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Register a new userThe best constant of discrete Sobolev inequality on the C60 fullerene buckyball
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The best constants of two kinds of discrete Sobolev inequalities on the C60 fullerene buckyball are obtained. All the eigenvalues of discrete Laplacian $A$ corresponding to the buckyball are found. They are roots of algebraic equation at most degree $4$ with integer coefficients. Green matrix $G(a)=(A+a I)^{-1} (0<a<infty)$ and the pseudo Green matrix $G_*=A^{dagger}$ are obtained by using computer software Mathematica. Diagonal values of $G_*$ and $G(a)$ are identical and they are equal to the best constants of discrete Sobolev inequalities.
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In this work we provide the best constants of the multiple Khintchine inequality. This allows us, among other results, to obtain the best constants of the mixed $left( ell_{frac{p}{p-1}},ell_{2}right) $-Littlewood inequality, thus ending completely a work started by Pellegrino in cite{pell}.
We compute the best constant in the Khintchine inequality under assumption that the sum of Rademacher random variables is zero.
We analyze using Poisson equation the spatial distributions of the positive charge of carbon atomic nuclei shell and negative charge of electron clouds forming the electrostatic potential of the C60 fullerene shell as a whole. We consider also the case when an extra positive charge appears inside C60 in course of e.g. photoionization of an endohedral A@C. We demonstrate that frequently used radial square-well potential U(r) simulating the C60 shell leads to nonphysical charge densities of the shell in both cases - without and with an extra positive charge inside. We conclude that the square well U(r) modified by adding a Coulomb-potential-like term does not describe the interior polarization of the shell by the electric charge located in the center of the C60 shell. We suggest another model potential, namely that of hyperbolic cosine shape with properly adjusted parameters that is able to describe the monopole polarization of C60 shell. As a concrete illustration, we have calculated the photoionization cross-sections of H@C60 taking into account the monopole polarization of the shell in the frame of suggested model. We demonstrate that proper account of this polarization does not change the photoionization cross-section.
We consider the imbedding inequality || f ||_{L^r(R^d)} <= S_{r,n,d} || f ||_{H^{n}(R^d)}; H^{n}(R^d) is the Sobolev space (or Bessel potential space) of L^2 type and (integer or fractional) order n. We write down upper bounds for the constants S_{r, n, d}, using an argument previously applied in the literature in particular cases. We prove that the upper bounds computed in this way are in fact the sharp constants if (r=2 or) n > d/2, r=infinity, and exhibit the maximising functions. Furthermore, using convenient trial functions, we derive lower bounds on S_{r,n,d} for n > d/2, 2 < r < infinity; in many cases these are close to the previous upper bounds, as illustrated by a number of examples, thus characterizing the sharp constants with little uncertainty.
We consider a version of the fractional Sobolev inequality in domains and study whether the best constant in this inequality is attained. For the half-space and a large class of bounded domains we show that a minimizer exists, which is in contrast to the classical Sobolev inequalities in domains.
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