Chairpersonyiorgos-Sokratis Smyrlis

CHAIRPERSONYiorgos-Sokratis Smyrlis

VICE CHAIRPERSON

PROFESSORS

Georgios Alexopoulos

Tasos Christofides

Pantelis Damianou

Georgios Georgiou

Andreas Karageorghis

Nicolas Papamichael

Efstathios Paparoditis

ASSOCIATE PROFESSORS

Alexandros Karagrigoriou

Stamatis Koumandos

George Kyriazis

Υiorgos-Sokratis Smyrlis

Christodoulos Sophocleous

Nikos Stylianopoulos

Alekos Vidras

ASSISTANT PROFESSORS

Konstantinos Fokianos

Christos Pallikaros

Evangelia Samiou

Theofanis Sapatinas

Nikos Tziolas

Filia Vonta

Christos Xenophontos

AIMS

The famous Platonic inscription “μηδείς αγεωμέτρητος εισίτω” (“let no one ignorant of geometry enter”) has been adopted, directly or indirectly, by all universities in the world and, appropriately, the Department of Mathematics and Statistics was one of the departments with which the University of Cyprus commenced its operation. The primary goal of the Department is the promotion, through scientific research and teaching, of the Mathematical Sciences.

The achievement of this goal is inextricably linked with the need to produce well-trained scientists who will contribute to the continuation of the cultural and economic progress of Cyprus. Because of the pivotal role of Mathematics and Statistics for Science, it is necessary to create a department of high calibre.

Following modern trends, the University of Cyprus encourages joint degree programmes between the Department of Mathematics and Statistics and other Departments (e.g., Computer Science, Economics) that offer various career possibilities. The ambition of the Department is to become a recognized research and teaching centre. Taking into consideration the high standards of secondary education in Cyprus, this should be regarded as an obligation.

Important steps in achieving this ambition are the development of links with corresponding institutions abroad and the creation of high-level programmes of studies. The undergraduate programme, which started in September 1992, was designed in such a way so as to be naturally extended into a graduate one, which was designed at a later stage and started in September 1997.

During its first years of operation, the Department has prepared the programme of studies, has established a seminar series in Mathematics and Statistics, has placed emphasis on the development of a modern and fully equipped library and has collaborated with other sectors of the University to develop the necessary institutional and technical (especially in electronic equipment) infrastructure.

PROGRAMME OF STUDY

The programme of studies contains the list of courses, the description of most courses, the regulations for obtaining the degrees in Mathematics and Mathematics and Statistics and the requirements for the minor in Mathematics.

The Department’s objective is to offer an up-to-date and realistic programme leading to degrees whose standards are equivalent to those of well-established universities. The programme has been designed so that it can be easily adapted to future needs.

COURSES

The curriculum courses are divided into four levels and six groups. The level 101-199 corresponds mainly to courses of the first year of studies. The level 201-299 corresponds mainly to courses of the second year of studies. The levels 301-399 and 401-499 are defined accordingly. The level 001-099 corresponds to service courses (see table Β) and are not open to Mathematics or Statistics majors (except MAS007, see Degree Requirements).

The six groups to which the courses are divided, correspond (approximately) to the following areas of Mathematics: Analysis, Algebra, Geometry, Probability/Statistics, Numerical Analysis and Applied Mathematics. The second digit of the course number determines the area of mathematics that the course belongs to.The characteristic digit (second digit of course number) of the six areas are 0 & 1, 2, 3, 5 & 6, 7 and 8 respectively and they appear in table A.

Classes meet for four hours per week, one of which may be a recitation class. Each class corresponds to a number of credit units (c.u.) as they appear in table A.

DEGREE REQUIREMENTS

The following are required for the degree in Mathematics or Mathematics and Statistics:

1)17 compulsory courses for all students (see table A).

MAS101 – Calculus I
MAS102 – Calculus II
MAS121 - Linear Algebra I
MAS122 - LinearAlgebraII
MAS131 –Basic Mathematics
MAS191- Mathematics with computers
MAS201 - Multivariate Differential calculus
MAS202 - Multivariate Integral calculus
MAS203- Ordinary Differential Equations
MAS251 – Probability I
MAS252 – Statistics I
MAS271 – Numerical Analysis I
MAS301 – RealAnalysis
MAS302 – Complex Analysis I
MAS331 – Classical Differential geometry
MAS303 – Partial Differential Equations
MAS304 – Functional Analysis
MAS371 – Numerical Analysis II

2)9 courses that are related to different areas of concentration (Pure Mathematics, Applied Mathematics and Statistics). For the degree in Mathematics one of these courses, is PHY111 – General Physics I. Some of these courses are “free electives” within the department (see tables A and C1, C2, C3).

