« DOCTORAL SCHOOL OF MATHEMATICS AND COMPUTER SCIENCE »

## Criteria and Topics for the Doctoral Studies Admission, Mathematics

approved by the Faculty Council on 06.02.2018

## The admission exam consists of

1. A written exam based on topics from the thematic area selected by the candidate.
2. An interview in which the scientific interests of the candidate, as well as the research topic proposed for the PhD thesis are analyzed.

The candidates will be admitted, based on their options and admission scores, on the available state budgeted places (full-time and part-time). The admission score is computed as follows:

• 40% the grade for the written exam;
• 40% the grade for the interview;
• 20% the average grade of promoting the years of study at the Master level/Advanced Studies or the average grade of promoting the years of study at the Bachelor level for the graduates of the long-time higher education studies from the period prior to the application of the three Bologna cycles (which do not have a Master diploma). In the case of equal admission scores, the grade from the written exam will be considered for ranking.

For admission to the doctoral studies, the admission score must be at least 7 (seven).

## Topics in Algebra

1. Groups, subgroups, homomorphisms. Lagrange’s theorem. Normal subgroups, quotient groups. Direct products and semidirect products of groups.
2. Rings, subrings, homomorphisms. Rings of polynomials. Rings of matrices. Ideals, quotient rings. Prime fields;
3. Rings of quotients. The field of fractions of integral domain. The construction of the ring Z of integers and of the field Q of rationals;
4. Divisibility in semigroups and rings. Factorial, principal and Euclidean domains. Congruences, the ring of residues modulo n. Arithmetic in polynomial rings. Prime and maximal ideals in commutative rings;
5. G-sets. p-groups and the Sylow theorems.
6. Characteristic and fully invariant subgroups. The center of a group. The derived subgroup. Nilpotent groups. Solvable groups.
7. Vector spaces, modules and algebras. Free modules. The matrix of a linear map. Direct sums of modules. Simple modules and indecomposable modules.
8. Finitely generated modules over principal ideal domains. The invariant factors theorem. The structure of finitely generated abelian groups. The Jordan normal form of a matrix.
9. Bilinear symmetric forms and Hermitian forms. Orthogonal and unitary groups.
10. Field extensions. Algebraic closed fields. The fundamental theorem of algebra.
11. Normal extensions. Separable extensions. The Galois group. The fundamental theorem of Galois theory;
12. Finite fields. Wedderburn’s theorem. Subfields of a finite field.
13. Tensor products of modules and algebras.
14. Categories, functors, natural transformations. Adjoint functors.

### Bibliography

1. P. Aluffi, Algebra: chapter 0. American Mathematical Society, 2009.
2. T.W. Hungerford, Algebra. Springer-Verlag, New York, 2003.
3. N. Jacobson, Basic Algebra I, II. W.H. Freeman & Comp., New York, 1985, 1989.
4. S. Lang, Algebra. Springer-Verlag, New York, 2002.
5. J.J. Rotman, Advanced Modern Algebra. Pearson Education, 2010.

## Topics in Analysis

1. Differentiability of real and vector functions of vector variables: the Fréchet differential, necessary and sufficient conditions of differentiability; relation between differentiability and continuity; partial derivatives, representation of the differential by partial derivatives, derivatives and differentials of higher order, Taylor’s formula;
2. Optimal points and local optimal points in the Euclidean space Rn: necessary conditions and sufficient conditions of optimality, constrained optimization problems;
3. Extensions of linear and continuous functions: Helly’s lemma, Hahn-Banach’s theorem, separation theorems of convex sets by hyperplanes;
4. Linear and continuous operators between normed spaces: characterizations of continuity of linear operators between normed spaces; the open mapping theorem, the closed graphs theorem, the normed space of linear and continuous operators between normed spaces

### Bibliography

1. W.W. Breckner, Analiză funcţională, Presa Universitară Clujeană, Cluj-Napoca, 2009
2. Şt. Cobzaş Analiză matematică (Calculul diferenţial), Presa Universitară Clujeană, Cluj-Napoca, 1997
3. Finta Zoltán, Matematikai Analízis I-II, Presa Universitară Clujeană, Cluj-Napoca, 2007
4. Kassay Gábor, Kolumbán József, Marchiş Julianna: Valós Számok és Metrikus Terek, Presa Universitară Clujeană, Cluj-Napoca, 2005
5. L. Lupşa, L. Blaga, Analiză matematică – Note de curs, Presa Universitară Clujeană, Cluj-Napoca, 2003
6. I. Muntean, Analiză funcţională, Universitatea Babeş-Bolyai, Cluj-Napoca, 1993
7. W. Rudin, Principles of Mathematical Analysis, third edition, McGraw-Hill, 1976
8. W. Rudin, Functional Analysis, second edition, McGraw-Hill, 1991

