On the representation of partially ordered sets

It is well known that any distributive poset (short for partially ordered set) has an isomorphic representation as a poset (Q, ⊆) such that the supremum and the infimum of any finite setF ofP correspond, respectively to the union and intersection of the images of the elements ofF. Here necessary and sufficient conditions are given for similar isomophic representation of a poset where however the supremum and infimum of also infinite subsetsI correspond to the union and intersection of images of elements ofI.     

Representation of partially ordered sets

It is well known [1] that any distributive poset (short for partially ordered set) has an isomorphic representation as a poset (Q, (–) such that the supremum and the infimum of any finite setF ofp correspond, respectively, to the union and intersection of the images of the elements ofF. Here necessary and sufficient conditions are given for similar isomorphic representation of a poset where however the supremum and infimum of also infinite subsetsI correspond to the union and intersection of images of elements ofI. Presented by R. Freese.     

Sui gruppi unitari finitari e sulla congettura di dieudonné

In this paper we consider some questions concerning unitary spaces (V, h), even though (V, h) is not finitely generated. Our main result is as follows: letF be an infinite field of characteristic ≠2 andD anm ²-dimensional central division algebra overF with an involutionj≠1. Let Σ_j(D) denote the subgroup of the multiplicative groupD ^* generated by the non zero symmetric elements. If (V, h) is an infinite dimensional regular unitary space of Witt index at least two overD, then the finitary unitary groupFU(h) is a simple group if and only ifD*=Σ_j(D)[D*, D*]. On the other hand, when (V, h) is not regular,FU(h) cannot be simple since it containsFU ₀(h), the subgroup of elements ofFU(h) acting trivially onrad(V, h), as a normal subgroup. In the non regular case we show that under the above assumptions evenFU ₀(h) is not a simple group.     

The picard group of the variety of plane cuspidal cubics of ℙ3

We give a compactification of the varietyU of non-degenerate plane cuspidal cubics of ?³. We construct this compactification by means of the projective bundleX of a suitable vector bundleE. We describe the intersection ring ofX and, as a consequence, we obtain the intersection numbers ofU that satisfy 10 conditions of the following kinds:ρ, that the plane determined by the cuspidal cubic go through a point;c, that the cusp be on a plane;q, that the cuspidal tangent intersect a line;μ, that the cuspidal cubic intersect a line. Moreover, we prove that the Picard group of the varietyU is a product of two infinite cyclic groups generated byρ andc?q.     

An algorithm to compute the number of points on elliptic curves ofj-invariant 0 or 1728 over a finite field

We present an algorithm to compute the number ofF _q -rational points on elliptic curves defined over a finite fieldF _q , withj-invariant 0 or 1728. This algorithm takesO(log³ p) bit operations, werep is the characteristic ofF _q .     

Les Algebres A-F-normees Advertiblement Completes

We introduce the class ofA —F-normed algebras. It lies between that of locally bounded algebras and the one ofF*-algebras considered by W. Zelazko. We examine the structure of these algebras; and give some spectral theory results under the assumption of advertible completeness as generalized by A. Mallios.     

Feebly associativeP-hypergroupoids

In this paper we study the hyperstructures, saidP-hypergroupoids, (H, P*) in whichH is a set andP* is one of the hyperoperations defined as follows: ?(x, y) εH ²,xP*y=xyP orxP*y=Pxy whereP is a subset ofH. In particular we give a general formula for to express the simple hyperproducts ofn elements and then we consider some cases in which (H, P*) is feebly associative. We study, in such cases theβ-relations.     

The coefficients of the Laurent series expansions of real analytic functions

Letf be a real analytic function of a real variable such that 0 is an isolated (possibly essential) singularity off. In the existing literature the coefficients of the Laurent series expansion off around 0 are obtained by applying Cauchy's integral formula to the analytic continuation off on the complex plane. Here by means of a conformal mapping we derive a formula which determines the Laurent coefficients off solely in terms of the values off and the derivatives off at a real point of analyticity off. Using a more complicated mapping, we similarly determine the coefficients of the Laurent expansion off around 0 where now 0 is a singularity off which is not necessarily isolated.     

Operateurs essentiellement reguliers dans les espaces de Banach

Given a closed operatorA acting in a Banach spaceX, we define the regular (respectively the essentialy regular) spectrum σ _r (A) (respectively σ _e,r (A)) ofA. We prove that σ _r (A) and σ _e,r (A) are a closed subsets of the classical spectrum σ(A) ofA. Morever ifA is bounded we prove that σ _r (A) and σ _e,r (A)) satisfies the spectral mapping theorem.     

