Modular forms and differential operators

In 1956, Rankin described which polynomials in the derivatives of modular forms are again modular forms, and in 1977, H Cohen defined for eachn ≥ 0 a bilinear operation which assigns to two modular formsf andg of weightk andl a modular form [f, g]_n of weightk +l + 2n. In the present paper we study these “Rankin-Cohen brackets” from two points of view. On the one hand we give various explanations of their modularity and various algebraic relations among them by relating the modular form theory to the theories of theta series, of Jacobi forms, and of pseudodifferential operators. In a different direction, we study the abstract algebraic structure (“RC algebra”) consisting of a graded vector space together with a collection of bilinear operations [,]_n of degree + 2n satisfying all of the axioms of the Rankin-Cohen brackets. Under certain hypotheses, these turn out to be equivalent to commutative graded algebras together with a derivationS of degree 2 and an element Φ of degree 4, up to the equivalence relation (∂,Φ) ~ (∂ - ϕE, Φ - ϕ² + ∂(ϕ)) where ϕ is an element of degree 2 andE is the Fuler operator (= multiplication by the degree). Dedicated to the memory of Professor K G Ramanathan     

Infrared spectra of 1-phosphapropyne,CH3CP,and its perdeuteride,CD3CP

The infrared spectra of 1-phosphapropyne, CH₃CP, and its perdeuteride, CD₃CP, have been measured in the gaseous and solid states. The Q_K branches of perpendicular bands have been analyzed in terms of the usual quadratic expression in K. Fermi resonances were identified for the ν₁, ν₂ + ν₃, 2ν₃, 2ν₆⁰; and ν₅, 2ν₃ + ν₈ band systems of CH₃CP and the ν₁, 2ν₃, 2ν₆⁰; ν₆, ν₇ + ν₈; and ν₇, 3ν₈¹ band systems of CD₃CP. The xy Coriolis interaction was also identified between the ν₃ and ν₆ bands of the two species. All the fundamentals were assigned and the normal coordinate treatment was carried out along with the Coriolis constants, ζ^z.     

Analysis of the click-evoked brainstem potentials in man unsing high-pass noise masking

Brainstem electrical responses (BSER) to 60-dB-SL click in noise high passed at various cutoff frequencies separated b 1/2-octave steps were recorded in normal-hearing adult subjects. By applying a derived response technique, narrow-band contributions to the BSER from specific portions of the basilar membrane were revealed. Latencies and amplitudes of the various waves in the derived BSER were recorded. Results indicate that nearly the whole cochlear partition can contribute to the brainstem response. The shifts in latency of waves I, III, and V and amplitude changes of waves I and III as a function of CF appear to be fully comparable to those of the AP. In contrast, the amplitude behavior of wave V as a function of CF is different from waves I and III depending upon frequency range. The discrepency in the behavior of wave V with respect to the earlier waves suggests some sort of neural reorganization at the level where was V is generated. The fact that there are contributions to the brainstem response from apical portions of the cochlea opens the possibility for extending the brainstem technique in assessing the higher cochlear turn function.     

Structure sensitive factors in molecular cluster formation by ion bombardment of single crystal surfaces

The dynamics of molecular cluster formation from a solid bombarded by a 600 eV Ar⁺ ion have been studied classically by computer simulation. The dimers and trimers are found to establish their identity as clusters within interaction range of the solid, but not by a direct ejection of a bound molecule. The Cu₂/Cu and Cu₃/Cu ratios are found to be strongly dependent on crystal orientation. The (111) face is 2–3 times more likely to produce multimers than the (100) face. We find 9 trimers from (111) but none from (110). The relationship between cluster composition and the original arrangement of those atoms on the surface is presented in detail. We find that each multimer forms from atoms that originate within a roughly circular region of area ～70 Ų or less. This region is not necessarily centered on the ion impact point. A consequence of this observation is that dimers can consist of atoms that were several Ångströms apart on the surface but that most trimers contain at least one nearest neighbor pair of atoms. The calculated energy distribution for the dimers matches well with similar experimental studies.     

Laboratory simulation of atmospheric turbulence-induced optical wavefront distortion

Real-time liquid crystal television-based technique for simulating optical wavefront distortion due to atmospheric turbulence is presented and demonstrated. A liquid crystal television (LCTV) operating in the “phase mostly” mode was used as an array of spatially correlated phase delays. A movie of the arrays in motion was then generated and displayed on the LCTV. The turbulence simulation system was verified by passing a collimated and doubled diode pumped Nd:YVO₄ laser beam (532 nm) through the transparent LCTV screen. The beam was then passed through a lens and the power spectra of the turbulence information carrying beam was detected as a measure of the far-field distribution. The same collimated laser beam, without the LCTV, was also transmitted down an open-air range and the power spectra detected as a measure of a real far-field distribution. Accepted turbulence parameters were measured for both arrangements and then compared.     

