Sets with a mode

LetM be a point andS be a compact set inR ² such thatS is the closure of its interior. The theorem desired says that ifM is a mode ofS thenS is convex and centrally symmetric with respect toM. Some conditions on the boundary ofS are needed for the proof given.     

ON STABLE LINE SEGMENTS IN ALL TRIANGULATIONS OF A PLANAR POINT SET

Abstract. Let S be a set of finite plauar points. A llne segment L(p, q) with p, q E Sis called a stable line segment of S, if there is no Line segment with two endpoints in S intersecting L(p, q). In this paper, some geometric properties of the set of all stable line segments     

The rational loewy series and nilpotent ideals of endomorphism rings

Sufficient conditions are given, in module-theoretic terms, for the idealN(S) of the endomorphism ringS of a moduleM consisting of the endomorphisms with essential kernel to be nilpotent. This extends in a natural way several known results on the nilpotency ofN(S). WhenM is a quasi-injective module such thatS is right noetherian, it is shown thatS is right artinian if and only ifM has a finite rational Loewy series whose length is, in this case, equal to the index of nilpotency ofN(S). The author has been partially supported by the CAICYT.     

Upper bounds of small Eigenvalues of the Dirac operator and isometric immersions

We show that a topologically determined number of eigenvalues of the Dirac operatorD of a closed Riemannian spin manifoldM of even dimensionn can be bounded by the data of an isometric immersion ofM into the Euclidian spaceR ^N . From this we obtain similar bounds of the eigenvalues ofD in terms of the scalar curvature ofM ifM admits a minimal immersion intoS ^N or,ifM is complex, a holomorphic isometric immersion intoPC ^N .     

The rigidity of Clifford torusS^1 \left( {\sqrt {\frac{1}{n}} } \right) \times S^{n - 1} \left( {\sqrt {\frac{{n - 1}}{n}} } \right)

In this paper, we prove that ifM is ann-dimensional closed minimal hypersurface with two distinct principal curvatures of a unit sphereS ⁿ⁺¹ (1), thenS=n andM is a Clifford torus ifn≤S≤n+[2n ²(n+4)/3(n(n+4)+4)], whereS is the squared norm of the second fundamental form ofM.     

A characterization of convexity and central symmetry for planar polygonal sets

LetS be a compact set inR ² with nonempty interior,L(u,k) be a line 〈u, x〉 =k, and ζ _u (k) be the linear Lebesgue measure ofS∩L(u,k). It is well known that for a convexS, ζ _u (k) is unimodal, that is, as a function ofk, it is first non-decreasing and then nonincreasing for everyu∈R ². Further, ifS is centrally symmetric with respect toM, ζ _u (k) achieves maximum whenL(u, k) passes throughM. Converse propositions are considered in this paper for polygonalS with Jordan boundary. It is shown that unimodality alone does not suffice for convexity. However, if ζ _u (k) achieves maximum wheneverL(u, k) passes through some fixed pointM then unimodality yields convexity as well as central symmetry. It is also shown that continuity of ζ _u (k) in the interior of its support implies convexity ofS. This last result, however, is false for non-polygonal sets. Research supported by National Science Foundation Grant GP-28154.     

On the normal bundle of a complex submanifold of a locally conformal Kaehler manifold

We investigate the existence of parallel sections in the normal bundle of a complex submanifold of a locally conformal Kaehler manifold with positive holomorphic bisectional curvature. Also, ifM is a quasi-Einstein generalized Hopf manifold then we show that any complex submanifoldM with a flat normal connection ofM is quasi-Einstein, too, provided thatM is tangent to the Lee field ofM. As an application of our results we study the geometry of the second fundamental form of a complex submanifold in the locally conformal Kaehler sphereQ _m (of a complex Hopf manifoldS ^2m+1 ×S ¹).     

