Algebra



  • On relation between quadratic and bilinear forms I

    • (around 2005)

  • On calculating of integral....

    • (around 2006)

  • On number of real roots (Silvester Theorem).

    • We discuss the famous theorem which explains where from the signature definition comes,(around 2007)

  • On a simple proof of Nullstelensatz.

    • We retell here in detail the simple "folklore" proof of of Nullstelensatz. (04.08.09)

  • Three identities for determinants.

    • We consider here two simple but beautiful formuale for determinants... (28.08.09)

  • Bernoulli numbers, Bernoulli polynomials....

    • We explain how Bernoulli numbers appear from basic formulae of calculus... (20.09.09)

  • Geometrical meaning of Crammer rule

    • Crammer rule seems to be annoying. But rightly viewed it is so beautiful... We expose here a simple and beautiful identity which implies Crammer rule. (12.10.11)

  • Irrational powers of integers

    • Here I expose proof of some fact. This proof is much more close to me than a problem itself. (may be because n-homomorphisms?...) (19.02.12)

  • Borwein integrals

    • Here I expose a funny sequence of integrals which I learnt yesterday... (11.04.12)

  • Chebyshev approximation and Helly's Theorem

    • Here I discuss one application of Helly's Theorem on convex bodies to a problem of approximating continous functions by polynomials(10.11.13)

  • Barnes functions and integrals

    • We calculate so called incomplete Gamma function and related integrals using analytical continuation which was used for Barnes functions. The content is essentially overlapped with older etude ``On calculating of integral...''(28.12.15)

  • On calculation of one determinant

    • I hesitated to put this elementary calculation, but on other hand this is very beautiful: to calculate something polynomial we consider limit. Does this happen often? (18.01.17)

  • On brute force combinatorial calculations in Lie groups

    • I put here calculations of exponents in Lie group. They appear elsewhere, and they are related with Campbell-Haussdorff...(18.01.18)

  • On one problem of linear algebra

    • The problem was suggested to me by my son David, and by my knowledge he heard it from Vladimir Dotzenko. I present first my solution of this problem. It is founded on one combinatorial fact which was considered in one of my etudes (see the etude "The number of permutations..." in the subsection "Arithmetics" on 18.11.18) and on the concept of even and odd numbers. Then I present the solution of James Montaldi (see the first Appendix in my text). The solution of James gives the clear understanding of the role of the finite field $Z/2Z$. My old friend, Sasha Karabegov, mathematician and the "old olimpiadnik" suggested another solution which reveals the fact that the roots of the question are in $2$-adic numbers. Here is the Solution of Sasha Karabegov. Finally I present the version of the question suggested by Sasha Borovik, Solution of Sasha Borovik,. This version is accessible for school pupils, and it convinces the importance of $2$-adic on the elementary level.

  • On Taylor Theorem

    • I tell here the short story about effectiveness of one indentity to prove the Taylor Theorem and how it was used in the beautiful statement proved by my friend Sasha Karabegov in PUTNAM competition (22 December 2018)

  • On Stirling formula and stationary phase method

    • Well-known Stirling formula is a statement which makes us to understand stationary phase method including its 'hard' part. ( June 2020)