Quadrature in Besov spaces on the Euclidean sphere

Let q1 be an integer, denote the unit sphere embedded in the Euclidean space , and μ_q be its Lebesgue surface measure. We establish upper and lower bounds for

where is the unit ball of a suitable Besov space on the sphere. The upper bounds are obtained for choices of x_k and w_k that admit exact quadrature for spherical polynomials of a given degree, and satisfy a certain continuity condition; the lower bounds are obtained for the infimum of the above quantity over all choices of x_k and w_k. Since the upper and lower bounds agree with respect to order, the complexity of quadrature in Besov spaces on the sphere is thereby established.     

Chemistry of lac resin-VIII: Synthesis of jalaric ester-I,possible key compound in the elaboration of lac resin by laccifer lacca kerr

Preparation of (E)- and (Z)-16-hydroxy-9-hexadecenoic acids and erythro-aleuritic acid starting from naturally occurring aleuritic acid, is described. 16-Hydroxy-(Z)-9-hexadecenoic acid and jalaric acid have been selectively condensed to furnish the naturally occurring jalaric xxxter-I. The possible importance of this compound in the elaboration of lac resin by the insect is pointed out.     

Entrapment of inert gas ions near molybdenum surface

The thermal desorption spectra of inert gas ions (neon, argon and krypton) injected with various energies (430–1950 eV) into a polycrystalline molybdenum target with various dosages (6.4×10¹²−3.9×10¹⁴ ions/cm²) are reported. At least four different states of binding of the trapped atoms corresponding to the activation energies for desorption have been observed from the spectra. The activation energies are found to be relatively insensitive to the species of the bombarding ion, incident ion energy and the dosage. The shapes of the spectra are strongly influenced by the depth of penetration of the ions into the solid. The activation energies deduced are in good agreement with those reported for the migration of atoms and defects in molybdenum.     

Weighted analogues of capacity,transfinite diameter,and Chebyshev constant

For an arbitrary closed subsetE of the complex plane, the notions of logarithmic capacity, transfinite diameter, and Chebyshev constant ofE with respect to an admissible weightw onE are introduced. For thew-modified capacity, an electrostatics problem for logarithmic potentials in the presence of an external field is analyzed. This leads to an extremal measure whose support is the “smallest” compact set where the sup norm of weighted polynomials “live.” The introduction of a weightw has the advantage that the classical quantities mentioned in the title can be considered for unbounded setsE. Some of the theorems presented are generalizations of the authors' previous results for the case whenE?R.     

MRA contextual-recovery extension of smooth functions on manifolds

In a recent paper, the first author introduced an MRA (multi-resolution or multi-level approximation) approach to extend an earlier work of Chan and Shen on image inpainting, from isotropic diffusion to anisotropic diffusion and from bi-harmonic extension to multi-level lagged anisotropic diffusion extension. The objective of the present paper is to extend and generalize this work to nonstationary smooth function extension to meet the goal of inpainting missing image features, while matching the existing image content without apparent visual artifact. Our result is formulated as an MRA contextual-recovery extension for the completion of smooth functions on manifolds by deriving an error formula, from which sharp error estimates can be derived. A novel estimate for the biharmonic operator derived in this paper is a formulation of the error bound in terms the volume, as opposed to the diameter, of the image hole.     

The free energy of a reaction coordinate at multiple constraints: a concise formulation

The free energy as a function of the reaction coordinate (rc) is the key quantity for the computation of equilibrium and kinetic quantities. When it is considered as the potential of mean force, the problem is the calculation of the mean force for given values of the rc. We reinvestigate the PMCF (potential of mean constraint force) method which applies a constraint to the rc to compute the mean force as the mean negative constraint force and a metric tensor correction. The latter allows for the constraint imposed to the rc and possible artefacts due to multiple constraints of other variables which for practical reasons are often used in numerical simulations. Two main results are obtained that are of theoretical and practical interest. First, the correction term is given a very concise and simple shape which facilitates its interpretation and evaluation. Secondly, a theorem describes various rcs and possible combinations with constraints that can be used without introducing any correction to the constraint force. The results facilitate the computation of free energy by molecular dynamics simulations.     

A unified method for super-resolution recovery and real exponential-sum separation

In this paper, motivated by diffraction of traveling light waves, a simple mathematical model is proposed, both for the multivariate super-resolution problem and the problem of blind-source separation of real-valued exponential sums. This model facilitates the development of a unified theory and a unified solution of both problems in this paper. Our consideration of the super-resolution problem is aimed at applications to fluorescence microscopy and observational astronomy, and the motivation for our consideration of the second problem is the current need of extracting multivariate exponential features in magnetic resonance spectroscopy (MRS) for the neurologist and radiologist as well as for providing a mathematical tool for isotope separation in Nuclear Chemistry. The unified method introduced in this paper can be easily realized by processing only finitely many data, sampled at locations that are not necessarily prescribed in advance, with computational scheme consisting only of matrix-vector multiplication, peak finding, and clustering.