We start by stating a version of the Perron–Frobenius theorem. Let A be a d × d stochastic matrix, where here we use this to mean that the entries of A are non-negative, and every column sums to 1: Aij ∈ [0, 1] for all i, j, and ∑d i=1Aij = 1 for all j. Thus the columns of A are probability vectors. Such a matrix A describes a weighted random walk on d sites: if the walker is presently at site ...

We study co-Frobenius and more generally quasi-co-Frobenius corings over arbitrary base rings and over PF base rings in particular. We generalize some results about co-Frobenius and quasi-co-Frobenius coalgebras to the case of noncommutative base rings and give several new characterizations for co-Frobenius and more generally quasi-co-Frobenius corings, some of them are new even in the coalgebr...

We investigate, in this article, a generalization of the Riemann–Roch theorem for graphs of Baker and Norine, with a view toward identifying new objects for which a two-variable zeta-function can be defined. To a lattice Λ of rank n − 1 in Z n and perpendicular to a positive integer vector R, we define the notions of g-number and of canonical vector , in analogy with the notions of genus and ca...

We describe an algorithm for automatic classification of idiomatic and literal expressions. Our starting point is that idioms and literal expressions occur in different contexts. Idioms tend to violate cohesive ties in local contexts, while literals are expected to fit in. Our goal is to capture this intuition using a vector representation of words. We propose two approaches: (1) Compute inner ...

we give a new proof of the well known wedderburn's little theorem (1905) that a finite‎ ‎division ring is commutative‎. ‎we apply the concept of frobenius kernel in frobenius representation theorem in finite group‎ ‎theory to build a proof‎.

We continue the investigation of Hopf-Galois extensions with central invariants started in [29]. Our objective is not to imitate algebraic geometry using Hopf-Galois extension but to understand their geometric properties. Let H be a finite-dimensional Hopf algebra over a ground field k. Our main object of study is an H-Galois extension U ⊇ O such that O is a central subalgebra of U . Let us bri...

It is known that the Sylow subgroups of a Frobenius complement are cyclic or generalized quaternion. In this paper it is shown that there are no restrictions at all on the structure of the Sylow subgroups of the FrobeniusWielandt complements that appear in the well-known Wielandt's generalization of Frobenius' Theorem. Some examples of explicit constructions are also given. 0. Introduction Let ...

In this paper, vectors are column vectors. Unless otherwise specified, the norm of a matrix ‖M‖ is the Frobenius norm, defined by ‖M‖ = √∑ i,jM 2 ij . The all ones vector is denoted 1 and the all zeros vector is denoted 0. The all ones matrix is denoted J and the all zeros matrix is denoted O. The n× n identity matrix is denoted In, or simply I if its size is understood from context. The vector...

The notion of a Frobenius coring is introduced, and it is shown that any such coring produces a tower of Frobenius corings and Frobenius extensions.

We extend the definition of involutivity to non-Lipschitz tangent subbundles using generalized functions. prove Frobenius Theorem with sharp regularity estimate when subbundle is log-Lipschitz: if $${\mathcal {V}}$$ a log-Lipschitz involutive rank r, then for any $$\varepsilon >0$$ , locally there homeomorphism $$\Phi (u,v)$$ such that ,\frac{\partial \Phi }{\partial u^1},\dots u^r}\in C^{0,1-\...