Stein extensions of Riemannian symmetric spaces and dualities of orbits on flag manifolds

It is known [M4] that K_ℂ-orbits S and G_ℝ-orbits S' on a complex flag manifold are in one-to-one correspondence by the condition that S ∩ S' is nonempty and compact. It is possible to replace K_ℂ by some conjugate xK_ℂx⁻¹ so that the correspondence is preserved. We investigate the sets C(S) of such x for various orbits S and their relations with each other. We prove that for classical groups the intersection C = ∩_S C(S) equals D₀Z where D₀ = D₀/K_ℂ is the universal domain in G_ℂ/K_ℂ introduced in [AG] and Z is the center of G. As a corollary we prove that D₀ is Stein for classical groups. Moreover we conjecture that C(S)₀ = D₀ for generic S where C(S)₀ is the connected component of C(S) containing the identity.     

On comparing two chains of numerical semigroups and detecting arf semigroups

If T is a numerical semigroup with maximal ideal N , define associated semigroups B(T):=(N-N) and L(T)= \cup { (hN-hN) \colon h \geq 1 } . If S is a numerical semigroup, define strictly increasing finite sequences { B _i (S) \colon 0 ≤ i ≤β (S) } and { L _i (S) \colon 0 ≤ i ≤λ (S) } of semigroups by B ₀ (S):=S=:L ₀ (S) , B _{β (S)} (S):= \Bbb N =: L _{λ (S)} (S) , B _i+1 (S):=B(B _i (S)) for 0<i< β (S) , L _i+1 (S):=L(L _i (S)) for 0<i< λ (S) . It is shown, contrary to recent claims and conjectures, that B ₂ (S) need not be a subset of L ₂ (S) and that β (S) - λ (S) can be any preassigned integer. On the other hand, B ₂ (S) \subseteq L ₂ (S) in each of the following cases: S is symmetric;S has maximal embedding dimension;S has embedding dimension e(S) ≤ 3 . Moreover, if either e(S)=2 or S is pseudo-symmetric of maximal embedding dimension, then B _i (S) \subseteq L _i (S) for each i , 0 ≤ i ≤λ (S) . For each integer n \geq 2 , an example is given of a (necessarily non-Arf) semigroup S such that β (S) = λ (S)=n , B _i (S) = L _i (S) for all 0 ≤ i ≤ n-2 , and B _n-1 (S) \subsetneqq L _n-1 (S) . April 4, 2000     

Approximate Amenability of Certain Semigroup Algebras

In the present paper, it is shown that a left cancellative semigroup S (not necessarily with identity) is left amenable whenever the Banach algebra ℓ¹(S) is approximately amenable. It is also proved that if S is a Brandt semigroup over a group G with an index set I, then ℓ¹(S) is approximately amenable if and only if G is amenable. Moreover ℓ¹(S) is amenable if and only if G is amenable and I is finite. For a left cancellative foundation semigroup S with an identity such that for every M_a(S)-measurable subset B of S and s ∈ S the set sB is M_a(S)-measurable, it is proved that if the measure algebra M_a(S) is approximately amenable, then S is left amenable. Concrete examples are given to show that the converse is negative.     

Regular approximations of singular Sturm-Liouville problems

Given any self-adjoint realization S of a singular Sturm-Liouville (S-L) problem, it is possible to construct a sequence {S_r{ of regular S-L problems with the properties (i) every point of the spectrum of S is the limit of a sequence of eigenvalues from the spectrum of the individual members of {S_r{ (ii) in the case when 5 is regular or limit-circle at each endpoint, a convergent sequence of eigenvalues from the individual members of {S_r{ has to converge to an eigenvalue of S (iii) in the general case when S is bounded below, property (ii) holds for all eigenvalues below the essential spectrum of S.     

Arens Regularity of Semigroup Algebras with Certain Locally Convex Topologies

The authors have recently introduced and studied a locally convex topology β¹(S) on the semigroup algebra M_a(S) of a locally compact semigroup S; as the main result, they showed that the strong dual of (M_a(S),β¹(S)) can be identified with the Banach space L^∞₀(S,M_a(S)) for a large class of locally compact semigroups S. Here, an application of this result is made to define and investigate an Arens multiplication on the second dual of (M_a(S),β¹(S)).     

