Symmetric Semigroup Research Articles

Dynamical stability of metastable states

We continue our study of symmetric Markov semigroups for which the zero eigenvalue is almost, but not quite, degenerate. We show that the metastable states of the semigroup are dynamically stable under small perturbations, without any of the technical restrictions or approximations of our earlier papers.

Supports of holomorphic convolution semigroups and densities of symmetric convolution semigroups on a locally compact group

Supports of holomorphic convolution semigroups and densities of symmetric convolution semigroups on a locally compact group

Construction of a unique self-adjoint generator for a symmetric local semigroup

A definition is given of a symmetric local semigroup of (unbounded) operators P( t) (0 ⩽ t ⩽ T for some T > 0) on a Hilbert space H , such that P( t) is eventually densely defined as t → 0. It is shown that there exists a unique (unbounded below) self-adjoint operator H on H such that P( t) is a restriction of e − tH . As an application it is proven that H 0 + V is essentially self-adjoint, where e − tH 0 is an L p -contractive semigroup and V is multiplication by a real measurable function such that V ∈ L 2 + ε and e − δV ∈ L 1 for some ε, δ > 0.

A general resolution for grade four Gorenstein ideals

Gorenstein rings occur in a multitude of different guises: as rings of invariants, as coordinate rings of certain determinantal varieties and symmetric semigroup curves, as representatives of linkage classes, and so on. In an attempt to unify this motley collection of examples (at least for finite projective dimension) one seeks a generic free resolution whose specializations yield all examples of given embedding codimension. The present paper describes a resolution for codimension four, not generic, but general enough to encompass many diverse examples. The structure of this resolution is intimately related to the differential, graded, commutative algebra that it supports, and to the deformation theory of codimension four Gorenstein algebras. These ideas are brought together in the determination of the singular locus of certain codimension four Gorenstein varieties. More generally they suggest a classification of codimension four Gorenstein rings that begins to impose some order on the examples.

Order relations on symmetric semigroups of transformations and on their homomorphic images

Order relations on symmetric semigroups of transformations and on their homomorphic images

Unbounded, symmetric semigroups on a separable Hilbert space are essentially selfadjoint

Unbounded, symmetric semigroups on a separable Hilbert space are essentially selfadjoint

Dissipators for symmetric quasi-free dynamical semigroups on the CAR algebra

Let L be a negative self-adjoint bounded operator on a Hilbert space H , and p a projection on H with pLp trace class, and let { T t : t ⩾ 0} be the extension of { e tL : t ⩾ 0} to a strongly continuous semigroup of completely positive quasi-free unital maps of Fock type on the fermion algebra A H built over H . Then it is shown that there exists a strongly continuous self-adjoint contraction semigroup { G t : t ⩾ 0} on the Hilbert space of the GNS decomposition of the quasi-free state gw p such that in the representation of that state: T t ⩾ G t (·) G t , t ⩾0.

Inverse elements and their average number in a finite symmetric semigroup

Inverse elements and their average number in a finite symmetric semigroup

Monomial Gorenstein ideals

The paper concerns itself with generating sets for monomial Gorenstein ideals in polynomial rings k[x1,..., xr], k an arbitrary field. For r=5 it is shown that for a certain class of these ideals, the number of generators is bounded by 13. To establish the sharpness of this bound an algorithm is established, to obtain all numerical symmetric semigroups with a fixed odd integer 2n+1 as last integer unattained. Finally, a short proof of the known fact is given, that for r=4 the number of elements in a generating set is 3 or 5.

Automaton program realization of substitutions of symmetric semigroup. II

Automaton program realization of substitutions of symmetric semigroup. II

Iterative procedures for factorizing elements of symmetric semigroups

The problem of representing an arbitrary mapping o f a set A = {a~, az, ...,aN} into itself by means of a product o f mappings in a given generating set R is known as the Glushkov problem [1 ]. The solution of this problem is dealt with in [2-6], where algorithms for its solution are proposed for certain special, in the main minimal, R and the number of factors in the solutions are estimated. The choice of R in this connection is not related to technical realizations.

Symmetric semigroups of integers generated by 4 elements

Let K be an arbitrary field, t transcendental over over K. It is shown that if the numerical semigroup of nonnegative integers, generated nonredundantly by 4 elements, is symmetric, then the prime ideal of all polynomials f(x1,x2,X3,x4) e K[x1,x2,x3,x4] such that\(f(t^{n_1 } ,t^{n_2 } ,t^{n_3 } ,t^{n_4 } ) = 0\) is generated by 3 or 5 elements. From this, arithmetic conditions for the generators are obtained.

The asymptotic distribution of the order of elements in symmetric semigroups

P. Erdös and P. Turán [8] ( Acta Math. Acad. Sci. Hungar., 18 (1967) , 309–320) have shown that, if K( n, x) is the number of elements P in S n , the symmetric group on n letters, whose order O( P) satisfies logO(P) ⩽ 1 2 log 2n + ( 1 3 ) x log 3 2 n then lim n→∞ K(n,x) n! = ( 2π ) −1 ∫ x −∞ e −t 2 2 dt. In this paper the analogous result for the symmetric semigroup is obtained. Let α ϵ T n , the symmetric semigroup on n letters (the set of all mappings of {1, 2,…, n} into {1, 2,…, n}) and let O( α) be the order of α. If L( n, x) is the number of α ϵ T n with log O(α)⩽ 1 8 log 2n +(− 1 24 ) x log 3 2 n, then lim n→∞ L(n,x) n n = ( 2π ) −1 ∫ x ∞ e −t 2 2 dt.

On complexity of representing permutations of symmetric semigroups in terms of the elements of the system of generators

On complexity of representing permutations of symmetric semigroups in terms of the elements of the system of generators

Upper bound of the informational redundancy measure of a symmetric semigroup

Upper bound of the informational redundancy measure of a symmetric semigroup

Algorithm of decomposition in finite symmetric semigroups

Algorithm of decomposition in finite symmetric semigroups

Idempotents in symmetric semigroups

We count the number of idempotent elements in a certain section of the s symmetric semigroup S n on n letters. As a corollary of our result we have that every maximal principal right ideal of S n contains ∑ i=1 n−1 i n−i−1 ( n−2 i−1 )+( n−1 i ) idempotent elements. Let T r (1 ⩽ r ⩽ n − 1) be the set of all elements of S n of rank less than or equal to r, and let D r denote the set of all elements of S n of rank r. Then T r is a semigroup generated by the idempotent elements of D r . We shall obtain a maximal mutant of T n−1 = S n D n .

Counting elements in semigroups

The objective of this communication is to show that Tainiter's technique of characterizing and counting idempotent elements in a symmetric semigroup can be strengthened and generalized to apply to the characterization and counting of all elements of any symmetric semigroup and in fact of any symmetric m-semigroup. Some of the counting formulas obtained are quite complicated. Their computational aspects and the combinatorial analysis of their generating functions will be left for a later paper.

Semigroups Corresponding to Algebroid Branches in the Plane

The symmetric semigroups of nonnegative integers and their generators, corresponding to algebroid branches of the plane, are determined.

Semigroups corresponding to algebroid branches in the plane

The symmetric semigroups of nonnegative integers and their generators, corresponding to algebroid branches of the plane, are determined.