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Found total 224 results. Displaying results from 201 to 220.

 
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Article Name
Broad subject area
Abstract & Keywords
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Number of downloads
201
Winding Numbers of Loops

Assuming the basics of covering spaces, the article defines the winding number of a loop and uses it to prove the classical results of the topology of the plane: no retraction, Brouwer fixed point, Invariance of domain, Borsuk-Ulam and ham-sandwich theorems. Could serve as an aperitif for a course in algebraic topology.

Keywords :

M.Sc./Ph.D
MTTS Level :
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(Downloaded 795 times last on 19 December 2024, 09:32 pm
202
Axiom of Choice and its Equivalents

Equivalence of the axiom of choice, Zorn's lemma and well-ordering principle are proved. Begining graduate students are fascinated by the equivalence.

Keywords :

M.Sc.
MTTS Level :
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(Downloaded 1069 times last on 21 December 2024, 11:22 am
203
A Problem Course in Metric Spaces
Toplogy, Analysis, Functional Analysis

This is an outline of a course on Metric spaces in the form of exercises. Whenever needed the hints are given. My book "Topology of Metric Spaces" has genesis in this article. In MTTS Level 1, the teacher is asked to follow outline as the book may have too many examples and problems.

Keywords : Metric space; Topology; Arzela-Ascoli; Banach Contraction; Norms

Final year BSc and first year MSc
MTTS Level :
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(Downloaded 2128 times last on 22 December 2024, 08:33 pm
204
Existence of continuous argument on C minus a closed half ray
Complex analysis and topology

We find an explicit expression for a continuous argument on the complex plane minus the closed negative real axis. One knows such a continuous argument exists (see my metric space book, also the article "Maps into the punctured plane") but no explicit expression is found.

Keywords : Argument of a complex number

MSc
MTTS Level : 2
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(Downloaded 612 times last on 19 December 2024, 09:32 pm
205
Row rank of a matrix equals it column rank
Linear Algeba, Matrices

We give a simple and visually appealing proof of the result of the title. While this proof is found in my book on Linear Algebra--A Geometric Approach, due my enhanced LaTeX skills, it is typeset better so that you can visualize the argument. It makes it easy to remember too.

Keywords : Row rank, column rank, rank of a matrix, matrix rank

BSc
MTTS Level : O
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(Downloaded 3181 times last on 22 December 2024, 09:29 pm
206
Q does not have the LUB Property
Real Analysis

We shall give a simple proof of the fact that the field of rational numbers Q does not enjoy the LUB property, that is, Q is not order-complete. The proof also gives the existence of a in R such that a^2=2, that is, square root of 2 exists in R. In particular, Q is a proper subset of R.

Keywords : LUB Property, Order completeness, Failure in Q

Final Year BSc onwards
MTTS Level :
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(Downloaded 656 times last on 19 December 2024, 09:32 pm
207
Arzela-Ascoli Theorem on R
Real Analysis, Differential Equations, Functional Analysis, Sobolev Spaces

We introduce the notion of equicontinuity, prove the theorem of the title using Cantor's diagonal method on closed and bounded intervals of R. We assume that the readers know notions of uniform convergence, Uniformly Cauchy.

Keywords : Uniform Convergence, Uniformly Cauchy, Equicontinuity, Cantor's diagonal trick

Final year BSc, MSc.
MTTS Level :
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(Downloaded 871 times last on 19 December 2024, 09:32 pm
208
Topological Groups
Analysis, Harmonic Analysis

This is a beamer presentation used for a series of 3 lectures on Topological Groups. It may need to be studied in a group as the details may be lacking.

Keywords : Topological groups

MSc
MTTS Level : 2
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(Downloaded 624 times last on 23 December 2024, 06:00 am
209
Minimal Counterexample Technique
Algebra

We illustrate the technique by means of easy examples in number theory, linear algebra and theory of finite groups.

Keywords : Minimal counterexample, Cauchy's theorem, Sylow's theorem

M.Sc.
MTTS Level : 2
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(Downloaded 628 times last on 22 December 2024, 04:09 pm
210
Dimension of a Vector Space
Linear Algebra

We give three proofs of the fact that any two bases of a finite dimensional vector space have the same number of elements. The first proof is the standard one which you will find in many textbooks. The second proof is the simplest one which you will find in our book. The third is a self-contained version of the second proof and is due to Jayanthan, Goa University.