On the Relation between the Scalar Moment Problem and the Matrix Moment Problem on *-Semigroups

There is a countable cancellative commutative *-semigroup S withzero (in fact, a *-subsemigroup of G × N₀ for some abelian group G carrying the inverse involution) such that the answer to the question “if f is a function on S , with values in M_d(C) (the square matrices of order d) and such that $\sum^{n}_{j,k=1} \lbrak f(s^*_k s_j)\xi_j, \xi_k \rbrak \ge 0$ for all n in N, s₁, . . . , s_n in S , and $\xi_1$, . . . , $\xi_n$ in C^d, does it follow that $f(s) = \int_{S^*}\sigma (s) d\mu(\sigma) (s \memb S)$ for some measure $\mu$ (with values in M_d(C)₊ , the positive semidenite matrices) on the space S of hermitian multiplicative functions on S?” is “yes” if d = 1 but “no” if d = 2 (hence also for d > 2).     

A counterexample in the cohomology of monoids

Let S be a commutative monoid, and 1et T be the maximal cancellative homomorphic image of S. The cohomology of S with coefficients in a left Z(S)-module D is given by Hⁿ(S,D)=Ext _Z(S) ⁿ (Z,D) as usual. [3] If from D we form the Z(T)-module D'=Hom_Z(S)(Z(T),D), then there is a natural homomorphism Hⁿ(T,D')→Hⁿ(S,D). If S is finite, then it follows from the results of [1] and [4] that this homomorphism is an isomorphism. The example presented below shows that this need no longer be an isomorphism if S is infinite. This research was supported in part by the National Science Foundation.     

Sublattices of the Lattices of Strong P-Congruences on P-Inversive Semigroups

In this paper, we describe strong P-congruences and sublattice-structure of the strong P-congruence lattice C_P(S) of a P-inversive semigroup S(P). It is proved that the set of all strong P-congruences C_P(S) on S(P) is a complete lattice. A close link is discovered between the class of P-inversive semigroups and the well-known class of regular ⋆-semigroups. Further, we introduce concepts of strong normal partition/equivalence, C-trace/kernel and discuss some sublattices of C_P(S). It is proved that the set of strong P-congruences, which have C-traces (C-kernels) equal to a given strong normal equivalence of P (C-kernel), is a complete sublattice of C_P(S). It is also proved that the sublattices determined by C-trace-equaling relation θ and C-kernel-equaling relation κ, respectively, are complete sublattices of C_P(S) and the greatest elements of these sublattices are given.     

On the brauer group of a designularization of a normal surface 1

If ‰ : ? S → is a desingularization of the norm3 surface S, then it is shown that the induced map H² _et:(S, G_m) → H^2: _et(?, G_m) is surjective. It

follows that if all of the singularities of S are rational, the Brauer group

map B(S) → B(?) is surjective. An example is given to show that this

property fails if the dimension of S ≥ 3.     

Genus number and l-rank of genus group of cyclic extensions of degree l

Let l be an odd prime, and k an algebraic number field of a finite degree. Let S be a finite product of distinct prime ideals g of k such that Ng1 (mod l). Let I(s) (resp. P(s)) denote the group of ideals (resp. principal ideals) of k prime to S, and let P_S denote the ray modulo S. In this paper we prove that the order (resp. the l-rank) of I(S)/P(S)^lP_S is expressed by the decomposition groups of prime factors of S in a Galois extension K_o (resp. K_r) over k. As an application of this, some results about genus theory are obtained.     

On the adjunction process over a surface in char.p

Let K_s be the canonical bundle on a non singular projective surface S (over an algebraically closed field F, char F=p) and L be a very ample line bundle on S. Suppose (S,L) is not one of the following pairs: (P²,O(e)), e=1,2, a quadric, a scroll, a Del Pezzo surface, a conic bundle. Then

1. (K_s?L)² is spanned at each point by global sections. Let \(\phi :S \to P^N _F \) be the map given by the sections Γ(K_s?L)², and let φ=s o r its Stein factorization.
2. r:S→S′=r(S) is the contraction of a finite number of lines, E_i for i=1,...r, such that E_i·E_i=K_S·E_i=?L·E_i=?1.
3. If h°(L)≥6 and L·L≥9, then s is an embedding.