### Topics in Geometry

1. Differentiable manifolds.
2. Vector fields and integration.
3. Riemannian manifolds.
4. The rank of a differentiable mapping. Properties.
5. The critical set and the bifurcation set of a mapping.Examples.
6. Submanifolds and the preimage theorem. Applications.
7. The fundamental group of a topological space.
8. The computation of the fundamental groupi.
9. The fundamental group of the circle. Applications.
10. The theorem of Seifert-van Kampen.

### Bibliography

1. D. Andrica, Critical Point Theory and Some Applications, Cluj University Press, Cluj-Napoca, 2005
2. D. Andrica, I.N. Casu,Lie Groups, exponential map, and geometric mecanics (Romanian), Presa Universitara Clujeana, Cluj-Napoca, 2008
3. D. Andrica, C. Pintea, Elements of homotopy theory with applications to the study of critical points (Romanian), Editura Mirton, Timisoara, 2002
4. M. Struwe, Variational Methods, Application to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, Berlin, Heidelberg, New-York, 1996

## Topics in Complex Analysis

• The complex integral. Cauchy’s fundmental theorem. Cauchy’s integral formulas. Applications.
• Isolated singularities. The residue theorem. Applications.
• Meromorphic functions. The study of zeros and poles of mermorphic functions by using the residue theory. Cauchy’s theorem related to zeros and poles of meromorphic functions. Applications.
• Univalent functions. General results. The class S.
• Conformal mappings in the complex plane. The Riemann mapping theorem. The conformal automorphisms of the unit disc, upper half-plane, and the complex plane C. Examples and applications.
• Harmonic functions, subharmonic functions, and holomorphy in the complex plane.

### Bibliography

1. P. Hamburg, P.T. Mocanu, N. Negoescu, Mathematical Analysis (Complex Functions), Editura Didacticã şi Pedagogică, Bucureşti, 1982 (in Romanian).
2. G. Kohr, P.T. Mocanu, Special Topics in Complex Analysis, Cluj University Press, Cluj-Napoca, 2005 (in Romanian).
3. I. Graham, G. Kohr, Geometric Function Theory in One and Higher Dimensions, Marcel Dekker Inc., New York, 2003.
4. P.T. Mocanu, T. Bulboacă, G.S. Sălăgean, Geometric Theory of Univalent Functions, House of the Book of Science Cluj-Napoca, 2006 (in Romanian).
5. J.B. Conway, Functions of One Complex Variable, vol. I, Graduate Texts in Mathematics, 159, Springer Verlag, New York, 1996.
6. S. Krantz, Handbook of Complex Variables, Birkhäuser Verlag, Boston, Basel, Berlin, 1999.
7. W. Rudin, Real and Complex Analysis, 3rd ed., Mc. Graw-Hill, 1987.
8. D. Gaşpar, N. Suciu, Complex Analysis, The Publishing House of the Romanian Academy, Bucharest, 1999 (in Romanian).

## Topics in Approximation, Numerical and Statistical Calculus

1. Orthogonal polynomials. General properties and orthogonal classes.
2. Interpolation methods. Polynomial, trigonometric and spline interpolation.
3. Numerical integration of functions. Newton quadrature formulas – Cotes, Gauss, Cebishev. Optimal formulas.
4. Numerical solutions of nonlinear equations. Examples of one-step methods.
5. Linear approximation operators. Popoviciu – Bohman – Korovkin theorem. Error evaluations by using moduli of smoothness. Uniform convergence and statistical convergence.
6. Probabilistic methods of generating linear operators. Random schemes. Semigroups of operators.

### Bibliography

1. T. Cătinaș, I. Chiorean, R. Trîmbițaș, Analiză numerică, Presa Universitară Clujeană, Cluj-Napoca, 2010.
2. D.D. Stancu, Gh. Coman (coordonatori), P. Blaga, Analiză numerică şi Teoria aproximării, vol. II, Presa Universitară Clujeană, Cluj-Napoca, 2002.
3. O. Agratini, P. Blaga, Gh. Coman, Lectures on wavelets, numerical methods and statistics, Casa Cărţii de Ştiinţă, Cluj-Napoca, 2005.

