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Course Learning Outcomes:

1. Computing eigenvalues and eigenvectors of endomorphisms.
2. Computing Jordan Normal Forms of endomorphisms.
3. Proving properties of homomorphisms between vector spaces, as well as properties of eigenvalues and eigenvectors of endomorphisms.
4. Working with the properties of inner-products and Hilbert spaces.

Course Learning Outcomes:

1. Proving basic results on relations and functions as well as basic results in set theory and number theory.Ìý
2. Proving basic results in group theory.
3. Showing an understanding of groups, subgroups, and group homomorphisms.
4. Proficiency in symbolic logic.
5. Using linear algebraic and group theoretic tools to model and analyze error-correcting codes.

Course Learning Outcomes:

1. Computing the sample mean and standard deviation from summary data.
2. Computing basic statistical quantities such as median and quantiles from data.
3. Verifying solutions of ordinary differential equations.
4. Using Euler's method to compute approximate solutions of ordinary differential equations.
5. Using probability theory to solve problems of quality control.
6. Modeling and analyzing physical processes using differential equations.
7. Analyzing empirical data for quality control purposes.

Course Learning Outcomes:

1. Solving basic initial value problems.
2. Solving linear constant coefficient differential equations.
3. Using the Laplace transform to solve differential equations.
4. Modeling a mass-spring-damper system or RLC circuit using differential equations.
5. Modeling interconnected fluid reservoirs using differential equations.

Course Learning Outcomes:

1. Computing path integrals.
2. Computing the flux of a vector field across a surface.
3. Computing the circulation of a vector field around a boundary.
4. Modeling the area of a region bounded by a simple closed curve as an integral.
5. Modeling the volume enclosed by a closed surface as an integral.

Course Learning Outcomes:

1. Evaluating contour integrals using residue theory.
2. Computing harmonic conjugates of harmonic functions.
3. Solving algebraic equations involving complex numbers.
4. Computing Taylor and Laurent expansions of complex analytic functions.
5. Proving basic properties of complex analytic functions.
6. Understanding basic properties of complex mappings.

Course Learning Outcomes:

1. Solving basic initial value problems.
2. Using the Laplace transform to solve differential equations.
3. Computing Laplace and inverse Laplace transforms.
4. Find general solutions of ordinary differential equations.
5. Model an RLC circuit using ordinary differential equations.

Course Learning Outcomes:

1. Compute the general or particular solutions to linear and nonlinear differential equations, and systems of differential equations.
2. Understand the implications of the existence and uniqueness theorem and linear independence/dependence properties of solutions to differential equations.
3. Develop mathematical models describing physical phenomena.
4. Use the method of Laplace transform to solve problems involving discontinuous forcing terms.
5. Determine stability of differential equations.

Course Learning Outcomes:

1. Demonstrate use of concepts of continuity and differentiability in higher dimensions, including the use of the chain rule.
2. Demonstrate ability to parametrize and describe curves, surfaces, and solid regions in higher dimensions.
3. Integrate functions over parametrized regions to obtain line integrals, surface integrals, volume integrals.
4. Describe the physical meaning and use the differential operators: gradient, curl, and divergence.
5. Describe the physical meaning and use the three main theorems of multivariate calculus: Green's Theorem, Gauss's theorem, and Stokes's theorem.

Course Learning Outcomes:

1. Determining convergence or divergence of a sequence of real numbers.
2. Determining uniform/pointwise convergence or divergence of a sequence of functions.
3. Proving properties of limits of sequences of functions.
4. Using definitions to prove relationships between types of subsets of Euclidean space.
5. Using the definition of continuity to prove properties of continuous functions.

Course Learning Outcomes:

1. Solving algebraic equations involving complex numbers.
2. Computing path integrals of analytic functions using residue theory.
3. Computing Taylor and Laurent expansions for analytic functions of a complex variable.
4. Proving results on roots of polynomials using complex function theory.
5. Proving results on analytic functions.
6. Precise formulation of basic definitions and results on analytic functions.
7. Precise use of mathematical definitions in proving results on analytic functions.