Conormal bundles with vanishing Maslov form

IfM is a Riemannian manifold, andL is a Lagrangian submanifold ofT ^* M, the Maslov class ofL has a canonical representative 1-form which we call theMaslov form ofL. We prove that ifL =v ^* N = conormal bundle of a submanifoldN ofM, its Maslov form vanishes iffN is a minimal submanifold. Particularly, ifM is locally flatv ^* N is a minimal Lagrangian submanifold ofT ^* M iffN is a minimal submanifold ofM. This strengthens a result of Harvey and Lawson [H L].     

Prime modules of a gamma ring

We define a prime ΓM-module for a Γ-ringM. It is shown that a subsetP ofM is a prime ideal ofM if and only ifP is the annihilator of some prime ΓM-moduleG. s-prime ideals ofM were defined by the first author. We defines-modules ofM, analogous to a concept defined by De Wet for rings. It is shown that a subsetQ ofM is ans-prime ideal ofM if and only ifQ is the annihilator of somes-moduleG ofM. Relationships between prime ΓM-modules and primeR-modules are established, whereR is the right operator ring ofM. Similar results are obtained fors-modules.     

Identities of semigroup rings over semilattices of completely (0-) simple semigroups

LetR be a ring with identity,S be a semigroup with the set of idempotentsE(S), and denote (E(S)) for the subsemigroup ofS generated byE(S). In this paper, we prove that ifS is a semilattice of completely 0-simple semigroups and completely simple semigroups, then the semigroup ringRS possesses an identity iff so doesR(E(S)); especially, the result is true forS being a completely regular semigroup.     

Universal coextensions within a variety

A category is universal if it contains a full subcategory isomorphic to the category Γ of all directed graphs without loops and isolated points. LetV be a universal semigroup variety,S a semigroup inV, andV _S = {T εV;S is a homomorphic image ofT} the full subcategory ofV of all coextensions ofS withinV. We establish the universality ofV _S in two cases:(a) ifV is the varietyS of all semigroups andS has an idempotent, and(b) ifV is an arbitrary universal semigroup variety andS has a kernel. The results of this paper had been presented at the Colloquium on Semigroups, Szeged (Hungary), August 1994. Support of the Grant Agency of the Czech Republic under Grant 201/93/950 is gratefully acknowledged.     

Perfect elements in Dubreil-Jacotin regular semigroups

IfS is a strong Dubreil-Jacotin regular semigroup thenx∈S is said to beperfect ifx=x(ξ∶x)x where ζ is the bimaximum element ofS. It is shown that the setP(S) of perfect elements is an ideal ofS, and is also a strong Dubreil-Jacotin subsemigroup. It is then proved that every element ofS is perfect if and only ifS is naturally ordered. Finally, necessary and sufficient conditions forP(S) to be orthodox are determined.     

Indecomposable finitely generated modules over valuation domains

Summary LetR be a valuation domain,S a maximal immediate extension ofR. We introduce the definition of unitary independence. We use units ofS, which are unitarily independent over an ideal ofR, to construct indecomposable finitely generatedR-modules with Goldie dimension greater than one. We prove that, ifR is archimedean, the endomorphism ring of an indecomposable finitely generatedR-module is local. On the other hand, we prove that, ifR is a suitable non archimedean valuation domain, there exist indecomposable finitely generatedR-modulesM such that End (M) is not local.

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Riassunto SiaR un dominio di valutazione,S un'estensione massimale immediate diR. Si introduce la definizione di indipendenza unitaria. Si usano unità diS unitariamente indipendenti su un ideale diR per costruireR-moduli finitamente generati indecomponibili con dimensione di Goldie maggiore di uno. Si dimostra che, seR è archimedeo, l'anello degli endomorfismi di unR-modulo finitamente generato indecomponibile è locale. Si prova altresì che, seR è un opportuno dominio di valutazione non archimedeo, esistonoR-moduliM finitamente generati indecomponibili, tali che End (M) non è locale.