## Topics in Nonlinear Operators and Differential Equations

1. Cauchy’s problem for first order differential equations: existence, uniqueness, data dependence.
2. Systems of differential equations. Dynamical system generated by a system of first order differential equations.
3. Elliptic partial differential equations: classical theory. Divergence formula and Green’s formulas. The fundamental solution of Laplace equation and Riemann-Green representation theorem of smooth functions. Mean-value theorem for harmonic functions. Maximum principle. Uniqueness and continuous data dependence of the classical solution of Dirichlet problem.
4. Evolution equations. Maximum principle for heat equation. Cauchy-Dirichlet problem for heat equation.
5. Contraction principle: existence, uniqueness, data dependence for the fixed point equation.
6. Applications of the contraction principle to integral and differential equations.

### Bibliography

1. I.A. Rus, Ecuaţii diferenţiale, ecuaţii integrale şi sisteme dinamice, Transilvania Press, 2001.
2. R. Precup, Ecuaţii diferenţiale, Editura Risoprint, 2011.
3. R. Precup, Lecţii de ecuaţii cu derivate parţiale, Presa Universitară Clujeană, 2004.
4. A. Granas, J. Dugundji, Fixed Point Theory, Springer-Verlag, Berlin, 2003.

## Topics in Mechanics

1. Kinematics of rigid body: Motion equations. Poisson’s formulas. The distribution of velocity and acceleration in rigid body.
2. Dynamics of material point and of systems of material points. General theorems of dynamics of systems of material points.
3. Dynamics of the motion of rigid body. Euler’s equations for the motion of a rigid body around a fixed point.
4. Kinematics of fluids: Continuous medium, fluid, configuration, motion.
5. Fluid dynamics: Principle of mass conservation. The continuity equation. The Cauchy equations.The constitutive equation of ideal fluid. The Euler
6. The theory of viscous Newtonian fluid: The constitutive equation and the Navier-Stokes equations. Special forms of the Navier-Stokes equations. The Stokes system. The fundamental solution of the Stokes system. Uniqueness theorems for the Dirichlet problem for the Stokes system.

### Bibliography

1. M. Kohr, I. Pop, Viscous Incompressible Flow for Low Reynolds Numbers, WIT Press (Wessex Institute of Technology Press), Southampton (UK) – Boston, 2004.
2. M. Kohr, Special Topics of Mechanics, Cluj University Press, Cluj-Napoca, 2005 (in Romanian).
3. M. Kohr, Modern Problems in Viscous Fluid Mechanics, Cluj University Press, Cluj-Napoca, 2 vols. 2000 (in Romanian).
4. L. Dragoş, Fluid Mechanics, Vol. I, The Publishing House of the Romanian Academy, Bucharest, 1999 (in Romanian).
5. H. Goldstein, C. Poole, J. Safko, Classical Mechanics, Reading, MA: Addison-Wessley Publ. Co. (3rd edition), 2014.
6. C. Truesdell, K.R. Rajagopal, An Introduction to the Mechanics of Fluids, Birkhäuser, Basel, 2000.
7. G.C. Hsiao, W.L. Wendland, Boundary Integral Equations, Springer-Verlag, Heidelberg, 2008.

## Topics in Probability and Stochastic Models

1. Cumulative distribution function, density function, characteristic function of a random variable, respectively of a random vector;
2. The n-dimensional normal distribution;
3. Independent random variables;
4. The Chebyshev inequality, the Jensen inequality for random variables;
5. Almost sure convergence, convergence in probability, convergence in distribution of a sequence of random variables;
6. Weak law of large numbers; strong law of large numbers;
7. Markov chains;
8. Poisson processes.

### Bibliography

1. A. Gut, An intermediate course in probability. Second edition. Springer Texts in Statistics. Springer, New York, 2009
2. A. Klenke, Probability Theory – A Comprehensive Course, Springer Verlag, London, 2008
3. H. Lisei, Probability Theory, Casa Cărţii de Ştiinţă, 2004
4. H. Lisei, W. Grecksch, M. Iancu, Probability: Theory, Examples, Problems, Simulations. World Scientific Publishing, Singapore, 2020
5. F. M. Dekking, C. Kraaikamp, H.P. Lopuhaä, L.E. Meester, A Modern Introduction to Probability and Statistics: Understanding Why and How, Springer, 2005