Synthesis and properties of a new polydiacetylene: Poly[1,6-di(N-carbazolyl)-2,4-hexadiyne]

The solid-state synthesis and properties are reported for a new polydiacetylene: poly[1,6-di(N-carbazolyl)-2,4-hexadiyne]. The monomer crystals polymerize quantitatively with γ irradiation or thermal annealing. An Autocatalytic effect is observed in both γ-ray polymerization and thermal polymerization and is attributed to an increase in chain propagation length at about 5% conversion. The activation energy for thermal polymerization is about 25 kcal/mole, independent of the degree of conversion to polymer. The exceptional thermal stability of the polymer crystals allowed a thermomechanical analysis over a large temperature range, ?50 to 300°C. With increasing temperature, the polymer contracts in the chain direction linearly with temperature over the entire range, yielding a thermal expansion coefficient of (?2.32 ± 0.02) × 10^?5°C^?1. Photoconductivity action spectra are reported for the polymer crystals. The energies for the photoconductivity onset (ca. 2.3 eV) and for the lowest energy optical transition (1.89 eV) are the lowest reported for the polydiacetylenes. The photoconduction onset is blue-shifted with respect to optical absorption—a result which is consistent with the excitonic assignment for the lowest energy optical transition in the polydiacetylenes.     

Rotational analysis of the 1Π-X1Σ+ system of phosphorous antimonide

Seven bands of the ¹Π-X¹Σ⁺ band system of P¹²¹Sb have been analyzed in detail. A large number of strong rotational perturbations in the v = 0 and 1 levels and a vibrational perturbation in the v = 0 level of the Π state have been observed. The term values of the rotational levels and rotational constants for the two isotopic species, P¹²¹Sb and P¹²³Sb, are presented.     

Solid-state synthesis of a ladder polymer

Solid-state synthesis of a cyclically bound ladder polymer from a cyclic tetradiyne (cyclodotriaconta-1,3,9,11,17,19,25,27-octayne) is described. Irradiation of the colorless, needlelike monomer crystals with interstitially incorporated chloroform with 50 Mrad of ⁶⁰Co γ-ray radiation results in red-brown polymer fibers in nearly quantitative yield. Infrared, Raman, and x-ray diffraction analyses of the polymer are consistent with polymerization by a 1,4-addition reaction at each diacetylene linkage to produce four fully conjugated chains joined together in pairs by a total of four ? (CH₂)₄? interchain linkages per 4.8-Å polymer repeat unit. Conformationally, it appears that the cyclic tetradiyne monomer polymerizes via a chair form. The results of the mechanical and thermal analyses indicate the presence of unreacted diacetylene functionality in the ladder polymer crystals.     

Ramanujan’s partial theta series and parity in partitions

A partial theta series identity from Ramanujan’s lost notebook has a connection with some parity problems in partitions studied by Andrews in Ramanujan J., to appear  where 15 open problems are listed. In this paper, the partial theta series identity of Ramanujan is revisited and answers to Questions 9 and 10 of Andrews are provided.     

Empirical study of the scaling behavior of the amplitude–frequency distribution of the Hilbert–Huang transform and its application in sunspot time series analysis

Investigating long-range correlation by the Hurst exponent, H$H$, is crucial in the study of time series. Recently, empirical-mode-decomposition-based arbitrary-order Hilbert spectral analysis (EMD-HSA) has been proposed to numerically obtain without proof a scaling relationship, generated from the amplitude–frequency distribution, related to H$H$. We propose a formalism to empirically study EMD-HSA, to deduce its scaling exponent ξ(q)$\xi \left(q\right)$ from the perspective of EMD-based arbitrary-order Hilbert marginal spectrum (EMD-HMS), and to numerically compare the results with the expected H$H$. EMD-HSA and EMD-HMS experiments show that, by incompletely removing (quasi-)periodic trends, the sunspot series should have an H$H$ value around 0.12.     

Evolutionary snowdrift game incorporating costly punishment in structured populations

The role of punishment and the effects of a structured population in promoting cooperation are important issues. Within a recent model of snowdrift game (SG) incorporating a costly punishing strategy (P), we study the effects of a population connected through a square lattice. The punishers, who carry basically a cooperative (C) character, are willing to pay a cost α$\alpha$ so as to punish a non-cooperative (D) opponent by β$\beta$. Depending on α$\alpha$, β$\beta$, the cost-to-benefit ratio r$r$ in SG, and the initial conditions, the system evolves into different phases that could be homogeneous or inhomogeneous. The spatial structure imposes geometrical constraint on how one agent is affected by neighboring agents. Results of extensive numerical simulations, both for the steady state and the dynamics, are presented. Possible phases are identified and discussed, and isolated phases in the r–β$r\text{–}\beta$ space are identified as special local structures of strategies that are stable due to the lattice structure. In contrast to a well-mixed population where punishers are suppressed due to the cost of punishment, the altruistic punishing strategy can flourish and prevail for appropriate values of the parameters, implying an enhancement in cooperation by imposing punishments in a structured population. The system could evolve to a phase corresponding to the coexistence of C, D, and P strategies at some particular payoff parameters, and such a phase is absent in a well-mixed population. The pair approximation, a commonly used analytic approach, is extended from a two-strategy system to a three-strategy system. We show that the pair approximation can, at best, capture the numerical results only qualitatively. Due to the improper way of including spatial correlation imposed by the lattice structure, the approximation does not give the frequencies of C, D, and P accurately and fails to give the homogeneous AllD and AllP phases.     

Generation of 287 W, 5.5 ps pulses at 78 MHz repetition rate from a cryogenically cooled Yb:YAG amplifier seeded by a fiber chirped-pulse amplification system

We generate linearly polarized, 287 W average-power, 5.5 ps pulses using a cryogenically cooled Yb:YAG amplifier at a repetition rate of 78 MHz. An optical-to-optical efficiency of 41% is obtained at 700 W pump power. A 6 W, 0.4 nm bandwidth picosecond seed source at 1029 nm wavelength is constructed using a chirped-pulse fiber amplification chain based on chirped volume Bragg gratings. The combination of a fiber amplifier system and a cryogenically cooled Yb:YAG amplifier results in good spatial beam quality at large average power. Low nonlinear phase accumulation as small as 5.1 x 10(-3) rad in the bulk Yb:YAG amplifier supports power scalability to a > 10 kW level without being affected by self-phase modulation. This amplification system is well suited for pumping high-power high-repetition-rate optical parametric chirped-pulse amplifiers.