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 .     

Some extensions of the Laplace problem

Let ?₁(a,b) be a regular lattice with the fundamental cell a rectangle with sides a,b, let ?₂(a,b,?) be a regular lattice with the fundamental cell a parallelogram with sides a,b and angle ? and let ?₃(a,b,c) be a regular lattice with the fundamental cell \(C_{0}^{( 3) }\) as in Fig. 7. In this paper we compute the probability that a random rectangle r of constant side l,m intersects a side of the lattice. In particular when the rectangle r becomes a segment of length l, (m=0) we obtain the Laplace probability.     

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.     

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.     

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.     

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).     

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.     

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.     

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.     

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.     

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

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

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.     

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;     
   
   

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.     

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.     

A faster pseudo-primality test

We propose a pseudo-primality test using cyclic extensions of ?/n?. For every positive integer k?logn, this test achieves the security of k Miller-Rabin tests at the cost of k ^1/2+o(1) Miller-Rabin tests.     

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.     

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:

     

20.

On singular quadratic complexes,quintic curves and Cremona transformations     

Dan Avritzer  Gerard Gonzalez-Sprinberg  Ivan Pan 《Rendiconti del Circolo Matematico di Palermo》1937,61(2):201-240

Let X be a quadratic complex given by the intersection of two nonsingular quadrics in a projective space of dimension five. Let L be a line contained in X, and π the projection from X to a projective three space with center L. When X is nonsingular the map π is birational and the base locus scheme of π ^?1 is a smooth quintic curve of genus 2. Now assume X is a singular irreducible and reduced quadratic complex and consider the same set up. The purpose of this work is to classify quintic curves arising as the base locus scheme of π ^?1 in the case where π is birational and the Cremona transformations obtained by composing π ^?1 with another projection of the same type.     

On singular quadratic complexes,quintic curves and Cremona transformations

Let X be a quadratic complex given by the intersection of two nonsingular quadrics in a projective space of dimension five. Let L be a line contained in X, and π the projection from X to a projective three space with center L. When X is nonsingular the map π is birational and the base locus scheme of π ^?1 is a smooth quintic curve of genus 2. Now assume X is a singular irreducible and reduced quadratic complex and consider the same set up. The purpose of this work is to classify quintic curves arising as the base locus scheme of π ^?1 in the case where π is birational and the Cremona transformations obtained by composing π ^?1 with another projection of the same type.