Reflecting the pascal matrix about its main antidiagonal

Let B denote either of two varieties of order n Pascal matrix, i.e., one whose entries are the binomial coefficients. Let B_R denote the reflection of B about its main antidiagonal. The matrix B is always invertible modulo n; our main result asserts that B^-1 ≡ B^R mod n if and only if n is prime. In the course of motivating this result we encounter and highlight some of the difficulties with the matrix exponential under modular arithmetic. We then use our main result to extend the "Fibonacci diagonal" property of Pascal matrices.     

Temperature and concentration quenching of mid-IR photoluminescence in iron doped ZnSe and ZnS laser crystals

Concentration dependences of the mid-IR kinetic of luminescence at ⁵E?⁵T₂ transition in Fe:ZnSe and Fe:ZnS laser samples were studied in 14–300 K temperature ranges. Radiation lifetime in Fe:ZnSe samples measured using low doped samples with iron concentration 0.1×10¹⁸ cm^?3 was estimated to be 57 μs. The magnetic susceptibility for higher doped (C_Fe=38 and 112×10¹⁸ cm^?3) Fe:ZnSe samples was found to consist of a paramagnetic Curie–Weiss behavior arising from the weakly interacting Fe²⁺ ions and a diamagnetic ZnSe contribution plus a temperature-independent, field-dependent contribution possibly arising from very small amounts of aggregated Fe.     

Density functional theory study of the beta-carotene radical cation and deprotonated radicals

The beta-carotene radical cation and deprotonated neutral radicals were studied at the density functional theory (DFT) level using different density functionals and basis sets: B3LYP/3-21G, SVWN5/6-31G*, BPW91/DGDZVP2, and B3LYP/6-31G**. The geometries, total energies, spin distributions, and isotropic and anisotropic hyperfine coupling constants of these species were calculated. Deprotonation of the methyl group at the double bond of the cyclohexene ring of the carotenoid radical cation at 5 or 5' produces the most stable neutral radical because of retention of the pi-conjugated system while less stable deprotonation at 9 or 9' and 13 or 13' of the chain methyl groups causes significant distortion of the conjugation. The predicted methyl hyperfine coupling constants of 13-16 MHz of the neutral radicals are in good agreement with the previous electron nuclear double resonance (ENDOR) spectrum of photolyzed beta-carotene on a solid support. DFT calculations on the beta-carotene radical cation in a polar water environment showed that the polar environment does not cause significant changes in the proton hyperfine constants from those in the isolated gas-phase molecule. DFT calculated methyl proton hyperfine coupling constants of less than 7.2 MHz are in agreement with those reported for the radical cation in photosystem II (PS II) and those found in the absence of UV light for the radical cation on a silica alumina matrix.     

Conservation of adapting components in auditory-nerve responses

The responses of single auditory-nerve fibers of Mongolian gerbil were studied using tonal stimuli. The peristimulatory adaptation of firing rate in response to tone bursts presented in quiet and during a background stimulus is described quantitatively. The total transient response which can be produced to the onset of a tone burst, whether presented in quiet or as an intensity increment, is limited and appears to demonstrate a form of conservation. Specifically, the total numbers of spikes produced by the rapidly adapting component, and the slower short-term adaptation component, are proportional at all intensities, and are limited for each fiber. Furthermore, when an incremental stimulus is presented on a background, the total transient response to the background and to the increment is limited and depends upon the final intensity, not the background intensity. When the presumed underlying synaptic drive is derived by removing the effects of refractoriness from the spike train, the same conservation of the transient response components, and proportionality between rapid and short-term components, are observed.     

Thermodynamics of the system benzene-carbon tetrachloride

The thermodynamic behavior of benzene-carbon tetrachloride solutions at temperatures between 30 and 70°C and over the entire concentration range has been examined in terms of the Maron theory of nonelectrolyte solutions. The interaction parameter calculated from vapor pressure data has been found to be independent of concentration and to be linearly dependent on temperature. Use of this parameter in the equations given by the theory allows complete correlation of the available heats, entropies, and heat capacities of mixing for this system. For all these properties very good agreement has been obtained between the calculated and observed quantities.     

Chemical analysis of vanadium pentoxide catalysts

Reliable procedures are described for the analysis of alkali metal sulphate-promoted vanadium pentoxide catalysts supported on silica gel. Vanadium and alkali metals are determined by flame atomic absorption spectrometry in suitable dilutions of solutions obtained by digestion with sulphuric and hydrofluoric acids. “Free acidity” is determined by dissolution of the samples in alkali and back-titration with acid ; corrections for V₂0₅ acidity are included. Relative standard deviations are about 2.5% for metal contents of3–5%($\text{w}\text{w}$).     

Dimensional renormalization of scalar field theory in curved space-time

The regularization and renormalization of an interacting scalar field φ in a curved spacetime background is performed by the method of continuation to n dimensions. In addition to the familiar counter terms of the flat-space theory, c-number, “vacuum” counter terms must also be introduced. These involve zero, first, and second powers of the Reimann curvature tensor R_αβψδ. Moreover, the renormalizability of the theory requires that the Lagrange function couple φ² to the curvature scalar R with a coupling constant η. The coupling η must obey an inhomogeneous renormalization group equation, but otherwise it is an arbitrary, free parameter. All the counter terms obey renormalization group equations which determine the complete structure of these quantities in terms of the residues of their simple poles in n ? 4. The coefficient functions of the counter terms determine the construction of φ² and φ⁴ in terms of renormalized composite operators 1, [φ²], and [φ⁴]. Two of the counter terms vanish in conformally flat space-time. The others may be computed from the theory in purely flat space-time. They are determined, in a rather intricate fashion, by the additive renormalizations for two-point functions of [φ²] and [φ⁴] in Minkowski space-time. In particular, using this method, we compute the leading divergence of the R² interaction which is of fifth order in the coupling constant λ.