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On Ramanujan’s Nested Radicals
Four Math Problems
-
Solve the equation
-
The function
has
derivatives. Show that if
and
then we can find
such that
.
-
Let
.
For, let
where the sum ranges over all pairs
of positive integer satisfying the indicated inequalities. Evaluate:
.
- This problem deals to elementary functional analysis and is taken from very old paper of Putnam Competitions.
has continuous second derivative and
,
for all
.
are integers such that
are all integers. Put
and
.
•Prove that
.
• Show that for, there are atmost
such indices
.
• Show that there are atmostlattice points on the curve
.
Free Online Algebra Books
Internet is full of knowledge. There are many professors who have shared their works online. I have listed a few books on Algebra and Related Mathematics in this article. I am not very serious, but I think these are very useful for Undergraduate and Graduate Students, including me.
- Abstract Algebra OnLine by Prof. Beachy
- Understanding Algebra by James Brennan
- Abstract Algebra : Theory and Applications by Tom Judson
- Elements of Abstract and Linear Algebra by E H Connell
- Linear Algebra by Jim Hefferon
- Elementary Linear Algebra by Keith Matthews
- Linear Algebra, Infinite dimensions and Mapleby James Herod
- Elementary Number Theory by William Stein
- Abstract Algebra: The Basic Graduate Year, A Course in Algebraic Number Theory, and, A Course in Commutative Algebra are three ebooks by Robert Ash and are available here on his website.
- A Course in Universal Algebra by Stanley Burris & H. P. Sankappanvar
- An Introduction to the Theory of Numbers by Leo Moser
- A Computational introduction to Number Theory and Algebra by Victor Shoup
- Sets, Relations, Functions by Ivo Duentsch and Günther Gedigo
- Group Theory by Pedrag Civitanovic
- Linear & Multilinear Algebra by C C Wang & R M Bowen
- Abelian Categories by Peter Freyd
- Categories and Groupoids by P. J. Higgins
- Lie Algebras by Prof. Sternberg
All books, except few are in Acrobat Portable Document Format. You may need acrobat pdf reader to read these. However, if you don’t have that- or hate pdf readers – try online pdf/ps reader at view.samurajdata.se.
This list is expandable. If you know any other book on Algebra which is available online for free, then please give a few seconds and put that into the Comment-Box below, with the link. (It supports HTML.)
Related articles
- Life after Differential Equations (worldscientists.wordpress.com)
- The Lie Algebra of a Lie Group (unapologetic.wordpress.com)
A General Problem on functions
Problem
Let denote the set of all integers (as usually it do
).
Consider a function with the following properties:
for all
. Is it possible that all positive divisors of
occur as values of
?
A happy note: is actually 1748 and it is written to retain symmetry in problem.
A Problem on Infinite Sum and Recurrence Relations
ProblemConsider the infinite sum |
Solution
As the recurrence relation states , we shall reform the infinite sum into the same pattern (have a deep look);
After Simplifying and arranging
Now, as {the recurrence relation is}
for all
, all terms except first three are zero in R.H.S.
Hence we have,
and substituting , we have
or
so,
From the Problem,
So, the desired ordered pair is .
A College Level Problem on Jensen’s Inequality
ProblemGiven, Prove that
|
Solution
It is natural to make trigonometric substitution for some
,
Note that the monotonicity of the cosine function combined with the given inequalities shows that the form a decreasing sequence. The expression on the left
Here we used a subtraction and a double-angle formula. The sine function is concave down on ; hence we can use Jensen’s Inequality to obtain
Hence,
or,
Since,
Using the fact that for all
yields