Steven Skiena's Discrete Mathematics (CSE 547)

Lectures recored in 1999. Stony Brook University.

Course topics:

Josephus problem. Manipulating sums. General methods for manipulating sums. Floors and ceilings. Mod: the binary operation. Divisibility and primes. Relative primality. Congruences. Basic number theory identities. Generating functions. Stirling/Harmonic numbers. Fibonacci numbers. Basic generating function maneuvers. Solving recurrences. Convolutions. Exponential generating functions. Mean and variance. Probability gen. fn's. Degree sequences & invariants. Trees and connectivity. Eulerian and Hamiltonian cycles. Planarity. Graph coloring. Matching. Project presentations.

Convex Optimization I (Stanford University)

Course description:

Concentrates on recognizing and solving convex optimization problems that arise in engineering. Convex sets, functions, and optimization problems. Basics of convex analysis. Least-squares, linear and quadratic programs, semidefinite programming, minimax, extremal volume, and other problems. Optimality conditions, duality theory, theorems of alternative, and applications. Interiorpoint methods. Applications to signal processing, control, digital and analog circuit design, computational geometry, statistics, and mechanical engineering.

Course topics:

1. Solving Optimization Problems, Least-Squares, Linear Programming, Convex Optimizations. 2. Logistics, Convex Set, Convex Cone, Polyhedra, Positive Semidefinite Cone, Operations That Preserve Convexity, Intersection, Affine Function, Generalized Inequalities, Minimum And Minimal Elements, Supporting Hyperlane Theorem, Minimum And Minimal Elements Via Dual Inequalities. 3. Logistics, Convex Functions, Examples, Restriction Of A Convex Function To A Line, First-Order Condition, Examples (FOC And SOC), Epigraph And Sublevel Set, Jensen’s Inequality, Operations That Preserve Convexity, Pointwise Maximum, Pointwise Maximum, Composition With Scalar Functions, Vector Composition. 4. Vector Composition, Perspective, The Conjugate Function, Quasiconvex Functions, Examples, Properties (Of Quasiconvex Functions), Log-Concave And Log-Convex Functions, Properties (Of Log-Concave And Log-Convex Functions), Examples (Of Log-Concave And Log-Convex Functions). 5. Optimal And Locally Optimal Points, Feasibility Problem, Convex Optimization Problem, Local And Global Optima, Optimality Criterion For Differentiable F0, Equivalent Convex Problems, Quasiconvex Optimization, Problem Families, Linear Program. 6. Linear-Fractional Program, Quadratic Program (QP), Quadratically Constrained Quadratic Program (QCQP), Second-Order Cone Programming, Robust Linear Programming, Geometric Programming, Example (Design Of Cantilever Beam), GP Examples (Minimizing Spectral Radius Of Nonnegative Matrix). 7. Generalized Inequality Constraints, Semidefinite Program (SDP), LP And SOCP As SDP, Eigenvalue Minimization, Matrix Norm Minimization, Vector Optimization, Optimal And Pareto Optimal Points, Multicriterion Optimization, Risk Return Trade-Off In Portfolio Optimization, Scalarization, Scalarization For Multicriterion Problems. 8. Lagrangian, Lagrange Dual Function, Least-Norm Solution Of Linear Equations, Standard Form LP, Two-Way Partitioning, Dual Problem, Weak And Strong Duality, Slater’s Constraint Qualification, Inequality Form LP, Quadratic Program, Complementary Slackness. 9. Complementary Slackness, Karush-Kuhn-Tucker (KKT) Conditions, KKT Conditions For Convex Problem, Perturbation And Sensitivity Analysis, Global Sensitivity Result, Local Sensitivity, Duality And Problem Reformulations, Introducing New Variables And Equality Constraints, Implicit Constraints, Semidefinite Program. 10. Applications Section Of The Course, Norm Approximation, Penalty Function Approximation, Least-Norm Problems, Regularized Approximation, Scalarized Problem, Signal Reconstruction, Robust Approximation, Stochastic Robust LS, Worst-Case Robust LS. 11. Statistical Estimation, Maximum Likelihood Estimation, Examples, Logistic Regression, (Binary) Hypothesis Testing, Scalarization, Experiment Design, D-Optimal Design. 12. Continue On Experiment Design, Geometric Problems, Minimum Volume Ellipsoid Around A Set, Maximum Volume Inscribed Ellipsoid, Efficiency Of Ellipsoidal Approximations, Centering, Analytic Center Of A Set Of Inequalities, Linear Discrimination. 13. Linear Discrimination (Cont.), Robust Linear Discrimination, Approximate Linear Separation Of Non-Separable Sets, Support Vector Classifier, Nonlinear Discrimination, Placement And Facility Location, Numerical Linear Algebra Background, Matrix Structure And Algorithm Complexity, Linear Equations That Are Easy To Solve, The Factor-Solve Method For Solving Ax = B, LU Factorization. 14. LU Factorization (Cont.), Sparse LU Factorization, Cholesky Factorization, Sparse Cholesky Factorization, LDLT Factorization, Equations With Structured Sub-Blocks, Dominant Terms In Flop Count, Structured Matrix Plus Low Rank Term. 15. Algorithm Section Of The Course, Unconstrained Minimization, Initial Point And Sublevel Set, Strong Convexity And Implications, Descent Methods, Gradient Descent Method, Steepest Descent Method, Newton Step, Newton’s Method, Classical Convergence Analysis, Examples. 16. Continue On Unconstrained Minimization, Self-Concordance, Convergence Analysis For Self-Concordant Functions, Implementation, Example Of Dense Newton System With Structure, Equality Constrained Minimization, Eliminating Equality Constraints, Newton Step, Newton’s Method With Equality Constraints. 17. Newton's Method (Cont.), Newton Step At Infeasible Points, Solving KKT Systems, Equality Constrained Analytic Centering, Complexity Per Iteration Of Three Methods Is Identical, Network Flow Optimization, Analytic Center Of Linear Matrix Inequality, Interior-Point Methods, Logarithmic Barrier. 18. Logarithmic Barrier, Central Path, Dual Points On Central Path, Interpretation Via KKT Conditions, Force Field Interpretation, Barrier Method, Convergence Analysis, Examples, Feasibility And Phase I Methods. 19. Interior-Point Methods (Cont.), Example, Barrier Method (Review), Complexity Analysis Via Self-Concordance, Total Number Of Newton Iterations, Generalized Inequalities, Logarithmic Barrier And Central Path, Barrier Method, Course Conclusion, Further Topics.

