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

 
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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 :
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(Downloaded 818 times last on 08 January 2025, 12:04 pm
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 $C$.

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B.Sc
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(Downloaded 1038 times last on 08 January 2025, 12:04 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 :
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(Downloaded 1252 times last on 10 January 2025, 12:41 pm
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 :
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(Downloaded 628 times last on 08 January 2025, 08:41 pm
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 $R^3$.

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B.Sc
MTTS Level :
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(Downloaded 1360 times last on 11 January 2025, 11:06 am
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 :
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(Downloaded 721 times last on 10 January 2025, 04:52 am
87
Jordan Canonical Form

Simpler treatment of JCF

Keywords :

M.Sc.
MTTS Level :
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(Downloaded 980 times last on 08 January 2025, 12:05 pm
88
Jordan Canonical Form

Gives four different proofs of JCF

Keywords :

M.Sc.
MTTS Level :
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(Downloaded 1459 times last on 10 January 2025, 07:45 pm
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.

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M.Sc./Ph.D
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(Downloaded 618 times last on 08 January 2025, 12:05 pm
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)!

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B.Sc onwards
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(Downloaded 800 times last on 08 January 2025, 12:05 pm
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 :
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(Downloaded 526 times last on 08 January 2025, 12:06 pm
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.

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Final year B.Sc, M.Sc
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(Downloaded 767 times last on 10 January 2025, 05:32 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.

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B.Sc./M.Sc
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(Downloaded 656 times last on 08 January 2025, 12:06 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 :
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(Downloaded 773 times last on 10 January 2025, 03:34 pm
95
Computation of Real Integrals using the Residue Theorem

Computation of Real Integrals using the residue calculus

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also applications to Laplace and Fourier transforms
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(Downloaded 589 times last on 08 January 2025, 12:06 pm
96
Lebesgue Covering Lemma

Gives 3 proofs

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also establishes equivalence of Lebesgue number property and unifrom continuity of any continuous function.
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(Downloaded 1739 times last on 10 January 2025, 09:29 pm
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 :
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(Downloaded 499 times last on 08 January 2025, 12:06 pm
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 :
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(Downloaded 879 times last on 09 January 2025, 06:47 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 :
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(Downloaded 3283 times last on 11 January 2025, 04:31 pm
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 :
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(Downloaded 696 times last on 08 January 2025, 01:52 pm