Basis Problem for Turbulent Actions II: c0-Equalities

Let (Xⁿ,dⁿ) be a sequence of finite metric spaces of uniformlybounded diameter. An equivalence relation D on the product defined by if and only if is a c₀-equality.A systematic study is made of c₀-equalities and Borel reductionsbetween them. Necessary and sufficient conditions, expressedin terms of combinatorial properties of metrics d_n, are obtainedfor a c₀-equality to be effectively reducible to the isomorphismrelation of countable structures. It is proved that every Borelequivalence relation E reducible to a c₀-equality D either reducesa c₀-equality D' additively reducible to D, or is Borel-reducibleto the equality relation on countable sets of reals. An appropriatelydefined sequence of metrics provides a c₀-equality which isa turbulent orbit equivalence relation with no minimum turbulentequivalence relation reducible to it. This answers a questionof Hjorth. It is also shown that, whenever E is an F-equivalencerelation and D is a c₀-equality, every Borel equivalence relationreducible to both D and to E has to be essentially countable.2000 Mathematics Subject Classification: 03E15.     

Densité des Points à Coordonnées Multiplicativement Indépendantes

Nous montrons que pour toute sous-variété algébrique d'un tore multiplicatif (non contenue dans un sous-groupe algébrique propre), on peut choisir un ensemble Zariski dense de points algébriques de hauteur contrôlée, dont toutes les coordonnées sont multiplicativement indépendantes. Cet énoncé précise et généralise un théorème de S. Zhang qui lie la hauteur projective d'une varété au minimum essentiel de la hauteur des points algébriques de celle-ci. En tenant compte d'un résultat précédent des auteurs sur le problème de Lehmer généralisé à un tore, nous en déduisons une minoration pour la hauteur normalisée d'une sous-variété d'un tore. Cette dernière est optimale à un «-prés» en le degré géométrique de la variété étudiée (confer une conjecture du second auteur avec P. Philippon).In this article, we prove that on any subvariety of a multiplicative torus which is not contained in a proper algebraic subgroup, one can find a Zariski dense set of algebraic points of small height whose coordinates are multiplicatively independent. This statement generalizes an earlier result of S. Zhang which links the projective height of a variety with the essential minimum of its algebraic points. Taking into account an earlier result of the authors on the Lehmer problem generalized to a multiplicative torus, one deduces a lower bound for the normalized height of subvarieties of multiplicative groups. This lower bound is optimal up to an in the geometric degree of the variety studied (confer a conjecture by the second author and P. Philippon).     

Recurrence relations for the coefficients of the Fourier series expansions with respect to q-classical orthogonal polynomials

We propose an algorithm to construct recurrence relations for the coefficients of the Fourier series expansions with respect to the q-classical orthogonal polynomials p_k(x;q). Examples dealing with inversion problems, connection between any two sequences of q-classical polynomials, linearization of ϑ_m(x) p_n(x;q), where ϑ_m(x) is x^mor (x;q)_m, and the expansion of the Hahn-Exton q-Bessel function in the little q-Jacobi polynomials are discussed in detail. This revised version was published online in June 2006 with corrections to the Cover Date.     

Linear algebraic groups and countable Borel equivalence relations

If is an equivalence relation on a standard Borel space , then we say that is Borel reducible to if there is a Borel function such that . An equivalence relation on a standard Borel space is Borel if its graph is a Borel subset of . It is countable if each of its equivalence classes is countable. We investigate the complexity of Borel reducibility of countable Borel equivalence relations on standard Borel spaces. We show that it is at least as complex as the relation of inclusion on the collection of Borel subsets of the real line. We also show that Borel reducibility is -complete. The proofs make use of the ergodic theory of linear algebraic groups, and more particularly the superrigidity theory of R. Zimmer.

     

Sheared algebra maps and operation bialgebras for mod 2 homology and cohomology

The mod 2 Steenrod algebra and Dyer-Lashof algebra have both striking similarities and differences arising from their common origins in ``lower-indexed' algebraic operations. These algebraic operations and their relations generate a bigraded bialgebra , whose module actions are equivalent to, but quite different from, those of and . The exact relationships emerge as ``sheared algebra bijections', which also illuminate the role of the cohomology of . As a bialgebra, has a particularly attractive and potentially useful structure, providing a bridge between those of and , and suggesting possible applications to the Miller spectral sequence and the structure of Dickson algebras.

