Discuss simple functions of matrices.

6. Apply the notion of condition number to discuss relative errors.

#### Math 433 Methods of Applied Mathematics IIUpon completion of this course, students should be able to

1. Define basic notion of Hilbert Space, convergence, and orthogonal systems.

2. Solve Fredholm and Volterra integral equations.

3. Apply methods of singular or regular perturbations to certain integral equations.

4. Discuss Green's function and its properties.

5. Solve some practical problems using one of the following: Asymptotic methods or Perturbation Theory or Integral Transforms.

#### MATH 434 Calculus of Variations and Optimal ControlUpon completion of this course, students should be able to

1. Derive from first principle necessary conditions for an extremum in specific cases including the multivariable case.

2. Solve Euler-Lagrange equation.

3. Use Bolza, Mayer and Lagrange formulations in optimal control problems.

4. Use the variational approach to optimal control problems.

5. Apply the Pontryagin maximum principle.

6. Use dynamic programming in continuous time: Hamilton-Jacobi-Bellman equation.

7. Solve the linear quadratic regulator problem.

Math 435 Ordinary Differential Equations
Upon Completion students should be able to

1. Apply existence and uniqueness theory for initial value problems.

2. Discuss asymptotic behavior of linear and almost linear systems and the theory of Lyapunov stability.

3. Solve linear systems of differential equations, including higher order equations with constant coefficients.

4. Calculate and classify critical points of autonomous systems.

#### Math 437 Partial Differential EquationsUpon completion of this course, students should be able to:

1. Solve linear and quasi-linear first order PDEs in two variables using the characteristic method.

2. Classify second-order equations in two variables by type (parabolic, hyperbolic, elliptic).

3. Use separation of variables to solve some PDEs.

4. Apply the maximum principle to the Laplace and heat equations.

#### After completion of the course, the students should be able to:

1. Recall basic geometry and topology of Euclidean space.

2. Discuss notion of limit of a function of several variables to state directional, partial and Frechet derivatives.

3. Discuss Inverse and Implicit function theorems.

4. Determine nature of critical points using Hessian matrix.

5. Apply method of Lagrange multipliers to extremum problems with constraints.

6. Use Fubini's theorem to compute multiple integrals.

7. Discuss convergence of improper integrals.

#### Upon completion of this course, the student should be able to:

1. Describe properties of functions of bounded variation.

2. Explain concept of Riemann-Stieltjes integral.

3. Apply inverse and implicit function theorems.

4. State main properties of vector functions and vector fields.

5. Use change of variables to evaluate multiple integrals.

6. Compute line integral along plane or space curves and surface integral over surfaces in 3-space.

7. Calculate different types of integrals using Stokes' and Divergence theorems.

#### Math 445 Introduction to Complex variablesUpon completion of this course, students should be able to:

1. Explain geometry of a complex plane.

2. State properties and examples of analytic functions.

3. Evaluate line integrals using parameterization.

4. Compute Taylor and Laurent expansions of standard functions.

5. Determine nature of singularities and calculate residues.

6. Use Residue theorem to evaluate integrals and series.

7. State main properties of conformal mappings.

#### Upon successful completion the student should be able to

1. Define parametric curves and surfaces.

2. Review the Frenet-Serret frame and Frenet-Serret apparatus.

3. Define normal and principal curvatures, Gauss and mean curvatures of surfaces.

4. Calculate curvature and torsion of parametric regular curves.

5. Use Frenet-Serret equation to characterize regular curves.

6. Explain normal and principal curvatures of regular surfaces.

7. Calculate Gauss curvature from Gauss equation.

#### A student who succeeded in this course, should be able to:

1. Define basic concepts of topology.

2. Distinguish between metric and nonmetrizable topologies.

3. Apply connectedness, compactness and Tychonoff theorem.

4. Distinguish between countability and separation axioms including countable basis, countable dense subsets, normal spaces, Urysohn lemma and Tietze extension theorem.

5. Explain metrization problem and Urysohn Metrization theorem.

6. Discuss properties and applications of complete metric spaces.

#### Upon successful completion the student should be able to

1. Recognize the fundamental concepts and techniques of Combinatorics.

2. Apply enumerative techniques in combinatorics.

3. Explain combinatorial proofs.

4. Use recurrence relations and generating functions for sequences arising from combinatorial problems.

#### Upon completion of this course, students should be able to

1. Define and describe basic concepts and graph theory terminology: induced subgraphs, cliques, matchings, covers in graphs, graph coloring.

2. Recognize different families of graphs and their properties such as Hamiltonian, Eulerian and planar Graphs.

3. Describe automorphism groups and different types of graph matrices and their use.

4. Solve problems involving vertex and edge connectivity, planarity and crossing numbers.

5. Construct spanning trees, matching, and different matrices.

6. Apply different proof techniques in theorems and exercises.

7. Apply Graph theory to  model and solve real world problems and networks.

#### Math 471 Numerical Analysis IUpon completion of this course, students should be able to:

1. Discuss Floating-Point arithmetic.

2. Solve linear systems using computer software.

3. Explain mathematical reasoning in algorithms.

4. Develop error analysis of numerical methods.

5. Recognize the role-play of singular value decomposition in solving least square problems.

6. Calculate eigenvalues and eigenvectors of matrices using numerical techniques.

#### Math 472 Numerical Analysis IIThe Course Learning Outcomes: After completion of the course, the student should be able to

1. Approximate functions and interpolate precise data using Taylor series and polynomials, polynomial approximations, and piecewise polynomial approximations.