Convex Optimization II (Stanford University)

Course description:

Continuation of Convex Optimization I. Subgradient, cutting-plane, and ellipsoid methods. Decentralized convex optimization via primal and dual decomposition. Alternating projections. Exploiting problem structure in implementation. Convex relaxations of hard problems, and global optimization via branch & bound. Robust optimization. Selected applications in areas such as control, circuit design, signal processing, and communications. Course requirements include a substantial project.

Course topics:

1. Subgradients, Basic Inequality, Subgradient Of A Function, Subdifferential, Subgradient Calculus, Some Basic Rules (For Subgradient Calculus), Pointwise Supremum, Weak Rule For Pointwise Supremum, Expectation, Minimization, Composition, Subgradients And Sublevel Sets, Quasigradients. 2. Subgradients, Subgradients And Sublevel Sets, Quasigradients, Optimality Conditions – Unconstrained, Example: Piecewise Linear Minimization, Optimality Conditions – Constrained, Directional Derivative And Subdifferential, Descent Directions, Subgradients And Distance To Sublevel Sets, Descent Directions And Optimality, Subgradient Method, Step Size Rules, Assumptions, Convergence Results, Aside: Example: Applying Subgradient Method To Abs(X). 3. Convergence Proof, Stopping Criterion, Example: Piecewise Linear Minimization, Optimal Step Size When F* Is Known, Finding A Point In The Intersection Of Convex Sets, Alternating Projections, Example: Positive Semidefinite Matrix Completion, Speeding Up Subgradient Methods, A Couple Of Speedup Algorithms, Subgradient Methods For Constrained Problems, Projected Subgradient Method, Linear Equality Constraints, Example: Least L_1-Norm. 4. Project Subgradient For Dual Problem, Subgradient Of Negative Dual Function, Example (Strictly Convex Quadratic Function Over Unit Box), Subgradient Method For Constrained Optimization, Convergence, Example: Inequality Form LP, Stochastic Subgradient Method, Noisy Unbiased Subgradient, Stochastic Subgradient Method, Assumptions, Convergence Results, Convergence Proof, Stochastic Programming. 5. Stochastic Programming, Variations (Of Stochastic Programming), Expected Value Of A Convex Function, Example: Expected Value Of Piecewise Linear Function, On-Line Learning And Adaptive Signal Processing, Example: Mean-Absolute Error Minimization, Localization And Cutting-Plane Methods, Cutting-Plane Oracle, Neutral And Deep Cuts, Unconstrained Minimization, Deep Cut For Unconstrained Minimization, Feasibility Problem, Inequality Constrained Problem, Localization Algorithm, Example: Bisection On R, Specific Cutting-Plane Methods, Center Of Gravity Algorithm, Convergence Of CG Cutting-Plane Method. 6. Hit-And-Run CG Algorithm, Maximum Volume Ellipsoid Method, Chebyshev Center Method, Analytic Center Cutting-Plane Method, Extensions (Of Cutting-Plane Methods), Dropping Constraints, Epigraph Cutting-Plane Method, PWL Lower Bound On Convex Function, Lower Bound, Analytic Center Cutting-Plane Method, ACCPM Algorithm, Constructing Cutting-Planes, Computing The Analytic Center, Infeasible Start Newton Method Algorithm, Properties (Of Infeasible Start Newton Method Algorithm), Pruning Constraints, PWL Lower Bound On Convex Function, Lower Bound In ACCPM, Stopping Criterion, Example: Piecewise Linear Minimization. 7. Piecewise Linear Minimization, ACCPM With Constraint Dropping, Epigraph ACCPM, Motivation (For Ellipsoid Method), Ellipsoid Algorithm For Minimizing Convex Function, Properties Of Ellipsoid Method, Example (Using Ellipsoid Method), Updating The Ellipsoid, Simple Stopping Criterion, Basic Ellipsoid Algorithm, Interpretation (Of Basic Ellipsoid Algorithm), Example (Of Ellipsoid Method). 8. Ellipsoid Method, Improvements (To Ellipsoid Method), Proof Of Convergence, Interpretation Of Complexity, Deep Cut Ellipsoid Method, Ellipsoid Method With Deep Objective Cuts, Inequality Constrained Problems, Stopping Criterion, Epigraph Ellipsoid Method, Epigraph Ellipsoid Example, Summary: Methods For Handling, Nondifferentiable Convex Optimization Problems Directly, Decomposition Methods, Separable Problem, Complicating Variable, Primal Decomposition, Primal Decomposition Algorithm, Example (Using Primal Decomposition), Aside: Newton's Method With A Complicating Variable, Dual Decomposition, Dual Decomposition Algorithm. 9. Latex Typesetting Style, Recap: Primal Decomposition, Dual Decomposition, Dual Decomposition Algorithm, Finding Feasible Iterates, Interpretation, Decomposition With Constraints, Primal Decomposition (With Constraints) Algorithm, Example (Primal Decomposition With Constraints), Dual Decomposition (With Constraints), Dual Decomposition (With Constraints) Algorithm, General Decomposition Structures, General Form, Primal Decomposition (General Structures), Dual Decomposition (General Structures), A More Complex Example, Aside: Pictorial Representation Of Primal And Dual Decomposition. 