A matrix is said to have the Perron-Frobenius property (strong Perron-Frobenius property) if its spectral radius is an eigenvalue (a simple positive and strictly dominant eigenvalue) with a corresponding semipositive (positive) eigenvector. It is known that a matrix A with the Perron-Frobenius property can always be the limit of a sequence of matrices A(ε) with the strong Perron-Frobenius prope...

In “Is the Frobenius Matrix Norm Induced?”, the authors ask whether the Frobenius and the norms are induced. There, they claimed that the Frobenius norm is not induced and, consequently, conjectured that the norm may not be induced. In this note, it is shown that the Frobenius norm is induced on particular matrix spaces. It is then shown that the norm is in fact induced on a particular matrix-v...

A dynamical system is a pairing between a set of states X ⊂ Rd and a map T : X which describes how the system evolves from state to state over time. The Perron– Frobenius, or transfer, operator is a natural extension of the point-by-point dynamics defined by T to an ensemble theory which describes the evolution of distributions of points. It features heavily in dynamical systems theory and in a...

Let X be a smooth projective curve of genus g ≥ 2 defined over an algebraically closed field k of characteristic p > 0. For p sufficiently large (explicitly given in terms of r, g) we construct an atlas for the locus of all Frobenius-destabilized bundles (i.e. we construct all Frobenius-destabilized bundles of degree zero up to isomorphism). This is done by exhibiting a surjective morphism from...

We construct a structure of a ring with local units on a co-Frobenius coalgebra. We study a special class of co-Frobenius coalgebras whose objects we call symmetric coalgebras. We prove that any semiperfect coalgebra can be embedded in a symmetric coalgebra. A dual version of Brauer's equivalence theorem is presented, allowing a characterization of symmetric coalgebras by comparing certain func...

We first review various known algebraic structures on the Hochschild (co)homology of a differential graded algebras under weak Poincaré duality hypothesis, such as CalabiYau algebras, derived Poincaré duality algebras and closed Frobenius algebras. This includes a BV-algebra structure on HH(A,A) or HH(A,A), which in the latter case is an extension of the natural Gerstenhaber structure on HH(A,A...

A matrix is said to have the Perron-Frobenius property (strong Perron-Frobenius property) if its spectral radius is an eigenvalue (a simple positive and strictly dominant eigenvalue) with a corresponding semipositive (positive) eigenvector. It is known that a matrix A with the Perron-Frobenius property can always be the limit of a sequence of matrices A(ε) with the strong Perron-Frobenius prope...

Starting from the quantum differential equation associated to a weighted projective space, which is given by Coates, Corti, Lee and Tseng, we construct a Frobenius manifold. We see that the Frobenius manifold coincides with the big quantum cohomology of the weighted projective space. The construction is based on Dubrovin’s reconstruction theorem.

It is well known that the category of finite sets and cospans, composed by pushout, contains the universal special commutative Frobenius algebra. In this note we observe that the same construction yields also general commutative Frobenius algebras, if just the pushouts are changed to homotopy pushouts.

In the last two years Frobenius-Euler polynomials have gained renewed interest and were studied by several authors. This paper presents a novel approach to these polynomials by treating them as Appell polynomials. This allows to apply an elementary matrix representation based on a nilpotent creation matrix for proving some of the main properties of Frobenius-Euler polynomials in a straightforwa...