We present a new approach of proving certain Carath\'{e}odory-type theorems using the Perron-Frobenius Theorem, classical result in matrix theory describing largest eigenvalue with positive entries. One problems left open this note is whether our may be extended to prove similar results area, particular Colourful Carath\'{e}odory Theorem.

We consider the Diophantine problem of Frobenius for semigroup S ( d ) where d denotes the tuple (d1, d2, d3), gcd(d1, d2, d3) = 1. Based on the Hadamard product of analytic functions [17] we have found the analytic representation for the diagonal elements akk ( d ) of the Johnson’s matrix of minimal relations [12] in terms of d1, d2, d3. Bearing in mind the results of the recent paper [10] thi...

in this paper, our purpose is twofold. firstly, the tensor andresiduum operations on $l-$nested systems are introduced under thecondition of complete residuated lattice. then we show that$l-$nested systems form a complete residuated lattice, which isprecisely the classical isomorphic object of complete residuatedpower set lattice. thus the new representation theorem of$l-$subsets on complete re...

We define generalized Frobenius-Schur indicators for objects in a linear pivotal category C. An equivariant indicator of an object is defined as a functional on the Grothendieck algebra of the quantum double Z(C) of C using the values of the generalized Frobenius-Schur indicators. In a spherical fusion category C with Frobenius-Schur exponent N , we prove that the set of all equivariant indicat...

Using the notion of Levi form of a smooth distribution, we discuss the local and the global problem of existence of one horizontal section of a smooth vector bundle endowed with a horizontal distribution. The analysis will lead to the formulation of a “one-leaf” analogue of the classical Frobenius integrability theorem in elementary differential geometry. Several applications of the result will...