Notes on the nuclearity of random subalgebras of O2{\mathcal {O}_2}

There exists a separable exact C*-algebra A which contains all separable exact C*-algebras as subalgebras, and for each norm-dense measure μ on A and independent μ-distributed random elements x ₁, x ₂, ... we have lim_{n ? ￥}\mathbb P(C^*(x₁,?,x_n) is nuclear)=0{\rm {lim}}_{n \rightarrow \infty}\mathbb {P}(C^*(x_1,\ldots,x_n) \mbox{ is nuclear})=0. Further, there exists a norm-dense non-atomic probability measure μ on the Cuntz algebra O₂{\mathcal {O}_2} such that for an independent sequence x ₁, x ₂, ... of μ-distributed random elements x _i we have lim inf_{n ? ￥}\mathbb P(C^*(x₁,?,x_n) is nuclear)=0{\rm {lim\, inf}}_{n \rightarrow \infty}\mathbb {P}(C^*(x_1,\ldots,x_n) \mbox{ is nuclear})=0. We introduce the notion of the stochastic rank for a unital C*-algebra and prove that the stochastic rank of C([0, 1] ^d ) is d.     

On pseudo-random subsets of \mathbb Z n{\mathbb Z _n}

The notion of pseudo-randomness of subsets of \mathbb Z_n{\mathbb Z_n} is defined, and the measures of pseudo-randomness are introduced. Then a construction (based on the use of hybrid character sums) will be presented for subsets of \mathbb Z_p{\mathbb Z_p} with strong pseudo-random properties.     

Low complexity functions and convex sets in \mathbbZk\mathbb{Z}^k

In 1938 Morse and Hedlund proved that a function is periodic if the number of different n-blocks with does not exceed n for some n. In 1940 they studied such functions f with for all positive integers n. These are closely related to Sturmian sequences, which occur in many branches of mathematics, computer science and physics. Recently the authors studied k-dimensional functions with , where is the set of different vectors with for a given configuration . In this paper, we characterize functions satisfying for all configurations . Our proof requires a separation theorem for convex sets of lattice points, which may be of independent interest. Received July 20, 1998     

On the existence of spines for ${\mathbb{Q}}${\mathbb{Q}}-rank 1 groups

Let X = Γ \G/ K be an arithmetic quotient of a symmetric space of non-compact type. In the case that G has -rank 1, we construct Γ-equivariant deformation retractions of D = G/K onto a set D₀. We prove that D₀ is a spine, having dimension equal to the virtual cohomological dimension of Γ. In fact, there is a (k − 1)-parameter family of such deformation retractions, where k is the number of Γ -conjugacy classes of rational parabolic subgroups of G. The construction of the spine also gives a way to construct an exact fundamental domain for Γ.     

Tameness of Pseudovariety Joins Involving \sf R{\sf R}

In this paper, we establish several decidability results for pseudovariety joins of the form \sf Vú\sf W{\sf V}\vee{\sf W} , where \sf V{\sf V} is a subpseudovariety of \sf J{\sf J} or the pseudovariety \sf R{\sf R} . Here, \sf J{\sf J} (resp. \sf R{\sf R} ) denotes the pseudovariety of all J{\cal J} -trivial (resp. ?{\cal R} -trivial) semigroups. In particular, we show that the pseudovariety \sf Vú\sf W{\sf V}\vee{\sf W} is (completely) κ-tame when \sf V{\sf V} is a subpseudovariety of \sf J{\sf J} with decidable κ-word problem and \sf W{\sf W} is (completely) κ-tame. Moreover, if \sf W{\sf W} is a κ-tame pseudovariety which satisfies the pseudoidentity x₁ ⋯ x_ry^ω+1zt^ω = x₁ ⋯ x_ryzt^ω, then we prove that \sf Rú\sf W{\sf R}\vee{\sf W} is also κ-tame. In particular the joins \sf Rú\sf Ab{\sf R}\vee{\sf Ab} , \sf Rú\sf G{\sf R}\vee{\sf G} , \sf Rú\sf OCR{\sf R}\vee{\sf OCR} , and \sf Rú\sf CR{\sf R}\vee{\sf CR} are decidable.     

