Essential Mathematics for Applied Fields

1. Sets, Sequences, Series, and Functions.- Basic Set Definitions.- Unions, Intersections (multiple).- Lim Inf, Lim Sup, Limit, Convergence of Set Sequences.- Sup, Inf, Max, Min of Real Sets.- Limit Point of Real Sets; Closure, Boundary.- Open, Closed Real Sets.- Bolzano-Weierstrass Theorem.- Limit Point of Real Sequences.- Lim Inf, Lim Sup, Limit, Convergence of Real Sequences.- Cauchy Criterion for Convergence.- Sup, Inf, Max, Min of Functions over Sets.- General Principle of Convergence for Real Series.- Properties of Convergent Series.- Bracketing and Reordering.- Non-negative Series, Absolute Convergence.- Tests for Convergence of Real Series.- Hints and Answers.- References.- 2. Doubly Infinite Sequences and Series.- Definitions and Notation: Sequences.- Lim Inf, Lim Sup, Limit, Convergence: Sequences.- Cauchy Criterion: Sequences.- Iterated Limits: Sequences.- Definition and Notation: Series.- Iterated Sums: Series.- Convergence: Series.- Non-Negative Series.- Absolute Convergence: Series.- Tests for Convergence: Series.- Interchange of Summation Order: Series.- Hints and Answers.- References.- 3. Sequences and Series of Functions.- Real Function Sequences: Definition, Notation.- Lim Inf, Lim Sup, Pointwise Convergence, Limit.- Pointwise Convergence: Shortcomings.- Uniform Convergence: Real Function Sequences.- Continuity of Limit Under Uniform Convergence.- Real Function Sequences: Monotone, Continuous.- Term-by-Term Integration.- Term-by-Term Differentiation.- Real Function Series: Definition, Notation.- Sum Function and Pointwide Convergence.- Interchanging Limit Operations: Dominated Convergence.- Interchanging Limit Operations: Fatou's Lemma.- Uniform Convergence: Real Function Series.- Real Function Series: Uniform Convergence Tests.- Continuity of Sum Function.- Term-by-Term Integration.- Term-by-Term Differentiation.- Multiply Infinite Case.- Hints and Answers.- References.- 4. Real Power Series.- Real Power Series about a Point.- Radius of Convergence.- Convergence.- Uniform Convergence of Real Power Series.- Interval of Convergence.- Continuity of Sum Function.- Term-by-Term Integration.- Term-by-Term Differentiation.- Taylor Series: Definition.- Real Geometric Series.- Hints and Answers.- References.- 5. Behavior of a Function Near a Point: Various Types of Limits.- Notation: Types of Limits.- Two-Sided Limit.- Continuity.- Left-Hand Limit.- Left-Continuity.- Right-Hand Limit.- Right-Continuity.- Extensions.- Operations With Limits.- L'Hospital's Rules.- Limit Infimum: Definition, Properties.- Limit Supremum: Definition, Properties.- Limit Infimum and Supremum: Combined Properties.- Applications: Generalized Inequalities.- Hints and Answers.- References.- 6. Orders of Magnitude: The 0, o, ~ Notation.- Comparing Asymptotic Magnitudes.- Same Order of Magnitude: the ~ Relation.- At Most Order of Magnitude: the 0 Relation.- Smaller Order of Magnitude: the o Relation.- Hints and Answers.- References.- 7. Some Abelian and Tauberian Theorems.- The Laplace Transform of a Function.- Nature of Abelian and Tauberian Theorems.- Classical Results.- Reformulations.- Functions of Slow and Regular Variation.- A General Abelian-Tauberian Theorem.- Infinite Series Version.- Hints and Answers.- References.- 8. 1-Dimensional Cumulative Distribution Functions and Bounded Variation Functions.- 1-C.D.F.: Definition, Properties.- 1-C.D.F.: Riemann-Continuous Case.- Functions of 1-C.F.F.'s.- Sequences of 1-C.D.F.'s: Complete, Weak Convergence.- Convergence Properties.- 1-B.V.F.'s: Definition, Relation to 1-C.D.F.'s.- 1-B.V.F.'s: Properties.- 1-B.V.F.'s: Alternate Definition by Variation Sums.- Combinations of 1-B.V.F.'s.- Sequences and Convergences of 1-B.V.F.'s.- Hints and Answers.- References.- 9. 1-Dimensional Riemann-Stieltjes Integral.- Approximating Sums: Partition of Bounded Interval [a,b).- Definition and Notation: Integral with 1-C.D.F. Integrator.- Sufficient Conditions for Existence.