On a conjecture of S. Chowla and of S. Chowla and H. Walum,II

In 1965, Chowla and Walum conjectured that, $\text{G}{}_{\mathrm{a,k}}\text{(x):=}\text{Σ}{}_{\mathrm{n\; \le \; \surd x}}\text{n}{}^{\mathrm{a}}\text{P}{}_{\mathrm{k}}\text{(}\text{x}\text{n}\text{) = O(x}{}^{\mathrm{<mtext>a</mtext><mtext>2</mtext><mtext>\; +\; </mtext><mtext>1</mtext><mtext>4</mtext><mtext>\; +\; \epsilon </mtext>}}\text{)}$ holds for each ε > 0 and x → ∞, where integers a ≥ 0 and k ≥ 1 are given and P_k is the periodic Bernoulli function of order k. Recently, the authors established this conjecture in case k ≥ 2 and $\text{a >}\text{1}\text{2}$ with ε = 0 while if $\text{a =}\text{1}\text{2}$, with a log factor. In this paper, it is proven that the conjecture, in case $\text{|a| <}\text{1}\text{2}$, k = 2, is true “on average”.     

Ortho effects in organic molecules on electron impact. Part 16. Novel oxygen transfers from the nitro group to acetylenic carbons in 2-nitrodiphenylacetylenes

Oxygen transfers to both acetylene carbons are noticed in parallel fragmentation pathways during the electron impact induced decompositions of 2-nitrodiphenylacetylene. The oxygen transfer to β-acetylenic carbon leads to the most abundant ion corresponding to benzoyl cation while transfer to the α-acetylenic carbon affords less intense fragments corresponding to [M? OH]⁺, [M? CO]^{+ ˙} and [M? CO₂]^{+ ˙}. The proposed fragmentation pathways and ion structures are supported by high-resolution data, linked-scan spectra and chemical substitution.     

Pre-equilibrium effects in (n, 2n) reactions at 14.2 MeV+

With a view to study the pre-equilibrium effects in neutron-induced reactions, the activation cross-sections for (n, 2n) reactions at 14.2±0.2 MeV in the heavy mass region have been measured using the versatile mixed powder technique and high resolution Ge(Li) detection. The experimental cross-sections are found to be consistently smaller than the predictions based on the statistical theory and this is attributed to the effect of pre-equilibrium decay in these reactions. The cross-sections due to pre-equilibrium decay were estimated using exciton, hybrid and unified models. When this cross-section was included in comparing the experimental cross-sections with theory, we obtained better agreement, within the limitations of the present-day preequilibrium theories. Preliminary results of this work were reported at the Nuclear Physics and Solid State Physics Symposium, Pune (1978).     

Poisson structures due to Lie algebra representations

Using a unitary solution of the classical Yang-Baxter equation on a Lie algebraG we describe a particular way of constructing homogeneous quadratic Poisson structures on the dual of aG-moduleV and study some local features of the symplectic foliation due to the involutive distribution of the Hamiltonian vector fields. We also give some examples where the symplectic leaves are explicitly calculated.     

On the geometry of nonequilibrium magneto gasdynamic flows

After formulation of the various dynamical and kinematical relations connecting the flow quantities with the geometrical parameters of the stream line trajectories, the expressions for the tangent, principal normal and binormal vectors and the curvature and torsion of the stream line have been obtained in-terms of the velocity components, the pressure, the density, the magnetic field and the relaxation variable. This is followed by expressing the equations governing the flow in the intrinsic forms. It has been shown that the non-equilibrium character of the gas decreases the total pressure gradient along the streamlines, but the total pressure remains constant along the binormals and if the stream lines are straight lines, the trajectories of the principal normals lie on the surface of the constant total pressure. Further, the expressions for vorticity components in terms of curvature of the stream line and the velocity gradients along the stream line and their principal normals and binormals have been obtained. Finally, a class of circular helical flows have been discussed.     

Physico-chemical studies on Gum Dhawa (Anogeissus latifoliaWall.)

Summary The p_H of an aqueous solution of Gum Dhawa (Anogeissus latifoliaWall.) is 2.68 and it is sensitive to p_H changes. The absence of the sulphate and the presence of a slight trace of phosphate, confirms the view that the acidity is due to the presence of carboxyl groups, which are formed by the hydrolysis of the gum. The hydrogen ion activity increases up to a concentration of 3% and beyond this point it attains a steady value. The sp. cond. is zero at the zero concentration and the relation between sp. cond. and up to 3% concentration is linear and afterwards it attains a steady value. It has been found by experimental observation that the gum solution has got no buffering capacity. Neutralization curves with NaOH and Ba(OH)₂ show that the combination with alkali takes place in equivalent proportion. The viscosity of the gum solution always increases with increasing p_H on the addition of HCl. The viscosity increase with NaOH is slow in the beginning, but after p_H 4.2 it rises rapidly, attains a maximum and steeply falls off near p_H 6.7. Now it decreases slowly and again a deviation is noted at 10.5. The viscosity is maximum near about the neutralization point. This is due to the stretched chain of the polymer unit. The fall in viscosity is explained by the folding chain theory in conjunction with mass action or common ion effect and due to the destruction of the proteins, by concentrated NaOH, which forms a complex with carbohydrate polymer.The authors wish to thank Dr.R. C. Mehrotra, Professor and the Head of the Chemistry Department, for providing all facilities for this work. Our thanks are also due to the Ministry of Education, Government of India, for the award of a research scholarship to one of us (V.K.S.).