On the Greatest Common Divisor of u ? 1 and v ? 1 with u and v Near S{\cal S}-units

In [2], it was shown that if a and b are multiplicatively independent integers and ɛ > 0, then the inequality gcd (aⁿ − 1,bⁿ − 1) < exp(ɛn) holds for all but finitely many positive integers n. Here, we generalize the above result. In particular, we show that if f(x),f₁(x),g(x),g₁(x) are non-zero polynomials with integer coefficients, then for every ɛ > 0, the inequality gcd (f(n)aⁿ+g(n), f₁(n)bⁿ+g₁(n)) < exp(ne){\rm gcd}\, (f(n)a^n+g(n), f_1(n)b^n+g_1(n)) < \exp(n\varepsilon) holds for all but finitely many positive integers n.     

On certain functional equations related to Jordan triple (q, f){(\theta, \phi)}-derivations on semiprime rings

The main purpose of this paper is to prove the following result. Let R be a 2-torsion free semiprime ring with symmetric Martindale ring of quotients Q _s and let q{\theta} and f{\phi} be automorphisms of R. Suppose T:R? R{T:R\rightarrow R} is an additive mapping satisfying the relation T(xyx)=T(x)q(y)q(x)-f(x)T(y)q(x)+f(x)f(y)T(x){T(xyx)=T(x)\theta (y)\theta (x)-\phi (x)T(y)\theta (x)+\phi (x)\phi (y)T(x)}, for all pairs x,y ? R{x,y\in R}. In this case T is of the form 2T(x)=qq(x)+f(x)q{2T(x)=q\theta (x)+\phi (x)q}, for all x ? R{x\in R} and some fixed element q ? Q_s{q\in Q_{s}}.     

Covering and packing in \Bbb Zn{\Bbb Z}^n and \Bbb Rn{\Bbb R}^n, (I)

Given a finite subset A{\cal A} of an additive group \Bbb G{\Bbb G} such as \Bbb Zⁿ{\Bbb Z}^n or \Bbb Rⁿ{\Bbb R}^n , we are interested in efficient covering of \Bbb G{\Bbb G} by translates of A{\cal A} , and efficient packing of translates of A{\cal A} in \Bbb G{\Bbb G} . A set S ì \Bbb G{\cal S} \subset {\Bbb G} provides a covering if the translates A + s{\cal A} + s with s ? Ss \in {\cal S} cover \Bbb G{\Bbb G} (i.e., their union is \Bbb G{\Bbb G} ), and the covering will be efficient if S{\cal S} has small density in \Bbb G{\Bbb G} . On the other hand, a set S ì \Bbb G{\cal S} \subset {\Bbb G} will provide a packing if the translated sets A + s{\cal A} + s with s ? Ss \in {\cal S} are mutually disjoint, and the packing is efficient if S{\cal S} has large density. In the present part (I) we will derive some facts on these concepts when \Bbb G = \Bbb Zⁿ{\Bbb G} = {\Bbb Z}^n , and give estimates for the minimal covering densities and maximal packing densities of finite sets A ì \Bbb Zⁿ{\cal A} \subset {\Bbb Z}^n . In part (II) we will again deal with \Bbb G = \Bbb Zⁿ{\Bbb G} = {\Bbb Z}^n , and study the behaviour of such densities under linear transformations. In part (III) we will turn to \Bbb G = \Bbb Rⁿ{\Bbb G} = {\Bbb R}^n .     

2-local superderivations on a superalgebra Mn(\mathbb C){{\rm M}_{\rm n}({\mathbb C})}

The aim of this paper is to show that every 2-local superderivation on an associative superalgebra M_n(\mathbb C){M_n({\mathbb C})} is a superderivation.     