Wijsman topology on function spaces

LetC(X,Y) be the space of continuous functions from a metric space (X,d) to a metric space (Y, e).C(X, Y) can be thought as subset of the hyperspaceCL(X×Y) of closed and nonempty subsets ofX×Y by identifying each element ofC(X,Y) with its graph. We considerC(X,Y) with the topology inherited from the Wijsman topology induced onCL(X×Y) by the box metric ofd ande. We study the relationships between the Wijsman topology and the compact-open topology onC(X,Y) and also conditions under which the Wijsman topology coincide with the Fell topology. Sufficient conditions under which the compactopen topology onC(X,Y) is weaker than the Wijsman topology are given (IfY is totally bounded, then for every metric spaceX the compactopen topology onC(X,Y) is weaker than the Wijsman topology and the same is true forX locally connected andY rim-totally bounded). We prove that a metric spaceX is boundedly compact iff the Wijsman topology onC(X, ?) is weaker than the compact-open topology. We show that ifX is a σ-compact complete metric space andY a compact metric space, then the Wijsman topology onC(X,Y) is Polish.     

The interior operation in aψ-density topology

We prove that for an arbitrary setA ? ? its interior in aψ-density topology equalsA ∩ φ^β(B), whereB is a measurable kernel ofA andβ is some countable ordinal. Moreover, eachβ, 1≤β<Ω, realizes the interior ofA for someA εS.     

Stochastic integration for abstract,two parameter stochastic processes I. Stochastic processes with finite semivariation in banach spaces

In this paper we define the stochastic integral for two parameter processes with values in a Banach spaceE. We use a measure theoretic approach. To each two parameter processX withX _st ∈L _E ^p we associate a measureI _X with values inL _E ^p . IfX isp-summable, i.e. ifI _X can be extended to aσ-additive measure with finite semivariation on theσ-algebra of predictable sets, then the integralε HdI _X can be defined and the stochastic integral is defined by (H·X) _z =ε _[0,z] HdI _X . We prove that the processes with finite variation and the processes with finite semivariation are summable and their stochastic integral can be computed pathwise, as a Stieltjes Integral of a special type.     

Derivations in prime rings

LetR be a prime ring andD a nonzero derivation ofR. If one of the four conditions holds inR, thenR is commutative:

1. (i)
   X ²D(X)?D(X)X²∈Z(R), CharR≠2;     
   
   

Lower bound for matrix operators on the Euler weighted sequence space e_{w,p}^{\theta} (0<p<1)

Let A=(a _n,k ) _n,k≥0 be a non-negative matrix. Denote by \(L_{l_{p} (w),~e_{w,q}^{\theta}}(A)\) the supremum of those L, satisfying the following inequality:

     

15.

Higher order secant varieties and blown up varieties     

E. Ballico 《Rendiconti del Circolo Matematico di Palermo》1921,45(1):93-104

Let π:X→Y be the blowing up of the projective varietyY at s general points. Here we study the higher order secant varieties of the linearly normal embeddings ofX andY into projective spaces. We give conditions on the embedding ofY which imply that the firstt secant varieties of a related embedding ofX have the expected dimension.     

16.

Refining Castelnuovo-Halphen bounds     

Vincenzo Di Gennaro  Davide Franco 《Rendiconti del Circolo Matematico di Palermo》1937,61(1):91-106

Fix integers r,d,s,π with r≥4, d?s, r?1≤s≤2r?4, and π≥0. Refining classical results for the genus of a projective curve, we exhibit a sharp upper bound for the arithmetic genus p _a (C) of an integral projective curve C?? ^r of degree d, assuming that C is not contained in any surface of degree <s, and not contained in any surface of degree s with sectional genus >π. Next we discuss other types of bound for p _a (C), involving conditions on the entire Hilbert polynomial of the integral surfaces on which C may lie.     

17.

Some remarks on spectral states ofLMC-algebras     

Michael Pannenberg 《Rendiconti del Circolo Matematico di Palermo》1915,39(1):79-98

We generalize the notion of a spectral state (as introduced for Banach algebras by Moore, Bonsall and Duncan) to the context of locally multiplicatively-convex (LMC) algebras by proceeding in a way analogous to the generalization of numerical range theory from Banach algebras toLMC-algebras carried out by Giles and Koehler. Among the results obtained in this note are integral representations of spectral states by probability measures on the structure space ofA and the determination of the extreme points of the convex set\(\Omega _A \) of all spectral states on a commutativeLMC-algebraA (which is related to different Choquet boundaries) as well as a characterization of symmetric involutions by the coincidence of the notions of positive state and spectral state and a characterization of theQ-property by the weak-^*-boundedness of\(\Omega _A \). The paper finishes with two elementary commutativity criteria involving spectral states and two Korovkin-type theorems for the approximation of unital algebra homomorphisms by σ-equicontractive nets of linear operators mapping anLMC-algebraA into theLMC-algebra of all continuous complex-valued functions on a completely regular spaceX.     

18.

The role of convexity in existence theorems for invariant and hyperinvariant subspaces in hilbert space     

George Maltese 《Rendiconti del Circolo Matematico di Palermo》1925,49(2):381-390

We use extreme point methods to obtain some existence theorems for invariant and hyperinvariant subspaces for certain classes of operators In Hilbert space. These convexity methods have, heretofore, not been used in this context, however it would appear that they provide useful alternatives to existing methods. Using the Krein-Milman theorem we prove several variants of the classical lemma of I. Schur. In one of the main results we prove the following: LetA ∈B (H), thenA andA* have a common non-trivial closed hyperinvariant subspace if and only if there exists a non-scalar operatorW εB(H) such that bothW andW* belong to the double commutant ofA.     

