##### Last Updated:
27/03/2024 - 17:40

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### Preliminary Exam Topics (Subtopic A, Subtopic B):

1. ALGEBRA (A. Groups and Rings, B. Modules and Fields)
2. ANALYSIS (A. Real Analysis, B. Complex Analysis)
3. DIFFERENTIAL EQUATIONS (A. Ordinary Diff. Eqns., B. Partial Diff. Eqns.)
4. GEOMETRY - TOPOLOGY (A. Geometry, B. Topology)
5. NUMERICAL ANALYSIS (A. Numerical Analysis 1, B. Numerical Analysis 2)

#### Topic 1 : ALGEBRA

A. Groups and Rings :

Groups, quotient groups, isomorphism theorems, alternating and dihedral groups, direct products, free groups, generators and relations, actions, Sylow theorems, nilpotent and solvable groups, normal and subnormal series. Rings, ring homomorphisms, ideals, factorization in commutative rings, rings of quotients, localization, principal ideal domains, Euclidean domains, unique factorization domains, polynomials and formal power series, factorization in polynomial rings.

B. Modules and Fields :

Modules, homomorphisms, exact sequences, free modules, vector spaces, tensor products, modules over a PID. Fields, field extensions, the fundamental theorem of Galois theory, splitting fields, algebraic closure and normality, the Galois group of a polynomial, finite fields.

Topic 2 : ANALYSIS

A. Real Analysis :

1. Measures: Sigma algebras, the Concept of Measure, Outer measures, Caratheodory's Theorem, Borel measures.

2. Integration: Measurable Functions, Integration of Non-Negative Functions and the Monotone Convergence Theorem, Integration of Complex Functions and the Dominated Convergence Theorem, Modes of Convergence, Egoroff's Theoerem, Product Measures, the Fubini-Tonelli Theoerem, the Lebesgue Integral in Rn.

3. Decomposition of Measures: Signed Measures, The Hahn Decomposition Theorem, the Lebesgue-Radon-Nikodym Theorem, Complex Measures.

4. Lp Spaces: Holder's and Minkowski's Inequalities, the Dual of Lp , Convolutions.

5. Radon Measures: Positive Linear Functions on Cc(X), the Riesz Representetion Theorem.

B. Complex Analysis :

1. Elementary Properties of Analytic Functions: Power series expansions, Complex line integrals, Complex differentiation, Cauchy-Riemann equations, Cauchy's theorem and Integral Formula, Open mapping theorem, Classification of isolated singularities, Laurent expansions, Calsulus of residues.

2. The Argument Principle: The index of a closed curve, The general form of Cauchy's theorem, Residue theorem, The Argument Principle, Rouche's theorem.

3. The Maximum Modulus Principle: The Maximum Modulus Principle, Schwarz Lemma, One-to-one holomorphic mappings of the unit disc onto itself, Mobius transformations.

4. Zeros and Poles of Analytic Functions: Runge's theorem, Meromorphic functions, Infinite products, Weierstrass Factorization theorem.

5. Analytic Continuation: Analytic continuation along a path, Monodromy theorem.

6. Riemann Mapping Thoerem: Normal Families, Riemann mapping theorem.

#### Topic 3 : DIFFERENTIAL EQUATIONS

A. Ordinary Differential Equations:

Please note that the contents and references of this section were changed on 29 June 2004.

1. Initial Value Problem: First order systems of equations; Peano's existence theorem; Euler's method of approximation; Uniqueness of solutions; The method of successive approximations; Differential inequalities and comparison method; Gronwall inequality; Continuous and differential dependence of solutions on parameters; The maximal interval of existence; Continuation of solutions.

2. Sturmian theory: Sturm-Picone theorem and its consequences.

3. Linear systems: Linear homogeneous and nonhomogeneous systems of differential equations with constant and variable coefficients; Fundamental matrices; Abel's formula; The matrix Exponential; Structure of solutions of systems with constant Coefficients; Floquet theory; Adjoint system.

4. Higher order linear differential equations: Fundamental set; Abel's formula; Adjoint equation.

5. Stability: Lyapunov stability and instability; Basic definitions on stability; Stability by linearization.

6. The topics in Math 254 are included.

• Main Reference 1:  R.K. Miller and A.N. Michel, Ordinary Differential Equations ( Chapters 1.1-1.2, 2.1-2.8, 3.1-3.3,3.4-3.6, 5.1-5.3, 5.5-5.6, 6.1,6.2).
• Main Reference 2: Boyce and DiPrima, Elementary Differential Equations and Boundary Value Problems.
• Other Reference 1: Corduneanu, Principles of Differential and Integral Equations (Chapters 1.1-1.5, 2.1-2.5, 3.1-3.5, 4.1-4.6, 5.1-5.3, 8.5, 9.3, 9.4).
• Other Reference 2: J. Cronin, Differential Equations: Introduction and Qualitative Theory , Chapters 1, Chapter 2 (except Inhomogeneous Linear Systems), Chapter 4 (except Some stability theory for autonomous nonlinear systems and Phase asymptotic stability for periodic solutions).
• Other Reference 3: Ş. Alpay, E. Akyıldız, A. Erkip, Lectures on Differential Equations.
• Past Exams : Spring 2024Fall 2017, Spring 2017, Spring 2016, Fall 2015, Fall 2014, Spring 2014, Spring 2013, Fall 2012, Spring 2011, Fall 2010, Fall 2009, Spring 2008, Fall 2006, Spring 2005, Spring 2004.

B. Partial Differential Equations :

1. First order equations: Introduction, quasi-linear equations, characteristic curves (method of Lagrange), Cauchy problem for quasi-linear equations.