- Integration Over a Single Point.- Physical Interpretation of the Integral.- Extension of Definition: all Bounded Intervals.- Properties of the Integral.- Integral Inequalities.- A Mean Value Theorem.- Extension of Definition: Unbounded Intervals.- Limit Properties:.- Varying Integrand.- Varying Integrator.- Varying Integration Interval.- Integration-by-Parts:.- Version A.- Version B.- Extension of the Integral to Case of:.- 1-B.V.F. Integrators.- Complex-Valued Integrands.- Discontinuous Integrands.- Change-of-Variables Formula.- Differentiation of the Indefinite Integral.- Hints and Answers.- References.- 10. n-Dimensional Cumulative Distribution Functions and Bounded Variation Functions.- n-Monotonicity: Definition.- n-Monotonicity: Characterization in Differentiable Case.- n-C.D.F.'s: Definition, Properties.- Functions of n-C.D.F.'s.- n-C.D.F.'s: Riemann Continuous Case.- Sequences of n-C.D.F.'s: Complete, Weak Convergence.- n-B.V.F.'s: Definition, Relation to n-C.D.F.'s.- n-B.V.F.'s: Case of Differentiability.- n-B.V.F.'s: Alternate Characterization; Variation Sums.- Hints and Answers.- References.- 11. n-Dimensional Riemann-Stieltjes Integral.- Approximating Sums: Partition of n-Rectangle [a',b').- Definition and Notation: n-C.D.F. Integrator.- Sufficient Conditions for Existence.- Extensions of Definition to:.- Various Bounded Rectangles.- Unbounded Rectangles.- Unions of Rectangles.- Properties of the Integral.- A Mean Value Theorem.- Factoring, Iterated Integrals.- Limit Properties:.- Varying Integrand.- Varying Integrator.- Varying Integration Set.- An Integration-by-Parts Formula.- Extension of the Integral to Case of:.- n-B.V.F. Integrators.- Complex Valued Integrands.- Discontinuous Integrands.- Hints and Answers.- References.- 12. Finite Differences and Difference Equations.- Definition: ? and E Operators.- Definition: First Unit Differences.- Definition: n-th Unit Difference (n > 1).- Simple Ascending and Descending Factorials.- Stirling Numbers: First Kind.- Stirling Numbers: Second Kind.- ? Operator: Properties.- General Ascending and Descending Factorials.- Definition: ?-Inverse Operator.- Anti-Differences and Properties of ?-Inverse Operators.- Anti-Differences: Techniques for Obtaining.- Application: Summation of Series.- Definition: Difference Equations.- n-th Order Linear Difference Equations.- General Solution: n-th Order Homogeneous; Constant Coefficient.- n-th Order Non-Homogeneous Case; Constant Coefficients.- Techniques of Solution.- Simultaneous Difference Equations.- Hints and Answers.- References.- 13. Complex Variables.- Basic Definitions.- Modulus.- Addition, Multiplication, Division.- Conjugate.- Polar Form.- Function of a Complex Variable.- Limit at a Point.- Properties of Limit.- Differentiability at a Point.- Cauchy-Riemann Equations.- Sufficient Condition for Differentiability.- Regularity.- Derivatives of Regular Functions.- Complex Power Series.- Contours.- Integral of a Complex Function.- Properties of the Integral.- Cauchy's Theorem and Goursat's Lemma.- Evaluation of Certain Integrals by Cauchy's Theorem.- Cauchy's Integral Formula.- Integral Formula for Derivatives of Regular Functions.- Morera's Theorem: a Converse of Cauchy's Theorem.- Taylor's Theorem: General Form.- Comparing Complex and Real Variables.- Integral Inequality due to Cauchy.- Liouville's Theorem.- Fundamental Theorem of Algebra.- Zeros of a Function.- Isolated Zeros.- Poles and Singularities.- Laurent's Theorem and Expansion.- Types of Singularities.- Residues.- Evaluation of Residues.- Method A.- Method B.- Fundamental Residue Theorem.- Applications: Evaluation of Contour Integrals.- Hints and Answers.- References.- 14. Matrices and Determinants.- Definitions.- Addition of Matrices.- Multiplication of Matrices.- Transpose of a Matrix.- Conjugate of a Matrix.- Determinant of a Square Matrix.- Submatrix, Minor, Principal Minor, Cofactor.- Evaluation of a Determinant.- Properties of a Determinant.- Inverse: Existence and Uniqueness.