Notes on the nuclearity of random subalgebras of O2{\mathcal {O}_2}

There exists a separable exact C*-algebra A which contains all separable exact C*-algebras as subalgebras, and for each norm-dense measure μ on A and independent μ-distributed random elements x ₁, x ₂, ... we have . Further, there exists a norm-dense non-atomic probability measure μ on the Cuntz algebra such that for an independent sequence x ₁, x ₂, ... of μ-distributed random elements x _i we have . We introduce the notion of the stochastic rank for a unital C*-algebra and prove that the stochastic rank of C([0, 1] ^d ) is d. B. Burgstaller was supported by the Austrian Schr?dinger stipend J2471-N12.     

Pointwise -convergence and -convergence in measure of sequences of functions

Let be an ideal. We say that a sequence of real numbers is -convergent to if for every neighborhood U of y the set of n's satisfying y_nU is in . Basing upon this notion we define pointwise -convergence and -convergence in measure of sequences of measurable functions defined on a measure space with finite measure. We discuss the relationship between these two convergences. In particular we show that for a wide class of ideals including Erdős–Ulam ideals and summable ideals the pointwise -convergence implies the -convergence in measure. We also present examples of very regular ideals such that this implication does not hold.     

Hyperflock determining line sets and totally regular parallelisms of PG (3, \mathbbR){(3,\,{\mathbb{R}})}

By a totally regular parallelism of the real projective 3-space P₃:=PG(3, \mathbb R){\Pi_3:={{\rm PG}}(3, \mathbb {R})} we mean a family T of regular spreads such that each line of Π ₃ is contained in exactly one spread of T. For the investigation of totally regular parallelisms the authors mainly employ Klein’s correspondence λ of line geometry and the polarity π ₅ associated with the Klein quadric H ₅ (for details see Chaps. 1 and 3). The λ-image of a totally regular parallelism T is a hyperflock of H ₅, i.e., a family H of elliptic subquadrics of H ₅ such that each point of H ₅ is on exactly one subquadric of H. Moreover, {p₅(span  l(X))|X ? T}=:H_T{\{\pi_5({{\rm span}} \,\lambda(\mathcal {X}))\vert\mathcal {X}\in\bf{T}\}=:\mathcal {H}_{\bf{T}}} is a hyperflock determining line set, i.e., a set Z{\mathcal {Z}} of 0-secants of H ₅ such that each tangential hyperplane of H ₅ contains exactly one line of Z{\mathcal {Z}} . We say that dim(span H_T)=:d_T{{{\rm dim}}({{\rm span}}\,\mathcal {H}_{\bf{T}})=:d_{\bf{T}}} is the dimension of T and that T is a d _T - parallelism. Clifford parallelisms and 2-parallelisms coincide. The examples of non-Clifford parallelisms exhibited in Betten and Riesinger [Result Math 47:226–241, 2004; Adv Geom 8:11–32, 2008; J Geom (to appear)] are totally regular and of dimension 3. If G{\mathcal{G}} is a hyperflock determining line set, then {l^-1 (p₅(X) ?H₅) | X ? G}{\{\lambda^{-1}\,{\rm (}\pi_5(X){\,\cap H_5)\,|\, X\in\mathcal{G}\}}} is a totally regular parallelism. In the present paper the authors construct examples of topological (see Definition 1.1) 4- and 5-parallelisms via hyperflock determining line sets.     