10. Decomposition Applications, Rate Control Setup, Rate Control Problem, Rate Control Lagrangian, Aside: Utility Functions, Rate Control Dual, Dual Decomposition Rate Control Algorithm, Generating Feasible Flows, Convergence Of Primal And Dual Objectives, Maximum Capacity Violation, Single Commodity Network Flow Setup, Network Flow Problem, Network Flow Lagrangian, Network Flow Dual, Recovering Primal From Dual, Dual Decomposition Network Flow Algorithm, Electrical Network Analogy, Example: Minimum Queueing Delay, Optimal Flow, Convergence Of Dual Function, Convergence Of Primal Residual, Convergence Of Dual Variables, Aside: More Complicated Problems. 11. Sequential Convex Programming, Methods For Nonconvex Optimization Problems, Sequential Convex Programming (SCP), Basic Idea Of SCP, Trust Region, Affine And Convex Approximations Via Taylor Expansions, Particle Method, Fitting Affine Or Quadratic Functions To Data, Quasi-Linearization, Example (Nonconvex QP), Lower Bound Via Lagrange Dual, Exact Penalty Formulation, Trust Region Update, Nonlinear Optimal Control, Discretization, SCP Progress, Convergence Of J And Torque Residuals, Predicted And Actual Decreases In Phi, Trajectory Plan, 'Difference Of Convex' Programming, Convex-Concave Procedure. 12. Recap: 'Difference Of Convex' Programming, Alternating Convex Optimization, Nonnegative Matrix Factorization, Comment: Nonconvex Methods, Conjugate Gradient Method, Three Classes Of Methods For Linear Equations, Symmetric Positive Definite Linear Systems, CG Overview, Solution And Error, Residual, Krylov Subspace, Properties Of Krylov Sequence, Cayley-Hamilton Theorem, Spectral Analysis Of Krylov Sequence. 13. Conjugate Gradient Method, Recap: Krylov Subspace, Spectral Analysis Of Krylov Sequence, A Bound On Convergence Rate, Convergence, Residual Convergence, CG Algorithm, Efficient Matrix-Vector Multiply, Shifting, Preconditioned Conjugate Gradient Algorithm, Choice Of Preconditioner, CG Summary, Truncated Newton Method, Approximate Or Inexact Newton Methods, CG Initialization, Hessian And Gradient, Methods, Convergence Versus Iterations, Convergence Versus Cumulative CG Steps, Truncated PCG Newton Method, Extensions. 14. Methods (Truncated Newton Method), Convergence Versus Iterations, Convergence Versus Cumulative CG Steps, Truncated PCG Newton Method, Truncated Newton Interior-Point Methods, Network Rate Control, Dual Rate Control Problem, Primal-Dual Search Direction (BV Section 11.7), Truncated Netwon Primal-Dual Algorithm, Primal And Dual Objective Evolution, Relative Duality Gap Evolution, Relative Duality Gap Evolution (N = 10^6), L_1-Norm Methods For Convex-Cardinality Problems, L_1-Norm Heuristics For Cardinality Problems, Cardinality, General Convex-Cardinality Problems, Solving Convex-Cardinality Problems, Boolean LP As Convex-Cardinality Problem, Sparse Design, Sparse Modeling / Regressor Selection, Estimation With Outliers, Minimum Number Of Violations, Linear Classifier With Fewest Errors, Smallest Set Of Mutually Infeasible Inequalities, Portfolio Investment With Linear And Fixed Costs, Piecewise Constant Fitting, Piecewise Linear Fitting, L_1-Norm Heuristic, Example: Minimum Cardinality Problem, Polishing, Regressor Selection. 15. Minimum Cardinality Problem, Interpretation As Convex Relaxation, Interpretation Via Convex Envelope, Weighted And Asymmetric L_1 Heuristics, Regressor Selection, Sparse Signal Reconstruction, L_1-Norm Methods For Convex-Cardinality Problems Part II, Total Variation Reconstruction, Total Variation Reconstruction, TV Reconstruction, L_2 Reconstruction, Iterated Weighted L_1 Heuristic, Sparse Solution Of Linear Inequalities, Detecting Changes In Time Series Model, Time Series And True Coefficients, TV Heuristic And Iterated TV Heuristic, Extension To Matrices, Factor Modeling, Trace Approximation Results, Summary: L_1-Norm Methods. 16. Model Predictive Control, Linear Time-Invariant Convex Optimal Control, Greedy Control, 'Solution' Via Dynamic Programming, Linear Quadratic Regulator, Finite Horizon Approximation, Cost Versus Horizon, Trajectories, Model Predictive Control (MPC), MPC Performance Versus Horizon, MPC Trajectories, Variations On MPC, Explicit MPC, MPC Problem Structure, Fast MPC, Supply Chain Management, Constraints And Objective, MPC And Optimal Trajectories, Variations On Optimal Control Problem. 17. Stochastic Model Predictive Control, Causal State-Feedback Control, Stochastic Finite Horizon Control, 'Solution' Via Dynamic Programming, Independent Process Noise, Linear Quadratic Stochastic Control, Certainty Equivalent Model Predictive Control, Stochastic MPC: Sample Trajectory, Cost Histogram, Simple Lower Bound For Quadratic Stochastic Control, Branch And Bound Methods, Methods For Nonconvex Optimization Problems, Branch And Bound Algorithms. 18. Branch And Bound Methods, Basic Idea, Unconstrained, Nonconvex Minimization, Lower And Upper Bound Functions, Branch And Bound Algorithm, Comment: Picture Of Branch And Bound Algorithm In R^2, Comment: Binary Tree, Example, Pruning, Convergence Analysis, Bounding Condition Number, Small Volume Implies Small Size, Mixed Boolean-Convex Problem, Solution Methods, Lower Bound Via Convex Relaxation, Upper Bounds, Branching, New Bounds From Subproblems, Branch And Bound Algorithm (Mixed Boolean-Convex Problem), Minimum Cardinality Example, Bounding X, Relaxation Problem, Algorithm Progress, Global Lower And Upper Bounds, Portion Of Non-Pruned Sparsity Patterns, Number Of Active Leaves In Tree, Global Lower And Upper Bounds.

