Systeme von längenmengen

LetH be a semigroup with divisor theory and finite divisor class groupG such that every class contains a prime divisor (e.g.H the multiplicative semigroup of the ring of integers of an algebraic number field). For an elementa εH / H ^× letL(a) =k a has a factorization into irreducibles of lengthk}. In this paper the system ?(H) = {L(a)|a εH / H ^× is studied.     

On uniformly repetitive semigroups

Letk be an integer greater than 1 andS be a finitely generated semigroup. The following propositions are equivalent: 1) the semigroup of non negative integers is not uniformlyk-repetitive; 2) any finitely generated and uniformlyk-repetitive semigroup is finite. As a consequence we prove that any finitely generated and uniformly 4-repetitive semigroup is finite.     

On the stability of differentiability of semigroups

LetA be the infinitesimal generator of aC ₀-semigroup. The semigroup generated byA is called differentiable ifA exp (At) is bounded for everyt>0. In this note, an example is given of an operatorA and a bounded operatorB such that the semigroup generated byA is differentiable but the semigroup generated byA+B is not. This gives a negative answer to a question of Pazy.     

Cusp forms

LetG andH ⊂G be two real semisimple groups defined overQ. Assume thatH is the group of points fixed by an involution ofG. Letπ ⊂L ²(H\G) be an irreducible representation ofG and letf επ be aK-finite function. Let Γ be an arithmetic subgroup ofG. The Poincaré seriesP _f(g)=Σ_H∩ΓΓ f(γ{}itg) is an automorphic form on Γ\G. We show thatP _f is cuspidal in some cases, whenH ∩Γ\H is compact. Partially supported by NSF Grant # DMS 9103608.     

A linearity theorem for group actions on spheres with applications to homotopy representations

LetG be a finite group andX an equivariantZ/|G|-homology sphere. By Smith-theory the fixed point setX ^H for ap-subgroupH is aZ/p-homology sphere of dimensiond(H).     

A note on the abstract cauchy problem

LetA be the infinitesimal generator of aC ₀ semigroup in a Banach spaceE. We obtain necessary conditions for a solution of the Cauchy problem {fx112-1} to be classical for arbitrary ϕ εC([0,T]) andf εE.     

Arithmetics of the numbers of orbits of the Fermat–Euler dynamical systems

Given positive integers a and n with (a,n)=1, we consider the Fermat–Euler dynamical system defined by the multiplication by a acting on the set of residues modulo n relatively prime to n. Given an integer M>1, the integers n for which the number of orbits of this dynamical system is a multiple of M form an ideal in the multiplicative semigroup of odd integers. We provide new results on the arithmetical properties of these ideals by using the topological properties of some directed graphs (the monads).      

Constraints on the angular distribution of the zeros of a polynomial of low complexity

LetP(z) be a polynomial of degreen with complex coefficients. Theorem.There exists a constant C such that if P has at most k terms then the number of zeros of P in any open sector of aperture π/n at the origin is no more than C ^k2. The main point of this bound is that it is independent both ofn and the coefficients ofP. The proof is a simple application of Khovanskii's real Bezout theorem to the system of real equations ReP=ImP=0. We also describe a measure of additive complexity for sets of integers and use it to estimate the angular distribution more finely.     

On the multiplicative representation of integers

Leta ₁<a ₂<··· be an infinite sequence of integers. Denote byg(n) the number of solutions ofn=a _i···a _j. Ifg(n)>0 for a sequencen of positive upper density then lim supg(n)=∞. Dedicated to my friend A. D. Wallace on the occasion of his 60th birthday.     

On Crawley Modules

A curve Xis said to be of type (iV, γ) if it is an iV-sheeted covering of a curve of genus γ with at least one totally ramified point. A numerical semigroup His said to be of type (iV, γ) if it has γ positive multiples of Nin [N, 2N_J] such that its γth element is 2Nγ and (2γ+1)NεH. If the genus of X is large enough and N is prime, X is of type (TV, γ) if and only if there is a point P6 X such that the Weierstrass semigroup at P is of type (N, γ) (this generalizes the case of double coverings of curves). Using the proof of this result and the Buchweitz's semigroup, we can construct numerical semigroups that cannot be realized as Weierstrass semigroups although they might satisfy Buchweitz's criterion     

Projective and indecomposableS-acts

LetS be a semigroup andE the set of all idempotents inS. LetS-Act be the category of allS-acts. LetC be a full subcategory ofS-Act which contains_s S and is closed under coproducts and summands. It is proved that, inC, anS-actP is projective and unitary if and only ifP≅ ∐ _{jϕ I} Se _i ,e _i ϕE. In particular,P is a projective, indecomposable and unitary object if and only ifP ≅Se for somee ∈E. These generalize some results obtained by Knauer and Talwar. Research partially supported by a UGC (HK) (Grant No. 2160092).     

