Generalized Symmetric (f,g) – Biderivations on Lattices

In this paper, we introduce the notion of generalized symmetric  -derivations on lattices, also some properties of generalized symmetric - derivations we studies

Differential Calculi on Associative Algebras and Integrable Systems

After an introduction to some aspects of bidifferential calculus on associative algebras, we focus on the notion of a "symmetry" of a generalized zero curvature equation and derive Backlund and (forward, backward and binary) Darboux transformations from it. We also recall a matrix version of the binary Darboux transformation and, inspired by the so-called Cauchy matrix approach, present an infinite system of equations solved by it. Finally, we sketch recent work on a deformation of the matrix binary Darboux transformation in bidifferential calculus, leading to a treatment of integrable equations with sources.Comment: 19 pages, to appear in "Algebraic Structures and Applications", S. Silvestrov et al (eds.), Springer Proceedings in Mathematics & Statistics, 202

Combinatorics of free vertex algebras

This paper illustrates the combinatorial approach to vertex algebra - study of vertex algebras presented by generators and relations. A necessary ingredient of this method is the notion of free vertex algebra. Borcherds \cite{bor} was the first to note that free vertex algebras do not exist in general. The reason for this is that vertex algebras do not form a variety of algebras, because the locality axiom (see sec 2 below) is not an identity. However, a certain subcategory of vertex algebras, obtained by restricting the order of locality of generators, has a universal object, which we call the free vertex algebra corresponding to the given locality bound. In [J. of Algebra, 217(2):496-527] some free vertex algebras were constructed and in certain special cases their linear bases were found. In this paper we generalize this construction and find linear bases of an arbitrary free vertex algebra. It turns out that free vertex algebras are closely related to the vertex algebras corresponding to integer lattices. The latter algebras play a very important role in different areas of mathematics and physics. Here we explore the relation between free vertex algebras and lattice vertex algebras in much detail. These results comply with the use of the word "free" in physical literature refering to some elements of lattice vertex algebras, like in "free field", "free bozon" or "free fermion". Among other things, we find a nice presentation of lattice vertex algebras in terms of generators and relations, thus giving an alternative construction of these algebras without using vertex operators. We remark that our construction works in a very general setting; we do not assume the lattice to be positive definite, neither non-degenerate, nor of a finite rank

Lattice Boltzmann simulations of 3D crystal growth: Numerical schemes for a phase-field model with anti-trapping current

A lattice-Boltzmann (LB) scheme, based on the Bhatnagar-Gross-Krook (BGK) collision rules is developed for a phase-field model of alloy solidification in order to simulate the growth of dendrites. The solidification of a binary alloy is considered, taking into account diffusive transport of heat and solute, as well as the anisotropy of the solid-liquid interfacial free energy. The anisotropic terms in the phase-field evolution equation, the phenomenological anti-trapping current (introduced in the solute evolution equation to avoid spurious solute trapping), and the variation of the solute diffusion coefficient between phases, make it necessary to modify the equilibrium distribution functions of the LB scheme with respect to the one used in the standard method for the solution of advection-diffusion equations. The effects of grid anisotropy are removed by using the lattices D3Q15 and D3Q19 instead of D3Q7. The method is validated by direct comparison of the simulation results with a numerical code that uses the finite-difference method. Simulations are also carried out for two different anisotropy functions in order to demonstrate the capability of the method to generate various crystal shapes

Flag-symmetry of the poset of shuffles and a local action of the symmetric group

We show that the poset of shuffles introduced by Greene in 1988 is flag-symmetric, and we describe a "local" permutation action of the symmetric group on the maximal chains which is closely related to the flag symmetric function of the poset. A key tool is provided by a new labeling of the maximal chains of a poset of shuffles, which is also used to give bijective proofs of enumerative properties originally obtained by Greene. In addition we define a monoid of multiplicative functions on all posets of shuffles and describe this monoid in terms of a new operation on power series in two variables.Comment: 34 pages, 6 figure