Expository articles
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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.
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B.Sc
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(Downloaded 1035 times last on 30 March 2025, 04:34 am
(Downloaded 1035 times last on 30 March 2025, 04:34 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
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(Downloaded 2672 times last on 31 March 2025, 08:34 am
(Downloaded 2672 times last on 31 March 2025, 08:34 am
163
Several Variable Differential Calculus
Differential Calculus of several variables leading up to Taylor's theorem, maxima-minima and bump functions
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Final year B.Sc, M.Sc
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(Downloaded 1725 times last on 29 March 2025, 07:25 am
(Downloaded 1725 times last on 29 March 2025, 07:25 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
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(Downloaded 832 times last on 01 April 2025, 02:25 am
(Downloaded 832 times last on 01 April 2025, 02:25 am
165
Singular Values and Spectral Theorem
Topological/analytical proofs of singular value decomposition, polar decomposition and spectral theorem for symmetric matrices.
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M.Sc.
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(Downloaded 490 times last on 26 March 2025, 07:46 pm
(Downloaded 490 times last on 26 March 2025, 07:46 pm
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
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it discusses also the set of generators and relations that could be read off from the SNF.
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(Downloaded 543 times last on 29 March 2025, 05:57 am
(Downloaded 543 times last on 29 March 2025, 05:57 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.
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M.Sc./Ph.D
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(Downloaded 529 times last on 30 March 2025, 08:37 pm
(Downloaded 529 times last on 30 March 2025, 08:37 pm
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.
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Final year B.Sc, M.Sc
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(Downloaded 1165 times last on 30 March 2025, 08:46 pm
(Downloaded 1165 times last on 30 March 2025, 08:46 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.
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M.Sc.
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(Downloaded 471 times last on 26 March 2025, 07:47 pm
(Downloaded 471 times last on 26 March 2025, 07:47 pm
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.
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Final year B.Sc, M.Sc
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(Downloaded 592 times last on 28 March 2025, 12:57 pm
(Downloaded 592 times last on 28 March 2025, 12:57 pm
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.
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M.Sc./Ph.D
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(Downloaded 655 times last on 26 March 2025, 07:47 pm
(Downloaded 655 times last on 26 March 2025, 07:47 pm
172
Gives a proof as graded exercises. As an application, Tietze extension theorem for $R^n$ is proved.
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M.Sc.
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(Downloaded 715 times last on 31 March 2025, 05:04 pm
(Downloaded 715 times last on 31 March 2025, 05:04 pm
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.
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Final year B.Sc, M.Sc
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(Downloaded 2287 times last on 31 March 2025, 08:01 pm
(Downloaded 2287 times last on 31 March 2025, 08:01 pm
174
Sturm’s Separation and Comparison Theorems
Sturm's separation and oscillation theorems.
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M.Sc.
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(Downloaded 4909 times last on 01 April 2025, 08:52 pm
(Downloaded 4909 times last on 01 April 2025, 08:52 pm
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.
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M.Sc./Ph.D
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(Downloaded 525 times last on 01 April 2025, 04:19 pm
(Downloaded 525 times last on 01 April 2025, 04:19 pm
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.
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(Downloaded 1006 times last on 26 March 2025, 07:48 pm
(Downloaded 1006 times last on 26 March 2025, 07:48 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.
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(Downloaded 574 times last on 26 March 2025, 07:48 pm
(Downloaded 574 times last on 26 March 2025, 07:48 pm
178
Sylow Theorems
Gives proofs( (with motivation) of all Sylow theorems employing group actions.
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Final year B.Sc, M.Sc
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(Downloaded 1385 times last on 01 April 2025, 11:36 pm
(Downloaded 1385 times last on 01 April 2025, 11:36 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
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(Downloaded 1264 times last on 31 March 2025, 12:07 pm
(Downloaded 1264 times last on 31 March 2025, 12:07 pm
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.
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M.Sc.
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(Downloaded 519 times last on 29 March 2025, 09:33 pm
(Downloaded 519 times last on 29 March 2025, 09:33 pm