Example Topics:

• Linear Equations and Functions
• Systems of Linear Equations
• Exponents and Polynomials
• Rational Expressions and Rational Equations
• Roots, Radicals, and Complex Numbers
• Quadratic Equations
• Circles
• Relations, Functions, and Their Graphs
• Polynomial Functions

Example Topics:

• Relations, Functions, and Their Graphs
• Polynomial Functions
• Exponential and Logarithmic Functions
• Trigonometric Functions
• Trigonometric Identities and Equations

Example Topics:

• Vectors in R
• Matrices
• Linear equations
• Determinants
• Eigenvalues and Eigenvectors
• Definitions and Terminology
• Initial-Value Problems
• Solution Curves Without a Solution
• Separable Equations
• Linear Equations
• Exact Equations
• Solutions by Substitution
• Nolinear Differential Equations
• Linear Models
• Nonlinear Models
• Preliminary Theory-Linear Equations
• Homogeneous Linear Equations with Constant Coefficients
• Nonhomogeneous Linear Equations- Undetermined Coefficients
• Nonhomogeneous Linear Equations-Variation of Parameters
• Cauchy-Euler Equations
• Linear Models: Initial-Value Problems
• Preliminary Theory-First Order Linear Systems
• Homogeneous Linear Systems
• Nonhomogeneous Linear Systems
• The Laplace Transform
• Inverse Transforms and Transforms of Derivatives
• Translation Theorems
• Derivative of a Transform, Transform of an Integral
• The Dirac Delta Function

Example Topics:

• Foundations of Probabilistic Modeling:
  • The Probability Space
  • Combinatorial Probability
  • Conditional Probability and Bayes Formula
  • Independent Events.
• Discrete Random Variables:
  • Moments
  • Calculation of the Expectation and Variance
  • Moment Generating Functions
• Continuous Distributions:
  • Probability Distribution Function (PDF)
  • Continuous Random Variables
  • Exponential Random Variable Revisited. Gamma Random Variable
  • Gaussian Random Variable
  • Jointly Distributed Random Variables
  • Marginal PDF's and Marginal Densities
  • Independent Random Variables
  • Sums of Independent Random Variables
  • Correlation.
• Reliability Analysis:
  • Reliability Measures
  • Reliability Measures of Special Distributions (Exponential, Weibull, Rayleigh)
  • Reliability of k‐out‐of‐n Systems
• Point and Interval Estimation:
  • Point Estimation. Maximum Likelihood Estimators. Estimation of Reliability Parameters
  • The Central Limit Theorem
  • Confidence Intervals
  • Approximate Confidence Intervals and other Ramifications of the Central Limit Theorem
  • The Difference in Means of Gaussian Populations
• Bayes Analysis:
  • Conditional distributions and densities. Conjugate Priors
  • Bayes estimators
• Nonparametric Methods:
  • Goodness‐of‐Fit‐Test
  • Testing Independence in Contingency Tables
• Simple Linear Regression

Example Topics:

• Root Finding Algorithm for Single/System of Equations: Bisection Method, Fixed Point Iterations, Newton’s Method and Variations, Convergence Analysis
• Polynomial Approximations: Lagrange Polynomial and Newton’s Divided Difference, Error Analysis and Chebyshev Points, Piecewise Interpolations and Cubic Splines
• Numerical Differentiation and Integration: Basic Finite Difference Formulas, Truncation Error Analysis, Newton-Cotes Formula for Numerical Integration, Composite and Adaptive Numerical Quadrature
• Numerical Solution of ODE/System of ODEs: General Single-Step First/Second Order ODE Solvers, Runge Kutta Methods, Multi-Step Methods and the Root Condition, Numerical Solution of a Stiff System
• Direct and Iterative Methods for Linear Systems: Gaussian Elimination and LU Factorization, General Residual Correction Schemes such as Jacobi, Gauss-Seidel and SOR, Convergence Analysis

Example Topics:

• Fundamental Principles of Counting
• Introduction to Logic
• Properties of the Integers: Mathematical Induction
• Relations and Functions
• The Principle of Inclusion and Exclusion
• Generating Functions
• Recurrence Relations
• An Introduction to Graph Theory
• Trees
• Optimization and Matching

Example Topics:

• Polynomial Approximations: Lagrange Polynomial and Newton’s Divided Difference; Chebyshev Points and Chebyshev Polynomial, Cubic Splines, WENO Interpolations.
• Numerical Solution of ODE/System of ODEs: General Single-Step First/Second Order ODE Solvers; Explicit/Implicit Runge Kutta Methods; Multi-Step Methods and the Root Condition; Stiff System and Stability Analysis
• Iterative Methods for Sparse Linear System: General Residual Correction Schemes; Conjugate Gradient Method and Preconditioner; QR Factorization and Solution of Linear Least Square Problems
• Trigonometric Interpolation and Fast Fourier Transformation
• Finite Difference and Finite Element Methods for Boundary Value Problems

Example Topics:

• Combinatorial Analysis. Set theoretic preliminaries. Probability axioms and basic theorems. Conditional probability, Bayes theorem, independence. Discrete random variables. Moments. Transforms: probability and moment generating functions.
• Continuous univariate distributions: Exponential, Poisson, Binomial, Gaussian, Uniform, Erlang. Moment generating functions. Random vectors, independent random variables, sums and transforms of sums of independent random variables. Correlation.
• Statistics of Stochastic processes: mean, subcovariance, autocovariance, complex‐valued random variables and stochastic processes, wide‐sense stationary processes. Voltage process and its statistics. White noise, Poisson process. Random telegraph signal process. Estimators of mean and autocovariance. Wiener process. Systems with stochastic inputs. The Dirac delta function as an impulse function. Fourier transform. Fourier transform of the Dirac delta function. Linear, time‐invariant, operators or filters (LTIO). System response under LTIO. Impulse response of a linear system. Examples with LRC circuits.
• Mean and autocorrelation of the system response with a wide‐sense stationary input. Cross‐correlation measurement system. The transfer function, unit sinusoid. The transfer function of a linear system. Examples of LR circuits. Spectral analysis, spectral density function. An illustration with the random telegraph process. White noise and an example with a voltage process. Spectral density response. Power transfer function. Example with an LR circuit.