19.

Calcul Fonctionnel Holomorphe Global Unicité et Invariance Par Les Homomorphismes D’algebres de Banach     

M. Akkar  R. El Harti 《Rendiconti del Circolo Matematico di Palermo》1925,49(1):103-114

LetA be a commutative Banach algebra with unit. Denote byX _A, the global spectrum ofA. There is a holomorphic functional calculusθ _A:O(X _A)→A such thatθ _A(â)=a. In this paper, we show the uniqueness of the global holomorphic functional calculus and we establish its compatibility with Banach algebra morphisms. We also extend this holomorphic functional calculus to the case ofImc algebras.     

20.

Incomplete Fibonacci and Lucas numbers     

Piero Filipponi 《Rendiconti del Circolo Matematico di Palermo》1921,45(1):37-56

A particular use of well-known combinatorial expressions for Fibonacci and Lucas numbers gives rise to two interesting classes of integers (namely, the numbersF _n(k) andL _n(k)) governed by the integral parametersn andk. After establishing the main properties of these numbers and their interrelationship, we study some congruence properties ofL _n(k), one of which leads to a supposedly new characterisation of prime numbers. A glimpse of possible generalisations and further avenues of research is also caught.     

Higher order secant varieties and blown up varieties

Let π:X→Y be the blowing up of the projective varietyY at s general points. Here we study the higher order secant varieties of the linearly normal embeddings ofX andY into projective spaces. We give conditions on the embedding ofY which imply that the firstt secant varieties of a related embedding ofX have the expected dimension.     

Refining Castelnuovo-Halphen bounds

Fix integers r,d,s,π with r≥4, d?s, r?1≤s≤2r?4, and π≥0. Refining classical results for the genus of a projective curve, we exhibit a sharp upper bound for the arithmetic genus p _a (C) of an integral projective curve C?? ^r of degree d, assuming that C is not contained in any surface of degree <s, and not contained in any surface of degree s with sectional genus >π. Next we discuss other types of bound for p _a (C), involving conditions on the entire Hilbert polynomial of the integral surfaces on which C may lie.     

Some remarks on spectral states ofLMC-algebras

We generalize the notion of a spectral state (as introduced for Banach algebras by Moore, Bonsall and Duncan) to the context of locally multiplicatively-convex (LMC) algebras by proceeding in a way analogous to the generalization of numerical range theory from Banach algebras toLMC-algebras carried out by Giles and Koehler. Among the results obtained in this note are integral representations of spectral states by probability measures on the structure space ofA and the determination of the extreme points of the convex set\(\Omega _A \) of all spectral states on a commutativeLMC-algebraA (which is related to different Choquet boundaries) as well as a characterization of symmetric involutions by the coincidence of the notions of positive state and spectral state and a characterization of theQ-property by the weak-^*-boundedness of\(\Omega _A \). The paper finishes with two elementary commutativity criteria involving spectral states and two Korovkin-type theorems for the approximation of unital algebra homomorphisms by σ-equicontractive nets of linear operators mapping anLMC-algebraA into theLMC-algebra of all continuous complex-valued functions on a completely regular spaceX.     

The role of convexity in existence theorems for invariant and hyperinvariant subspaces in hilbert space

We use extreme point methods to obtain some existence theorems for invariant and hyperinvariant subspaces for certain classes of operators In Hilbert space. These convexity methods have, heretofore, not been used in this context, however it would appear that they provide useful alternatives to existing methods. Using the Krein-Milman theorem we prove several variants of the classical lemma of I. Schur. In one of the main results we prove the following: LetA ∈B (H), thenA andA* have a common non-trivial closed hyperinvariant subspace if and only if there exists a non-scalar operatorW εB(H) such that bothW andW* belong to the double commutant ofA.     

Calcul Fonctionnel Holomorphe Global Unicité et Invariance Par Les Homomorphismes D’algebres de Banach

LetA be a commutative Banach algebra with unit. Denote byX _A, the global spectrum ofA. There is a holomorphic functional calculusθ _A:O(X _A)→A such thatθ _A(â)=a. In this paper, we show the uniqueness of the global holomorphic functional calculus and we establish its compatibility with Banach algebra morphisms. We also extend this holomorphic functional calculus to the case ofImc algebras.     

Incomplete Fibonacci and Lucas numbers

A particular use of well-known combinatorial expressions for Fibonacci and Lucas numbers gives rise to two interesting classes of integers (namely, the numbersF _n(k) andL _n(k)) governed by the integral parametersn andk. After establishing the main properties of these numbers and their interrelationship, we study some congruence properties ofL _n(k), one of which leads to a supposedly new characterisation of prime numbers. A glimpse of possible generalisations and further avenues of research is also caught.