The Frobenius problem for numerical semigroups

In this paper, we characterize those numerical semigroups containing 〈n₁,n₂〉. From this characterization, we give formulas for the genus and the Frobenius number of a numerical semigroup. These results can be used to give a method for computing the genus and the Frobenius number of a numerical semigroup with embedding dimension three in terms of its minimal system of generators.     

Some properties of Frobenius numbers and a fraction of symmetric semigroups in the weak limit for <Emphasis Type="Italic">n</Emphasis>=3

We generalize and prove a conjecture by V.I. Arnold on the parity of Frobenius numbers. For the case of symmetric semigroups with three generators we give an exact formula for Frobenius numbers, which is, in a sense, a sum of two Sylvester’s formulae. We prove that a fraction of symmetric semigroups vanishes in the weak limit.     

Constructing Almost Symmetric Numerical Semigroups from Irreducible Numerical Semigroups

Every almost symmetric numerical semigroup can be constructed by removing some minimal generators from an irreducible numerical semigroup with its same Frobenius number.     

Atoms of the set of numerical semigroups with fixed Frobenius number

Given a positive integer g, we denote by F(g) the set of all numerical semigroups with Frobenius number g. The set (F(g),∩) is a semigroup. In this paper we study the generators of this semigroup.     

Frobenius Problem and the Covering Radius of a Lattice

Let and let be relatively prime integers. The Frobenius number of this N-tuple is defined to be the largest positive integer that cannot be expressed as are non-negative integers. The condition that implies that such a number exists. The general problem of determining the Frobenius number given N and is NP-hard, but there have been a number of different bounds on the Frobenius number produced by various authors. We use techniques from the geometry of numbers to produce a new bound, relating the Frobenius number to the covering radius of the null-lattice of this N-tuple. Our bound is particularly interesting in the case when this lattice has equal successive minima, which, as we prove, happens infinitely often.     

A short proof of Fel’s theorems on 3-D Frobenius problem

The Frobenius number and the generating function for numerical additive semigroups with three generators is obtained in a way shorter with respect to the original proof by L. Fel.     

The Frobenius problem for numerical semigroups with multiplicity four

In this paper, we study the gender, Frobenius number and pseudo-Frobenius number for numerical semigroups with multiplicity four, embedding dimension three and minimal generators pairwise relatively prime.     

Finitude de la dimension homologique d'algèbres d'opérateurs différentiels faiblement complètes et à coefficients surconvergents

We prove that the sheaf of arithmetic differential operators with overconvergent coefficients, introduced by P. Berthelot, has finite cohomological dimension. A similar geometrical proof shows that the weak p-adic completion of the Weyl algebra has also finite cohomological dimension. Moreover, this algebra can be naturally endowed with a filtration which is compatible with the Frobenius.     

A novel dynamic model of pseudo random number generator

An interesting hierarchy of random number generators is introduced in this paper based on the review of random numbers characteristics and chaotic functions theory. The main objective of this paper is to produce an ergodic dynamical system which can be implemented in random number generators. In order to check the efficacy of pseudo random number generators based on this map, we have carried out certain statistical tests on a series of numbers obtained from the introduced hierarchy. The results of the tests were promising, as the hierarchy passed the tests satisfactorily, and offers a great capability to be employed in a pseudo random number generator.     

Frobenius–Perron theory of modified ADE bound quiver algebras

The Frobenius–Perron dimension for an abelian category was recently introduced in [5]. We apply this theory to the category of representations of the finite-dimensional radical square zero algebras associated to certain modified ADE graphs. In particular, we take an ADE quiver with arrows in a certain orientation and an arbitrary number of loops at each vertex. We show that the Frobenius–Perron dimension of this category is equal to the maximum number of loops at a vertex. Along the way, we introduce a result which can be applied in general to calculate the Frobenius–Perron dimension of a radical square zero bound quiver algebra. We use this result to introduce a family of abelian categories which produce arbitrarily large irrational Frobenius–Perron dimensions.     

求矩阵方程AXB+CXD=F的中心对称最小二乘解的迭代算法

该文建立了求矩阵方程AXB+CXD=F的中心对称最小二乘解的迭代算法.使用该算法不仅可以判断该矩阵方程的中心对称解的存在性,而且无论中心对称解是否存在,都能够在有限步迭代计算之后得到中心对称最小二乘解.选取特殊的初始矩阵时,可求得极小范数中心对称最小二乘解.同时,也能给出指定矩阵的最佳逼近中心对称矩阵.     