3)The course CS031-Introduction to Programming.

4)25 c.u. (5 courses of 5 c.u. each) must be free electives from other departments.

5)The student is required to take two foreign language courses (any language).

At most two “free electives” within the Department could be substituted, in exceptional cases and during the last year of studies, by graduate courses. For that, a grade average of at least 7 is required in the departmental courses as well as the approval of the instructor and the academic advisor.

One of the free electives from other Departments could be substituted by MAS007-History of Mathematics (5 c.u.). In this case the free electives from other Departments correspond to 20 c.u.

Students can complete their studies with more than 240 c.u. depending on the elective courses, within and out of the Department, that they choose to take.

Typical programmes of study for the Degree in Mathematics with emphasis in Pure Mathematics, with emphasis in Applied Mathematics and for the Degree in Mathematics and Statistics are given in tables C1, C2 and C3 respectively.

COURSES FOR OTHER DEPARTMENTS

The courses that are offered exclusively for other departments and their corresponding credit units are given in table B.

TABLEA

Course Titles and Numbers for MAS Students

Numbers and Titles / Credit Units (c.u.) / Pure Mathematics / Applied Mathematics / Statistics
MAS101 – Calculus I / 8 / ▲ / ▲ / ▲
MAS102 – Calculus II / 8 / ▲ / ▲ / ▲
MAS121 - Linear Algebra I / 8 / ▲ / ▲ / ▲
MAS122 - LinearAlgebraII / 8 / ▲ / ▲ / ▲
MAS131 –Basic Mathematics / 8 / ▲ / ▲ / ▲
MAS191- Mathematics with computers / 8 / ▲ / ▲ / ▲
MAS201 - Multivariate Differential calculus / 8 / ▲ / ▲ / ▲
MAS202 - Multivariate Integral calculus / 8 / ▲ / ▲ / ▲
MAS203- Ordinary Differential Equations / 8 / ▲ / ▲ / ▲
MAS223 – Number Theory / 7 / ▲ / ▲ / ▲
MAS251 – Probability I / 8 / ▲ / ▲ / ▲
MAS252 – Statistics I / 8 / ▲ / ▲ / ▲
MAS271 – Numerical Analysis I / 8 / ▲ / ▲ / ▲
MAS301 – RealAnalysis / 8 / ▲ / ▲ / ▲
MAS302 – Complex Analysis I / 8 / ▲ / ▲ / ▲
MAS303 – Partial Differential Equations / 7 / + / ▲ / +
MAS304 – Functional Analysis / 7 / + / ▲ / +
MAS321 – Introduction to Algebra / 7 / ▲
MAS331 – Classical Differential geometry / 8 / ▲ / ▲ / ▲
MAS350 – Stochastic Processes / 7 / ▲
MAS351 – Probability II / 8 / ▲
MAS352 – Statistics II / 7 / ▲
MAS371 – Numerical Analysis II / 7 / + / ▲ / +
MAS401 – Measure Theory and Integration / 7
MAS402 – Complex Analysis II / 7
MAS403 – Stability of Dynamical Systems / 7
MAS418 – Introduction to Fourier Analysis / 7
MAS419 – Topics in Analysis / 7
MAS422 - Introduction to Coding Theory / 7
MAS424 - Theory of Rings and Modules / 7
MAS425 – Theory of Groups / 7
MAS426 – Group Representation Theory / 7
MAS427 - Galois Theory / 7
MAS429 – Topics in Algebra / 7
MAS431 – Introduction to differentiable manifolds / 7
MAS432 – Introduction to Riemannian Geometry / 7
MAS433 – Introduction to Algebraic Topology / 7 / ▲
MAS434 – Algebraic Topology / 7
MAS439 – Topics in Geometry / 7
MAS451 – Linear Models I / 7 / ▲
MAS452 – Linear Models II / 7 / ▲
MAS454 – Nonparametric Statistics / 7 / ■
MAS455 – Sampling Theory / 7 / ■
MAS456 – Time Series / 7 / ■
MAS458 – Statistical Data Analysis / 7 / ■
MAS459 – Multivariate Analysis / 7 / ■
MAS466 - Survival Analysis / 7 / ■
MAS468 – Topics in Probability / 7 / ■
MAS469 – Topics in Statistics / 7 / ■
MAS471- Numerical solution of ordinary differential equations / 7 / ●
MAS472 - Numerical solution of partial differential equations / 7 / ●
MAS473 – Finite Element Method / 7
MAS481 – Applied Mathematical Analysis / 7 / ●
MAS482 – Classical Mechanics / 7 / ●
MAS483 – Fluid Mechanics / 7 / ●
MAS484 – Introduction to Mathematical Modelling / 7
MAS499 – Independent Study / 7

