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     

Structures and properties of functional metal iodates

Metal iodates with a lone-pair containing I(V) that is in an asymmetric coordination geometry can form a diversity of unusual structures and many of them are promising new second homonic generation (SHG) materials. They exhibit wide transparency wavelength regions, large SHG coefficients and high optical-damage thresholds as well as moderately high thermal stability. In this paper, the structures and properties of the metal iodates are reviewed. The combination of d0 transition-metal cations with the iodate...     

Structure–reactivity relationships of alkyl α‐hydroxymethacrylate derivatives

A series of alkyl α‐hydroxymethacrylate derivatives with various secondary functionalities (ether, ester, carbonate, and carbamate) and terminal groups (alkyl, cyano, oxetane, cyclic carbonate, phenyl and morpholine) were synthesized to investigate the effect of intermolecular interactions, H‐bonding, π–π interactions, and dipole moment on monomer reactivity. All of the monomers except one ester and one ether derivative are novel. The polymerization rates, determined by using photo‐DSC, showed the average trend (aromatic carbamate > hydroxyl > ester > carbonate ～ aliphatic carbamate ～ ether), with several exceptions due to the differences in terminal groups. There is a correlation between the chemical shift differences of the double bond carbons, the calculated dipole moments, and the reactivities only for nonhydrogen bonded monomers. © 2011 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2011