Multiple solutions for a superlinear and periodic elliptic system on \mathbbRN{\mathbb{R}^N}

This paper is concerned with the following periodic Hamiltonian elliptic system

Idempotent multipliers for LP (\Bbb R)(\Bbb R)

The algebra B_p(\Bbb R){\cal B}_p({\Bbb R}), p ? (1,￥)\{2}p\in (1,\infty )\setminus \{2\}, consisting of all measurable sets in \Bbb R{\Bbb R} whose characteristic function is a Fourier p-multiplier, forms an algebra of sets containing many interesting and non-trivial elements (e.g. all intervals and their finite unions, certain periodic sets, arbitrary countable unions of dyadic intervals, etc.). However, B_p(\Bbb R){\cal B}_p({\Bbb R}) fails to be a s\sigma -algebra. It has been shown by V. Lebedev and A. Olevskii [4] that if E ? B_p(\Bbb R)E\in {\cal B}_p({\Bbb R}), then E must coincide a.e. with an open set, a remarkable topological constraint on E. In this note we show if $2 < p < \infty $2 < p < \infty , then there exists E ? B_p(\Bbb R)E\in {\cal B}_p({\Bbb R}) which is not in B_q(\Bbb R){\cal B}_q({\Bbb R}) for any q > pq>p.     

Quasilinear elliptic equations on \mathbbRN{\mathbb{R}^{N}} with singular potentials and bounded nonlinearity

We study the existence and multiplicity of nontrivial radial solutions of the quasilinear equation

A graph associated with the p\pi-character degrees of a group

Let G be a group and $\pi$ be a set of primes. We consider the set ${\rm cd}^{\pi}(G)$ of character degrees of G that are divisible only by primes in $\pi$. In particular, we define $\Gamma^{\pi}(G)$ to be the graph whose vertex set is the set of primes dividing degrees in ${\rm cd}^{\pi}(G)$. There is an edge between p and q if pq divides a degree $a \in {\rm cd}^{\pi}(G)$. We show that if G is $\pi$-solvable, then $\Gamma^{\pi}(G)$ has at most two connected components.     

Orthogonal complex structures on domains in \mathbb R4{\mathbb {R}^4}

An orthogonal complex structure on a domain in is a complex structure which is integrable and is compatible with the Euclidean metric. This gives rise to a first order system of partial differential equations which is conformally invariant. We prove two Liouville-type uniqueness theorems for solutions of this system, and use these to give an alternative proof of the classification of compact locally conformally flat Hermitian surfaces first proved by Pontecorvo. We also give a classification of non-degenerate quadrics in under the action of the conformal group SO _°(1, 5). Using this classification, we show that generic quadrics give rise to orthogonal complex structures defined on the complement of unknotted solid tori which are smoothly embedded in . To Nigel Hitchin on the occasion of his 60th birthday. This work was partially supported by MIUR (Metriche Riemanniane e Varietà Differenziabili, PRIN05) and NSF Grant DMS-0503506.     

Homological Invariants for pro-p Groups and Some Finitely Presented pro- C{\cal C} Groups

Let G be a finitely presented pro- C{\cal C} group with discrete relations. We prove that the kernel of an epimorphism of G to [^(\Bbb Z)]_C\hat{\Bbb Z}_{\cal C} is topologically finitely generated if G does not contain a free pro- C{\cal C} group of rank 2. In the case of pro-p groups the result is due to J. Wilson and E. Zelmanov and does not require that the relations are discrete ([15], [17]).For a pro-p group G of type FP_m we define a homological invariant C{\cal C} groups, pro-p groups, homological type FP_m, finite presentabilityBoth authors are partially supported by CNPq, Brazil.     

The unit ball of the complex P(3H){{\mathcal P}(^3H)}

Let H be a two-dimensional complex Hilbert space and P(³H){{\mathcal P}(^3H)} the space of 3-homogeneous polynomials on H. We give a characterization of the extreme points of its unit ball, B_P(³H){{\mathsf B}_{{\mathcal P}(^3H)}} , from which we deduce that the unit sphere of P(³H){{\mathcal P}(^3H)} is the disjoint union of the sets of its extreme and smooth points. We also show that an extreme point of B_P(³H){{\mathsf B}_{{\mathcal P}(^3H)}} remains extreme as considered as an element of B_L(³H){{\mathsf B}_{{\mathcal L}(^3H)}} . Finally we make a few remarks about the geometry of the unit ball of the predual of P(³H){{\mathcal P}(^3H)} and give a characterization of its smooth points.