Average condition number for solving linear equations

We study an average condition number and an average loss of precision for the solution of linear equations and prove that the average case is strongly related to the worst case. This holds if the perturbations of the matrix are measured in Frobenius or spectral norm or componentwise. In particular, for the Frobenius norm we show that one gains about log₂n+0.9 bits on the average as compared to the worst case, n being the dimension of the system of linear equations.     

Pseudo-random numbers for comparative Monte Carlo calculations

To compare the results of two slightly different situations in a given problem solved by the Monte Carlo method it is convenient to use the same random numbers in both cases. To facilitate the reproducibility of the random numbers used a method based on two pseudo-random number generators may be adopted. In this paper the method is discussed with reference to multiplicative congruential generators, the influence of the initial value on a random sequence obtained by a given multiplier being examined first.     

A Method of Systematic Search for Optimal Multipliers in Congruential Random Number Generators

This paper presents a method of systematic search for optimal multipliers for congruential random number generators. The word-size of computers is a limiting factor for development of random numbers. The generators for computers up to 32 bit word-size are already investigated in detail by several authors. Some partial works are also carried out for moduli of 2⁴⁸ and higher sizes. Rapid advances in computer technology introduced recently 64 bit architecture in computers. There are considerable efforts to provide appropriate parameters for 64 and 128 bit moduli. Although combined generators are equivalent to huge modulus linear congruential generators, for computational efficiency, it is still advisable to choose the maximum moduli for the component generators. Due to enormous computational price of present algorithms, there is a great need for guidelines and rules for systematic search techniques. Here we propose a search method which provides ‘fertile’ areas of multipliers of perfect quality for spectral test in two dimensions. The method may be generalized to higher dimensions. Since figures of merit are extremely variable in dimensions higher than two, it is possible to find similar intervals if the modulus is very large. This revised version was published online in July 2006 with corrections to the Cover Date.     

Twisted Frobenius Extensions of Graded Superrings

We define twisted Frobenius extensions of graded superrings. We develop equivalent definitions in terms of bimodule isomorphisms, trace maps, bilinear forms, and dual sets of generators. The motivation for our study comes from categorification, where one is often interested in the adjointness properties of induction and restriction functors. We show that A is a twisted Frobenius extension of B if and only if induction of B-modules to A-modules is twisted shifted right adjoint to restriction of A-modules to B-modules. A large (non-exhaustive) class of examples is given by the fact that any time A is a Frobenius graded superalgebra, B is a graded subalgebra that is also a Frobenius graded superalgebra, and A is projective as a left B-module, then A is a twisted Frobenius extension of B.     

The maximum proportionally modular numerical semigroup with given multiplicity and ratio

We prove that the set of all proportionally modular numerical semigroups with fixed multiplicity and ratio has a maximum (with respect to set inclusion). We show that this maximum is a maximal embedding dimension numerical semigroup, for which we explicitly calculate its minimal system of generators, Frobenius number and genus.     

On the discrepancy of inversive congruential pseudorandom numbers with prime power modulus,II

One of the alternatives to linear congruential pseudorandom number generators with their known deficiencies is the inversive congruential method with prime power modulus. Recently, it was proved that pairs of inversive congruential pseudorandom numbers have nice statistical independence properties. In the present paper it is shown that a similar result cannot be obtained fork-tuples withk≥3 since their discrepancy is too large. The method of proof relies on the evaluation of certain exponential sums. In view of the present result the inversive congruential method with prime power modulus seems to be not absolutely suitable for generating uniform pseudorandom numbers.     

Enumeration of labelled threshold graphs and a theorem of frobenius involving eulerian polynomials

We enumerate labelled threshold graphs by the number of vertices, the number of isolated vertices, and the number of distinct vertex-degrees and we give the exact asymptotics for the number of labelled threshold graphs withn vertices. We obtain the appropriate generating function and point out a combinatorial interpretation relating its coefficients to the Stirling numbers of the second kind. We use these results to derive a new proof of a theorem of Frobenius expressing the Eulerian polynomials in terms of the Stirling numbers.     

Generic initial ideals of Artinian ideals having Lefschetz properties or the strong Stanley property

For a standard Artinian k-algebra A=R/I, we give equivalent conditions for A to have the weak (or strong) Lefschetz property or the strong Stanley property in terms of the minimal system of generators of gin(I). Using the equivalent condition for the weak Lefschetz property, we show that some graded Betti numbers of gin(I) are determined just by the Hilbert function of I if A has the weak Lefschetz property. Furthermore, for the case that A is a standard Artinian k-algebra of codimension 3, we show that every graded Betti number of gin(I) is determined by the graded Betti numbers of I if A has the weak Lefschetz property. And if A has the strong Lefschetz (respectively Stanley) property, then we show that the minimal system of generators of gin(I) is determined by the graded Betti numbers (respectively by the Hilbert function) of I.