Docking positrophilic electrons into molecular attractive potential of fluorinated methanes

The present study shows that the positrophilic electrons of a molecule dock into the positron attractive potential region in the annihilation process under the plane-wave approximation. The positron-electron annihilation processes of both polar and non-polar fluorinated methanes (CH4-nFn, n=0, 1,..., 4) are studied under this role. The predicted gamma-ray spectra of these fluorinated methanes agree well with the experiments. It further indicates that the positrophilic electrons of a molecule docking at the negative end of a bond dipole are independent from the molecular dipole moment in the annihilation process.Comment: 11 pages, 5 figure

Density of Eigenvalues of Random Normal Matrices with an Arbitrary Potential, and of Generalized Normal Matrices

Following the works by Wiegmann-Zabrodin, Elbau-Felder, Hedenmalm-Makarov, and others, we consider the normal matrix model with an arbitrary potential function, and explain how the problem of finding the support domain for the asymptotic eigenvalue density of such matrices (when the size of the matrices goes to infinity) is related to the problem of Hele-Shaw flows on curved surfaces, considered by Entov and the first author in 1990-s. In the case when the potential function is the sum of a rotationally invariant function and the real part of a polynomial of the complex coordinate, we use this relation and the conformal mapping method developed by Entov and the first author to find the shape of the support domain explicitly (up to finitely many undetermined parameters, which are to be found from a finite system of equations). In the case when the rotationally invariant function is $\beta |z|^2$, this is done by Wiegmann-Zabrodin and Elbau-Felder. We apply our results to the generalized normal matrix model, which deals with random block matrices that give rise to *-representations of the deformed preprojective algebra of the affine quiver of type $\hat A_{m-1}$. We show that this model is equivalent to the usual normal matrix model in the large $N$ limit. Thus the conformal mapping method can be applied to find explicitly the support domain for the generalized normal matrix model.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

On universal Lie nilpotent associative algebras

We study the quotient Q_i(A) of a free algebra A by the ideal M_i(A) generated by relation that the i-th commutator of any elements is zero. In particular, we completely describe such quotient for i=4 (for i<=3 this was done previously by Feigin and Shoikhet). We also study properties of the ideals M_i(A), e.g. when M_i(A)M_j(A) is contained in M_{i+j-1}(A) (by a result of Gupta and Levin, it is always contained in M_{i+j-2}(A)).Comment: 7 page