In this paper, a method for finding an approximate solution of a class of two-dimensional nonlinear Volterra integral equations of the first-kind is proposed. This problem is transformedto a nonlinear two-dimensional Volterra integral equation of the second-kind. The properties ofthe bivariate shifted Legendre functions are presented. The operational matrices of integrationtogether with the produ...

Given any rectangular polyhedron 3-manifold $P$ tiled with unit cubes, we find infinitely many explicit directions related to cubic algebraic numbers such that all half-infinite geodesics in these are uniformly distributed $P$.

We study a class of growth processes in which clusters evolve via exchange of particles. We show that depending on the rate of exchange there are three possibilities: (I) Growth-clusters grow indefinitely, (II) gelation-all mass is transformed into an infinite gel in a finite time, and (III) instant gelation. In regimes I and II, the cluster size distribution attains a self-similar form. The la...

In this article we consider a lattice system of unbounded continuous spins. Otto and Reznikoff used the two-scale approach to show that exponential decay of correlations yields a logarithmic Sobolev inequality (LSI) with uniform constant in the system size. We improve their statement by weakening the assumptions, for which a more detailed analysis based on two new ingredients is needed. The two...

We consider a new plane problem of the potential theory outside symmetric T-shaped profile. In view symmetry profile, we reduce analyzed to mixed boundary-value in half with perpendicular cut whose solution is constructed by method partial domains use polar coordinates and Mellin integral transformation. The original reduced system two Wiener–Hopf equations completely regular infinite linear al...

This paper introduces semi-explicit multilinear modeling to ac power systems, including voltage source converters. Firstly, the novel class is explained. Then this applied a small example of PV inverter connected an infinite bus. Some part nonlinear current reference calculation expressed as algebraic constraint. The formulation constraint allowed whole model. approach compared with common time...

Lattice defects in crystalline materials create long-range elastic fields which can be modelled on the atomistic scale using an infinite system of discrete nonlinear force balance equations. Starting with these equations, this work rigorously derives a novel far-field expansion fields: The is computable and expressed as sum continuum correctors multipole terms decay increasing algebraic rate or...

For a matroid M, define the algebraic characteristic set x4(M) to be the set of field characteristics over which M can be algebraically represented. We construct many examples of rank three matroids with finite, non-singleton algebraic characteristic sets. We also determine x,,(PG(Z, p)) and xa(AG(2, p)). An infinite family of rank three matroids with empty algebraic characteristic set is const...