Compute various types of limits of functions of one variable.

Determine the region of continuity and types of discontinuity of a function.

Compute the slope of the tangent line at a point.

Calculate derivatives of polynomial, rational, trigonometric, inverse trigonometric, exponential, logarithmic, hyperbolic, piecewise, and related functions.

Find extreme values, regions of monotonicity and concavity, asymptotes of a function of one variable.

Apply derivatives in estimating errors, approximating roots of equations via Newton's method and in solving optimization problems.

Recover some basic functions from their derivatives.

Estimate areas and definite integrals by Riemann sums.

Apply the Fundamental Theorem of Calculus

Evaluate Integrals using various techniques of integration.

Calculate the average value of a function, areas between curves, length of curves, volumes and surface areas of solids of revolutions.

Evaluate improper integrals and limits of sequences.

Apply convergence tests of series and evaluate sum of some selected convergent series.

Find interval and radius of convergence of a power series and express a function as a power series (Taylor and Maclaurin).

Formulate and solve business related problems using equations and inequalities.

Solve system of linear equations using matrices.

Solve linear programing problems graphically and by the simplex method.

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

Demonstrate ability to count and use descriptive statistics and basic probability concepts.

Recognize the Binomial and Normal distributions and their applications in business.

Apply the Binomial and Normal distributions in business.

Compute the derivative of various functions using the appropriate technique.

Use the concepts of relative minima and/or maxima, absolute minimum and/or maximum and the inflection point.

Solve problems about optimization and exponential growth and decay.

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

Find the area between two curves

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

Employ the calculus concepts in business and economics.

Describe curves given by parametric and polar equations in the plane and calculate areas, slopes, surface area, arc length for such curves.

Describe regions and quadric surfaces in space.

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

Determine the limits and continuity of multivariable functions.

Calculate partial and directional derivatives, tangent planes, and gradient vectors.

Find extreme values of functions of two/three variables with constraints (Lagrange multipliers) or without constraints.

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

Solve different types of first-order differential equations, including separable, exact, homogeneous, linear and Bernoulli equations.

Discuss basic theory of linear differential equations.

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

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

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

Use series method to solve a second order differential equation.

Solve systems of linear homogeneous and nonhomogeneous differential equations.

Find bases of vector spaces.

Use linear algebra in systems of linear equations.

Solve eigenvalue problem.

Perform diagonalization and compute the Jordan form of matrices.

Solve first order differential equations and related models.

Solve linear ordinary differential equations.

Solve systems of ordinary differential equations.

Use basic results of set theory involving concepts such as intersection and union, indexed sets, relations, functions, and cardinality.

Use basic results on divisibility and congruences, including the fundamental theorem of arithmetic.

Use basic results of group theory, including Lagrange's theorem.

Use concepts of elementary logic such as negation, implication, quantifiers and other logical terminology.

Construct mathematical proofs using rigorous methods such as induction and contradiction

Explain fundamental concepts such as, elementary operations, vector spaces, subspaces, linear independence, spanning sets, bases, dimension, and linear transformations.

Discuss inner product spaces and orthonormal bases.

Solve linear systems and compute determinants and matrix inverses.

Apply the Gram-Schmidt process to construct orthonormal bases.

Determine matrix representations of linear transformations.

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

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

Compute eigenvalues, eigenvectors and inverse of a square matrix and rank of a matrix.

Construct an orthogonal matrix using eigenvectors of a symmetric matrix.

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

Explain the geometry of the complex plane and state the main properties and examples of analytic functions.

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

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

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

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

Describe Godel's Completeness Theorem and Compactness Theorem for First-Order logic.

Explain the system ZFC as a formalization of set theory.

Investigate nonstandard models of arithmetic.

Use the Axiom of Choice in discussing cardinality.

Calculate limit of Goodstein sequences.

Recall the history of numeration.

Demonstrate acquisition of basic knowledge of arithmetic.

Recall the beginning development of fractions.

Describe the beginning of algebra.

