The Gelfand–Kirillov dimension of the universal algebras of M a,b (E)⊗E in positive characteristic

In this paper we calculate the Gelfand–Kirillov dimension of the relatively free (also called universal) algebra of rank m, U _m (M _a,b (E)?E), in the variety generated by M _a,b (E)?E, in positive characteristic p>2.     

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.     

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.     

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.     

Everywhere Hölder continuity of the spatial gradient of weak solutions to nonlinear parabolic systems

We consider weak solutions to the parabolic system ?u ⁱ?t?D _α A _i ^α (?u)=B _i(?u) in (i=1,...,) (Q=Ω×(0,T), R ⁿ a domain), where the functionsB _i may have a quadratic growth. Under the assumptionsn≤2 and ?u ?L _loc ^4+δ (Q; R ^nN ) (δ>0) we prove that ?u is locally Hölder continuous inQ.     

Subvarieties of the matrix variety of second order

LetR _s be the subalgebra ofM ₂(K[t]/(t ^s )) generated bye ₁₁,e ₂₂,te ₁₂ andte ₂₁, whereK is a field of characteristic 0,K[t] is the polynomial algebra in one variablet and (t ^s ) is the principal ideal inK[t], generated byt ^s . The main result of this paper is that we have described theT-idealT(R _s ). Besides the two matrix polynomial identities — the standart identityS ₄ and the identity of Hall, thisT-ideal is generated by one more explicitly given identity. The algebrasR _s are interesting due to the fact that the proper identities of any subvarietyu of the variety ?=varM ₂(K), generated by the matrix algebraM ₂(K) of second order overK, asymptoticaly coincide with the proper identities of someR _s .     

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.     

Numerical range and compact convex sets

Let K be a compact convex subset of the plane, μ be a regular Borel measure with support K and N _μ be the multiplication operator on L ²(μ). In this article we show that \(\overline{W}(N_{\mu})\), the closure of numerical range of N _μ , is K. Also we prove that if K has uncountable many extreme points then the Berberian Hilbert space extension of L ²(μ) is non separable.     

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.     

On a class of hyperbolic 3-orbifolds of small volume and small heegaard genus associated to 2-bridge links

We consider a class of hyperbolic 3-orbifoldsO(α/β); the underlying topological space of such an orbifold is the 3-sphere and the singular set is obtained by adding the two standard (upper and lower) unknotting tunnels to a 2-bridge linkL(α/β) (and associating branching order two to both unknotting tunnels). These 3-orbifolds are extremal with respect to the notion of Heegaard genus or Heegaard number of 3-orbifolds; it is to be expected that they are also extremal with respect to the volume, that is the smallest volume hyperbolic 3-orbifolds should belong to this or some closely related class. We show that an orbifoldO(α/β) has a uniqueD ₂-covering by an orbifold? _n(α/β) wose space is the 3-sphere and whose singular set is the same 2-bridge linkL(α/β) used for the construction ofO(α/β); moreoverO(α/β) is hyperbolic if and only if? _n(α/β) is hyperbolic. As the volumes of the orbifolds? _n(α/β) are known resp. can be computed, this allows to compute the volumes of the orbifoldsO(α/β). The problem of computation of volumes remains open for some closely related classes of 3-orbifolds which are also extremal with respect to the Heegaard genus (for example associating a branching order bigger than two to one or both unknotting tunnels).     

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 .     

Distance and fractional isomorphism in Steiner triple systems

In [8], Quattrochi and Rinaldi introduced the idea ofn ^?1-isomorphism between Steiner systems. In this paper we study this concept in the context of Steiner triple systems. The main result is that for any positive integerN, there existsv ₀(N) such that for all admissiblev≥v ₀(N) and for each STS(v) (sayS), there exists an STS(v) (sayS′) such that for somen>N, S is strictlyn ^?1-isomorphic toS′. We also prove that for all admissiblev≥13, there exist two STS(v)s which are strictly 2^?1-isomorphic. Define the distance between two Steiner triple systemsS andS′ of the same order to be the minimum volume of a tradeT which transformsS into a system isomorphic toS′. We determine the distance between any two Steiner triple systems of order 15 and, further, give a complete classification of strictly 2^?1-isomorphic and 3^?1-isomorphic pairs of STS(15)s.     

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:

     

14.

An upper bound for the zeros of the derivative of Bessel functions     

Árpád Elbert  Andrea Laforgia 《Rendiconti del Circolo Matematico di Palermo》1922,46(1):123-130

Letj _vk ^′ denotes thekth positive zero of the derivativeJ _v ^′ (x)=dJ _v (x)/dx of Bessel functionJ _v (x) fork=1, 2,…. We establish the upper bound

$$j'_{\nu k}< \nu + a_k \left( {\nu + \frac{{{\rm A}_k^3 }}{{a_k^3 }}} \right)^{\frac{1}{3}} + \frac{3}{{10}}a_k^2 \left( {\nu + \frac{{A_k^3 }}{{a_k^3 }}} \right)^{\frac{1}{3}} , \nu \geqslant 0, k = 1,2, \ldots $$     

15.

