Expository articles


 

Search for an article

Part of title, description, or keywords. Minimum 4 characters required.

 Reset search and show all articles

 
Found total 224 results. Displaying results from 81 to 100.

 
SL
Article Name
Broad subject area
Abstract & Keywords
Targetted Audience
Number of downloads
81
Principles of Induction and Well-ordering

Equivalence of the principles of induction and well-ordering is proved. Includes problems on induction for practice.

Keywords :

B.Sc
MTTS Level :
Download now
(Downloaded 1011 times last on 14 March 2026, 11:52 am
82
Integers and Polynomials

The aim of this article is to bring to the attention of undergraduate students of mathematics the striking similarity between the set of integers and the set of polynomials, say, over $Q$, $R$ or $C$.

Keywords :

B.Sc
MTTS Level :
Download now
(Downloaded 1632 times last on 15 March 2026, 08:19 pm
83
Lebesgue Integral without Measure Theory

Construction of Lebesgue integral without measure theory.This article gives a very brief account of the theory of Lebesgue integral which avoids a lengthy treatment of measure theory. The omitted proofs are either standard or easy. This article Follows closely Van Daele and Riesz-Nagy' treatise on Functional Analysis.

Keywords :

M.Sc./Ph.D
MTTS Level :
Download now
(Downloaded 1588 times last on 15 March 2026, 11:10 am
84
Similarity of Matrices via Module Theory

Structure theorem for finitely generated $F[x]$-modules and canonical forms. The aim of this article is to give an introduction to the structure theorem of finitely generated modules over a PID confining ourselves only to $F[X]$-modules. The treatment acquaints the reader with all basic results such as invariant factor theorem, elementary divisor theorem, rational and Jordan canonical forms.

Keywords :

M.Sc.
MTTS Level :
Download now
(Downloaded 944 times last on 16 March 2026, 05:24 am
85
Isometries of R n

I first review some basic facts on isometries of $R^n$ and $R^2$ and then end up with the study of isometries of $R^3$.

Keywords :

B.Sc
MTTS Level :
Download now
(Downloaded 2103 times last on 15 March 2026, 09:05 pm
86
Isometries of R n and Sylvester Criterion for Positive Definite Matrices

This article deals with two topics: one is the isometries of $R^3$ and the other is the criterion for a symmetric matrix to be positive definite. I first review some basic facts on isometries of $R^n$ and $R^2$ and then end up with the study of isometries of $R^3$.

Keywords :

B.Sc
MTTS Level :
Download now
(Downloaded 986 times last on 16 March 2026, 12:13 am
87
Jordan Canonical Form

Simpler treatment of JCF

Keywords :

M.Sc.
MTTS Level :
Download now
(Downloaded 1354 times last on 15 March 2026, 03:46 am
88
Jordan Canonical Form

Gives four different proofs of JCF

Keywords :

M.Sc.
MTTS Level :
Download now
(Downloaded 2388 times last on 16 March 2026, 08:05 am
89
Jordan Curve Theorem

The aim of this article is to prove that if $Gamma$ is a subset of $R^2$ that is homeomorphic to $S^1$, then $R^2setminus Gamma$ has two connected components. The proof uses some basic results in homotopy theory and covering spaces. It has a strong analytic flavour.

Keywords :

M.Sc./Ph.D
MTTS Level :
Download now
(Downloaded 817 times last on 15 March 2026, 08:18 am
90
A Dignostic Test in Logical Thinking/Writing

Gives a sample test which tests the logical reasoning of students in a familiar context. This is an eye-opener for many students (who score high)!

Keywords :

B.Sc onwards
MTTS Level :
Download now
(Downloaded 986 times last on 16 March 2026, 12:38 am
91
Banach Mapping Theorem

We prove Knaster fixed point theorem and Banach mapping theorem. As a corollary, we obtain simple proof of Schroeder-Bernstein theorem;

Keywords :

B.Sc
MTTS Level :
Download now
(Downloaded 693 times last on 15 March 2026, 07:39 am
92
Convex Functions on R and Inequalities

A near complete treatment of convex functions with applications to inequalities theorems. A msu read for any budding analyst.

Keywords :

Final year B.Sc, M.Sc
MTTS Level :
Download now
(Downloaded 2111 times last on 15 March 2026, 10:15 pm
93
A Lemma from Linear Algebra – Lagrange Mutipliers

Statement and proof of a lemma from Linear Algebra required to deal with vector constraints in the theory of constrained exterma.

Keywords :

B.Sc./M.Sc
MTTS Level :
Download now
(Downloaded 838 times last on 14 March 2026, 09:23 pm
94
The Method of Lagrange Multipliers — A Geometric Treatment

Geometric meaning and Proof of Lagrange Multiplier Theorem. Applications to diagonalization of symmetric matrices and Hadamard's inequality.

Keywords :

M.Sc.
MTTS Level :
Download now
(Downloaded 1019 times last on 15 March 2026, 08:24 am
95
Computation of Real Integrals using the Residue Theorem

Computation of Real Integrals using the residue calculus

Keywords :

also applications to Laplace and Fourier transforms
MTTS Level :
Download now
(Downloaded 900 times last on 15 March 2026, 11:00 am
96
Lebesgue Covering Lemma

Gives 3 proofs

Keywords :

also establishes equivalence of Lebesgue number property and unifrom continuity of any continuous function.
MTTS Level :
Download now
(Downloaded 3450 times last on 16 March 2026, 09:01 am
97
A Lebesgue Measurable Set which is not Borel

Proves the existence of a Lebesgue measurable set which is not Borel ? uses the cantor function and the existence of a non-measurable set.

Keywords :

M.Sc.
MTTS Level :
Download now
(Downloaded 722 times last on 15 March 2026, 09:18 am
98
How to draw the graphs of functions on R 2 ?

How to draw the graphs of functions from $R^2$ to $R$ using the level sets?

Keywords :

B.Sc
MTTS Level :
Download now
(Downloaded 1122 times last on 15 March 2026, 10:35 am
99
Limit Inferior and Limit Superior

A thorough treatment of limsup and liminf. This topic is a source of concern for many students. It culminates in a proof of Cauchy completeness of R.

Keywords :

B.Sc, M.Sc
MTTS Level :
Download now
(Downloaded 4398 times last on 16 March 2026, 09:08 am
100
Linear ODE: Systems of First Order and n-th Order

Givs all basic results abou a inear systems of ODE. Included some unusual material. See Remark 19.

Keywords :

M.Sc.
MTTS Level :
Download now
(Downloaded 1047 times last on 16 March 2026, 06:48 am