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This note is purely expository and is in Russian. We show how to prove interesting combinatorial results using the local Lovasz lemma. The note is accessible for students having basic knowledge of combinatorics; the notion of independence is defined and the Lovasz lemma is stated and proved. Our exposition follows `Probabilistic methods of N. Alon and J. Spencer. The main difference is that we show how the proof could have been invented. The material is presented as a sequence of problems, which is peculiar not only to Zen monasteries but also to advanced mathematical education; most problems are presented with hints or solutions.
This is a collection of definitions, notations and proofs for the Bernoulli numbers $B_n$ appearing in formulas for the sum of integer powers, some of which can be found scattered in the large related historical literature in French, English and Germ
We give a purely combinatorial proof of the positivity of the stabilized forms of the generalized exponents associated to each classical root system. In finite type A_{n-1}, we rederive the description of the generalized exponents in terms of crystal
We obtain a necessary and sufficient condition for the linear independence of solutions of differential equations for hyperlogarithms. The key fact is that the multiplier (i.e. the factor $M$ in the differential equation $dS=MS$) has only singulariti
In probability theory, the independence is a very fundamental concept, but with a little mystery. People can always easily manipulate it logistically but not geometrically, especially when it comes to the independence relationships among more that tw
For every constant c > 0, we show that there is a family {P_{N, c}} of polynomials whose degree and algebraic circuit complexity are polynomially bounded in the number of variables, that satisfies the following properties: * For every family {f_n}