Site–bond percolation on triangular lattices: Monte Carlo simulation and analytical approach

A generalization of the pure site and pure bond percolation problems called site–bond percolation on a triangular lattice is studied. Motivated by considerations of cluster connectivity, two distinct schemes (denoted as S∩B$S\cap B$ and S∪B$S\cup B$) for site–bond percolation are used. In S∩B$S\cap B$ (S∪B$S\cup B$), two points are said to be connected if a sequence of occupied sites and (or  ) bonds joins them. By using finite-size scaling theory, data from S∩B$S\cap B$ and S∪B$S\cup B$ are analyzed in order to determine (i$i$) the phase boundary between the percolating and non-percolating regions and (ii$ii$) the numerical values of the critical exponents of the phase transition occurring in the system. A theoretical approach, based on exact calculations of configurations on finite triangular cells, is applied to study the site–bond percolation on triangular lattices. The percolation processes have been monitored by following the percolation function, defined as the ratio between the number of percolating configurations and the total number of available configurations for a given cell size and concentration of occupied elements. A comparison of the results obtained by these two methods has been performed and discussed.     

Diffusion-limited aggregates grown on nonuniform substrates

In the present paper, patterns of diffusion-limited aggregation (DLA) grown on nonuniform substrates are investigated by means of Monte Carlo simulations. We consider a nonuniform substrate as the largest percolation cluster of dropped particles with different structures and forms that occupy more than a single site on the lattice. The aggregates are grown on such clusters, in the range the concentration, p$p$, from the percolation threshold, _pc$p_c$ up to the jamming coverage, _pj$p_j$. At the percolation threshold, the aggregates are asymmetrical and the branches are relatively few. However, for larger values of p$p$, the patterns change gradually to a pure DLA. Tiny qualitative differences in this behavior are observed for different k$k$ sizes. Correspondingly, the fractal dimension of the aggregates increases as p$p$ raises in the same range _pc≤p≤_pj$p_c\le p\le p_j$. This behavior is analyzed and discussed in the framework of the existing theoretical approaches.     

On σδ-Picard–Vessiot Extensions

We study the differential Galois theory of difference equations under weaker hypothesis on the field of σ-constants. This framework yields a new approach to results by C. Hardouin and M. Singer, which answers positively a question by M. Singer: under the classical hypothesis, the known results are still valid. In particular, our Galois group is isomorphic to theirs over a suitable field. We also explicitly calculate the number of connected components of the Galois group.     

Greenʼs function for first-order multipoint boundary value problems and applications to the existence of solutions with constant sign

We consider a first-order linear differential equation subject to boundary value conditions which take into account the values of the function at multiple points in the interval of interest. For this problem, we calculate the Green?s function which allows to express in integral form the exact expression of the unique solution to the multipoint boundary value problem under the appropriate conditions. From this study, some results are derived concerning the existence of solutions with a constant sign (that is, some comparison results for first-order multipoint boundary value problems).     

Mathematical modelling and study of the consolidation of an elastic saturated soil with an incompressible fluid by FEM

The main aim of this paper is to validate and to solve a model for consolidation of an elastic saturated soil with incompressible fluid. Firstly, we prove the existence and uniqueness of the solution of the variational problem corresponding to an initial and boundary value problem (IBVP): a special case of the Biot’s ‘consolidation of clay’ model (where the applied forces depend on time). Secondly, we prove the stability of the method as well as the estimation of the error by using semi-discretization in time. Finally, we then solved this one by the finite element method (FEM) employing repeated fixed point techniques in order to obtain the results for displacement and pore water pressure. The pore fluid is considered incompressible. The results of the numerical experiments are compared with analytical solutions and, in cases where such solutions do not exist, with experimental data.     

Epidemic model of COVID-19 outbreak by inducing behavioural response in population

Nonlinear Dynamics - COVID-19 has spread around the world since December 2019, creating one of the greatest pandemics ever witnessed. According to the current reports, this is a situation when...     

A theoretical NMR study of polymorphism in crystal structures of azoles and benzazoles

The NMR chemical shifts of two azoles and one benzazole whose crystal structures present polymorphism have been computed using the GIPAW approach. ¹⁵N and ¹³C nuclei have been studied. Statistical analysis of the computed ¹³C and ¹⁵N chemical shifts indicates that the GIPAW chemical shifts reproduce with a high degree of accuracy those experimentally reported. This methodology can be used to identify other polymorphic crystal structures.     

Anti-periodic solutions for evolution equations associated with maximal monotone mappings

In this work, we study the anti-periodic problem for a nonlinear evolution inclusion where the nonlinear part is an odd maximal monotone mapping and the forcing term is an anti-periodic mapping. Several existence results are obtained under suitable conditions. An example is presented to illustrate the results.     

Existence results for a nondensely-defined impulsive neutral differential equation with state-dependent delay

In this paper, we prove the existence of integral solutions for a first order nondensely-defined impulsive neutral differential equation with state-dependent delay. The results are obtained by using suitable fixed point theorems. As applications of these main theorems, some consequences are obtained for the sub-linear growth cases.