2. Second order equations: Linear (almost linear) second order equations, auxiliary conditions, normal (canonical) forms, Cauchy problem for second order equations, Cauchy-Kowalewski theorem, Green's identity.

3. Elliptic equations (Laplace equation): Harmonic functions, fundamental solutions, maximum principle and its applications, solution of Dirichlet problem (by use of Green's function and of seperation of variables), smoothness of solutions.

4. Hyperbolic Equations (Wave equation): Initial value problems, d'Dalembert's solution, domain of dependence and influence, well-posedness, n dimensional wave equation (use of spherical means), initial and boundary value problems.

5. Parabolic equations (Heat equation): Initial value problems, initial-boundary value problem, maximum principle.

#### Topic 4 : GEOMETRY - TOPOLOGY

A. Geometry :

1. Differentiable Manifolds, Differentiable Functions and Mappings: Differentiable manifolds, differentiable functions and mappings, rank of a mapping, immersions, submersions, submanifolds and imbeddings

2. Vector Fields on Manifolds: Tangent space at a point of amanifold, the differential of a differentiable mapping, vector fields, Lie bracket of vector fields

3. Tensors and Tensor Fields on Manifolds: Tensors, tensor fields and differential forms, pull back of a differentiable mapping by differentable mapping, exterior differentiation, Riemannian metric on manifolds, orientation on manifolds, volume element

4. Integration on Manifolds: Integration on manifolds, manifolds with boundary, boundary orientation of the boundary of a manifold, Stokes's Theorem.

B. Topology :

1. Topological spaces and continuous functions: Topological spaces, basis and subbasis, subspace topology, continuous functions, product topology, metric topology, quotient topology.

2. Compactness : Compact spaces, compact sets in Rn, Heine Borel Theorem, Tychonoff Theorem, limit point compactness, sequential compactness, compactness in metric spaces, local compactness and one-point compactification.

3. Connectedness: Connected spaces, path connected spaces, components, local connectedness, local path connectedness.

4. Separation and Countability Properties: T0, Hausdorff, regular, normal spaces, Uryshon Lemma, Tietze Extension Theorem, countability properties, Lindelöf, separable, countably compact spaces.

#### Topic 5 : NUMERICAL ANALYSIS

A. Numerical Analysis 1:

1. Computational Preliminaries: Absolute and relative errors, loss of significance, roundoff errors, stable and unstable computations, conditioning.

2. Solution of Linear Systems of Equations: Gaussian elimination, LU-decomposition, pivoting and scaling in Gaussian elimination, condition numbers, error analysis and stability in Gaussian elimination. Basic iterative methods (Jacobi, Gauss-Seidel and Successive over relaxation methods), convergence of Jacobi, Gauss-Seidel and successive over relation methods. Conjugate gradient method.

3. Linear Least Square Problems: Matrix factorizations that solve the linear least-squares problems, normal equations, QR decomposition and solving least square problems using QR decomposition, orthogonal matrices, Householder transformation.

4. Singular Value Decomposition.

5. The Algebraic Eigenvalue Problem: The power method, the inverse power method, localization of eigenvalues, reduction to Hessenberg form, Householder transformation, QR algorithm for eigenvalue problems, estimation of eigenvalues.

• Reference 1: J.W. Demmel, Applied Numerical Linear Algebra, SIAM, 1997. Sections: 1.7, 2.1, 2.2, 2.3, 2.4, 3.1, 3.2 (3.2.1, 3.2.2), 3.2.3.,3.4 (3.4.1), 3.5 (3.5.2), 4.4 (4.4.1, 4.4.2, 4.4.3, 4.4.4, 4.4.5, 4.4.6), 6.1, 6.4, 6.5 (6.5.1, 6.5.2, 6.5.3, 6.5.4, 6.5.5 (up to Definition 6.12)), 6.6.3.
• Reference 2: J. Stoer and R. Bulirsh, Introduction to Numerical Analysis, Third Edition, Springer-Verlag, 2002. Sections: 1.2, 1.3, 1.4, 4.1, 4.2, 4.3, 4.4, 4.7, 4.8 (4.8.1, 4.8.2, 4.8.3), 6.0, 6.1, 6.5.1, 6.5.4, 6.6.1, 6.6.2, 6.6.3, 6.6.6, 6.9 (up to corollary 6.9.5), 8.0, 8.1, 8.2, 8.3. , 8.7.1.
• Past Exams: Spring 2024Spring 2023, Spring 2017, Fall 2016, Fall 2015, Spring 2015, Spring 2014, Spring 2012, Spring 2011, Spring 2010, Fall 2009, Fall 2007, Spring 2007.

B. Numerical Analysis 2 :

1. Interpolation and Approximation: Polynomial interpolation (Lagrange and Hermite interpolations), divided difference and the Newton form of the interpolating polynomial, error of polynomial interpolation, interpolation by cubic splines. Bsplines. Trigonometric interpolation. Interpolation at the zeros of orthogonal polynomials. Orthogonal polynomials and least-squares approximations. Interpolation using Chebyshev polynomials.

2. Numerical Differentiation: Numerical differentiation based on polynomial interpolation.

3. Numerical Integration (Quadrature): Interpolatory numerical integration, Newton-cotes formulas, Gaussian Quadrature, errors of quadrature formulas. Extrapolation, adaptive quadrature.

4. Solution of Nonlinear Equations: Newton's method, fixed-point algorithm, convergence of the fixed-point algorithm and Newton's method. Root finding algorithms for polynomials.