     

The cleanness of (symbolic) powers of Stanley-Reisner ideals

Let Δ be a pure simplicial complex on the vertex set [n] = {1,..., n} and I _Δ its Stanley-Reisner ideal in the polynomial ring S = K[x ₁,..., x _n]. We show that Δ is a matroid (complete intersection) if and only if S/I _Δ ^(m) (S/I _Δ ^(m)) is clean for all m ∈ N and this is equivalent to saying that S/I _Δ ^(m) (S/I _Δ ^(m), respectively) is Cohen-Macaulay for all m ∈ N. By this result, we show that there exists a monomial ideal I with (pretty) cleanness property while S/I ^m or S/I ^m is not (pretty) clean for all integer m ≥ 3. If dim(Δ) = 1, we also prove that S/I _Δ ⁽²⁾ Δ (S/I _Δ ²) is clean if and only if S/I _Δ ⁽²⁾ (S/I _Δ ², respectively) is Cohen-Macaulay.     

两个模糊子半群集合之间的同态

设S,T是半群,F(S)和F_s(S)分别表示S的所有模糊子集的集合和所有模糊子半群的集合。文中,讨论了F(S)(F_s(S))和F(T)(F_s(T))之间的模糊同态,建立了模糊商子半群的概念,把分明半群的基本同态定理推广到模糊子半群。     

几乎优越扩张与同调维数

设环S是环R的几乎优越扩张．本文证明了R和S具有相同的f．f．P．维数以及finitistic维数．若M_S是右S－模,则FP－id（M_S）＝FP－id（M_R）．若G是有限群,R是G分次环且|G|^－1∈R,则Smash积R＃G^*和R具有相同的f．f．P．维数,finitistic维数,以及FP－整体维数．     

Star partitions of graphs

Let G be a graph and n ≥ 2 an integer. We prove that the following are equivalent: (i) there is a partition (V₁,…,V_m) of V (G) such that each V_i induces one of stars K_1,1,…,K_1,n, and (ii) for every subset S of V(G), G\ S has at most n|S| components with the property that each of their blocks is an odd order complete graph. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 185–190, 1997     

Sull'omotopia uniforme

Summary Following in the classical theory's footsteps, it is possible to construct a ? rational homotopy theory ?. To this purpose, we take all the uniform maps f: I _Q ⁿ →S (where I_Q is the closed unit interval of the rational line equipped with the standard metric uniformity) as paths of a uniform space S. This brings us to the definition of the rational homotopy groups Q_n(S, U_Q, x) and enables us to consider the related exact sequences. All these objects are uniform invariants. The main problem which arises now is to find a suitable space S* in order that its classical homotopy groups Π_n(S*,x) are isomorphic to the classical ones of S. We are able to reach an answer to this question if S is a metrizable uniform space. Since considering the completion Ŝ of S serves no useful purpose (as it is shown by a simple example), we prove that the required space is the ? rational path completion ? S* of S, with S⊆S*⊆Ŝ. We finally recall that the rational uniform homotopy is a special case of regular homotopy, which has been defined and widely investigated in[1].

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Entrata in Redazione il 5 gennaio 1978.

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Lavoro svolto nell'ambito del gruppo GNSAGA del CNR.     

GEODETIC GAMES FOR GRAPHS

ABSTRACT

Let S be a subset of the vertex set V(G) of a nontrivial connected graph G. The geodetic closure (S) of S is the set of all vertices on geodesics between two vertices in S. The first player A chooses a vertex v₁ of G. The second player B then picks v₂ ≠ v₁ and forms the geodetic closure (S₂) = ({v₁, v₂}). Now A selects v₃ ε V—S₂ and forms (S₃) = ({v₁, v₂, v₃}), etc. The player who first selects a vertex v_n such that (S_n) = V wins the achievement game, but loses the avoidance game. These geodetic achievement and avoidance games are solved for several families of graphs by determining which player is the winner.     

General symbolic powers: A homological point of view

Let I be an ideal of a Noetherian ring R and let S be a multiplicatively closed subset of R. We define the n-th (S)-symbolic power of 7 as S(Iⁿ) = IⁿR_s ∩R. The purpose of this paper is to compare the topologies defined by the adic {I_n}_n≤0 and the (S)-symbolic filtration {S(Iⁿ)}n≥o using the direct system {Extⁱ _R(R/Iⁿ,R)}_n≥0