The Fourier Transform and its Applications

Course description:

The goals for the course are to gain a facility with using the Fourier transform, both specific techniques and general principles, and learning to recognize when, why, and how it is used. Together with a great variety, the subject also has a great coherence, and the hope is students come to appreciate both.

Brief overview of course topics:

The Fourier transform as a tool for solving physical problems. Fourier series, the Fourier transform of continuous and discrete signals and its properties. The Dirac delta, distributions, and generalized transforms. Convolutions and correlations and applications; probability distributions, sampling theory, filters, and analysis of linear systems. The discrete Fourier transform and the FFT algorithm. Multidimensional Fourier transform and use in imaging. Further applications to optics, crystallography. Emphasis is on relating the theoretical principles to solving practical engineering and science problems.

Course topics:

1. The Fourier Series, Analysis V. Synthesis, Periodic Phenomena And The Fourier Series - Periodicity In Time And Space - Reciprocal Relationship Between Domains, The Reciprocal Relationship Between Frequency And Wavelength. 2. Periodicity; How Sine And Cosine Can Be Used To Model More Complex Functions, Example Of Periodizing A Signal, Discussion Of How To Model Signals With Sinusoids, "One Period, Many Frequencies" Idea In Modeling Signals, Modeling A Signal As The Sum Of Modified Sinusoids (Formula), Complex Exponential Notation, Symmetry Property Of The Complex Coefficients In The Fourier Series, Discussion Of The Generality Of The Fourier Series Representation For Modeling A Periodic Function. 3. Summary Of Previous Lecture (Analyzing General Periodic Phenomena As A Sum Of Simple Periodic Phenomena), Fourier Coefficients; Discussion Of How General The Fourier Series Can Be (Examples Of Discontinuous Signals), Discontinuity And Its Impact On The Generality Of The Fourier Series, Infinite Sums To Represent More General Periodic Signals, Summary Of Convergence Issues, Convergence: Continuous Case, Smooth Case (Fourier Series Converges To The Signal), Convergence: Jump Discontinuity, Convergence: General Case (Convergence On Average/In Mean/In Energy). 4. Wrapping Up Fourier Series; Making Sense Of Infinite Sums And Convergence, Integrability Of A Function (Implies Existence Of Fourier Coefficients, Convergence), Orthogonality Of Complex Exponentials (Review), The Inner Product, Norm Of F Related To The Inner Product (+ Pythagorean Theorem), Complex Exponentials As Orthonormal Functions, Fourier Coefficients As Projections Onto Complex Exponentials, Rayleigh's Identity, Application Of Fourier Series To Heat Flow. 5. Continued Discussion Of Fourier Series And The Heat Equation, Transition From Fourier Series To Fourier Transforms (Periodic To Nonperiodic Phenomena), Fourier Series Analysis And Synthesis; Relation To Fourier Transform And Inverse Fourier Transform, Fourier Series/ Coefficients With Period T, Spectrum Picture For Fourier Series With Period T, Effects Of A Change In T, The Complications Of Finding The Fourier Transform By Letting T Go To Infinity (Fourier Coefficients Go To 0). 6. Correction To Heat Equation Discussion, Setup For Fourier Transform Derivation From Fourier Series, Results Of The Derivation: Fourier Transform And Inverse Fourier Transform, Definition Of The Fourier Transform (Analysis), Definition Of Fourier Inversion (Synthesis), Major Secret Of The Universe: Every Signal Has A Spectrum, Which Determines The Signal, Fourier Notation, Example: Rect Function, Example: Triangle Function. 7. Review Of Fourier Transform (And Inverse) Definitions, Notation, Review Of Rect And Triangle Transforms, Example: Fourier Transform Of A Gaussian, The Duality Property Of The Fourier Transform, Example Of An Application Of The Duality Property. 