1. Let A = [aii] be an n-th order irreducible non-negative matrix. As is very well-known, the matrix A has a positive characteristic root p (provided that n> I), which is simple and maximal in the sense that every characteristic root A satisfies I A I ~ p, and the characteristic vector x belonging to p may be chosen positive. These results, originally due to Frobenius, have been proved by Wiela...

First we introduce some of the basic notations. For any vector u = (u1, . . . , up) T ∈ R, denote by |u|q the vector `q-norm defined by |u|q = (∑p k=1 |uk| )1/q for q ≥ 1 and write |u|0 = ∑p k=1 I(uk 6= 0). For any set S, denote by S its complement. For a matrix A = (ak`) ∈ Rp×p, we denote by ‖A‖2 the spectral norm, ‖A‖F the Frobenius norm, and ‖A‖1 = ∑p k,`=1 |ak`| the elementwise `1-norm. Rec...

We revisit the problem of analytically computing one-point functions for scalar fields in planar AdS black holes arbitrary dimension, which are sourced by Weyl squared tensor. analyze terms power series expansions around boundary using method Frobenius. clarify pole structure final answer operator mixing, as argued previously Grinberg and Maldacena. generalize techniques to also obtain analytic...

New family of flat potential (Darboux-Egoroff) metrics on the Hurwitz spaces and corresponding Frobenius structures are found. We consider a Hurwitz space as a real manifold. Therefore the number of coordinates is twice as big as the number of coordinates used in the construction of Frobenius manifolds on Hurwitz spaces found by B.Dubrovin more than 10 years ago. The branch points of a ramified...

A first kind Frobenius graph is a Cayley graph Cay(K,S) on the Frobenius kernel of a Frobenius group K o H such that S = a for some a ∈ K with 〈a〉 = K, where H is of even order or a is an involution. It is known that such graphs admit ‘perfect’ routing and gossiping schemes. A circulant graph is a Cayley graph on a cyclic group of order at least three. Since circulant graphs are widely used as ...

In [1], the multiple Frobenius-Euler numbers and polynomials were constructed. In this paper we give some interesting formulae which are related to the multiple Frobenius-Euler polynomials. The main purpose of this paper is to give the Kummer type congruences for the multiple Frobenius-Euler numbers. §

Iskander Aliev A sharp lower bound for the Frobenius number Ferdinand Georg Frobenius (1849–1917) raised the following problem: given N positive integers a1, . . . , aN with gcd(a1, . . . , aN ) = 1, find the largest natural number gN = gN (a1, . . . , aN) (called the Frobenius

Introduction Lecture 1. Algebraic properties of correlators in 2D topological field theory. Moduli of a 2D TFT and WDVV equations of associativity. Lecture 2. Equations of associativity and Frobenius manifolds. Deformed flat connection and its monodromy at the origin. Lecture 3. Semisimplicity and canonical coordinates. Lecture 4. Classification of semisimple Frobenius manifolds. Lecture 5. Mon...

This is the introductory chapter to this issue. We review the main idea of the localization technique and its brief history both in geometry and in QFT. We discuss localization in diverse dimensions and give an overview of the major applications of the localization calculations for supersymmetric theories. We explain the focus of the present issue.