Let i : A → R be a ring morphism, and χ : R → A a right R-linear map with χ(χ(r)s) = χ(rs) and χ(1 R) = 1 A. If R is a Frobenius A-ring, then we can define a trace map tr : A → A R. If there exists an element of trace 1 in A, then A is right FBN if and only if A R is right FBN and A is right noetherian. The result can be generalized to the case where R is an I-Frobenius A-ring. We recover resul...

Frobenius splittings were introduced by V. B. Mehta and A. Ramanathan in [6] and refined further by S. Ramanan and Ramanathan in [9]. Frobenius splittings have proven to be a amazingly effective when they apply. Proofs involving Frobenius splittings tend to be very efficient. Other methods usually require a much more detailed knowledge of the object under study. For instance, while showing that...

Given a manifold M of dimension n + k, attach to every p ∈ M an ndimensional subspace of the tangent space Tp(M). It is natural to ask if this collection of subspaces is the collection of tangent spaces to a family of submanifolds that cover M . In this paper we prove Frobenius’ Theorem, which gives a necessary and sufficient condition for the answer to be yes.

Let N ≥ 2 and let 1 < a1 < · · · < aN be relatively prime integers. Frobenius number of this N-tuple is defined to be the largest positive integer that cannot be expressed as P N i=1 aixi where x1, ..., xN are non-negative integers. The condition that gcd(a1 , ..., aN ) = 1 implies that such number exists. The general problem of determining the Frobenius number given N and a1, ..., aN is NP-har...

Let a and b be positive, relatively prime integers. We show that the following are equivalent: (i) d is a dead end in the (symmetric) Cayley graph of Z with respect to a and b, (ii) d is a Frobenius value with respect to a and b (it cannot be written as a non-negative or non-positive integer linear combination of a and b), and d is maximal (in the Cayley graph) with respect to this property. In...

The theory of monads on categories equipped with a dagger (a contravariant identity-on-objects involutive endofunctor) works best when all structure respects the dagger: the monad and adjunctions should preserve the dagger, and the monad and its algebras should satisfy the so-called Frobenius law. Then any monad resolves as an adjunction, with extremal solutions given by the categories of Kleis...

A matrix is said to have the Perron-Frobenius property if it has a positive dominant eigenvalue that corresponds to a nonnegative eigenvector. Matrices having this and similar properties are studied in this paper. Characterizations of collections of such matrices are given in terms of the spectral projector. Some combinatorial, spectral, and topological properties of such matrices are presented...

Deformations of Dubrovin’s Hurwitz Frobenius manifolds are constructed. The deformations depend on g(g+1)/2 complex parameters where g is the genus of the corresponding Riemann surface. In genus one, the flat metric of the deformed Frobenius manifold coincides with a metric associated with a one-parameter family of solutions to the Painlevé-VI equation with coefficients (1/8,−1/8, 1/8, 3/8) . A...

For each Frobenius algebra there is defined a skein module of surfaces embedded in a given 3manifold and bounding a prescribed curve system in the boundary. The skein relations are local and generate the kernel of a certain natural extension of the corresponding topological quantum field theory. In particular the skein module of the 3-ball is isomorphic to the ground ring of the Frobenius algeb...

This paper presents a new methodology to compute the number of numerical semigroups of given genus or Frobenius number. We apply generating function tools to the bounded polyhedron that classifies the semigroups with given genus (or Frobenius number) and multiplicity. First, we give theoretical results about the polynomial-time complexity of counting these semigroups. We also illustrate the met...