On the Frobenius problem for Beatty sequences

We study the solvability of linear diophantine equations in two variables within the context of Beatty sequences.     

Congruences modulo powers of 3 for 3- and 9-colored generalized Frobenius partitions

Let $c{?}_k\left(n\right)$ be the number of $k$-colored generalized Frobenius partitions of $n$. We establish some infinite families of congruences for $c{?}_3\left(n\right)$ and $c{?}_9\left(n\right)$ modulo arbitrary powers of 3, which refine the results of Kolitsch. For example, for $k\ge 3$ and $n\ge 0$, we prove that

$c{?}_3\left(3^{2k}n+\frac{7?3^{2k}+1}{8}\right)\equiv 0(mod{3}^{4k+5}).$

We give two different proofs to the congruences satisfied by $c{?}_9\left(n\right)$. One of the proofs uses a relation between $c{?}_9\left(n\right)$ and $c{?}_3\left(n\right)$ due to Kolitsch, for which we provide a new proof in this paper.     

On Groups with Frobenius Elements

It is proved that group G contains an Abelian normal periodic complement to C_G(a²) if a is an H-Frobenius element a of order 4 of G. Mathematics Subject Classification (2000) 20E25.     

The Frobenius problem, sums of powers of integers, and recurrences for the Bernoulli numbers

In the Frobenius problem with two variables, one is given two positive integers a and b that are relative prime, and is concerned with the set of positive numbers NR that have no representation by the linear form ax+by in nonnegative integers x and y. We give a complete characterization of the set NR, and use it to establish a relation between the power sums over its elements and the power sums over the natural numbers. This relation is used to derive new recurrences for the Bernoulli numbers.     

On the Frobenius Problem of Numerical Semigroups

We define the minimal transversal of a numerical semigroup with respect to one of its elements and use it to calculate the Frobenius number, genus, and the Hilbert function of the semigroup. We give various examples for the use of this method.     

On the Edge-forwarding Indices of Frobenius Graphs

A G-Frobenius graph F, as defined by Fang, Li, and Praeger, is a connected orbital graph of a Frobenius group G = K × H with Frobenius kernel K and Frobenius complement H. F is also shown to be a Cayley graph, F = Cay（K, S） for K and some subset S of the group K. On the other hand, a network N with a routing function R, written as （N, R）, is an undirected graph N together with a routing R which consists of a collection of simple paths connecting every pair of vertices in the graph. The edge-forwarding index π（N） of a network （N, R）, defined by Heydemann, Meyer, and Sotteau, is a parameter to describe tile maximum load of edges of N. In this paper, we study the edge-forwarding indices of Frobenius graphs. In particular, we obtain the edge-forwarding index of a G-Frobenius graph F with rank（G） ≤ 50.     

On Finite Simple Groups with the Set of Element Orders as in a Frobenius Group or a Double Frobenius Group

It is proved that a finite simple group with the set of element orders as in a Frobenius group (a double Frobenius group, respectively) is isomorphic to L₃(3) or U₃(3) (to U₃(3) or S₄(3), respectively).     

Some algebraic examples of Frobenius manifolds

Using analytic methods of finite-gap integration, we construct quasihomogeneous algebraic solutions of the WDVV associativity equations and the nonsemisimple Frobenius manifolds associated with them. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 2, pp. 195–206, May, 2007.     

Givental symmetries of Frobenius manifolds and multi-component KP tau-functions

We establish a link between two different constructions of the action of the twisted loop group on the space of Frobenius structures. The first construction (due to Givental) describes the action of the twisted loop group on the partition functions of formal (axiomatic) Gromov-Witten theories. The explicit formulas for the corresponding tangent action were computed by Y.-P. Lee. The second construction (due to van de Leur) describes the action of the same group on the space of Frobenius structures via the multi-component KP hierarchies. Our main theorem states that the genus zero restriction of the Y.-P. Lee formulas coincides with the tangent van de Leur action.     

On quantum stabilizer codes derived from local Frobenius rings

In this paper we consider stabilizer codes over local Frobenius rings. Firstly, we study the relative minimum distances of a stabilizer code and its reduction onto the residue field. We show that for various scenarios, a free stabilizer code over the ring does not underperform the according stabilizer code over the field. This leads us to conjecture that the same is true for all free stabilizer codes. Secondly, we focus on the isometries of stabilizer codes. We present some preliminary results and introduce some interesting open problems.     

余半单Hopf代数的Frobenius性质

设k是特征为0的代数闭域,H为其上的余半单Hopf代数,本文证明了当H有型:l:1 m:p 1:q(其中p~2相似文献   

一类四度Frobenius图

本文给出了Frobenius群Z_(2n~2 2n 1)■Z_4的一类四度Frobenius图．沿用方、李和Praeger的方法,计算出了这一类图的直径和型(定理3．2)．     

高维的Frobenius型半单Hopf代数

设H是特征为零的代数闭域k上的半单Hopf代数.本文证明了如果dimкH 是小于351的奇数,则H是Frobenius型Hopf代数.     

L-invariants and logarithm derivatives of eigenvalues of Frobenius

Let K be a p-adic local field.We study a special kind of p-adic Galois representations of it.These representations are similar to the Galois representations occurred in the exceptional zero conjecture for modular forms.In particular,we verify that a formula of Colmez can be generalized to our case.We also include a degenerated version of Colmez’s formula.     

Morphism spaces in stable categories of Frobenius algebras

We present an explicit formula computing morphism spaces in the stable category of a Frobenius algebra.     

低维的Frobenius 型半单Hopf 代数

Let k be an algebraically closed field of characteristic zero.This paper proves that semisimple Hopf algebras over k of dimension 66,70 and 78 are of Frobenius type.     

Cocyclic Hadamard matrices from forms over finite Frobenius rings

We give a construction of cocyclic Butson-Hadamard matrices from bilinear forms over finite Frobenius rings, which generalizes previous constructions. We then prove a classification result, showing that the resulting matrices depend only on the additive group of the underlying module of the form.     

Some properties of Schur complements and diagonal-Schur complements of diagonally dominant matrices

In this paper, we prove that the diagonal-Schur complement of a strictly doubly diagonally dominant matrix is strictly doubly diagonally dominant matrix. The same holds for the diagonal-Schur complement of a strictly generalized doubly diagonally dominant matrix and a nonsingular H-matrix. We point out that under certain assumptions, the diagonal-Schur complement of a strictly doubly (doubly product) γ-diagonally dominant matrix is also strictly doubly (doubly product) γ-diagonally dominant. Further, we provide the distribution of the real parts of eigenvalues of a diagonal-Schur complement of H-matrix. We also show that the Schur complement of a γ-diagonally dominant matrix is not always γ-diagonally dominant by a numerical example, and then obtain a sufficient condition to ensure that the Schur complement of a γ-diagonally dominant matrix is γ-diagonally dominant.