A Sharp Inequality of Markov Type for Polynomials Associated with Laguerre Weight

The best possible constant A_n in an inequality of Markov type

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, where ·_[0, ∞) denotes the sup-norm on the half real line [0, ∞) and p_n is an arbitrary polynomial of degree at most n, is determined in terms of the weighted Chebyshev polynomials associated with the Laguerre weight e^−x on [0, ∞).     

Oblique wave scattering by an impermeable ocean-bed of variable depth in a two-layer fluid with ice-cover

Within the framework of linearized theory, obliquely incident water wave scattering by an uneven ocean-bed in the form of a small bottom undulation in a two-layer fluid, where the upper layer has a thin ice-cover while the lower one has the undulation, is investigated here. In such a two-layer fluid, there exist two modes of time-harmonic waves—the one with lower wave number propagating just below the ice-cover and the one with higher wave number along the interface. An incident wave of a particular mode gets reflected and transmitted by the bottom undulations into waves of both the modes. Assuming irrotational motion, a perturbation technique is employed to solve the first-order corrections to the velocity potentials in the two-layer fluid by using Fourier transform appropriately and also to calculate the reflection and transmission coefficients in terms of integrals involving the shape function representing the bottom undulation. For a sinusoidal bottom topography, these coefficients are depicted graphically against the wave number. It is observed that when the oblique wave is incident on the ice-cover surface, we always find energy transfer to the interface, but for interfacial oblique incident waves, there are parameter ranges for which no energy transfer to the ice-cover surface is possible.     

Understanding neutrino masses and mixings

We discuss ways to understand large neutrino mixings using new symmetries of quarks and leptons beyond the standard model for the three allowed patterns of neutrino masses: normal, inverted hierarchy and degenerate masses.     

Extraction of $$ {\text{Np}}^{4+} $$ and $$ {\text{NpO}}_{2}^{2+} $$NpO22+ from Nitric Acid Medium Using TODGA in Room Temperature Ionic Liquids

Extraction of Np⁴⁺ and \( {\text{NpO}}_{2}^{2 + } \) was carried out from nitric acid feeds using solutions of N,N,N′,N′-tetra-n-octyldiglycolamide (TODGA) in two imidazolium-based room temperature ionic liquids, viz., 1-butyl-3-methylimidazolium bis(trifluoromethanesulphonyl) imide ([C₄mim][NTf₂]) and 1-octyl-3-methylimidazolium bis(trifluoromethanesulphonyl) imide ([C₈mim][NTf₂]). The extraction equilibrium was attained within 2 h for both the metal ions in both the ionic liquids. While a cation exchange mechanism is proposed for the extraction of \( {\text{NpO}}_{2}^{2 + } , \) an ion-pair mechanism of extraction is proposed for the Np⁴⁺ ion. The nature of the extracted species was determined by carrying out experiments at varying concentrations of TODGA, and species of the type Np(L)₂(NO₃)₄ and NpO₂(L)²⁺ were found to be extracted in 3 mol·dm^?3 HNO₃. The identification of these extracted species was also supported from the variable nitrate and C₄mim⁺ ion concentration experiments.     

A protecting-group-free synthesis of arbusculone,andirolactone, pinnatolide,ipomolactone, cyclocapitelline and isocyclocapitelline

A general approach for a collective synthesis of natural products containing substituted THF ring is described. In this paper, Arbusculone, a small molecule natural product accomplished using a short route, is used as the key intermediate to achieve the total synthesis of Andirolactone, Pinnatolide, Ipomolactone, Cyclocapitelline, Isocyclocapitelline and their two isomers in less than ten steps. The present effort highlights protecting-group-free total syntheses and the shortest route to access these natural products from commercially available cheap starting materials.