8. Effect On Fourier Transform Of Shifting A Signal, Resulting Delay Formula (Shift Theorem), Effect Of Scaling The Time Signal, Stretch Theorem Formula/ Interpretation, Convolution In Context Of Fourier Transforms; Multiplying Two Signals In Frequency, Resulting Convolution Formula. 9. Continuing Convolution: Review Of The Formula, Situiation In Which It Arose, Example Of Convolution: Filtering, The Ideas Behind Filtering, Terminology, Interpreting Convolution In The Time Domain, General Properties Of Convolution In The Time Domain, Derivative Theorem For Fourier Transforms, Heat Equation On An Infinite Rod. 10. Central Limit Theorem And Convolution; Main Idea, Introduction, Normalization Of The Gaussian, The Gaussian In Probability; Pictorial Demonstration With Convolution, The Setup For The CLT, Key Result: Distribution Of Sums And Convolution (With Proof), Other Assumptions Needed To Set Up CLT, Statement Of The Central Limit Theorem, Using The Fourier Transform To Prove The CLT. 11. orrection To The End Of The CLT Proof, Discussion Of The Convergence Of Integrals; Approaches To Making A More Robust Definition Of The Fourier Transform, Examples Of Problematic Signals, How To Approach Solving The Problem; Choosing Basic Phenomena To Use To Explain Others, Identifying The Best Class Of Signals For Fourier Transforms; + Their Properties, The Definition Of The Class Of Rapidly Decreasing Functions, Rationale For Why These Properties Are Used (Derivative Theorem, Parseval's). 12. Cop Story, Review Of Rapidly Decreasing Functions, Generalized Functions (Distributions) (Delta Function, Etc.), Viewing Delta As A Limit V. Operationally, Definition Of A Distribution, Delta As A Distribution, Discussion Of How To Consider Ordinary Functions In This Space; Pairing Through Integration. 13. Setting Up The Fourier Transform Of A Distribution, Example Of Delta As A Distribution, Distributions Induced By Functions (Includes Many Functions), The Fourier Transform Of A Distribution, The Class Of Tempered Distributions, FT Of A Tempered Distribution, Definition Of The Fourier Transform (By How It Operates On A Test Function), The Inverse Fourier Transform (Proof), Calculations Of Fourier Transforms Using This Definition (Distributions). 14. Derivative Of A Distribution, Example: Derivative Of A Unit Step, Example: Derivative Of Sgn(X), Applications To The Fourier Transform (Using The Derivative Theorem), Caveat To Distributions: Multiplying Distributions, Distributions*Functions, Special Case: The Delta Function And Sampling, Convolution In Distributions, Special Case: Convolution When T = Delta, The Scaling Property Of Delta. 15. Application Of The Fourier Transform: Diffraction: Setup, Representation Of Electric Field, Approach Using Huyghens' Principle, Discussion Of The Phase Change Associated With Different Paths, Use Of The Fraunhofer Approximation, Aperture Function, Result; In General And For Single/Double Slits. 16. More On Results From Last Lecture (Diffraction Patterns And The Fourier Transforms), Setup For Crystallography Discussion (History, Concepts), 1-Dimensional Version, The Fourier Transform Of The Shah Function, Trick: Poisson Summation Formula, Proof Of The Poisson Summation Formula, Fourier Transform Of The Shah Function: Result, Fourier Transform Of The Shah Function With Spacing P, Application To Crystals. 17. Review Of Main Properties Of The Shah Function, Setup For The Interpolation Problem, Bandwidth Assumption, Solving For Exact Interpolation For Bandlimited Signals, Periodizing The Signal By Convolution With The Shah Function, Solution Of The Interpolation Problem. 18. Review Of Sampling And Interpolation Results, Terminology: Sampling Rate, Nyquist Rate, Issues With The Interpolation Formula In Practical Applications, Aliasing And Interpolation, Main Argument In Aliasing, Example Of Aliasing: Cosine. 