▲ = Compulsory Courses

● = At least 3 out of 5 courses are to be selected

■ = 2 out of 6 courses are to be selected

+ = 2 out of 3 courses are to be selected.

Courses with no symbols are considered free electives within the Department.

TABLE B

Service Courses

Number / Title / Department / Credit Units (c.u.)
ΜAS001 / Mathematics I / ECO, PBA / 6
ΜAS002 / Mathematics II / PBA / 6
ΜAS004 / Introductory Mathematics for Physics I / PHY / 8
ΜAS005 / Introductory Mathematics for Physics II / PHY / 7,5
ΜAS006 / Complex Analysis for Physics Majors / PHY / 7,5
ΜAS007 / History of Mathematics / MAS, EDU, «Ε»* / 5
ΜAS014 / Introductory Mathematics I / CHE, CS / 6
ΜAS015 / Introductory Mathematics II / CHE / 6
ΜAS021 / Calculus I / ECE / 6
ΜAS022 / Calculus II / ECE / 6
ΜAS023 / Linear Algebra and Topics in Multivariate Calculus / ECE / 6
ΜAS024 / Ordinary Differential Equations / ECE / 6
MAS031 / Calculus I / MME, CEE / 5
MAS032 / Linear Algebra / MME, CEE / 5
ΜAS033 / Engineering Mathematics / MME, CEE / 5
MAS034 / Probability and Statistics for Engineers / MME, CEE / 5
ΜAS051 / Statistical Methods / EDU, SPS, PSY, «Ε»* / 5
ΜAS055 / Introduction to Probability and Statistics / CS / 5
ΜAS061 / Statistical Analysis I / ECO, PBA / 6
ΜAS062 / Statistical Analysis II / PBA / 6

* «Ε» = Free Elective Courses

MINOR PROGRAMME OF STUDY

The requirements for the minor in Mathematics is the successful completion of eight courses which must include the courses MAS101, MAS102, MAS121, MAS131, MAS251 or MAS252, MAS271, MAS007 and an additional course of 7 c.u.

OTHER REQUIREMENTS

Students cannot register simultaneously for more than 5 departmental courses.

ADDITIONAL INFORMATION

The Department offers sufficient number of courses to allow for the completion of the requirements for obtaining the degree in Mathematics or in Mathematics and Statistics in eight semesters with regular attendance. Regular attendance is considered to be the successful completion of an average of three courses in the Department of Mathematics and Statistics and one course offered by other departments, per semester. It should be noted that the students of the Department, after the completion of their studies, receive only one of the degrees offered by the Department.

RESEARCH INTERESTS AND PROJECTS

There are several research projects in the Department, and some of these are in cooperation with overseas universities. The projects which are currently in progress involve research in the following areas of Mathematics and Statistics:

• Algebraic and Discrete Geometry.

• Algebraic Topology.

• Applications of Lie Groups to Differential Equations, Integrable and Hamiltonian Systems.

• Applications of Statistics to Biomedical Sciences.

• Asymptotic Efficiency of Model Selection Criteria for Autoregressive Processes.

• Computational Complex Analysis, Numerical Conformal Mapping, Spline Functions and Applications.

• Computational Oceanography.

• Differential Geometry.

• Efficiency in Parametric-Semiparametric Models in Survival Analysis.

• Finite Elements.

• Group Representation Theory.

• Harmonic Analysis, Orthogonal Polynomials, Special Functions.

• Lie Groups/Algebras

• Numerical Simulation of Newtonian and Viscoelastic Flow Problems. Stability Analysis.

