We define a notion of “Frobenius pair”, which is a mild generalization of the notion of “Frobenius object” in a monoidal category. We then show that Atiyah duality for smooth manifolds can be encapsulated in the statement that a certain collection of structure obtained from a manifold forms a “commutative Frobenius pair” in the stable homotopy category of spectra.

Frobenius algebras play an important role in the representation theory of finite groups. In the present work, we investigate the (quasi) Frobenius property of n-group algebras. Using the (quasi-) Frobenius property of ring, we can obtain some information about constructions of module category over this ring ([2], p. 66–67).

a finite group g is said to be a pos-group if for each x in g the cardinality of the set {y in g | o(y) = o(x)} is a divisor of the order of g. in this paper we study the structure of pos-groups with some cyclic sylow subgroups.

a finite group g is said to be a pos-group if for each x in g the cardinality of the set {y in g | o(y) = o(x)} is a divisor of the order of g. in this paper we study the structure of pos-groups with some cyclic sylow subgroups.

A new randomized algorithm is presented for computing the Frobenius form of an n×n matrix over a sufficiently large field K. Let 2 < ω ≤ 3 be such that two matrices in Kn×n can be multiplied together with O(n) operations in K. If #K ≥ 2n, the new algorithm uses an expected number of O(n) operations in K to compute the Frobenius form of A, matching the lower bound for the cost of this problem. A...

We give a formula for the number of rational points of projective algebraic curves de ned over a nite eld, and a bound \ a la Weil" for connected ones. More precisely, we give the characteristic polynomials of the Frobenius endomorphism on the etale `-adic cohomology groups of the curve. Finally, as an analogue of Artin's holomorphy conjecture, we prove that, if Y ! X is a nite at morphism betw...

Given two numerical semigroups S and T and a positive integer d, S is said to be one over d of T if S = {s ∈ N | ds ∈ T} and in this case T is called a d-fold of S. We prove that the minimal genus of the d-folds of S is g + ⌈ (d−1)f 2 ⌉, where g and f denote the genus and the Frobenius number of S. The case d = 2 is a problem proposed by Robles-Pérez, Rosales, and Vasco. Furthermore, we find th...

We compare several quasi-Frobenius-type properties for corings that appeared recently in literature and provide several new characterizations for each of these properties. By applying the theory of Galois comodules with a firm coinvariant ring, we can characterize a locally quasi-Frobenius (quasi-co-Frobenius) coring as a locally projective generator in its category of comodules.

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...