19. Aliasing Demonstration With Music, Transition To Discrete! The DFT, The Plan For Transitioning To Discrete Time, Creating A Discrete Signal From F(T) Creating A Discrete Version Of The Fourier Transform Of The Sampled Version Of F(T), Summary Of What We Just Did, Summary Of Results (Formulas), Moving From Continuous To Discrete Variables, Final Result: The DFT. 20. Review: Definition Of The DFT, Sample Points, Relationship Between N And Spacing In Time/Frequency, Complex Exponentials In The Discrete DFT, DFT Written With Discrete Complex Exponential Vector, Periodicity Of Inputs And Outputs In The DFT (More On This In Next Lecture), Orthogonality Of The Vector Of Discrete Complex Exponentials, Note On Orthonormality Of Discrete Complex Exponential Vector (Or Lack Thereof), Consequence Of Orthogonality: Inverse DFT. 21. Review Of Basic DFT Definitions, Special Case: Value Of The DFT At 0, Two Special Signals: One Vector, Delta Vector, DFT Of Deltas, Complex Exponentials, DFT As Nxn Matrix Multiplication, Periodicity Of Input/Output Signals In The DFT, Result Of Periodicity: Indexing, Result Of Periodicity: Duality. 22. FFT Algorithm: Setup: DFT Matrix Notation, One Intuition Behind FFT: Factoring Matrix, Our Approach: Split Order N Into Two Order N/2, Iterate, Notation (To Keep Track Of Powers Of Complex Exponentials), Plugging New Notation Into DFT; Split Into Even And Odd Indices, Result For Indices 0 To N/2-1, Result For Indices N/2 To N-1, Summary Of Results (DFT As Combination Of 2 Half Order Dfts). 23. Linear Systems: Basic Definitions, Direct Proportionality As Example, Special Cases Of Linear Systems, Eigenvectors And Eigenvalues, The Spectral Theorem And Finding A Basis Of Eigenvectors, Matrix Multiplication = Only Example Of Finite Dimensional Linear Systems, Integration Against A Kernel Generalizing Matrix Multiplication, Example: The Fourier Transform. 24. eview Of Last Lecture: Discrete V. Continuous Linear Systems, Cascading Linear Systems, Derivation Of The Impulse Response, Schwarz Kernel Theorem, Example: Impulse Response For Fourier Transform, Example: Switch, Special Case: Convolution, Time Invariance, Result: If A System Is Given By Convolution, It Is Time Invariant; Converse True As Well, Two Main Ideas Sumarized (Linear->Integration Against Kernel, Time Invariant If Given By Convolution). 25. Review Of Last Lecture: LTI Systems And Convolution, Comment On Time Invariant Discrete Systems, The Fourier Transform For LTI Systems; Complex Exponentials As Eigenfunctions, Discussion Of Sine And Cosine V. Complex Exponentials As Eigenfunctions (Generally They Are Not), Discrete Version (Discrete Complex Exponentials Are Eigenvectors), Discrete Results From A Matrix Perspective. 26. Approaching The Higher Dimensional Fourier Transform, Notation: Thinking In Terms Of Vectors, Definition Of The Higher Dimensional Fourier Transform, Inverse Fourier Transform, Reciprocal Relationship Between Spatial And Frequency Domain, One Dimensional Case: Reciprocal Relationship, 2-D Case: Visualizing Higher Dimensional Complex Exponentials, Results: Visualizing 2-D Complex Exponentials. 27. Higher Dimensional Fourier Transforms- Review, Fourier Transforms Of Seperable Functions (Ex: 2-D Rect), Result: Formula For Fourier Transform Of A Seperable Function, Example: 2-D Gaussian, Radial Functions, Proof That The Fourier Transform Of A Radial Function Is Also Radial, Convolution In Higher Dimensions. 28. Shift Theorem In Higher Dimensions, Shift Theorem: Result, Stretch Theorem Derivation, Stretch Theorem Result, Special Case: Scaling, Special Case: Rotation, What Reciprocal Means In Higher Dimensions (Inverse Transpose), Deltas In Higher Dimensions (Basic Properties, Scaling). 29. Shahs, Lattices, And Crystallography, 2-D Shah, Crystals As Lattices, The Fourier Transform Of The Shah Function Of An Oblique Lattice, Relation To Crystals; Notation, Concepts, And Results, Application To Medical Imaging: Tomography. 30. Tips For Filling Out Evals, Tomography And Inverting The Radon Transform; Setup, Introducing Coordinates, Delta Along A Line, The Integral Of U Along A Line, Inverting The Radon Transform.