Keywords : Basis, Dimension

B.Sc, M.Sc.
MTTS Level :
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(Downloaded 596 times last on 19 December 2024, 09:32 pm
211
An Example illustrating the Concepts of Basic Linear Algebra
Linear Algebra

The article teaches you how to "construct" linear maps satisfying certain requirements. It is a concrete illustration of the concepts encountered in a basic course in linear algebra till Rank-Nullity theorem.

Keywords : Linear maps, Kernel, Range, Rank-Nullity theorem

B.Sc., M.Sc.
MTTS Level :
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(Downloaded 727 times last on 23 December 2024, 07:48 am
212
Characterization of Compact Metric Spaces (Aliter)
Metric Spaces, Topology

In this article, we offer a proof of the standard results on various characterizations of compact metric spaces. The proof here is different from the one given in my book ``Topology of Metric Spaces" and the one in another article of mine with the same title cite{sk2} as this one. I believe that this proof is somewhat easier for students to understand and retain than the one in the book, which is perhaps a little more demanding.

Keywords : Compact, cluster point, totally bounded, complete metric space

3 year BSc onwards
MTTS Level :
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(Downloaded 585 times last on 19 December 2024, 09:32 pm
213
L'Hospital's Rules
Real Analysis, Calculus

We give proofs for three typical cases L'Hospital's Rules.

Keywords : Cauchy's mean value theorem, indeterminate forms

3rd BSc, MSc
MTTS Level :
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(Downloaded 414 times last on 19 December 2024, 09:32 pm
214
Linear Maps and Matrices
Linear Algebra

In this article, we start with setting up a system of coordinates associated with an ordered basis and then talk of matrix representation of linear maps. We shall give some typical examples as well as some unusual examples. We prove all the standard results of this theme.

Keywords : Ordered basis, coordinate system, transition matrix, matrix representation of a linear map

2nd year BSc and MSc
MTTS Level :
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(Downloaded 559 times last on 22 December 2024, 04:09 pm
215
Grassman Manifold with Complete Details
Differential Manifolds, Differential Topology, Differential Geometry, Algebaric Topology

We give two descriptions of Grassmann manifolds. One is via Echelon forms some thing very concrete and the other is through Linear Algebra, a bit more abstract but quite needed for proper understanding and higher learning.

Keywords : Echelon forms, canonical forms, Grassmann manifolds

MSc and PhD students
MTTS Level : 2
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(Downloaded 653 times last on 21 December 2024, 06:01 pm
216
Riemann Stieltjes Integration
Real Analysis

The article gives a complete treatment of all the results with a good set of examples. A novel feature of this article is that we prove the equivalence of Darboux sum and Riemann sum approaches very early in the development (for an increasing integrator). This allows us to use any of these which is likely to simplify the proofs.

Keywords : Riemann integral, Riemann-Stieltjes integral, Riemann sum, Darboux sum

Final year BSc, 1st year MSc
MTTS Level :
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(Downloaded 1450 times last on 23 December 2024, 01:42 pm
217
Differentiability of Bilinear Maps
Calcuslus of Severall Variables

We prove that multilinear maps on euclidean spaces are differentiable and compute their derivatives. We then apply the result to compute the derivative of the determinant function on M(n, R). For the sake of simplicity and clear exposition of the ideas, we restrict ourselves to bilinear maps.

Keywords : Bilinear Maps, Multilinear maps

MSc
MTTS Level : 2
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(Downloaded 1602 times last on 23 December 2024, 01:04 pm
218
List of Videos on my YouTube Channel
All

The file gives a list of almost all my videos on my YouTube Channel with a brief description and URL. It will be updated periodically. Updated on Pongal, 14 Jan 2022 .

Keywords : Videos, YouTube, List

All
MTTS Level :
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(Downloaded 1997 times last on 23 December 2024, 11:28 am
219
Linear Maps from R^m to R^n
Linear Algebra

We classify/find all linear maps from R^m to R^n.

Keywords : Linear maps

BSC and BTech students
MTTS Level :
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(Downloaded 293 times last on 20 December 2024, 09:41 pm
220
Playlist for Complex Analysis
Complex Analysis

This is a playlist for the videos on complex analysis

Keywords : complex analysis, videos, playlist

MSc
MTTS Level : 2
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(Downloaded 681 times last on 23 December 2024, 02:35 pm