# Dirichlet's principle.

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German mathematician Lejeune Dirichlet Peter Gustav (02.13.1805 - 05.05.1859) is known as the principle of the founder, the name of his name.But in addition to the theory, traditionally explained by the example of "birds and cages", on account of a foreign corresponding member of the St. Petersburg Academy of Sciences, a member of the Royal Society of London, the Paris Academy of Sciences, the Berlin Academy of Sciences, Professor of Berlin and the University of Gottingen many works on mathematical analysis and number theory.

He not only introduced into mathematics well-known principle, Dirichlet also could prove a theorem on an infinite number of prime numbers that exist in any arithmetic progression of integers with certain conditions.A condition for this is that the first term of her and the difference - the number of relatively prime.

He received a thorough study of the law of distribution of prime numbers, which are peculiar to arithmetic progressions.Dirichlet introduced a series of functions that have a particular view, he succeeded in part of mathematical analysis for the first time accurately articulate and explore the concept of conditional convergence and to establish the convergence of a number, give a rigorous proof of the expanded in the Fourier series, which has a finite number, as the highs and lows.I do not leave unattended in the works of Dirichlet questions of mechanics and mathematical physics (Dirichlet's principle in the theory of harmonic functions).

uniquely designed by the German scientist of the method lies in its visual simplicity, which allows us to study the Dirichlet principle in grade school.The universal tool for solving a wide range of applications, which are used as evidence for the simple theorems in geometry and to solve complex logical and mathematical problems.

availability and simplicity of the method has allowed to use to explain it clearly playing the way.The complex and slightly confused expression, formulating the Dirichlet principle, is: "For a set of N elements are divided into a certain number of non-overlapping parts - n (common elements are missing), provided N & gt; n, at least one portion will contain more than oneelement. "He decided to successfully paraphrase, this in order to obtain clarity, had to replace the N in "hare", and n in the "cage" and abstruse expression to get the look: "Provided that the birds at least one greater than the cell, there is always atto a single cell, which gets more than two and a hare. "

This method of reasoning is called More to the contrary, he was widely known as the Dirichlet principle.Problems are solved when it is used, a wide variety.Without going into a detailed description of the decision, the principle of the Dirichlet problem with equal success for both simple geometric proofs and logical tasks and lays down the basis for conclusions in dealing with problems of higher mathematics.

Proponents of this method states that the main difficulty of the method is to determine what data are covered under the definition of "hare", and which should be regarded as "cells."

The problem of direct and triangle lying in the same plane, if necessary, to prove that it can not cross the three sides at once, as a constraint uses one condition - the line does not pass through any height triangle.As a "hare" is considered the height of the triangle, and "cells" are the two half-planes, which lie on either side of the line.Obviously, at least two will be in the height of one of the half-plane, respectively, the length of which they limit is not directly suppressed, as required.

also simply and succinctly the principle of the Dirichlet problem in the logic of the ambassador and pennants.The round table is located downstream of the various states, but the flags of their countries located around the perimeter so that each ambassador was close to the symbol of another country.It is necessary to prove the existence of such a situation, when at least two flags will be located near the representatives of the countries concerned.If you received the Ambassador of the "birds" and "cells" to designate the remainder of the rotation at the table (they will have one less), then the problem comes to a decision by itself.

These two examples are given to illustrate how easy to solve intricate problems when using the method developed by the German mathematician.