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  2. Dirichlet's principle - Wikipedia

    en.wikipedia.org/wiki/Dirichlet's_principle

    The name "Dirichlet's principle" is due to Bernhard Riemann, who applied it in the study of complex analytic functions. [1]Riemann (and others such as Carl Friedrich Gauss and Peter Gustav Lejeune Dirichlet) knew that Dirichlet's integral is bounded below, which establishes the existence of an infimum; however, he took for granted the existence of a function that attains the minimum.

  3. Dirichlet energy - Wikipedia

    en.wikipedia.org/wiki/Dirichlet_energy

    Since it is the integral of a non-negative quantity, the Dirichlet energy is itself non-negative, i.e. E[u] ≥ 0 for every function u. Solving Laplace's equation () = for all , subject to appropriate boundary conditions, is equivalent to solving the variational problem of finding a function u that satisfies the boundary conditions and has minimal Dirichlet energy.

  4. Pigeonhole principle - Wikipedia

    en.wikipedia.org/wiki/Pigeonhole_principle

    Although the pigeonhole principle appears as early as 1624 in a book attributed to Jean Leurechon, [2] it is commonly called Dirichlet's box principle or Dirichlet's drawer principle after an 1834 treatment of the principle by Peter Gustav Lejeune Dirichlet under the name Schubfachprinzip ("drawer principle" or "shelf principle"). [3]

  5. Peter Gustav Lejeune Dirichlet - Wikipedia

    en.wikipedia.org/wiki/Peter_Gustav_Lejeune_Dirichlet

    Johann Peter Gustav Lejeune Dirichlet (/ ˌ d ɪər ɪ ˈ k l eɪ /; [1] German: [ləˈʒœn diʁiˈkleː]; [2] 13 February 1805 – 5 May 1859) was a German mathematician. In number theory , he proved special cases of Fermat's last theorem and created analytic number theory .

  6. Differential forms on a Riemann surface - Wikipedia

    en.wikipedia.org/wiki/Differential_forms_on_a...

    Since χ D can be approximated by bump functions in L 2, γ 1 − γ lies in the real Hilbert space of 1-forms Re H; similarly α 1 − α lies in H. Dirichlet's principle states that the distance function F(ξ) = ||γ 1 − γ – ξ|| on Re H 1 is minimised by a smooth 1-form ξ 0 in Re H 1. In fact −du coincides with the minimising 1-form ...

  7. Dirichlet character - Wikipedia

    en.wikipedia.org/wiki/Dirichlet_character

    In analytic number theory and related branches of mathematics, a complex-valued arithmetic function: is a Dirichlet character of modulus (where is a positive integer) if for all integers and : [1] χ ( a b ) = χ ( a ) χ ( b ) ; {\displaystyle \chi (ab)=\chi (a)\chi (b);} that is, χ {\displaystyle \chi } is completely multiplicative .

  8. Dirichlet distribution - Wikipedia

    en.wikipedia.org/wiki/Dirichlet_distribution

    This particular distribution is known as the flat Dirichlet distribution. Values of the concentration parameter above 1 prefer variates that are dense, evenly distributed distributions, i.e. all the values within a single sample are similar to each other. Values of the concentration parameter below 1 prefer sparse distributions, i.e. most of ...

  9. List of incomplete proofs - Wikipedia

    en.wikipedia.org/wiki/List_of_incomplete_proofs

    Dirichlet's principle. This was used by Riemann in 1851, but Weierstrass found a counterexample to one version of this principle in 1870, and Hilbert stated and proved a correct version in 1900. Cayley incorrectly claimed that there are three different groups of order 6. This mistake is strange because in an earlier 1854 paper he correctly ...