On convergence of sections of sequences in Banach spaces     

Shem Aywa  Jan H. Fourie 《Rendiconti del Circolo Matematico di Palermo》1925,49(1):141-150

An elementary proof of the (known) fact that each element of the Banach space? _w ^p (X) of weakly absolutelyp-summable sequences (if 1≤p<∞) in the Banach spaceX is the norm limit of its sections if and only if each element of? _w ^p (X) is a norm null sequence inX, is given. Little modification to this proof leads to a similar result for a family of Orlicz sequence spaces. Some applications to spaces of compact operators on Banach sequence spaces are considered.     

16.

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

Nicolae Dinculeanu 《Rendiconti del Circolo Matematico di Palermo》1925,49(2):259-306

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.     

17.

Factorization in Hurwitz series domain     

Ali Benhissi 《Rendiconti del Circolo Matematico di Palermo》1936,60(1-2):69-74

Let A be a commutative ring with unit and HA the set of formal expressions of the type \(f=\sum_{i:0}^{\infty}a_{i}X^{i}\) where a _i ∈A. When \(g=\sum_{i:0}^{\infty}b_{i}X^{i}\) then \(f+g=\sum_{i:0}^{\infty}(a_{i}+b_{i})X^{i}\) and \(f*g=\sum_{n:0}^{\infty}c_{n}X^{n}\) with \(c_{n}=\sum_{i:0}^{n}C_{n}^{i}a_{i}b_{n-i}\), where \(C_{n}^{i}={n!\over i!(n-i)!}\). With these two operations HA is a commutative ring with identity. It was introduced and studied by Keigher in 1997. In this note we continue the investigation and we focus on factorization in HA and its sub-ring hA of Hurwitz polynomials. We recall from Benhissi (Contrib. Algebra. Geom. 48(1):251–256, 2007, Proposition 1.1) and Keigher (Commun. Algebra 25(6):1845–1859, 1997, Corollary 2.8) that HA is an integral domain if and only if A is an integral domain with zero characteristic. Let π ₀:HA?A be the natural ring homomorphism that assigns to each series its constant term. The key property is that a series f∈HA is a unit in HA if and only if π ₀(f) is a unit in A, Keigher (Commun. Algebra 25(6):1845–1859, 1997, Proposition 2.5).     

18.

Notes on (m,n) bi-Γ-ideals in Γ-semigroups     

Moin A. Ansari  M. Rais Khan 《Rendiconti del Circolo Matematico di Palermo》1936,60(1-2):31-42

The notions of quasi-ideals of rings and semigroups were introduced by Steinfeld (Acta Math. Acad. Sci. Hung. 4:289–298, 1953 and Publ. Math. (Debr.) 4:262–275, 1956) respectively. The notion of Γ-semigroups was introduced by Sen (Proceeding of International Symposium on Algebra and Its Applications, Decker Publication, New York, 1981). Further the notion of (m,n) ideals of semigroups was introduced by Lajos (Acta Sci. Math. 22:217–222, 1961). Later on (m,n) quasi-ideals and (m,n) bi-ideals were widely studied in various algebraic structures viz. semigroups, rings and near-rings etc. In this paper we have defined (m,n) quasi-Γ-ideal and (m,n) bi-Γ-ideal in Γ-semigroup. Including other results, we have shown that if Q is a minimal (m,n) quasi-Γ-ideal in Γ-semigroup S then intersection of minimal m-left Γ-ideals and minimal n-right Γ-ideals is again a minimal (m,n) quasi-Γ-ideals.     

19.

Applicabilità proiettiva di due superficie     

Guido Fubini 《Rendiconti del Circolo Matematico di Palermo》1916,42(1):135-154

Let Ω ? ? ⁿ be a convex bounded open set, of class\(C^2 ,Q_\tau = \Omega \times \left[ {\tau ,\tau + T} \right],\tau \in \mathbb{R},T > 0.\). LetB be a linear continuous operator ofL ²Ω ? ? ^N inL ²Ω ? ? ^N . It is shown that if\(f \in L^2 (Q_\tau ,\mathbb{R}^N )\) then there exists a unique solution of the problem:\(u \in W^{2,1} (Q_\tau ,\mathbb{R}^N ),\alpha (x,t,H(u)) - \frac{{\partial u}}{{\partial t}} = f(x,t)\), in\(Q_\tau \), such thatu(x,t)=B u(x, τ+T) in Ω, wherea(x, t, ζ) is misurable in(x,t), continuous in ζ,a(x,t, 0)=0, and verifies condition (A). IfB=Id this is the classical periodic problem. If moreovera(x,t,ζ)=a(x,t+T, ζ) anda(x,t, H (Bu))=B a(x,t,H (u)) ?t ∈ ?, the analogous problem in Ω × ? is studied.     

20.

