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

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.     

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

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.     

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 .     

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.     

Rational holomorphic maps from \mathbbB2 \mathbb{B}^2 into \mathbbBN \mathbb{B}^N with degree 2

Rational proper holomorphic maps from the unit ball in ℂ² into the unit ball ℂ ^N with degree 2 are studied. Any such map must be equivalent to one of the four types of maps. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

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.     

Anomalous dimensions of Wilson operators in the N = 4\mathcal{N} = 4 supersymmetric Yang-Mills theory

We present results for the universal anomalous dimension γ_un(j) of Wilson twist-2 operators in the supersymmetric Yang-Mills theory in the first three orders of the perturbation theory. We obtain these expressions by extracting the most complicated terms from the corresponding anomalous dimensions in QCD. The result obtained agrees with the hypothesis of the integrability of the supersymmetric Yang-Mills theory in the context of the AdS/CFT correspondence. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 2, pp. 249–262, February, 2007.     

The first Cohomology of Affine $ \mathbb{Z}^{p} $ \mathbb{Z}^{p} -actions on Tori and applications to rigidity*

Let a minimal affine -action on the torus T ^q , p 2 and q 1. The cohomology of (see definition below) depends on both the algebraic properties of the induced action on H ¹(T ^q , ) and the arithmetical properties of the translation cocycle. We give a Diophantine condition that characterizes those affine actions whose first cohomology group is finite dimensional. In this case it is necessarily isomorphic to . Thus the -action F obtained by suspension of is parameter rigid, i.e., any other -action with the same orbit foliation is smoothly conjugate to a reparametrization of F by an automorphism of .*Partially supported by CNPq fellowship by Fondecyt Grant 1000047 and DGICT-UCN and fundación Andes, Chile.     

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.     

Morse and semistable stratifications of K?hler spacesby \Bbb C*{\Bbb C}^{\ast}-actions

We prove that the Morse decomposition in the sense of Kirwan and semistable decomposition in the sense of GIT of a \Bbb C^*{\Bbb C}^{\ast} -K?hler manifold coincide if the moment map is proper and if the fixed points set X^{\Bbb C*}X^{{\Bbb C}^{\ast}} has a finite number of connected components. For general K?hler space with holomorphic action of a complex reductive group G, if every component of the moment map is proper, the two decompositions also coincide if each semistable piece is Zariski open in its topological closure and the moment map square is minimal degenerate Morse function in the sense of Kirwan.     

On the Posets ( \user1Wk2 , < ){\left( {{\user1{\mathcal{W}}}^{k}_{2} , < } \right)} and their Connections with Some Homogeneous Inequalities of Degree 2

A class of ranked posets {(D _h ^k , ≪)} has been recently defined in order to analyse, from a combinatorial viewpoint, particular systems of real homogeneous inequalities between monomials. In the present paper we focus on the posets D ₂ ^k , which are related to systems of the form {x _a x _b * _abcd x _c x _d : 0 ≤ a, b, c, d ≤ k, * _abcd ∈ {<, >}, 0 < x ₀ < x ₁ < ...< x _k}. As a consequence of the general theory, the logical dependency among inequalities is adequately captured by the so-defined posets . These structures, whose elements are all the D ₂ ^k 's incomparable pairs, are thoroughly surveyed in the following pages. In particular, their order ideals – crucially significant in connection with logical consequence – are characterised in a rather simple way. In the second part of the paper, a class of antichains is shown to enjoy some arithmetical properties which make it an efficient tool for detecting incompatible systems, as well as for posing some compatibility questions in a purely combinatorial fashion.     

Harmonic mean curvature lines on surfaces immersed in $ \mathbb{R}^{3} $ \mathbb{R}^{3}

Consider oriented surfaces immersed in . Associated to them, here are studied pairs of transversal foliations with singularities, defined on the Elliptic region, where the Gaussian curvature , given by the product of the principal curvatures k ₁, k ₂ is positive. The leaves of the foliations are the lines of harmonic mean curvature, also called characteristic or diagonal lines, along which the normal curvature of the immersion is given by , where is the arithmetic mean curvature. That is, is the harmonic mean of the principal curvatures k ₁, k ₂ of the immersion. The singularities of the foliations are the umbilic points and parabolic curves, where k ₁ = k ₂ and , respectively.Here are determined the structurally stable patterns of harmonic mean curvature lines near the umbilic points, parabolic curves and harmonic mean curvature cycles, the periodic leaves of the foliations. The genericity of these patterns is established.This provides the three essential local ingredients to establish sufficient conditions, likely to be also necessary, for Harmonic Mean Curvature Structural Stability of immersed surfaces. This study, outlined towards the end of the paper, is a natural analog and complement for that carried out previously by the authors for the Arithmetic Mean Curvature and the Asymptotic Structural Stability of immersed surfaces, [13, 14, 17], and also extended recently to the case of the Geometric Mean Curvature Configuration [15].The first author was partially supported by FUNAPE/UFG. Both authors are fellows of CNPq. This work was done under the project PRONEX/FINEP/MCT - Conv. 76.97.1080.00 - Teoria Qualitativa das Equações Diferenciais Ordinárias and CNPq - Grant 476886/2001-5.     

Operator and matrix representation for the generalized inverse AT,S(2)A_{T,S}^{(2)}

This paper presents the matrix representation for extension of inverse of restriction of a linear operator to a subspace, on the basis of which we establish useful representations in operator and matrix form for the generalized inverse A_T,S⁽²⁾A_{T,S}^{(2)} and give some of their applications.     

A note on plurisubharmonic defining functions in \mathbbCn{\mathbb{C}^{n}}

Let , n ≥ 3, be a smoothly bounded domain. Suppose that Ω admits a smooth defining function which is plurisubharmonic on the boundary of Ω. Then a Diederich–Forn?ss exponent can be chosen arbitrarily close to 1, and the closure of Ω admits a Stein neighborhood basis. Research of J. E. Forn?ss was partially supported by an NSF grant. Research of A.-K. Herbig was supported by FWF grant P19147.     

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}}.     

Constant angle surfaces in \Bbb S2×\Bbb R {\Bbb S}^2\times {\Bbb R}

In this article we study surfaces in \Bbb S²×\Bbb R {\Bbb S}^2\times {\Bbb R} for which the unit normal makes a constant angle with the \Bbb R {\Bbb R} -direction. We give a complete classification for surfaces satisfying this simple geometric condition.     

K-Theory for Triangulated Categories 3 $\frac{3}{4}$\frac{3}{4}: A Direct Proof of the Theorem of the Heart

Let be a triangulated category, and assume it admits at least one model. In this article, we define a K-theory for . The main theorem is that, given any bounded i-structure on , the K-theory of the heart agrees with the K-theory of . An immediate consequence tells us that, if two Abelian categories occur as hearts of a triangulated category for two different t-structures, then their K-theories must be isomorphic.The proof was also sketched in previous articles in this series. The virtue of this article is in the careful detail in which it is written down.     

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.