Course Learning Outcomes:

1. Understand and distinguish various topological notions for subsets of the real line and of the n-dimensional Euclidean space, such as interior and boundary points, open/closed sets, cluster points, isolated points, nowhere dense sets, compact sets, connected sets, G-delta and F-sigma sets.
2. Understand and apply the concept of limit, continuity for functions of several real variables, and their ramifications, such as the intermediate value theorem and the extreme value theorem. Understand and apply the concept of uniform continuity and its ramification, such as the Heine-Cantor theorem. Understand and distinguish sets of continuity and their properties.
3. Understand and apply the concept of differentiability for (possibly vector-valued) functions of several real variables and its ramifications. For instance, apply the chain rule, the inverse function theorem, and the implicit function theorem in concrete examples. Understand differentiability and gradients in terms of partial derivatives.
4. Understand and apply the concept of relative extrema for functions of several variables and their relation to the Jacobian and the Hessian. Understand and apply the concept of constrained extrema and the Lagrange Multiplier Theorem in concrete examples.
5. Understand and apply the Picard-Lindelöf theorem to study the existence and uniqueness of solutions of systems of ordinary differential equations.

Course Learning Outcomes:

1. Linearizing a nonlinear control system around an equilibrium point.
2. Computing the controllability and observability matrices of a linear time invariant system.
3. Determining internal stability of a linear time invariant system.
4. Making rigorous use of notions from linear algebra and differential equations in proving results on linear time invariant systems.
5. Using tools from linear algebra and differential equations to determine properties of linear time invariant systems.
6. Control of an electromechanical system.
7. Formulates clear problem specifications for the design of control systems.
8. Develops metrics for comparing designs of controllers.

Course Learning Outcomes:

1. Constructing or verifying discrete- and continuous-times signals with prescribed behaviour in the time-domain.
2. Constructing or verifying discrete- and continuous-times signals with prescribed behaviour in the frequency-domain.
3. Understanding discrete- and continuous-time signal spaces with ∞- and Lp-norms.
4. Understanding the four Fourier transforms and how they are related and not related.
5. Understanding and explaining the role of Lebesgue measure in continuous-time signal analysis.
6. Proving relationships between spaces of discrete- and continuous-time signals

Course Learning Outcomes:

1. Understand input-output systems and their properties such as stability and time-invariance.
2. Solve a difference or differential equation using the z-transform or the Laplace transform.
3. Prove results on the Fourier transform.
4. Prove results on distributions.
5. Investigate the possibility of signal representations through polynomials, Haar wavelets and harmonic signals and their use in system theoretic analysis.
6. Understand mathematical analysis of signal sampling.
7. Design control systems via frequency domain (Fourier/Laplace/Z transform) methods.
8. Formulate and design lowpass filters and noise removal algorithms.
9. Use mathematics to develop algorithms for noise removal.

Course Learning Outcomes:

1. Compute optimal policies via dynamic programming in applications such as inventory control.
2. Understand stability of queuing models.
3. Establish optimal decision and planning via various methods.
4. Rigorously prove results in Markov chains and optimal planning/control using tools from mathematical analysis.

Course Learning Outcomes:

1. Computing Fourier series expansions of functions.
2. Computing solutions to boundary value problems.
3. Proving that a given Sturm-Liouville problem has only positive eigenvalues.
4. Selection of Fourier integral or Fourier series for representation of harmonic functions.

Course Learning Outcomes:

1. Computing expected payoffs of games.
2. Using backward induction to find solutions to games in extensive form.
3. Using iterated elimination of dominated strategies to find a solution to games in normal form.
4. Using the theorem of mixed strategy Nash equilibria to find solutions to games in normal form.
5. Finding the Nash equilibria of a game.

Course Learning Outcomes:

1. Computing the probability density function of a function of a random variable.
2. Computing probabilities of events defined by random variables and computing moments of random variables.
3. Computing joint probability density functions of random variables.
4. Computing joint probability density functions of random variables.
5. Reasoning probabilistically in order to solve problems of quality control.
6. Rigorously establishing relations between probabilities or conditional probabilities of various events.

Course Learning Outcomes:

1. Computing probabilities of events defined by random variables.
2. Reasoning probabilistically in order to solve problems of sampling with or without replacement.
3. Using mathematics in order to estimate the probability of certain events.
4. Rigorously proving results in probability theory using tools from mathematical analysis.

Course Learning Outcomes:

1. Performing calculations for confidence intervals and hypothesis tests in various situations involving one, two, and three or more samples, including simple linear regression and one-way analysis of variance.
2. Computing values of random variables or probabilities associated with random variables.
3. Conversion of physical problem into appropriate statistical model.
4. Solving statistics problems using probability or Poisson or Normal or Exponential distributions or simple linear regression or analysis of variance.
5. Recognizing the design of experiments performed and/or performing appropriate statistical analysis to solve engineering problems.
6. Analysis of data using statistical tools and derivation of suitable conclusions from empirical data.

