Parity in partition identities

This paper considers a variety of parity questions connected with classical partition identities of Euler, Rogers, Ramanujan and Gordon. We begin by restricting the partitions in the Rogers-Ramanujan-Gordon identities to those wherein even parts appear an even number of times. We then take up questions involving sequences of alternating parity in the parts of partitions. This latter study leads to: (1) a bi-basic q-binomial theorem and q-binomial series, (2) a new interpretation of the Rogers-Ramanujan identities, and (3) a new natural interpretation of the fifth-order mock theta functions f ₀(q) along with a new proof of the Hecke-type series representation.     

Amalgamation of partitions from multiple segmentation bases: A comparison of non-model-based and model-based methods

The segmentation of customers on multiple bases is a pervasive problem in marketing research. For example, segmentation service providers partition customers using a variety of demographic and psychographic characteristics, as well as an array of consumption attributes such as brand loyalty, switching behavior, and product/service satisfaction. Unfortunately, the partitions obtained from multiple bases are often not in good agreement with one another, making effective segmentation a difficult managerial task. Therefore, the construction of segments using multiple independent bases often results in a need to establish a partition that represents an amalgamation or consensus of the individual partitions. In this paper, we compare three methods for finding a consensus partition. The first two methods are deterministic, do not use a statistical model in the development of the consensus partition, and are representative of methods used in commercial settings, whereas the third method is based on finite mixture modeling. In a large-scale simulation experiment the finite mixture model yielded better average recovery of holdout (validation) partitions than its non-model-based competitors. This result calls for important changes in the current practice of segmentation service providers that group customers for a variety of managerial goals related to the design and marketing of products and services.     

Two-color polarization spectroscopy of rubidium

We report on two-color, two-photon polarization spectroscopy in a room-temperature rubidium vapor. We use two separate lasers, a strong pump at 780 nm to induce an anisotropy in the atomic polarization and a weak probe at 776 nm to interrogate this anisotropy. The lasers are resonant with the 5²S_1/2 → 5²P_3/2 and 5²P_3/2 → 5²D_5/2 transitions in rubidium, respectively. Finally, we have used our polarization spectroscopy signal as an error signal to lock the 776 nm laser. This modulation-free locking scheme allows us to detune the lock point of the second laser by adjusting the detuning of the laser used for the first transition.     

Inter-particle interaction induced by broadband radiation

The recent fabrication of optical traps using super-continuum light impresses the need for a theory of inter-particle interactions under such conditions. Development of the theory provides a basis for calculation of the observations expected under experimental conditions. An expression for the inter-particle potential energy induced by continuum states of light is first derived using quantum electrodynamics. This energy expression is cast as a function of the spectral irradiance of the light, and the electric susceptibility of the interacting particles. Specific results are derived for light with a Lorentzian spectrum. It is shown that by filtering part of the spectrum, it is possible to exert control over the length of linear particle chains organized along the Poynting vector. The results exhibit scope for the optical fabrication of moldable structures using broadband light.     

Infrared spectrum of the novel electron-deficient BH(4) radical in solid neon

Laser-ablated boron reacts with hydrogen on condensation in excess neon to give BH4 radical, BH4- anion, and B2H6 as the major products. Identifications are based on 10B and D substitution, DFT frequency calculations, and comparison to previous spectra. Infrared spectra of BH4 support the C2v structure deduced from previous ESR spectra and theoretical calculations with two normal B-H bonds and two long B-H bonds for this novel electron-deficient radical. NBO analysis suggests that the two long B-H bonds and the H- -H bond are one-electron bonds.     

Argon matrix absorption spectra of ClO, BrO, and IO and emission spectra of IO

Absorption spectra of ClO, BrO, and IO and the emission spectrum of IO have been observed from argon matrix samples prepared by microwave discharging the reagent mixture before condensation. Vibronic progressions were observed for each system. The spectroscopic scopic constants T_e, ω′_e, ω_ex′_e, ω″_e, and ω_ex″_e were evaluated from the absorption and emission data for comparison with gas-phase constants. Very good agreement is found for ClO. The argon matrix observations dictate a revision of the gas-phase vibronic assignments for BrO. The ground-state vibrational fundamental and T_e for argon matrix isolated IO are similar to the gas-phase values, but a lower excited-state spacing is found in the matrix.