Introduction to Linear Dynamical Systems

Course description:

Introduction to applied linear algebra and linear dynamical systems, with applications to circuits, signal processing, communications, and control systems.

Brief course topics:

Least-squares approximations of over-determined equations and least-norm solutions of underdetermined equations. Symmetric matrices, matrix norm and singular value decomposition. Eigenvalues, left and right eigenvectors, and dynamical interpretation. Matrix exponential, stability, and asymptotic behavior. Multi-input multi-output systems, impulse and step matrices; convolution and transfer matrix descriptions. Control, reachability, state transfer, and least-norm inputs. Observability and least-squares state estimation.

Course topics:

1. Overview Of Linear Dynamical Systems, Why Study Linear Dynamical Systems?, Examples Of Linear Dynamical Systems, Estimation/Filtering Example, Linear Functions And Examples. 2. Linear Functions (Continued), Interpretations Of Y=Ax, Linear Elastic Structure, Example, Total Force/Torque On Rigid Body Example, Linear Static Circuit Example, Illumination With Multiple Lamps Example, Cost Of Production Example, Network Traffic And Flow Example, Linearization And First Order Approximation Of Functions. 3. Linearization (Continued), Navigation By Range Measurement, Broad Categories Of Applications, Matrix Multiplication As Mixture Of Columns, Block Diagram Representation, Linear Algebra Review, Basis And Dimension, Nullspace Of A Matrix. 4. Nullspace Of A Matrix(Continued), Range Of A Matrix, Inverse, Rank Of A Matrix, Conservation Of Dimension, 'Coding' Interpretation Of Rank, Application: Fast Matrix-Vector Multiplication, Change Of Coordinates, (Euclidian) Norm, Inner Product, Orthonormal Set Of Vectors. 5. Orthonormal Set Of Vectors, Geometric Interpretation, Gram-Schmidt Procedure, General Gram-Schmidt Procedure, Applications Of Gram-Schmidt Procedure, 'Full' QR Factorization, Orthogonal Decomposition Induced By A, Least-Squares. 6. Least-Squares, Geometric Interpretation, Least-Squares (Approximate) Solution, Projection On R(A), Least-Squares Via QR Factorization, Least-Squares Estimation, Blue Property, Navigation From Range Measurements, Least-Squares Data Fitting. 7. Least-Squares Polynomial Fitting, Norm Of Optimal Residual Versus P, Least-Squares System Identification, Model Order Selection, Cross-Validation, Recursive Least-Squares, Multi-Objective Least-Squares. 8. Multi-Objective Least-Squares, Weighted-Sum Objective, Minimizing Weighted-Sum Objective, Regularized Least-Squares, Laplacian Regularization, Nonlinear Least-Squares (NLLS), Gauss-Newton Method, Gauss-Newton Example, Least-Norm Solutions Of Undetermined Equations. 9. Least-Norm Solution, Least-Norm Solution Via QR Factorization, Derivation Via Langrange Multipliers, Example: Transferring Mass Unit Distance, Relation To Regularized Least-Squares, General Norm Minimization With Equality Constraints, Autonomous Linear Dynamical Systems, Block Diagram. 10. Examples Of Autonomous Linear Dynamical Systems, Finite-State Discrete-Time Markov Chain, Numerical Integration Of Continuous System, High Order Linear Dynamical Systems, Mechanical Systems, Linearization Near Equilibrium Point, Linearization Along Trajectory. 11. Solution Via Laplace Transform And Matrix Exponential, Laplace Transform Solution Of X_^ = Ax, Harmonic Oscillator Example, Double Integrator Example, Characteristic Polynomial, Eigenvalues Of A And Poles Of Resolvent, Matrix Exponential, Time Transfer Property. 12. Time Transfer Property, Piecewise Constant System, Qualitative Behavior Of X(T), Stability, Eigenvectors And Diagonalization, Scaling Interpretation, Dynamic Interpretation, Invariant Sets, Summary, Markov Chain (Example). 13. Markov Chain (Example), Diagonalization, Distinct Eigenvalues, Digaonalization And Left Eigenvectors, Modal Form, Diagonalization Examples, Stability Of Discrete-Time Systems, Jordan Canonical Form, Generalized Eigenvectors. 14. Jordan Canonical Form, Generalized Modes, Cayley-Hamilton Theorem, Proof Of C-H Theorem, Linear Dynamical Systems With Inputs & Outputs, Block Diagram, Transfer Matrix, Impulse Matrix, Step Matrix. 15. DC Or Static Gain Matrix, Discretization With Piecewise Constant Inputs, Causality, Idea Of State, Change Of Coordinates, Z-Transform, Symmetric Matrices, Quadratic Forms, Matrix Nom, And SVD, Eigenvalues Of Symmetric Matrices, Interpretations Of Eigenvalues Of Symmetric Matrices, Example: RC Circuit. 16. RC Circuit (Example), Quadratic Forms, Examples Of Quadratic Form, Inequalities For Quadratic Forms, Positive Semidefinite And Positive Definite Matrices, Matrix Inequalities, Ellipsoids, Gain Of A Matrix In A Direction, Matrix Norm, Properties Of Matrix Norm. 17. Gain Of A Matrix In A Direction, Singular Value Decomposition, Interpretations, Singular Value Decomposition (SVD) Applications, General Pseudo-Inverse, Pseudo-Inverse Via Regularization, Full SVD, Image Of Unit Ball Under Linear Transformation, SVD In Estimation/Inversion, Sensitivity Of Linear Equations To Data Error. 18. Sensitivity Of Linear Equations To Data Error, Low Rank Approximations, Distance To Singularity, Application: Model Simplification, Controllability And State Transfer, State Transfer, Reachability, Reachability For Discrete-Time LDS. 19. Reachability, Controllable System, Lest-Norm Input For Reachability, Minimum Energy Over Infinite Horizon, Continuous-Time Reachability, Impulsive Inputs, Least-Norm Input For Reachability. 20. Continuous-Time Reachability, General State Transfer, Observability And State Estimation, State Estimation Set Up, State Estimation Problem, Observability Matrix, Least-Squares Observers, Some Parting Thoughts, Linear Algebra, Levels Of Understanding, What's Next.