• Numerical Solution of Partial Differential Equations.

• Riemannian Geometry.

• Sampling.

• Several Complex Variables.

• Theory and Applications of Bootstrap Methods.

• Theory and Applications of U-Statistics.

• Theory and Practice of Wavelets in Statistics and Time Series Analysis, Functional Linear Models and Functional Data Analysis.

TABLE C1

Indicatory Undergraduate Programme – Pure Mathematics
Semesters / Courses Titles / Credit Units (c.u.) / Total Credit Units / Semesters
1st Semester / MAS101 –Calculus I / 8
MAS131 – Basic Mathematics / 8
MAS121 – Linear Algebra I / 8
Foreign Language I / 5 / 29
2nd Semester / MAS102 – Calculus II / 8
MAS122 – Linear Algebra II / 8
CS031 – Introduction to Programming / 7
MAS191 – Mathematics with Computers / 8 / 31
3rd Semester / MAS201 – Multivariate Differential Calculus / 8
MAS251 – Probability I / 8
MAS271 – Numerical Analysis I / 8
Foreign Language II / 5 / 29
4th Semester / MAS202 – Multivariate Integral Calculus / 8
MAS203 – Ordinary Differential Equations / 8
MAS252 – Statistics I / 8
ΜASΧΧ ( Free electives within the Department) / 7 / 31
5th Semester / MAS301 – Real Analysis / 8
ElectiveΙ* (e.g. MAS303 – Partial Differential Equations) / 7
MAS321 – Introduction to Algebra / 7
PHY111 – General Physics I / 8 / 30
6th Semester / MAS302 – Complex Analysis I / 8
MAS331 – Classical Differential Geometry / 8
ΜASΧΧ ( Free electives within the Department) / 7
ΜASΧΧ ( Free electives within the Department) / 7 / 30
7th Semester / MAS433 – Introduction to Algebraic Topology / 7
ElectiveΙΙ* (e.g. MAS304 – Functional Analysis) / 7
Free electives from other Departments / 5
Free electives from other Departments / 5
Free electives from other Departments / 5 / 29
8th Semester / ΜASΧΧ ( Freeelectives within the Department) / 7
ΜASΧΧ ( Free electives within the Department) / 7
ΜASΧΧ ( Free electives within the Department) / 7
Free electives from other Departments / 5
Free electives from other Departments / 5 / 31
Total Credit Units / 240

Explanations:

ΜASΧΧ = Free electives within the Department

* Selection of at least 2 courses from the list below:

a)MAS304 – Functional Analysis

b)MAS303 – Partial Differential Equations

c)MAS371 – Numerical Analysis II

TABLE C2

Indicatory Undergraduate Programme – Applied Mathematics
Semesters / Courses Titles / Credit Units (c.u.) / Total Credit Units / Semesters
1st Semester / MAS101 – Calculus I / 8
MAS131 – Basic Mathematics / 8
MAS121 – Linear Algebra I / 8
Foreign Language I / 5 / 29
2nd Semester / MAS102 – Calculus II / 8
MAS122 – Linear Algebra II / 8
CS031 – Introduction to Programming / 7
MAS191 – Mathematics with Computers / 8 / 31
3rd Semester / MAS201 – Multivariate Differential Calculus / 8
MAS251 – Probability I / 8
MAS271 – Numerical Analysis I / 8
Foreign Language II / 5 / 29
4th Semester / MAS202 – Multivariate Integral Calculus / 8
MAS203 – Ordinary Differential Equations / 8
MAS252 – Statistics I / 8
ΜASΧΧ ( Free electives within the Department) / 7 / 31
5th Semester / MAS301 – Real Analysis / 8
MAS303 – Partial Differential Equations / 7
MAS371 – Numerical Analysis II / 7
PHY111 – General Physics I / 8 / 30
6th Semester / MAS302 – Complex Analysis I / 8
MAS331 – Classical Differential Geometry / 8
MAS** / 7
ΜASΧΧ ( Free electives within the Department) / 7 / 30
7th Semester / MAS** / 7
MAS304 – Functional Analysis / 7
Free electives from other Departments / 5
Free electives from other Departments / 5
Free electives from other Departments / 5 / 29
8th Semester / MAS** / 7
ΜASΧΧ ( Free electives within the Department) / 7
ΜASΧΧ ( Free electives within the Department) / 7
Free electives from other Departments / 5
Free electives from other Departments / 5 / 31
Total Credit Units / 240

