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Found total 224 results. Displaying results from 161 to 180.

 
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161
Schröeder Bernstein Theorem

Gives a simple geometric proof of Schroder-Bernstein theorem.

Keywords :

B.Sc
MTTS Level :
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(Downloaded 1040 times last on 04 April 2025, 07:33 am
162
Problems - Set Theory and Foundations

Introduces quantifiers, negation of statements, basic set theory, functions, direct and inverse images, relations, equivalence relations and partial orders. Many concrete examples which are geometric and meaningful in modern mathematics are given. It attends to a severe lacuna found among mathematics students and enablesthem to work with the language of mathematics.

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B.Sc
MTTS Level :
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(Downloaded 2676 times last on 03 April 2025, 09:16 pm
163
Several Variable Differential Calculus

Differential Calculus of several variables leading up to Taylor's theorem, maxima-minima and bump functions

Keywords :

Final year B.Sc, M.Sc
MTTS Level :
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(Downloaded 1729 times last on 03 April 2025, 09:52 am
164
Signed Measures Theorems of Hahn, Jordan and Radon-Nikodym

Theorems of Hahn, Jordan, Radon-Nikodym.

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M.Sc./Ph.D
MTTS Level :
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(Downloaded 834 times last on 03 April 2025, 09:52 am
165
Singular Values and Spectral Theorem

Topological/analytical proofs of singular value decomposition, polar decomposition and spectral theorem for symmetric matrices.

Keywords :

M.Sc.
MTTS Level :
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(Downloaded 492 times last on 03 April 2025, 09:53 am
166
Matrices over a Euclidean DomainGenerators and Relations of a F.G. Abelain Group

Smith normal form of matrices over a Euclidean domain and its application to finitely generated abelian groups

Keywords :

it discusses also the set of generators and relations that could be read off from the SNF.
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(Downloaded 546 times last on 03 April 2025, 09:53 am
167
Frenet-Serre Formulas for Space Curves

Starts with very elementary treatment of Frenet formulas of space curves and then interprets them in terms of Maurer-Cartan forms of orthogonal groups, uses it to deduce the existence, uniqueness and rigidity results for space curves. Needs a bit of maturity as it quickly transcends the conventional treatment. It is good training ground for aspirants of modern differential geometry.

Keywords :

M.Sc./Ph.D
MTTS Level :
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(Downloaded 533 times last on 04 April 2025, 05:24 am
168
Finite Dimensional Spectral Theory

Gives a (topological/calculus/analytic) proof of the spectral theorem for self-adjoint operators, and then extends it ti normal operators (in the finite dimensional case). It addresses some of the questions not usually raised in linear algebra textbooks.

Keywords :

Final year B.Sc, M.Sc
MTTS Level :
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(Downloaded 1170 times last on 03 April 2025, 10:26 pm
169
pi_ 1 (S n ) is trivial for n greater than or equal to 2

Proves that the fundamental group of $S^n$ is trivial for $ngeq 2$. A rigorous geometric proof. Of course, there are simpler proofs which use a little sophisticated machinery.

Keywords :

M.Sc.
MTTS Level :
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(Downloaded 473 times last on 03 April 2025, 09:53 am
170
Stokes Theorem–An Outline of a Proof

Gives the 3-dimensional version of the Stokes theorem and deduces it from Green's theorem. Brings out the essential simplicity of the proof if one understands the underlyng concepts well.

Keywords :

Final year B.Sc, M.Sc
MTTS Level :
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(Downloaded 595 times last on 03 April 2025, 09:54 am
171
Stokes Theorem — A simple and intuitive proof

The proof uses two ingredients: (i) the integral definition of the exterior derivative of a differential form and (ii) a generalized Riemann integral (called the Heinstock-Kurzweil integral). The integral definition of exterior derivative is quite appealing for physical reasons. Suitable for advanced students who are serious about hard analysis.

Keywords :

M.Sc./Ph.D
MTTS Level :
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(Downloaded 657 times last on 03 April 2025, 09:53 am
172

Gives a proof as graded exercises. As an application, Tietze extension theorem for $R^n$ is proved.

Keywords :

M.Sc.
MTTS Level :
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(Downloaded 717 times last on 03 April 2025, 09:52 am
173
Structure of Linear Maps

Gives a motivated and clutter free proofs of many theorems about the structure of linear maps. Proves Jordan canonical form, spectral theorems for self-adjoint,normal, unitary and orthogonal maps, singular value decompositions, triangular forms.

Keywords :

Final year B.Sc, M.Sc
MTTS Level :
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(Downloaded 2290 times last on 03 April 2025, 09:52 am
174
Sturm’s Separation and Comparison Theorems

Sturm's separation and oscillation theorems.

Keywords :

M.Sc.
MTTS Level :
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(Downloaded 4936 times last on 04 April 2025, 06:08 am
175
Geometric Applications of Sturm Comparison Theorem

After recalling the Sturm's comparison theorem, we illustrate some of its typical applications in the theory of surfaces. Even though some acquaintance with Gauss theory of surfaces will be advantageous, it is not absolutely necessary, as I have striven to illustrate the uses in the case of a sphere in 3-space.

Keywords :

M.Sc./Ph.D
MTTS Level :
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(Downloaded 527 times last on 03 April 2025, 09:52 am
176
Subspace Topology

Starts with the definition of subspace topology and provides a lot of illuminating examples in the form of exercises to gain confidence in the concept.

Keywords :

M.Sc.
MTTS Level :
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(Downloaded 1009 times last on 03 April 2025, 07:00 pm
177
Theory of Surfaces—A Rapid Introduction

A very quick and geometric introduction to the theory of surfaces. Has an outline of a proof of Gauss theorema egregium and basic notions of geodesics.

Keywords :

M.Sc.
MTTS Level :
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(Downloaded 576 times last on 03 April 2025, 09:52 am
178
Sylow Theorems

Gives proofs( (with motivation) of all Sylow theorems employing group actions.

Keywords :

Final year B.Sc, M.Sc
MTTS Level :
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(Downloaded 1388 times last on 03 April 2025, 06:31 pm
179
Sylvester Criterion for Positive Definiteness

Gives a more traditional proof (which is computational in nature) and a conceptual proof of the Sylvester criterion of positive definiteness of symmetric matrices.

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Final year B.Sc, M.Sc
MTTS Level :
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(Downloaded 1268 times last on 04 April 2025, 01:21 am
180

Gives the details of the description of tangent spaces $T_pS$ where $S$ is a graph or $S$ is a level surface. To be read in conjunction with the article tangents.

Keywords :

M.Sc.
MTTS Level :
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(Downloaded 522 times last on 03 April 2025, 02:22 pm