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For example, the largest amount that cannot be obtained using only coins of 3 and 5 units is 7 units. The solution to this problem for a given set of coin denominations is called the Frobenius number of the set. The Frobenius number exists as long as the set of coin denominations is setwise coprime.
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.
Some solutions of a differential equation having a regular singular point with indicial roots = and .. In mathematics, the method of Frobenius, named after Ferdinand Georg Frobenius, is a way to find an infinite series solution for a linear second-order ordinary differential equation of the form ″ + ′ + = with ′ and ″.
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; Frobenius's theorem (group theory) about the number of ...
In mathematics, more specifically in abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite-dimensional associative division algebras over the real numbers. According to the theorem, every such algebra is isomorphic to one of the following: R (the real numbers) C (the complex numbers) H ...
The Frobenius theorem can be restated more economically in modern language. Frobenius' original version of the theorem was stated in terms of Pfaffian systems, which today can be translated into the language of differential forms. An alternative formulation, which is somewhat more intuitive, uses vector fields.
The Frobenius formula states that if χ is the character of the representation σ, given by χ(h) = Tr σ(h), then the character ψ of the induced representation is given by ψ ( g ) = ∑ x ∈ G / H χ ^ ( x − 1 g x ) , {\displaystyle \psi (g)=\sum _{x\in G/H}{\widehat {\chi }}\left(x^{-1}gx\right),}
The following theorem will be presented in a more general way, as it provides a very beautiful result about representations of compact – and therefore also of finite – groups: Theorem. Every linear representation of a compact group over a field of characteristic zero is a direct sum of irreducible representations.