Feebly associativeP-hypergroupoids     

Renato Migliorato  Stefanos Spartalis 《Rendiconti del Circolo Matematico di Palermo》1925,49(1):27-40

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.     

An upper bound for the zeros of the derivative of Bessel functions

Letj _vk ^′ denotes thekth positive zero of the derivativeJ _v ^′ (x)=dJ _v (x)/dx of Bessel functionJ _v (x) fork=1, 2,…. We establish the upper bound

$$j'_{\nu k}< \nu + a_k \left( {\nu + \frac{{{\rm A}_k^3 }}{{a_k^3 }}} \right)^{\frac{1}{3}} + \frac{3}{{10}}a_k^2 \left( {\nu + \frac{{A_k^3 }}{{a_k^3 }}} \right)^{\frac{1}{3}} , \nu \geqslant 0, k = 1,2, \ldots $$     

On convergence of sections of sequences in Banach spaces

An elementary proof of the (known) fact that each element of the Banach space? _w ^p (X) of weakly absolutelyp-summable sequences (if 1≤p<∞) in the Banach spaceX is the norm limit of its sections if and only if each element of? _w ^p (X) is a norm null sequence inX, is given. Little modification to this proof leads to a similar result for a family of Orlicz sequence spaces. Some applications to spaces of compact operators on Banach sequence spaces are considered.     

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.     

Factorization in Hurwitz series domain

Let A be a commutative ring with unit and HA the set of formal expressions of the type \(f=\sum_{i:0}^{\infty}a_{i}X^{i}\) where a _i ∈A. When \(g=\sum_{i:0}^{\infty}b_{i}X^{i}\) then \(f+g=\sum_{i:0}^{\infty}(a_{i}+b_{i})X^{i}\) and \(f*g=\sum_{n:0}^{\infty}c_{n}X^{n}\) with \(c_{n}=\sum_{i:0}^{n}C_{n}^{i}a_{i}b_{n-i}\), where \(C_{n}^{i}={n!\over i!(n-i)!}\). With these two operations HA is a commutative ring with identity. It was introduced and studied by Keigher in 1997. In this note we continue the investigation and we focus on factorization in HA and its sub-ring hA of Hurwitz polynomials. We recall from Benhissi (Contrib. Algebra. Geom. 48(1):251–256, 2007, Proposition 1.1) and Keigher (Commun. Algebra 25(6):1845–1859, 1997, Corollary 2.8) that HA is an integral domain if and only if A is an integral domain with zero characteristic. Let π ₀:HA?A be the natural ring homomorphism that assigns to each series its constant term. The key property is that a series f∈HA is a unit in HA if and only if π ₀(f) is a unit in A, Keigher (Commun. Algebra 25(6):1845–1859, 1997, Proposition 2.5).     

Notes on (m,n) bi-Γ-ideals in Γ-semigroups

The notions of quasi-ideals of rings and semigroups were introduced by Steinfeld (Acta Math. Acad. Sci. Hung. 4:289–298, 1953 and Publ. Math. (Debr.) 4:262–275, 1956) respectively. The notion of Γ-semigroups was introduced by Sen (Proceeding of International Symposium on Algebra and Its Applications, Decker Publication, New York, 1981). Further the notion of (m,n) ideals of semigroups was introduced by Lajos (Acta Sci. Math. 22:217–222, 1961). Later on (m,n) quasi-ideals and (m,n) bi-ideals were widely studied in various algebraic structures viz. semigroups, rings and near-rings etc. In this paper we have defined (m,n) quasi-Γ-ideal and (m,n) bi-Γ-ideal in Γ-semigroup. Including other results, we have shown that if Q is a minimal (m,n) quasi-Γ-ideal in Γ-semigroup S then intersection of minimal m-left Γ-ideals and minimal n-right Γ-ideals is again a minimal (m,n) quasi-Γ-ideals.     

Applicabilità proiettiva di due superficie

Let Ω ? ? ⁿ be a convex bounded open set, of class\(C^2 ,Q_\tau = \Omega \times \left[ {\tau ,\tau + T} \right],\tau \in \mathbb{R},T > 0.\). LetB be a linear continuous operator ofL ²Ω ? ? ^N inL ²Ω ? ? ^N . It is shown that if\(f \in L^2 (Q_\tau ,\mathbb{R}^N )\) then there exists a unique solution of the problem:\(u \in W^{2,1} (Q_\tau ,\mathbb{R}^N ),\alpha (x,t,H(u)) - \frac{{\partial u}}{{\partial t}} = f(x,t)\), in\(Q_\tau \), such thatu(x,t)=B u(x, τ+T) in Ω, wherea(x, t, ζ) is misurable in(x,t), continuous in ζ,a(x,t, 0)=0, and verifies condition (A). IfB=Id this is the classical periodic problem. If moreovera(x,t,ζ)=a(x,t+T, ζ) anda(x,t, H (Bu))=B a(x,t,H (u)) ?t ∈ ?, the analogous problem in Ω × ? is studied.     

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.