Course Learning Outcomes:

1. Construct or select appropriate mathematical models.
2. Generate a traceable and defensible record of technical projects using an appropriate records system.
3. Consider all factors in design, including economic, environmental, social, and regulatory considerations, as appropriate.
4. Develop mathematical tools to solve engineering problems.
5. Demonstrate a capacity for leadership and decision-making.
6. Interpret communication from a variety of sources and responds to instructions and questions while displaying a full understanding of the topic.
7. Demonstrate professional bearing.
8. Consider social and environmental factors and/or impacts in decisions.
9. Consider ethical factors and matters of equity as appropriate.
10. Perform economic analysis at an appropriate level.
11. Demonstrate skills needed for self-education.
12. Demonstrate proficiency in using sophisticated mathematical models in analysis of engineering problems.
13. Experimentally validate mathematical models and techniques.
14. Formulate clear problem specifications.
15. Understand limitations of mathematical tools.
16. Work effectively as a member of a group.
17. Demonstrate accurate use of technical vocabulary.
18. Integrate standards, codes of practice, and legal and regulatory factors into decision- making processes, as appropriate.
19. Critically evaluate procured information for authority, currency, and objectivity.
20. Conduct investigations to test hypotheses related to complex problems.
21. Use of a variety of tools appropriate for the problem.
22. Use graphics appropriately to explain, interpret, and assess information.
23. Develop metrics for comparison of designs.
24. Write and revise documents using appropriate discipline-specific conventions.
25. Present oral communication that is well thought out, well-prepared, and delivered in a convincing manner.

Course Learning Outcomes:

1. Understanding the structure of finite fields.
2. Rigorously proving results on error correction and detection.
3. Deriving bounds and asymptotic bounds for codes.
4. Probabilistic or combinatorial analysis for coding and decoding.
5. Designing linear codes given real-world parameters.

Course Learning Outcomes:

1. Computations and constructions in finite fields.
2. Reasoning on elliptic curves.
3. Understanding of the algebraic structure of a group.
4. Rigorously proving results in number theory.
5. Probability analysis of primality tests.
6. Analysis of cryptographic protocols.
7. Designing a cryptographic system with given real-world parameters.
8. Testing various designs to determine which performs the best.

Course Learning Outcomes:

1. Linearize a nonlinear system around a given trajectory.
2. Construct LTI realizations of given transfer functions.
3. Determine the controllability/observability properties of a linear time varying system.
4. Rigorously use concepts from linear algebra/differential equations in proving results on linear control systems.
5. Use tools from optimal control theory in order to solve optimization problems.
6. Use tools from linear algebra in order to construct stabilizing controllers.
7. Experimentally study linear approximations to nonlinear systems and the limits of validity of linearization.
8. Experimentally study finite dimensional linear approximations to infinite dimensional linear systems and the limits of validity of that approximation.

Course Learning Outcomes:

1. Understand definitions and techniques of tensor algebra and tensor calculus.
2. Use the methods of tensor algebra and calculus for the description of the deformation of a body.
3. Use the methods of tensor algebra and calculus to describe the motion of a body.
4. Compute tractions, principal stresses, and principal axes of stress of a body.
5. Formulate the differential equations governing the motion of a body.
6. Develop engineering solutions for real-world problems concerned with stress of materials, deformation of elastic bodies and motion of fluids.

Course Learning Outcomes:

1. Computing necessary conditions for optimality.
2. Solving constrained optimization problems.
3. Understanding the mathematical properties of convex sets and convex functions.
4. Rigorously using separation theorems for solving optimization problems.
5. Using numerical methods in the study of optimization problems.
6. Solving resource allocation problems using duality theory.

Course Learning Outcomes:

1. CLOs coming soon; please refer to your course syllabus in the meantime.

Course Learning Outcomes:

1. Using the Ito calculus in the study of stochastic processes.
2. Determining (local) martingale properties of stochastic processes defined through stochastic integration.
3. Rigorous understanding of Brownian motion and its main properties.
4. Rigorous understanding of the Ito stochastic integral and its properties.
5. Rigorously using results from martingale theory in establishing properties of stochastic processes.
6. Using tools from stochastic analysis in the study of uniformly elliptic partial differential equations.

Course Learning Outcomes:

1. Understand the definitions and constructions from differential geometry.
2. Translate physical concepts to differential geometric concepts.
3. Use the methods of differential geometry.
4. Model physical systems using methods of geometric mechanics.

Course Learning Outcomes:

1. CLOs coming soon; please refer to your course syllabus in the meantime.

Course Learning Outcomes:

1. Compute an expectation using conditioning.
2. Convert a process description into a Markov chain model.
3. Understand the mathematical structure of a Markov chain.
4. Identify the stationary distribution of Markov chains.
5. Prove results about Markov chains.
6. Compute an expectation using Markov Chain Monte Carlo.