Demonstrate basic understanding of numbers, coding theories and geometry.

Recognize the beginning of trigonometry.

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

Define normal subgroups, factor groups, homomorphisms.

Discuss the fundamental theorem of finite Abelian groups.

Explain integral domains and fields.

Define ideals, factor rings and ring homomorphisms.

Explain factorization of polynomials over a field, factor rings of polynomials over a field.

Define irreducible elements and unique factorization.

Discuss principal ideal domains.

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

Discuss inner product spaces and orthogonality.

Explain dual spaces and bilinear forms.

Explain Hermitian and unitary operators.

Discuss Polynomials of matrices.

Discuss Triangulation of matrices and Cayley-Hamilton Theorem.

Demonstrate the computational skills required to manipulate such concepts.

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

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

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

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

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

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

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

Identify different classes of real numbers.

Recall the concepts of limit and continuity.

Apply the concepts of limit and continuity.

Distinguish between the concepts of continuity and uniform continuity.

Apply properties of differentiation of functions of one variable.

Compute Riemann sums and apply them to evaluate integrals.

Interpret and apply the fundamental theorem of calculus.

Explain basic ideas of spherical and hyperbolic geometry.

Analyze statements in a formal mathematical system.

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

Explain the relation between matrices and geometric transformations.

Formulate Taylor Series to approximate functions, errors, and their upper bounds.

Devise algorithms to locate approximated roots of equations and numerically solve linear systems of equations.

Analyze engineering data using the least squares method.

Use polynomials to interpolate collected precise engineering data or approximate function.

Program algorithms to compute the derivative and the integral of a given function, estimate the approximation error involved and upper bound, and interpret engineering ordinary and partial differential equations.

Identify relationships among methods, algorithms, and computer errors.

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

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

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

Communicate effectively in oral, written, and graphical format.

Link theory to practice in the real-life workplace.

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

Apply linear algebra concepts to probability and statistics.

Apply linear algebra to optimization problems.

Use linear algebra and statistics in selected machine learning algorithms.

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

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

Explain field extensions, finite fields, and Galois groups.

Write proofs of statements and construct examples concerning important classes of groups, rings, and fields.

Describe basic notions of semigroups, monoids, and groups.

Discuss Boolean algebra

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

Explain automata theory and error correction codes.

Apply various concepts of Algebra in natural and computer science.

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Solve questions about divisibility and primes both theoretically and computationally.

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

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

Find the order of integers and primitive roots modulo primes.

Use Quadratic Reciprocity Law in computing and proving some statements in number theory.

Solve problems involving arithmetic functions.

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

Apply matrix theory to solve a system of linear ordinary differential equations.

Recognize different classes of matrices and exploit their properties.

Minimize/Maximize quotients of quadratic functions using the Rayleigh Principle.

Perform the Gram-Schmidt othogonalization process.

Manipulate simple functions of matrices.

Apply the notion of condition number to discuss relative errors.

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

Solve Fredholm and Volterra integral equations.

Use idea of distributions and Green's function.

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

Apply asymptotic approximation in simple cases or integral transforms (one of the two).

Solve some practical problems using Green's function, perturbation, or asymptotic methods (or integral transforms).

** **

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

Solve Euler-Lagrange equation.

Formulate an optimal control problem and work with the Bolza, Mayer and Lagrange formulations.

Use the variational approach to optimal control.

State and apply the Pontryagin maximum principle.

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

Solve the linear quadratic regulator problem.

Discuss the existence and uniqueness theory for initial value problems.

Apply the existence and uniqueness theory for initial value problems.

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

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

Calculate and classify critical points of autonomous systems.

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Solve linear and quasi-linear first order PDE's in two variables using the characteristic method.

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

Use separation of variables to solve some PDE's.

Write down the formula for the exact solution of the one-dimensional Heat and Wave equations in the real line.

Use the maximum principle for the Laplace and heat equations.

** **

Recall basic geometry and topology of Euclidean space.

Describe the notion of the limit of a function of several variables to state directional, partial and Frechet derivatives of this function and recognize the differences for these notions.

