This paper provides motivation as well as a method of construction for Hopf algebras, starting from an associative algebra. The dualization technique involved relies heavily on the use of Sweedlers dual.
We construct explicit solutions of a number of Stieltjes moment problems based on moments of the form ${rho}_{1}^{(r)}(n)=(2rn)!$ and ${rho}_{2}^{(r)}(n)=[(rn)!]^{2}$, $r=1,2,...$, $n=0,1,2,...$, textit{i.e.} we find functions $W^{(r)}_{1,2}(x)>0$ sa
tisfying $int_{0}^{infty}x^{n}W^{(r)}_{1,2}(x)dx = {rho}_{1,2}^{(r)}(n)$. It is shown using criteria for uniqueness and non-uniqueness (Carleman, Krein, Berg, Pakes, Stoyanov) that for $r>1$ both ${rho}_{1,2}^{(r)}(n)$ give rise to non-unique solutions. Examples of such solutions are constructed using the technique of the inverse Mellin transform supplemented by a Mellin convolution. We outline a general method of generating non-unique solutions for moment problems generalizing ${rho}_{1,2}^{(r)}(n)$, such as the product ${rho}_{1}^{(r)}(n)cdot{rho}_{2}^{(r)}(n)$ and $[(rn)!]^{p}$, $p=3,4,...$.
We report on the synthesis and on basic superconducting properties of a completely new Mo_2Re_3B ternary boride. The crystal structure of the Mo_2Re_3B compound is characterised by Pmmm space group and the cell parameters: a=11.626 A, b=8.465 A and c
=8.026 A. The critical temperature is Tc=8.5 K, whereas the lower and the upper-critical fields at zero temperature are equal to Hc1(0)=19.2 mT and to Hc2(0)=3.7 T, respectively. The corresponding Ginzburg-Landau parameter is equal to k=16.5 and the superconducting gap is estimated to be 2delta/kTc=3.2.