Course Learning Outcomes:

1. Understand regression problems and algorithms.
2. Understand the bias variance trade-off.
3. Understand classification problems and algorithms.
4. Perform supervised learning tasks on real-world datasets.
5. Understand concepts in unsupervised learning problems.

Course Learning Outcomes:

1. Understand stochastic stability notions for Markov chains, including recurrence, positive Harris recurrence, and transience.
2. Establish the existence and structure of optimal policies for finite horizon, discounted infinite horizon and average cost infinite horizon problems.
3. Investigate the applicability of computational or learning theoretic methods for stochastic control problems.
4. Understand linear and non-linear filtering theory and their use in engineering systems.
5. Arrive at solutions for optimality, near-optimality, or stability in decision making under uncertainty via alternative analytical, numerical or simulation methods.
6. Document, present, and critically review a scientific paper, method or algorithm on a subject involving stochastic control theory or applications.

Course Learning Outcomes:

1. Computing Shannon's information measures (entropy, Kullback-Leibler distance and mutual information).
2. Computing the capacity of communication channels.
3. Reasoning about the properties of Shannon's information measures (entropy, Kullback-Leibler distance and mutual information).
4. Using mathematical tools to infer properties of coding and communication systems.
5. Probabilistic modeling of communication systems for source and channel coding purposes.
6. Using tools from probability theory to analyze communication systems.
7. Metric assessment of data compression code designs.

Course Learning Outcomes:

1. Prove rigorously the optimality of lossless and lossy source codes.
2. Design of quantizers and evaluation of distortion.
3. Mathematically formulate source coding problems.
4. Use probabilistic tools to understand the effect of data compression on random sources.

Course Learning Outcomes:

1. CLOs coming soon; please refer to your course syllabus in the meantime.

Course Learning Outcomes:

1. CLOs coming soon; please refer to your course syllabus in the meantime.

Course Learning Outcomes:

1. Have a rigorous understanding of the key elements of the theory of martingales (sub- and super-).
2. Rigorously use key results from martingale theory in order to establish properties of stochastic processes, and, in particular, of Brownian motion.
3. Rigorously construct the Itô stochastic integral of a predictable process with respect to a right-continuous square-integrable martingale.
4. Rigorously apply the Itô stochastic calculus to solve problems in stochastic analysis.
5. Have a rigorous understanding of stochastic differential equations.
6. Rigorously apply tools from stochastic analysis to solve problems in mathematical finance.

Course Learning Outcomes:

1. Demonstrate appropriate referencing to cite previous work.
2. Integrate standards, codes of practice, and legal and regulatory factors into decision- making processes, as appropriate.
3. Evaluate trade-offs among goals and concepts.
4. Have ability to understand the scope and a project, and how to manage the day-to-day and long-term facets of a project.
5. Critically evaluate procured information for authority, currency, and objectivity.
6. Conduct investigations to test hypotheses related to complete problems.
7. Independently acquire new knowledge and skills for ongoing personal and professional development.
8. Use graphics appropriately to explain, interpret, and assess information.
9. Develop metrics for comparison of designs.
10. Demonstrate accurate use of technical vocabulary.
11. Present oral communication that is well thought out, well-prepared, and delivered in a convincing manner.
12. Select appropriate mathematical models and tools.
13. Generate a traceable and defensible record of a technical project using an appropriate records system.
14. Consider all factors in design, including economic, environmental, social, and regulatory considerations, as appropriate.
15. Develop mathematical tools to solve engineering problems.
16. Demonstrate a capacity for leadership and decision-making.
17. Write and revises documents using appropriate discipline-specific conventions.
18. Demonstrate professional bearing.
19. Consider social and environmental factors and/or impacts in decisions.
20. Consider ethical factors and matters of equity as appropriate.
21. Perform economic analysis at an appropriate level.
22. Demonstrate skills needed for self-education.
23. Demonstrate proficiency in using sophisticated mathematical models in analysis of engineering problems.
24. Experimentally validate mathematical models and techniques.
25. Formulate clear problem specifications.
26. Understand limitations of mathematical tools.
27. Work effectively as a member of a group.

Course Learning Outcomes:

1. CLOs coming soon; please refer to your course syllabus in the meantime.

Course Learning Outcomes:

1. Mathematically define a research problem.
2. Study and critically evaluate the literature.
3. Perform independent research, acquiring new knowledge via scientific and mathematical reasoning and fairly placing them in the general literature.
4. Rigorously and professionally report the research in a well-organized scientific document.
5. Communicate effectively in an oral presentation.
6. Consider environmental effects, ethical considerations, cultural and equity issues throughout the research problem as appropriate.