2. Fit the best curve in least-squares sense for data exhibiting a significant degree of error or scatter.

3. Approximate derivatives and definite integrals of functions.

4. Approximate solutions to IVPs and BVPs for ODEs.

5. Determine region of absolute stability for one and multi-step methods to solve IVPs for stiff ODEs.

6. Write numerical routines based on algorithms for different methods.

#### MATH 474 Linear & Nonlinear ProgrammingUpon completion of the course the student should be able to:

1. Discuss basic properties of linear programs and convex functions.

2. Discuss duality theory.

3. Discuss necessary and sufficient conditions for unconstrained problems.

4. Solve linear programs by simplex method.

5. Use Lagrange multipliers method and Kuhn-Tucker conditions to solve constrained problems.

6. Apply computational method to solve unconstrained and constrained problems.

#### Math 475 Wavelets and ApplicationsUpon completion of this course, students should be able to

1. Explain concept of a Multiresolution analysis.

2. Explain concept of a scaling function and its corresponding wavelet.

3. Explain concept of fast (discrete) wavelet transform.

4. Explain decomposition of a signal to its fast and slow modes using wavelet transforms.

5. Use available software to implement wavelet decomposition and reconstruction.

6. Use wavelet transform for signal compression, noise reduction and for solving simple boundary value problems.

#### Math490 Seminar in MathematicsUpon completion of this course, students should be able to:

1. Search mathematical literature.

2. Strengthen knowledge in the selected area.

3. Analyze assigned problem and formulate a similar problem.

4. Work independently and in a team.

5. Communicate mathematical ideas.

6. Present work orally and defend publicly.

#### STAT201 Introduction to StatisticsUpon completion of this course, students should be able to

1. Recall techniques of data analysis.

2. Explain basic elements of probability.

3. Discuss assumptions, methods, and implications associated with various methods of statistical inference.

4. Use MINITAB and interpret the associated output.

#### By completing this course, students should be able to:

1. Distinguish between a sample and a population and between a statistic and a parameter and classify business data into the most appropriate type and measurement levels.

2. Organize, manage, and present data.

3. Analyze statistical data graphically and analyze statistical data using measures of central tendency, dispersion, and location manually and by MINITAB.

4. Explain basic concepts of probability and random variables. and explain the basic probability rules, including additive and multiplicative laws, using the terms, independent and mutually exclusive events and calculate expected values for continuous and discrete probability distribution models.

5. Use the correct probability distribution model for a particular business application manually and by MINITAB.

6. Explain the concept of the sampling distribution of a statistic, and in particular describe the behavior of the sample mean.

7. Discuss foundations for classical inference involving confidence intervals manually and by MINITAB.

#### Stat 212 Statistics for Business-IIUpon completion of this course, students should be able to:

1. Recall the correspondence between levels of measurement and statistical procedures.

2. Recognize the assumptions underlying statistical procedures.

3. Select the appropriate statistical procedure for various applied business situations.

4. Compute procedures manually and by MINITAB and interpret the results of these statistical procedures. Finally, make the right decision.

#### Stat 310 Linear RegressionUpon completion of this course, students will be able to:

1. Find and interpret least square estimates of parameters.

2. Recall the single linear and multiple linear regressions model.

3. Build and use the single linear and multiple linear regressions model.

4. Perform hypothesis tests and construct confidence intervals in linear regression models.

5. Test the appropriateness of models and analyze data.

#### Stat319 Probability and Statistics for Engineers and Scientists By the end of this course, students will be able to:

1. Summarize data using common graphical and numerical tools.

2. Calculate the probabilities of operations on events based on sample space for random experiments.

3. Calculate the mean, the variance, and the probabilities for discrete and continuous distributions.

4. Discuss the concept of sampling distribution of a sample mean and proportion and apply the Central Limit Theorem to problems involving sums and averages of variables from arbitrary distributions.

5. Estimate the unknown population mean and proportion using confidence interval technique and testing of hypothesis.

6. Recognize the meaning of sample correlation coefficient and model real life problems using simple and multiple linear regression including estimation and testing of model parameters.

7. Use a statistical package to compute descriptive statistics, construct confidence intervals and build regression model.

#### Stat 460 Time SeriesBy the end of this course, students will be able to:

1. Define and explain the concepts and components of stochastic time series processes, including stationarity and autocorrelation.

2. Describe specific time series models, including random walk, exponential smoothing, autoregressive, and autoregressive conditionally heteroskedastic.

3. Interpret predicted values and confidence and prediction intervals.

4. Explain uses of time series models.