- Special Types of Square Matrices and Their Properties.- Singular, Non-Singular.- Symmetric.- Hermetian.- Skew-Symmetric.- Unitary.- Normal.- Orthogonal.- Hints and Answers.- References.- 15. Vectors and Vector Spaces.- Row Vectors of Complex Numbers.- Independence, Dependence of Sets of Row Vectors.- Vector Space and Vector Subspace.- Subspace Generated by Rows of a Matrix.- Basis of a Subspace.- Row Operations on a Matrix.- Existence of a Basis.- Unique Representation in Terms of a Fixed Basis.- Transformations of Basis Vectors.- Ranks of Subspaces.- Inner (dot) Product of Vectors.- Length of a Vector.- Orthogonality of Vectors.- Orthogonal Subspaces.- Ortho-Normal Basis: the Gram-Schmidt Procedure.- Conjugate Subspaces.- Hints and Answers.- References.- 16. Systems of Linear Equations and Generalized Inverse.- m Homogeneous Linear Equations in n(? m) Unknowns.- General Solution.- "Sweep Out" Technique for Finding General Solution.- Vector-Space Interpretation of General Solution.- Rank of Matrix and Rank of Subspace.- Properties of Rank.- m Non-Homogeneous Equations in n(? m) Unknowns.- Consistency and Inconsistency.- "Sweep Out" Technique for Finding General Solution.- Vector-Space Interpretation of General Solution.- Generalized Inverse of a Matrix.- Special Case of Non-Singular Matrix.- Technique for Obtaining a Generalized Inverse.- Hints and Answers.- References.- 17. Characteristic Roots and Related Topics.- Characteristic Root.- Characteristic Vector.- Characteristic Polynomial.- Characteristic Equation.- Determinant and Characteristic Roots.- Characteristic Roots of:.- Markov Matrices.- Transpose and Conjugate.- Similar Matrices.- Inverse.- Scalar Multiple.- Triangular Matrices.- Real, Symmetric Matrices.- Trace.- Characteristic Roots and Trace.- Characteristic Roots of AB and BA.- Characteristic Roots of Powers of a Matrix.- Ortho-Normal Characteristic Vectors.- Characteristic Roots of Orthogonal and Unitary Matrices.- Representation Theorem: Real, Symmetric Matrices.- Rank and Characteristic Roots.- Real Quadratic Forms.- Positive Definite, Positive Semi-Definite Forms.- Principal Axes Theorem.- Inverse of a P.D. Matrix.- Characteristic Roots and P.D., P.S.D. Matrices.- Submatrices of a P.D., P.S.D. Matrix.- Test for P.D. or P.S.D. Matrix.- Characterization of P.D. Matrices.- Largest Characteristic Root.- Applications of Largest Characteristic Root.- Hints and Answers.- References.- 18. Convex Sets and Convex Functions.- Definition: Convex Set in En.- Convex Linear Combination of Points in En.- Convex Hull.- Inner Product of Points in En.- Hyperplane in En.- Separating Hyperplanes.- Supporting Hyperplanes.- Characterizations for Convex Subsets of En:.- Separation Theorem.- Support Theorem.- Representation Theorem.- Convex Functions of n=1 Variable.- Characterization: Case When Second Derivative Exists.- Properties.- Convex Functions of n>1 Variables.- Sections of a Convex Subset of En.- Restrictions of Convex Functions n.- Jensen's Inequality.- Concave Functions.- Arithmetic-Geometric-Harmonic Mean Inequality.- Hints and Answers.- References.- 19. Max-Min Problems.- Statement of Problem.- Relative vs. Global Max-Min.- Critical Points.- Unconstrained Max-Min: n=1 Variable.- Unconstrained Max-Min: n>1 Variables.- Constrained Max-Min.- Rationale of Lagrange Multipliers.- Lagrange Function.- Locating Critical Points.- Testing Critical Points.- Limitations.- Generalizations: Linear and Non-Linear Programming.- Hints and Answers.- References.- 20. Some Basic Inequalities.- Cauchy Inequality:.- Finite Series Version.- Infinite Series Version.- Complex Series Version.- Riemann-Stieltjes Integral Version.- p-q Inequality.- Holder Inequality:.- Finite Series Version.- Infinite Series Version.- Complex Series Version.- Riemann-Stieltjes Integral Version.- Triangle Inequality: Finite Series Version.- Minkowski Inequality:.- Finite Series Version.- Infinite Series Version.- Riemann-Stieltjes Integral Version.- cr Inequality.- Hints and Answers.- References.