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 1 to 20.

 
SL
Article Name
Broad subject area
Abstract & Keywords
Targetted Audience
Number of downloads
1
Four Applications of Inverse Function Theorem

Gives 4 typical applications of the inverse function theorem. A must read for any budding differential topologist or differential geometer!

Keywords :

Senior undergraduates, M.Sc
MTTS Level :
Download now
(Downloaded 2982 times last on 22 December 2024, 07:50 am
2
Summary of Differential Geometry (January-April 2008)

A summary of the course on differential geometry of curves and surface as it unfolded. More was covered and we used other articles on these themes.

Keywords :

Senior undergraduates, M.Sc
MTTS Level :
Download now
(Downloaded 2360 times last on 23 December 2024, 10:40 am
3
Insolvability of Quintic

It attempts to give a direct proof of Abel's theorem on the insolubility of the quintic by avoiding a detailed discussion on the standard Galois theory. Needs to provide more details and polish up the writing. But could be of service to students to get into the spirit of Galois theory faster.

Keywords :

M.Sc.
MTTS Level :
Download now
(Downloaded 1304 times last on 20 December 2024, 08:34 pm
4
Problems in Algebraic Topology

Assignments in a first course in algebraic topology. A good set of exercises to face your pre-Ph.D.exams.

Keywords :

2nd year M.Sc.
MTTS Level :
Download now
(Downloaded 2264 times last on 23 December 2024, 11:00 am
5
Arithmetic-Geometric Mean Inequality Proof by Induction and Calculus

A proof of the classic inequality on AM-GM using calculus and induction.

Keywords :

2nd /final year BSc
MTTS Level :
Download now
(Downloaded 24951 times last on 20 December 2024, 09:33 pm
6
Approximate Identities

Starting with a definition of an approximate identity, it gives examples of Fejer's kernel, Landau's kernel, Poisson kernel and Weierstrass kernel and their uses. A must read to master one of the basic tricks in harmonic analysis.

Keywords :

M.Sc.
MTTS Level :
Download now
(Downloaded 1792 times last on 23 December 2024, 10:54 am
7
Arzela-Ascoli Theorem and Applications

Gives a proof of Arzela-Ascoli using the characterization of compact metric spaces via completeness and total boundedness. Different from the standard proof which uses separability and digonal trick.

Keywords :

M.Sc.
MTTS Level :
Download now
(Downloaded 4021 times last on 23 December 2024, 10:42 am
8
Applications of Baire Category Theorem

Gives typical applications of Baire category theorem. A must read for all budding analysts.

Keywords :

Senior undergraduates, M.Sc.
MTTS Level :
Download now
(Downloaded 4703 times last on 21 December 2024, 08:07 pm
9
Functions of Bounded Variation

Motivated account of the study of functions of bounded variation. Complete details of all the results of this topic which one needs in his career is given.

Keywords :

M.Sc.
MTTS Level :
Download now
(Downloaded 2055 times last on 23 December 2024, 11:04 am
10
Brouwer Fixed Pint Theorem a la Milnor

Starts with Milnor's proof of Hairy-ball theorem, adapts his argument to prove non-retraction theorem, proves Brouwer fixed point theorem. Ends up with a proof of non-retraction using Stokes theorem.

Keywords :

M.Sc., Ph.D.
MTTS Level : 2
Download now
(Downloaded 1142 times last on 19 December 2024, 09:29 pm
11
Bishop-Stone-Weierstrass Theorem

Proves Machado's version of Bishop-Stone-Weierstrass theorem.

Keywords :


MTTS Level :
Download now
(Downloaded 850 times last on 19 December 2024, 09:29 pm
12
Burnside Theorem on Matrix Algebras and Its Application to Group Theory

Gives a simple proof of Burnside's theorem on subrings of matrix rings. A good training for students who want to specialize in groups theory, representation theory of any sort.

Keywords :

M.Sc.
MTTS Level :
Download now
(Downloaded 1446 times last on 19 December 2024, 09:29 pm
13
A Review of Differential Calculus

The article reviews several variable calculus with emphasis on concepts, theorems and their geometric contents. It gives a lot of examples which will wean the reader away from Jacobian computations!

Keywords :

M.Sc.
MTTS Level :
Download now
(Downloaded 2653 times last on 21 December 2024, 10:43 pm
14
Every compact metric space is a continuous image of the cantor set

Proves that any compact metric space is a continuous image of the Cantor's middle thirds set. A good illustration of the principle of reducing to simpler problems and building from them.

Keywords :

M.Sc.
MTTS Level :
Download now
(Downloaded 5303 times last on 22 December 2024, 09:05 pm
15
Cantor Set

Gives a detailed analytic as well as geometric construction of Cantor set. More detailed than most of the cursory treatments found in books.

Keywords :

Senior undergraduates, M.Sc.
MTTS Level :
Download now
(Downloaded 2006 times last on 21 December 2024, 12:37 am
16
Cardinality and Countability

A quick review of cardinality, countability and uncountability via problems. Readers are asked to refer to Munkres for details. Hints are given to some where our proofs are different from those in Munkres. Some explicit bijections are given so as to train a student in the art of setting up such functions.

Keywords :

1st/2nd year B.Sc
MTTS Level :
Download now
(Downloaded 3392 times last on 23 December 2024, 10:54 am
17
Completeness of R

Gives perhaps the simplest proof of Cauchy completeness of $R$. Does not use anything other than the definitions of convergence and Cauchy sequences and LUB property of $R$. This needs wider publicity. Included in out book on Real analsyis.

Keywords :

2nd year B.Sc
MTTS Level :
Download now
(Downloaded 2299 times last on 23 December 2024, 10:38 am
18
Cayley Hamilton Theorem A Topological Proof

Gives a topological proof of Cayley-Hamilton theorem for matrices over $C$. A good training ground for budding mathematicians who aspire for higher levels of understanding.

Keywords :

M.Sc.
MTTS Level :
Download now
(Downloaded 1419 times last on 23 December 2024, 10:44 am
19
Change of Variable Formula for Multiple Integrals

Adapts Lax' proof of a version of change of variable formula to obtain the standard version of the result. Needs wider publicity as there had been attempts by others to modify lax proof to obtain the classical statement but none of them is satisfactory in our opinion.

Keywords :

M.Sc.
MTTS Level :
Download now
(Downloaded 1294 times last on 23 December 2024, 10:56 am
20
Chinese Remainder Theorem

Proves various versions of CRT with typical applications. Suitable for Masters students.

Keywords :

M.Sc.
MTTS Level :
Download now
(Downloaded 1777 times last on 22 December 2024, 10:56 pm