     

Stability of a Special Class of $$q_{ij}$$ -CCR and Extensions of Higher-Dimensional Noncommutative Tori

The -algebras A_{{q i}}, generated by generalised quon commutation relations are considered. The nuclearity of these algebras is proved. It is shown that A_{{q i}}, is isomorphic to the extension of a higher-dimensional noncommutative torus. Irreducible representations of A_{{q i}}, are considered. It is shown that the Fock representation is faithful.     

三维各向同性谐振子波函数的递推关系(英文)

本文应用拉普拉斯变换得到了三维各向同性谐振子波函数边界的精确解,同时,利用同种方法还得到了利用产生算符和湮灭算符表达的该波函数的递推关系.     

Positive-, negative-, and orthogonal-phase-velocity propagation of electromagnetic plane waves in a simply moving medium

Planewave propagation in a simply moving, dielectric-magnetic medium that is isotropic in the co-moving reference frame, is classified into three different categories: positive-, negative-, and orthogonal-phase-velocity (PPV, NPV, and OPV). Calculations from the perspective of an observer located in a non-co-moving reference frame show that, whether the nature of planewave propagation is PPV or NPV (or OPV in the case of non-dissipative mediums) depends strongly upon the magnitude and direction of that observer's velocity relative to the medium. PPV propagation is characterized by a positive real wavenumber, NPV propagation by a negative real wavenumber. OPV propagation only occurs for non-dissipative mediums, but weakly dissipative mediums can support nearly OPV propagation.     

Perylene diimide copolymers with dithienothiophene and dithienopyrrole: Use in n‐channel and ambipolar field‐effect transistors

Solution‐processable polymers consisting of perylene diimide (PDI) acceptor moieties alternating with dithienothiophene (DTT), N‐dodecyl‐dithienopyrrole (DTP), or oligomers of these donor groups have been synthesized. We have, in addition to varying the donor, varied the N,N′ substituents of the PDIs. The thermal, optical, electrochemical, and charge‐transport properties of the polymers have been investigated. The polymers show broad absorption extending from 300 to 1000 nm with optical band gaps as low as 1.2 eV; the band gap decreases with increasing the conjugation length of donor block, or by replacement of DTT by DTP. The electron affinities of the polymers, estimated from electrochemical data, range from ?3.87 to ?4.01 eV and are slightly affected by the specific choice of donor moiety, while the estimated ionization potentials (?5.31 to ?5.92 eV) are more sensitive to the choice of donor. Bottom‐gate top‐contact organic field‐effect transistors based on the polymers generally exhibit n‐channel behavior with electron mobilities as high as 1.7 × 10^–2 cm²/V/s and on/off ratios as high as 10⁶; one PDI‐DTP polymer is an ambipolar transport material with electron mobility of 4 × 10^–4 cm²/V/s and hole mobility of 4 × 10^–5 cm²/V/s in air. There is considerable variation in the charge transport properties of the polymers with the chemical structures. © 2013 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2013     

Correlation of charge transport with structural order in highly ordered melt‐crystallized poly(3‐hexylthiophene) thin films

A series of well‐defined poly(3‐hexylthiophene)s (P3HT) of different molecular weight (MW) and high regioregularity was investigated for charge transport properties in as‐cast and melt‐crystallized films. The semicrystalline structure of the P3HT was characterized by X‐ray scattering and Atomic force microscopy. Crystallization by cooling from the melt led to a substantial increase in crystallinity and a stronger alignment of the crystals in comparison to as‐cast films. The increase in crystallinity went along with an increase in hole mobility of up to an order of magnitude as measured by the space charge limited current method. Additionally, the hole mobility depended on the long period of P3HT lamellae and consequently on the MW. In compliance with the long period, the charge carrier mobility first increased with the MW before decreasing again at the onset of chain folding. © 2013 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2013 , 51, 943–951