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Only lines with n = 1 or 3 have no points (red). In mathematics, the coin problem (also referred to as the Frobenius coin problem or Frobenius problem, after the mathematician Ferdinand Frobenius) is a mathematical problem that asks for the largest monetary amount that cannot be obtained using only coins of specified denominations. [1]
A more general version of Frobenius's theorem states that if C is a conjugacy class with h elements of a finite group G with g elements and n is a positive integer, then the number of elements k such that k n is in C is a multiple of the greatest common divisor (hn,g) (Hall 1959, theorem 9.1.1).
The postage stamp problem (also called the Frobenius Coin Problem and the Chicken McNugget Theorem [1]) is a mathematical riddle that asks what is the smallest postage value which cannot be placed on an envelope, if the latter can hold only a limited number of stamps, and these may only have certain specified face values.
Ray E. Artz (2009) Scalar Algebras and Quaternions, Theorem 7.1 "Frobenius Classification", page 26. Ferdinand Georg Frobenius (1878) "Über lineare Substitutionen und bilineare Formen", Journal für die reine und angewandte Mathematik 84:1–63 (Crelle's Journal). Reprinted in Gesammelte Abhandlungen Band I, pp. 343–405.
Frobenius reciprocity theorem in group representation theory describing the reciprocity relation between restricted and induced representations on a subgroup Perron–Frobenius theorem in matrix theory concerning the eigenvalues and eigenvectors of a matrix with positive real coefficients
The 1-form dz − y dx. on R 3 maximally violates the assumption of Frobenius' theorem. These planes appear to twist along the y-axis.It is not integrable, as can be verified by drawing an infinitesimal square in the x-y plane, and follow the path along the one-forms.
He also posed the following problem: If, in the above theorem, k = 1, then the solutions of the equation x n = 1 in G form a subgroup. Many years ago this problem was solved for solvable groups. [3] Only in 1991, after the classification of finite simple groups, was this problem solved in general.
The necessary and sufficient conditions for complete integrability of a Pfaffian system are given by the Frobenius theorem. One version states that if the ideal I {\displaystyle {\mathcal {I}}} algebraically generated by the collection of α i inside the ring Ω( M ) is differentially closed, in other words