Explanations:

MASXX = Free electives within the Department

MAS** = Selection of at least 3 courses from the list below:

(a)MAS471 – Numerical Solution of Ordinary Differential Equations

(b)MAS472 – Numerical Solution of Partial Differential Equations

(c) MAS481 – Applied Mathematical Analysis

(d)MAS482 – Classical Mechanics

(e)MAS483 – Fluid Mechanics

TABLE C3

Indicatory Undergraduate Programme - Statistics
Semesters / Courses Titles / Credit Units (c.u.) / Total Credit Units / Semesters
1st Semester / MAS101 – Calculus I / 8
MAS131 – Basic Mathematics / 8
MAS121 – Linear Algebra I / 8
Foreign Language I / 5 / 29
2nd Semester / MAS102 – Calculus II / 8
MAS122 – Linear Algebra II / 8
CS031 – Introduction to Programming / 7
MAS191 – Mathematics with Computers / 8 / 31
3rd Semester / MAS201 – Multivariate Differential Calculus / 8
MAS251 – Probability I / 8
MAS271 – Numerical Analysis I / 8
Foreign Language II / 5 / 29
4th Semester / MAS202 – Multivariate Integral Calculus / 8
MAS203 – Ordinary Differential Equations / 8
MAS252 – Statistics I / 8
ΜASΧΧ ( Free electives within the Department) / 7 / 31
5th Semester / MAS301 – Real Analysis / 8
MAS352 – Statistics II / 7
Elective Ι* / 7
MAS351 – Probability II / 8 / 30
6th Semester / MAS302 – Complex Analysis I / 8
MAS331 – Classical Differential Geometry / 8
MAS350 – Stochastic Processes / 7
ΜASΧΧ ( Free electives within the Department) / 7 / 30
7th Semester / MAS451 – Linear Models I / 7
Elective ΙΙ* / 7
Free electives from other Departments / 5
Free electives from other Departments / 5
Free electives from other Departments / 5 / 29
8th Semester / MAS452 – Linear Models II / 7
MAS (Stat) *** / 7
MAS (Stat)*** / 7
Free electives from other Departments / 5
Free electives from other Departments / 5 / 31
Total Credit Units / 240

Explanations:

ΜASΧΧ= Free electives within the Department.

MAS (Stat) ***= Selection from the list below:

a)MAS454 – Nonparametric Statistics

b)MAS455 – Sampling Theory

c)MAS456 – Time Series

d)MAS458 – Statistical Data Analysis

e)MAS459 – Multivariate Analysis

f)MAS466 – Survival Analysis

g)MAS468 – Topics in Probability

h)MAS469 – TopicsinStatistics

* Selection of at least 2 courses from the list below:

a)MAS303 – Partial Differential Equations

b)MAS304 – Functional Analysis

c)MAS371 – Numerical Analysis II
COURSES DESCRIPTION

Α.Description of Courses

ΜAS101 - CalculusΙ

Properties of real numbers. The basic properties of supA, infA. Sequences of real numbers, limits of sequences. Real valued functions, the inverse of a function, limits of functions, continuous functions, uniform continuity, the Intermediate Value Theorem, the Extreme Value Theory. Derivatives, graphs of functions, the Mean value theorem, L’Hopital’s rule.

ΜAS102 - Calculus II

Riemann integral, integrability of continuous, monotone functions.The fundamental Theorem of calculus. Areas of regions in the plane, volumes of solids of revolution. The indefinite integral, integration by parts, integration by change of variables, integration of rational functions. Taylor’s formula. Infinite series, tests of convergence, absolutely convergent series, conditionally convergent series, Leibniz’s Theorem, product of series.

MAS121 - Linear Algebra I

Numbers, equivalence relations. Groups, Examples (symmetric, cyclic, dihedral). Isomorphism. Rings and Fields. Examples. Vector spaces, basis, dimension. Linear maps. Matrices and linear maps. Rank, change of basis matrix. Determinant. Linear systems.