Apply the Inverse Function Theorem and the Implicit Function Theorem.

Determine the nature of critical points using the Hessian matrix.

Apply the method of Lagrange multipliers to extremum problems with a constraint.

Practice changes of variables and use Fubini's theorem to compute multiple integrals.

Discuss convergence of improper integrals.

Describe the properties of functions of bounded variation.

Explain the concept of Riemann-Stieltjes integral.

Discuss the inverse and implicit function theorems.

State the main properties of vector functions and vector fields.

Use change of variables to evaluate multiple integrals.

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

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

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

Explain the geometry of the complex plane.

State the main properties and examples of analytic functions.

Evaluate line integrals using parameterization.

Compute the Taylor and Laurent expansions of standard functions.

Determine the nature of singularities and calculate residues.

Use Residue Theorem to evaluate integrals and series.

State main properties of conformal mappings.

Define parametric curves and surfaces.

Recall the Frenet-Serret frame and Frenet-Serret Apparatus.

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

Calculate curvature and torsion of parametric regular curves.

Use Frenet-Serret equation to characterize regular curves.

Explain normal and principal curvatures of regular surfaces.

Calculate Gauss curvature from Gauss equation.

Define basic concepts of topology such as set theory, open, closed, closure, interior and boundary of a set.

Distinguish between a metric topology a nonmetrizable topology.

Decide whether a given function is continuous.

Define and apply connectedness, compactness and Tychonoff theorem.

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

Explain the metrization problem and Urysohn Metrization theorem.

Recognize some properties and applications of complete metric spaces.

Recognize the fundamental concepts and techniques of Combinatorics.

Apply enumerative techniques in combinatorics.

Explain combinatorial proofs.

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

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

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

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

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

Construct spanning trees, matching, and different matrices.

Apply different proof techniques in theorems and exercises.

Apply Graph Theory to solve and model real world problems and Networks.

Program numerical methods

Discuss Floating-Point Arithmetic.

Solve linear systems numerically using advanced programming software such as MATLAB.

Explain the mathematical reasoning behind the algorithms developed in the course.

Develop error analysis of the numerical methods considered.

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

Calculate the eigenvalues and eigenvectors of matrices using numerical techniques.

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

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

Approximate the derivatives and definite integrals of functions.

Approximate the solutions to IVPs and BVPs of ODEs.

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

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Discuss basic properties of linear programs and convex functions

Discuss duality theory.

Discuss necessary and sufficient conditions for unconstrained problems.

Solve linear programs by simplex method.

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

Apply computational methods to solve unconstrained problems.

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Explain the concept of a Multiresolution Analysis.

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

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

Explain the decomposition of a signal into its fast and slow modes (approximation and detail) using wavelet transforms.

Use available software such as MATLAB to implement wavelet decomposition and reconstruction.

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

Search mathematical literature.

Strengthen the knowledge in the selected area.

Analyze the assigned problem.

Formulate a similar problem to the assigned problem.

Work independently and in a team.

Communicate mathematical ideas.

Present his work orally, and defend it publicly.

Recall the techniques of data analysis studied.

Explain the basic elements of probability studied.

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

Use MINITAB and interpret the associated output.

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.

Organize, manage, and present data.

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

Demonstrate an understanding of the 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.

Recognize and use the correct probability distribution model for a particular business application manually and by MINITAB.

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

Understand the foundations for classical inference involving confidence intervals manually and by MINITAB.

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

Recall the correspondence between levels of measurement and statistical procedures.

Recognize the assumptions underlying statistical procedures.

Select the appropriate statistical procedure for various applied business situations.

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

Find and interpret least square estimates of parameters.

Recall the single linear and multiple linear regressions model.

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

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

Test the appropriateness of models and analyze data.

By the end of this course, students will be able to:

Summarize data using common graphical and numerical tools.

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

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

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.

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

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.

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

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

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

Interpret predicted values and confidence and prediction intervals.

Explain uses of time series models.