On numerical semigroups and the order bound

Let S={s₀=0<s₁<?<s_i…}⊆N be a numerical non-ordinary semigroup; then set, for each . We find a non-negative integer m such that d_ORD(i)=ν_i+1 for i≥m, where d_ORD(i) denotes the order bound on the minimum distance of an algebraic geometry code associated to S. In several cases (including the acute ones, that have previously come up in the literature) we show that this integer m is the smallest one with the above property. Furthermore it is shown that every semigroup generated by an arithmetic sequence or generated by three elements is acute. For these semigroups, the value of m is also found.     

Trace theorems for holomorphic semigroups and the second order Cauchy problem

We use the theory of boundary values (also called traces) of holomorphic semigroups as developed by Boyadzhiev-deLaubenfels (1993) and El-Mennaoui (1992) to study the second order Cauchy problem for certain generators of holomorphic semigroups. Our results contain in particular the result of Hieber (Math. Ann. 291 (1991), 1--16) for the Laplace operator on .

     

Groups of order 24 and their endomorphism semigroups

It is proved that among the finite groups of order 24 only the binary tetrahedral group is not determined by its endomorphism semigroup in the class of all groups. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 3, pp. 155–172, 2005.     

Minimal order semigroups with specified commuting probability

We determine the minimal order of a semigroup whose commuting probability equals any specified rational value in (0,1].     

The order bound for general algebraic geometric codes

The order bound gives an in general very good lower bound for the minimum distance of one-point algebraic geometric codes coming from curves. This paper is about a generalization of the order bound to several-point algebraic geometric codes coming from curves.     

The semigroups of order 9 and their automorphism groups

We report the number of semigroups with 9 elements up to isomorphism or anti-isomorphism to be 52?989?400?714?478 and up to isomorphism to be 105?978?177?936?292. We obtained these results by combining computer search with recently published formulae for the number of nilpotent semigroups of degree 3. We further provide a complete account of the automorphism groups of the semigroups with at most 9 elements. We use this information to deduce that there are 148?195?347?518?186 distinct associative binary operations on an 8-element set and 38?447?365?355?811?944?462 on a 9-element set.     

Centralizers in symmetric inverse semigroups: Structure and order

The representation [5] of the centralizerC(x) of a permutationx in (a symmetric inverse semigroup)C _n involves direct products of wreath products. Indeed, this semigroup case extends its group theory counterpart. Here, the last case (forx nilpotent) is addressed: A quotient of a wreath product is introduced and used to obtain a representation of the correspondingC(x). It follows that, for anyx∈C _n ,C(x) can be imbedded in a direct product of wreath products with a quotient of a wreath product. A formula for calculating the order ofC(x) is given. The independent parameters in the formula are precisely those that define the path structure ofx∈C _n . Part of this research was supported by a Mary Washington College faculty development grant.     

Growth order and stability of semigroups and cosine operator functions

Equivalent conditions for a trajectory of a C₀-semigroup T(⋅) (resp. cosine function C(⋅)) of operators to have the growth order O(tα) or o(tα) are expressed in terms of Cesàro and Abel means of the norm of the trajectory. We then deduce characterizations of growth order and stability for T(⋅) and C(⋅). It is also shown that under some Tauberian condition the uniform boundedness (resp. strong convergence) of T(⋅) is equivalent to the uniform boundedness (resp. strong convergence) of its Abel means.     

The number of commutative semigroups of order N

An explicit upper bound is given for the number of commutative semigroups withn elements.     

Dual semigroups and second order linear elliptic boundary value problems

It is shown that general second order elliptic boundary value problems on bounded domains generate analytic semigroups onL ₁. The proof is based on Phillips’ theory of dual semigroups. Several sharp estimates for the corresponding semigroups inL _p, 1≦p<∞, are given.     

An upper bound for the order of primitive permutation groups

Let G be a primitive permutation group of order |G| and degree n. Then |G|≤nd^m, where d is the minimal size of a nontrivial orbit of a one-point stabilizer of G and m is the minimal degree of a nonprincipal irreducible representation of G entering its permutation representation. Bibliography: 8 titles. Dedicated to L. D. Faddeev on occasion of his 60th birthday Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 215, 1994, pp. 256–263. Translated by I. Ponomarenko.     

An improved upper bound for the order of mixed graphs

A mixed graph $G$ can contain both (undirected) edges and arcs (directed edges). Here we derive an improved Moore-like bound for the maximum number of vertices of a mixed graph with diameter at least three. Moreover, a complete enumeration of all optimal $(1,1)$-regular mixed graphs with diameter three is presented, so proving that, in general, the proposed bound cannot be improved.     

On the spectral and growth bound of semigroups associated with hyperbolic equations

Some new and quite general conditions are presented to ensure equality of the spectral bound and the growth bound for those strongly continuous semigroups governing second-order (in time) equations in a Banach (or Hilbert) space. A counterexample arising from higher-order PDEs is given to show that the growth bound could be arbitrarily large while the spectral bound remains zero.