Parameśvara's rule for the circumradius of a cyclic quadrilateral

The expression for the circumradius of a cyclic quadrilateral in terms of its sides, usually attributed to L'Huilier in 1782, was known in India to Parame?vara (circa 1430). The present paper contains the original Sanskrit text of the rule, its English translation, and a discussion of its derivation as given by Sankara Váriar in his Kriy$\text{a}$kramakar$\text{i}$ (16th Century) along with relevant historical remarks.     

Carboncarbon bond cleavage during lithium aluminum hydride reduction of a terminal propargyl alcohol

Reduction of dehydronerolidol with lithium aluminum hydride in the presence of sodium methoxide gave, in addition to the expected nerolidol, geranyl acetone. Labeling experiments established that the C-1 methyl of geranyl acetone is derived from one of the carbons of the acetylene.     

An Indian form of third order Taylor series approximation of the sine

The paper describes an approximation formula for sine (x + h) that differs from the first four terms of the Taylor expansion only by having 4 in place of 6 in the denominator of the fourth term. It appears in Sanskrit stanzas quoted in a work of about the fifteenth century and given here with translation and explanation.     

Fourier multipliers for Sobolev spaces on the Heisenberg group

In this paper, it is shown that the class of right Fourier multipliers for the Sobolev space W ^k,p (H ⁿ ) coincides with the class of right Fourier multipliers for L ^p (H ⁿ ) for k ∈ ?, 1 < p < ∞. Towards this end, it is shown that the operators R _j $ \bar R $ _j ?^?1 and $ \bar R $ _j R _j ?^?1 are bounded on L ^p (H ⁿ ), 1 < p < ∞, where $$ R_j = \frac{\partial } {{\partial z_j }} - \frac{i} {4}\bar z_j \frac{\partial } {{\partial t}}, \bar R_j = \frac{\partial } {{\partial \bar z_j }} + \frac{i} {4}z_j \frac{\partial } {{\partial t}} $$ and ? is the sublaplacian on H ⁿ . This proof is based on the Calderon-Zygmund theory on the Heisenberg group. It is also shown that when p = 1, the class of right multipliers for the Sobolev space W ^k,1(H ⁿ ) coincides with the dual space of the projective tensor product of two function spaces.     

On the images of Sobolev spaces under the heat kernel transform on the Heisenberg group

The aim of this paper is to obtain certain characterizations for the image of a Sobolev space on the Heisenberg group under the heat kernel transform. We give three types of characterizations for the image of a Sobolev space of positive order $H^m(\mathbb {H}^n), m\in \mathbb {N}^n,$ under the heat kernel transform on $\mathbb {H}^n,$ using direct sum and direct integral of Bergmann spaces and certain unitary representations of $\mathbb {H}^n$ which can be realized on the Hilbert space of Hilbert‐Schmidt operators on $L^2(\mathbb {R}^n).$ We also show that the image of Sobolev space of negative order $H^{-s}(\mathbb {H}^n), s(>0) \in \mathbb {R}$ is a direct sum of two weighted Bergman spaces. Finally, we try to obtain some pointwise estimates for the functions in the image of Schwartz class on $\mathbb {H}^n$ under the heat kernel transform.     

Studies towards the stereoselective total synthesis of Gliomasolide A

The first synthetic approach towards the Gliomasolide A is described in 15 linear steps with 4% overall yield. The key steps in this approach are RCM protocol for the construction of 14-member macrolide, a Sharpless kinetic resolution, Keck and Brown’s allylations for the installation of desired stereocenters.     

Role of hot-electron preheating in the compression of direct-drive imploding targets with cryogenic D2 ablators

The compression of direct-drive, spherical implosions is studied using cryogenic D2 targets on the 60-beam, 351-nm OMEGA laser with intensities ranging from approximately 3x10(14) to approximately 1x10(15) W/cm2. The hard-x-ray signal from hot electrons generated by laser-plasma instabilities increases with laser intensity, while the areal density decreases. Mitigating hot-electron production, by reducing the laser intensity to approximately 3x10(14) W/cm2, results in areal density of the order of approximately 140 mg/cm2, in good agreement with 1D simulations. These results will be considered in future direct-drive-ignition designs.     

EPR and Optical Studies of Mo<Superscript>5+</Superscript> Ions in Lithium Molybdoborate Glasses

Electron paramagnetic resonance (EPR) and optical absorption studies of Li₂O–MoO₃–B₂O₃ with varying concentrations of Li₂O, MoO₃ and B₂O₃ have been carried out at room temperature. Two series of glasses, one with constant MoO₃ (CM) and another with constant borate (CB), have been investigated. Characteristic EPR spectra of Mo⁵⁺ have been observed centered around g ≅ 2.00, which are attributed to Mo⁵⁺ ion in an octahedral coordination sphere with an axial distortion. The spectra also show strong dependence on the concentration of Li₂O and B₂O₃. Spin concentrations (N) and magnetic susceptibilities (χ) have been calculated. In the CM series, the N values decrease with increasing Li₂O content up to 30 mol%, while in the CB series variation of N is found to increase initially up to 20 mol%, and with further increase in the Li₂O content the N values tend to decrease. The variation of magnetic susceptibilities is almost similar to that observed with the variation of N. From the optical absorption spectra, an absorption edge (α) has been evaluated. In the CM series, the values of α show a blueshift. On the other hand, in the CB series a redshift is observed. The observed variations in spectral parameters are explained by considering the molybdoborate network. Addition of Li₂O to the CM and CB series results in modification of [MoO_6/2]⁰ → [MoOO_5/2]⁻ and [BO_3/2]⁰ → [BO_4/2]⁻ → [BOO_2/2]⁻ groups, respectively, leading to creation of nonbridging oxygens. The optical basicity of the glasses has been evaluated in both the CM and the CB glasses. The optical basicity can be used to classify the covalent-to-ionic ratios of the glass, since an increasing optical basicity indicates decreasing covalency. It is observed that the covalency between Mo⁵⁺ ions and oxygen ligands increases in the CB series, whereas in the CM series the covalency between Mo⁵⁺ ions and oxygen ligands decreases. Authors' address: R. P. Sreekanth Chakradhar, Glass Technology Laboratory, Central Glass and Ceramic Research Institute, Kolkata 700032, India