Projective and indecomposableS-acts

LetS be a semigroup andE the set of all idempotents inS. LetS-Act be the category of allS-acts. LetC be a full subcategory ofS-Act which contains_s S and is closed under coproducts and summands. It is proved that, inC, anS-actP is projective and unitary if and only ifP≅ ∐ _{jϕ I} Se _i ,e _i ϕE. In particular,P is a projective, indecomposable and unitary object if and only ifP ≅Se for somee ∈E. These generalize some results obtained by Knauer and Talwar.     

Generators for the derivation modules and the relation ideals of certain curves

LetO be a curve in the affine algebroide-space over a fieldK of characteristic zero. LetD be the module ofK-derivations andP the relation ideal ofO. Generators forD andP are computed in several cases. It is shown in particular that in the case of a monomial curve defined by a sequence ofe positive integers somee−1 of which form an arithmetic sequence, μO ≤ 2e - 3 and μ(P)≤e(e−1)/2.     

Ordered semigroups having the <Emphasis Type="Italic">P</Emphasis>-property

In this paper we obtain the following main results. The ordered semigroups which have the P-property are decomposable into archimedean semigroups. Moreover, they are decomposable into semigroupswith the P-property. Conversely, if an ordered semigroup S is a complete semilattice of semigroups which have the P-property, then S itself also has the P-property. An ordered semigroup is CS-indecomposable and has the P-property if and only if it is archimedean. If S is an ordered semigroup, then the relation N:= {(a, b) | N(a) = N(b)} (here N(a) is a filter of S generated by a (a ∈ S)) is the least complete semilattice congruence on S and the class (a) _N is a CS-indecomposable subsemigroup of S for each a ∈ S. We introduce the notion of the P _m -property and describe it in terms of the P-property. Our approach simplifies the proofs of the corresponding results about unordered semigroups. The text was submitted by the authors in English.     

Retractability and the fixed point property for products

LetP, Q be ordered sets and letaP. IfP \ {a} is a retract ofP and setsP and {xP:x>p} (or its dual) have the fixed point property then, for each chain complete setP,P×Q has the fixed point property if and only if (P\{a})×Q has this property. This establishes the fixed point property for some products of ordered sets which are beyond the reach of all known product theorems.The work of the first of authors was supported in part by the K.B.N. Grant No. 2 2037 92 03.     

On K-theoretic invariants of semigroup C*-algebras attached to number fields

We show that semigroup C*-algebras attached to ax+b$ax+b$-semigroups over rings of integers determine number fields up to arithmetic equivalence, under the assumption that the number fields have the same number of roots of unity. For finite Galois extensions, this means that the semigroup C*-algebras are isomorphic if and only if the number fields are isomorphic.     

Proof of a conjecture of T. Gallai concerning connectivity properties of colour-critical graphs

Tibor Gallai made the following conjecture. LetG be ak-chromatic colour-critical graph. LetL denote the set of those vertices ofG having valencyk−1 and letH be the rest ofV(G). Then the number of components induced byL is not less than the number of components induced byH, providedL ≠ 0. We prove this conjecture in a slightly generalized form. Dedicated to Tibor Gallai on his seventieth birthday     

Amitsur cohomology of cubic extensions of algebraic integers

LetK be the rational fieldQ or a complex quadratic number field other than . LetL be a normal three-dimensional field extension onK. IfR andS are the rings of algebraic integers ofK andL respectively, then the Amitsur cohomology groupH ² (S/R, U) is trivial. Inflation and class numbers give information about cohomology arising from certain nonnormal cubic extensions. This work was supported in part by NSF Grant GP-28409.     

No distribution is prime

Summary Letℱ denote the convolution semigroup of probability distributions on the real line. We prove thatno element of ℱ is prime in the sense that given anℱ one can always find two distributionsG,H∈ℱ such thatF is a convolution factor ofG#x002A;H but neither ofG nor ofH. In contrast,ℱ is known to possess many irreducible elements.     

Symmetrizations of Markov processes

Two methods for symmetrizing Markov processes are discussed. Letu ^a(x, y) be the potential density of a Lévy process on a compact Abelian groupG. A general condition is given that guarantees thatv(x, y)=u^a(x, y)+u^a(y, x) is the potential density of a symmetric Lévy process onG. The second method arises by considering the linear space of one-potentialsU ¹ f, withf inL ², endowed with the inner product (U ¹ f,U ¹ g)=fU ¹ g+gU ¹ f. If the semigroup ofX(t) is normal, then the completionH of this space is the Dirichlet space of a symmetric processY(t). A set that is semipolar forX(t) is polar forY(t).