Keio University Math Video Archive

Videos include:

Algebraic and Symplectic Geometry. Number Theory. Geometry and Dynamical Systems. Singularities. Noncommutativity. Poisson Geometry in Mathematics and Physics. Nonlinear PDE in Analysis and Geometry. Dynamics and Arithmetics. Noncommutative Geometry, K-theory and Physics. Geometric, Spectral, and Stochastic Analysis. Noncommutative Geometry and Physics. Geometry and Analysis Towards Quantum Theory. Topological Quantum Field Theories and Some Modern Problems of Mathematics. Equivariant topological quantum field theory and stringy invariants of orbifolds. New formulas for π (pi) and other classical constant. A geometric proof that e is irrational and a new measure of its irrationality. An elementary reformulation of the Riemann Hypothesis. Deformed Diffeomorphisms as a basis of a deformed group theory and a deformed theory of gravity. Algebraic cycles and special values of zeta-functions. Momentum maps and reduction in the symplectic and Poisson categories. Determining the DNA sequence, a billion dollar logic puzzle. The HIV/AIDS Epidemic: When is HIV most infectious? Chaos. Some aspects of deformation quantization. Matter Wave Solitons in Bose-Einstein Condensate of Ultra Cold Atoms. Combinatorial study of algebro-geometric objects via monomial ideals. Combinatorial-Algebraic Cryptosystems and Polynomial-based Cryptography. Algebraic approach to multidimensional systems theory. Submanifold geometry and critical point theory. Atiyah-Singer Index Theory and Its Applications. Introduction to the theory of the Navier-Stokes equations. Localization operators based on Weyltransforms. Recent topics on the Navier-Stokes equations. Poisson Geometry and Lie Theory. Topics in Analysis, Geometry, and Probability. An introduction to measure preserving systems and Poincare' recurrence. Aperiodicity and periodic approximation; Rokhlin's lemma. Orbit equivalence; Dye's theorem. Capacity and spectral properties of Schrödinger operators. Moment maps and moduli spaces. An Introduction to K-theory. The biggest mysteries of our physical world: why is there quantum mechanics, and why is the vacuum stable? Lie groupoids and differential stacks. Amenable Groups in Ergodic Theory. Microlocal Analysis: An Overview. Global Attractors for Semiflows without Uniqueness. Mathematical Models of Crystal Microstructure. Open Problems in Elasticity. Topological string and gravity/gauge theory correspondence. Chaos Games and Fractal Images. The Mandelbros Set, the Farey Tree, and the Fibonacci Sequence. Exponential Dynamics and Topology. Sierpinski Curve and Triangle Julia Sets. Elementary Mechanics From a Mathematician's Viewpoint. How Newton actually analyzed planetary motion. Rigid bodies. Kählerian Geometry, a Crossroad of Geometries.

Russell Towle's 4D Star Polytope Animations

First animations ever made of the solid sections of four-dimensional star polytopes.

The Monty Hall Problem Explained

Problem definition:

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

Solving IMO (International Mathematics Olympiad) Problems 2006

Video description:

The 2006 US IMO team members describe the steps they took to solve problems 1-3 of the International Mathematical Olympiad.

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This month I present to you new mathematics video lectures.They include: Discrete Math, Convex Optimization, Fourier Transform and its Applications, Linear Dynamical Systems, Keio University Math Videos, Videos from IMO (International Mathematics Olympiad), Explanation of Monty Hall Problem and 4D Star Polytope Animations.Have fun watching these math lectures!