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Lavoro eseguito nell'ambito del GNSAGA.     

Metrics on products of surfaces with non-positive sectional curvature

Let (S _i, g_i),i=1, 2 be two compact riemannian surfaces isometrically embedded in euclidean spaces. In this paper we show that ifM=S ₁×S₂,then for any functionF: M→R, the graph ofF, i.e. the manifold {(x, F(x)): x∈M}, does not have positive sectional curvature.     

On the commutant of certain multiplication operators on spaces of analytic functions

Let ℬ be a Banach space of analytic functions defined on the open unit disk. We characterize the commutant ofM _Z ² (the operator of multiplication by the square of independent variable defined on ℬ) and show that for an operatorS in the commutantM _Z ² ifSM _Z ^2k+1−M _Z ^2k+1 S is compact for some nonnegative integerk, thenS=M _ϕ whereϕ is a multiplier of ℬ. Letn be a positive integer andS be an operator in the commutant ofM _Z ⁿ defined on a functional Hilbert spaces of analytic functions. We show that under certain conditionsS has the formM _ϕ. Research supported by the Shiraz University Grant 78-SC-1188-657.     

Implicative commutative semigroups are equivalent to a class of BCK algebras

Chan and Shum [2] introduced the notion of implicative semigroups and obtained some of its important properties. BCK algebras with condition (S) were introduced by Iséki [4] and extensively investigated by several authors. In this note, we prove that implicative commutative semigroups are equivalent to BCK algebras with condition (S), that is, given an algebra <S;≤,·,*,1> of type (2,2,0), define ⊗ by stipulatingx⊗y=y*x and ≺ by puttingx≺y if and only ify≤x, then <S≤,·,*,1> is an implicative commutative semigroup if and only if <S;≺,·,⊗, 1> is a BCK algebra with condition (S); a nonempty subsetF ofS is an ordered filter of <S;≤,·,*, 1> if and only ifF is an ideal of <S;≺,·, ⊗, 1>. The author would like to thank the referee for his valuable comments which helped in the modification of this paper.     

Dolbeault cohomology of compact complex homogeneous manifolds

We show that ifM is the total space of a holomorphic bundle with base space a simply connected homogeneous projective variety and fibre and structure group a compact complex torus, then the identity component of the automorphism group ofM acts trivially on the Dolbeault cohomology ofM. We consider a class of compact complex homogeneous spacesW, which we call generalized Hopf manifolds, which are diffeomorphic to S¹ ×K/L whereK is a compact connected simple Lie group andL is the semisimple part of the centralizer of a one dimensional torus inK. We compute the Dolbeault cohomology ofW. We compute the Picard group of any generalized Hopf manifold and show that every line bundle over a generalized Hopf manifold arises from a representation of its fundamental group.     

The quotient field as a torsion-free covering module

R will denote a commutative integral domain with quotient fieldQ. A torsion-free cover of a moduleM is a torsion-free moduleF and anR-epimorphism σ:F→M such that given any torsion-free moduleG and λ∈Hom _R (G, M) there exists μ∈Hom _R (G,F) such that σμ=λ. It is known that ifM is a maximal ideal ofR, R→R/M is a torsion-free cover if and only ifR is a maximal valuation ring. LetE denote the injective hull ofR/M thenR→R/M extends to a homomorphismQ→E. We give necessary and sufficient conditions forQ→E to be a torsion-free cover.     

Cycles through specified vertices of a graph

We prove that ifS is a set ofk−1 vertices in ak-connected graphG, then the cycles throughS generate the cycle space ofG. Moreover, whenk≧3, each cycle ofG can be expressed as the sum of an odd number of cycles throughS. On the other hand, ifS is a set ofk vertices, these conclusions do not necessarily hold, and we characterize the exceptional cases. As corollaries, we establish the existence of odd and even cycles through specified vertices and deduce the existence of long odd and even cycles in graphs of high connectivity.