The effects of sensory hearing loss on cochlear filter times estimated from auditory brainstem response latencies.

Derived-band auditory brainstem responses (ABRs) were obtained in 43 normal-hearing and 80 cochlear hearing-impaired individuals using clicks and high-pass noise masking. The response times across the cochlea [the latency difference between wave V's of the 5.7- and 1.4-kHz center frequency (CF) derived bands] were calculated for five levels of click stimulation ranging from 53 to 93 dB p.-p.e. SPL (23 to 63 dB nHL) in 10-dB steps. Cochlear response times appeared to shorten significantly with hearing loss, especially when the average pure tone (1 to 8 kHz) hearing loss exceeded 30 dB. Examination of derived-band latencies indicates that this shortening is due to a dramatic decrease of wave V latency in the lower CF derived band. Estimates of cochlear filter times in terms of the number of periods to maximum response (Nmax) were calculated from derived-band latencies corrected for gender-dependent cochlear transport and neural conduction times. Nmax decreased as a function of hearing loss, especially for the low CF derived bands. The functions were similar for both males and females. These results are consistent with broader cochlear tuning due to peripheral hearing loss. Estimating filter response times from ABR latencies enhances objective noninvasive diagnosis and allows delineation of the differential effects of pathology on the underlying cochlear mechanisms involved in cochlear transport and filter build-up times.     

Implantation of the 111In/Cd Probe as InO− Ion for Radioisotope Tracer Studies

A radioisotope ion implanter has been developed using a cesium-sputtering, negative ion source, which offers versatility and sustained operation. Employing the molecular ¹¹¹In¹⁶O⁻ ion, μCi activities of the radioisotope probe ¹¹¹In/Cd have been implanted into different material hosts. The implanted tracer activity has been shown to be sufficient for LTNO, NMRON and PAC. A new NMRON resonance for ¹¹¹InAg was observed at 75.08 MHz. In₂O₃ powder performed well as the radioisotope carrier in the ion source, with the ratio of radioisotope and parasitic ion current being typically 4 × 10⁻⁴.     

640 × 512 pixel narrow-band, four-band, and broad-band quantum well infrared photodetector focal plane arrays

A 9 μm cutoff 640 × 512 pixel hand-held quantum well infrared photodetector (QWIP) camera has been demonstrated with excellent imagery. A noise equivalent differential temperature (NEDT) of 10.6 mK is expected at a 65 K operating temperature with f/2 optics at a 300 K background. This focal plane array has shown background limited performance at a 72 K operating temperature with the same optics and background conditions. In this paper, we discuss the development of this very sensitive long-wavelength infrared camera based on a GaAs/AlGaAs QWIP focal plane array and its performance in quantum efficiency, NEDT, uniformity, and operability. In the second section of this paper, we discuss the first demonstration of a monolithic spatially separated four-band 640 × 512 pixel QWIP focal plane array and its performance. The four spectral bands cover 4–5.5, 8.5–10, 10–12, and 13.5–15 μm spectral regions with 640 × 128 pixels in each band. In the last section, we discuss the array performance of a 640 × 512 pixel broad-band (10–16 μm full-width at half-maximum) QWIP focal plane.     

Adaptive behaviour of the spinal cord in the transition from quiet stance to walking

Background

Parkinson??s disease is characterized by a continuous loss of neurons within the substantia nigra (SN) leading to a depletion of dopamine. Within the adult SN as a non-neurogenic region, cells with mainly oligodendrocytic precursor characteristics, expressing the neuro-glial antigen-2 (NG2) are continuously generated. Proliferation of these cells is altered in animal models of Parkinson??s disease (PD). Exercise and environmental enrichment re-increase proliferation of NG2⁺ cells in PD models, however, a possible mechanistic role of dopamine for this increase is not completely understood. NG2⁺ cells can differentiate into oligodendrocytes but also into microglia and neurons as observed in vitro suggesting a possible hint for endogenous regenerative capacity of the SN. We investigated the role of dopamine in NG2-generation and differentiation in the adult SN stimulated by physical activity and environmental enrichment.

Results

We used the 1-methyl-4-phenyl-1,2,3,6-tetrahydropyridine (MPTP)-model for dopamine depletion and analysed newborn cells in the SN at different maturation stages and time points depending on voluntary physical activity, enriched environment and levodopa-treatment. We describe an activity- induced increase of new NG2-positive cells and also mature oligodendrocytes in the SN of healthy mice. Running and enriched environment refused to stimulate NG2-generation and oligodendrogenesis in MPTP-mice, an effect which could be reversed by pharmacological levodopa-induced rescue.

Conclusion

We suggest dopamine being a key regulator for activity-induced generation of NG2-cells and oliogodendrocytes in the SN as a potentially relevant mechanism in endogenous nigral cellular plasticity.