MAS122 - Linear Algebra II

Polynomial Ring. Eigenvalues, eigenvectors. Diagonalization and applications. Theorem of Cayley – Hamilton, minimal polynomial. Generalized eigenspaces, nilpotent endomorphisms Jordan canonical form. Inner product spaces (Gram – Schmidt). Orthogonal, self dual endomorphisms. Bilinear, quadratic forms.

MAS131 – Basic Mathematics

Methods and applications of differentiation. Methods of integration and applications. Improper Integrals. Power series. Fourier series. Elements of analytic geometry on the place and in space. Functions and surfaces. Polar coordinates. Partial derivatives and Lagrange multipliers. Multiple integration and Jacobien.

MAS191 - Mathematics with computers

Preliminaries: Basic Matlab commands. Matlab as a programming language. Real and complex numbers, vectors, matrices. Representation of numbers, vectors& matrices. Simple Matlab programs. Matrices: General notions. Matrix operations with Matlab. Computation of determinants and inverses. Eigenvalues and Eigenvectors: General notions of Eigenvalues and Eigenvectors. Computation of them with Matlab. Special emphasis on the complex case. Diagonal table matrices. Plots with Matlab: Simple plot, two- and three- dimensional plots. Special plots: Phase planes, contour plots, flows. Linear on OPES. Special topics on differential equations. Multivariate calculus. Fast Fourier Transform.

MAS201 - Multivariate Differential calculus

Spaces with norm (examples, n- dimensional Euclidean space, equivalent norms, Cauchy – Schwarz inequality).

Open, closed sets, limits, continuity. Compactness (Theorem of Heine – Borel, Bolzano – Weierstrass). Vector valued functions of one real variable. Partial derivates. Total differential. Mean value theorem, Taylor’s Theorem. Implicit and inverse function theorems. Lagrange multipliers.

MAS202 - Multivariate Integral Calculus

Integration of continuous functions with compact support. Transformation theorem. Integrable functions and sets, properties. Volumes. Theorem of Fubini. Convergence theorems. Transformation Theorem, applications. Parameterized surfaces, partition of unity. Surface and curve – integrals. Differential forms. Theorem of Stokes, applications.

MAS203 - Ordinary Differential Equations

Basic notions. Solutions techniques for first – order equations& physical applications. Theorems of Existence and Uniqueness. Linear systems& exponential of matrices. Higher order linear equations. Method of power series: Smooth and singular solutions. Smooth dependence of solutions on parameters.

MAS 223 - Number Theory

Divisibility theory in the integers. The Euclidean algorithm. Primes and their distribution. The fundamental theorem of arithmetic The theory of consequences. Fermat´ s little theorem. The quadratic reciprocity law. Perfect numbers. Representation of integers as sums of squares. Fibonacci numbers, Continued fractions.

Pell´ s equation.

MAS 251 – Probability I

Probability, random variables, distribution functions, independence, expected value, moment generating functions, modes of convergence of sequences of random variables, laws of large numbers, introduction to central limit theorems.

MAS252 - Statistics Ι

Statistics. Sufficiency and completeness. Exponential families of distributions. Unbiasedness, unbiased estimators. Cramer – Rao inequality. Method of moments, maximum likelihood estimators. Asymptotic properties of estimators. Bayes estimation. Introduction to confidence intervals and to hypothesis testing problems.

.

MAS271 - Numerical Analysis I

Propagation and estimation of errors: Floating-point arithmetic - Rounding error analysis - Loss of significance - Stability and condition of problems and algorithms - The symbolism - Richardson extrapolation.The solution of nonlinear equations:Fixed-point iteration - Order of convergence and asymptotic error constant - The Newton and the secant methods - Multiple roots - Always convergent methods (the bisection and the regula falsi methods) - Aitken´s acceleration process. Solution of linear systems: Direct methods (Gauss elimination and LU-decomposition) - The need for partial pivoting and for scaling - Cholesky´s method for symmetric and positive definite systems - The computation of the determinant and the inverse of an nxn matrix - the least squares method for over-determined systems. Interpolation and quadrature: Lagrange interpolation (Existence and uniqueness - Cardinal and Newton representations of the interpolating polynomial - The error of the interpolating polynomial) - Hermite interpolation (Existence and uniqueness - Cardinal representation of the interpolating polynomial - The error of the interpolating polynomial) - Newton-Cotes quadrature rules - The precision of a quadrature rule - Detailed description and analysis of the trapezoidal and the Simpson rules - Composite rules.