A collection of some of my notes. Some of them were written for teaching purposes, while others were written for entertainment. In case you find any blunders/errors/typos/grammatical mistakes (however small or big it might be), I would greatly appreciate if you could let me know.

#### Combinatorics

• A note on 2048 game
• Scoreline in a 'n' match cricket test series
• Inclusion exclusion principle
• Constructing a $$3 \times 3$$ normal magic square
• Josephus Problem
• #### Geometry

• Gauss Lucas theorem
• Baudhayana Apastamba theorem (Neo-historians call it as Pythagoras theorem)
• Brahmagupta area formula
• Ptolemy's theorem
• Bhaskara's proof of Baudhayana-Apastamba Theorem
• Isoperimetric inequality
• Ceva's and Menelaus' theorem and their converses
• Determining center of a circle using only compass (an equivalent version of the Napoleon's problem)
• Angle bisector theorem
• Stewart theorem
• Lengths of median, altitude and angle bisector
• Intercept theorem
• #### Number Theory

• Bezout's Lemma
• Pell's equation: Ramanujan and Mahalanobis
• Characterization of even perfect numbers
• Divisors
• Euler Totient function
• Euler's theorem (generalized Fermat's little theorem)
• Product of $$k$$ consecutive integers divisible by $$k!$$
• Characterizing Pythagorean triplets
• Wilson's Theorem
• Lagrange's theorem (Number Theory)
• Characterizing Three Squares in Arithmetic Progression
• Fermat's Right Triangle Theorem
• Some properties of Pythagorean triplets
• No four squares are in non-trivial Arithemtic Progression
• Numbers as difference of two squares
• Thue's Lemma
• The harmonic number $$H_n$$ is never an integer for $$n > 1$$
• Infinitude of the set of primes
• Divergence of sum of reciprocal of primes
• Goldbach-Euler theorem
• Highest power of a prime dividing $$n!$$
• Abel summation lemma/Method of partial summation
• Euler-Maclaurin summation
• Bertrand's postulate
• Chebyshev bounds
• #### Numerical Analysis and Scientific Computing

• Catastrophic round-off in finite arithmetic. The file catastrophic_round_off.m.
• Solving $$u_{xx}=f$$ numerically
• Fundamental theorem of polynomial interpolation
• Orthogonal polynomials
• Chebyshev polynomials
• Legendre polynomials
• Maximum eigenvalue ratio doesn't always indicate conditioning of the matrix
• Gram-Schmidt QR decomposition
• Pivoting in Linear systems
• Error in computing integrals recursively
• Backward stability of triangular systems
• Backward stability of inner and outer products
• #### Set Theory

• Cantor's theorem
• Denumerability of $$\mathbb{N} \times \mathbb{N}$$
• #### Differential equations

• Poisson's Equation in 1D
• Variation of parameters
• Adjoint of second order ODE, Lagrange's identity and Green's second identity
• Sturm-Liouville operator
• Uniqueness for Laplace equation

• Tautochrone
• Projectile