The aim of the course is to provide an
introduction to fundamental skills and ideas for students who require some
mathematics in their degree. The major aim of the course is not to introduce
the students with the procedural mathematics but to expose them to the skills
that the professional mathematicians use to solve the real world problems. In
simple words the this is an attempt to develop the valuable skills of thinking
outside the box. We will explore a variety of interesting ideas from diverse areas
of applied mathematics

- Teacher: Dr. Javed Hussain

- Teacher: Idress Azad

- Teacher: Dr. Hyder Ali Muttaqi Shah Syed

The course covers the basic principles of the *theory of probability
* and its *applications*.
Topics include combinatorial analysis used in computing
probabilities, the axioms of probability, conditional probability and
independence of events; discrete and continuous random variables; joint,
marginal, and conditional densities, moment generating function; laws of
large numbers; Central limit Theorem, binomial, Poisson, gamma, univariate, and bivariate normal
distributions, probability spaces, sigma-fields, probability measure and measure theoretic introduction to probability, different modes of convergence

- Teacher: Dr. Javed Hussain

This is an introductory course on Algebra. The students will learn about Groups, Rings and Modules in detail. Moreover a short amount of time will be spent on covering Representation theory of finite groups.

- Teacher: Fazeel Anwar

In this course the students will learn about system of linear equations and their applications in real world problems. The whole course will primarily focus on finding solutions of systems of linear equations. Along the way students will also learn key concepts such as echelon form and reduced echelon form, matrices and their properties, inverse of matrices, factorization of matrices, linear transformations, vector spaces and their subspaces, eigenvalues and eigenvectors for real matrices and inner product spaces.

- Teacher: Fazeel Anwar

This course provides an introduction to basic computer programming concepts and techniques useful for Computer Scientists, Mathematicians and Engineers. The course exposes students to practical applications and complexities in computing and commonly used tools within these domains. It introduces techniques for problem solving, program design and algorithm development using MATLAB and/or PYTHON programming language. The content of this course gives an introduction to the MATLAB and/or PYTHON environment and the help system, data types and scalar variables, arithmetic and mathematical functions, input and output, selection and iteration statements. Functions: user defined functions, function files, passing information to and from functions, function design and program decomposition, recursion. Arrays: vectors, arrays and matrices, array addressing, vector, matrix and element-by-element operations. Graphics: 2-D and 3-D plotting. Mathematical modelling: dynamical systems, linear systems of equations, numerical differentiation and integration.

- Teacher: Dr. Hyder Ali Muttaqi Shah Syed

The modern notion of measure, developed in the late 19th century, is an extension of the notions of length, area or volume. A *measure* *m* is a law which assigns a number to certain subsets *A*
of a given space and is a natural generalization of the following
notions: 1) length of an interval, 2) area of a plane figure, 3) volume
of a solid, 4) amount of mass contained in a region, 5) probability that
an event from *A* occurs, etc.

It originated in the real analysis and is used now in many areas of mathematics like, for instance, geometry, probability theory, dynamical systems, functional analysis, etc.

Given a measure *m*, one can define the *integral* of suitable real valued functions with respect to *m*.
Riemann integral is applied to continuous functions or functions with
``few`` points of discontinuity. For measurable functions that can be
discontinuous ``almost everywhere'' Riemann integral does not make
sense. However it is possible to define more flexible and powerful *Lebesgue's integral* (integral with respect to *Lebesgue's measure*) which is one of the key notions of modern analysis.

The course will cover the following topics: Definition of a measurable space and -additive
measures, Construction of a measure form outer measure, Construction of
Lebesgue's measure, Lebesgue-Stieltjes measures, Examples of
non-measurable sets, Measurable Functions, Integral with respect to a
measure, Lusin's Theorem, Egoroff's Theorem, Fatou's Lemma, Monotone
Convergence Theorem, Dominated Convergence Theorem, Product Measures and
Fubini's Theorem. Selection of advanced topics such as Radon-Nikodym
theorem, covering theorems, differentiability of monotone functions
almost everywhere, descriptive definition of the Lebesgue integral,
description of Riemann integrable functions, *k*-dimensional measures in *n*-dimensional spaces, divergence theorem, Riesz representation theorem, etc.

- Teacher: Dr. Javed Hussain

Much of mathematics relies on our ability to be able to solve equations,
if not in explicit exact forms, then at least in being able to
establish the existence of solutions. To do this requires a knowledge of
so-called "analysis", which in many respects is just Calculus in very
general settings. The foundations for this work are commenced in Real
Analysis, a course that develops this basic material in a systematic and
rigorous manner in the context of real-valued functions of a real
variable. Topics covered are: Basic set theory. The real numbers and
their basic properties. Sequences: convergence, sub-sequences, Cauchy
sequences. Open, closed, and compact sets of real numbers. Continuous
functions and uniform continuity. The Riemann integral. Differentiation
and Mean Value theorems. The Fundamental Theorem of Calculus. Series.
Power series and Taylor series. Convergence of sequences and series of
functions.

- Teacher: Dr. Javed Hussain

- Teacher: Dr. Javed Hussain