Existence theorems for some abstract nonlinear non-autonomous systems with delays

For some abstract classes of nonlinear non-autonomous systems with variable and state-dependent delays existence, non-existence and multiplicity of periodic solutions are discussed. To illustrate the efficiency of the method, we obtain some well-known results for applied systems as corollaries of our existence theorems.     

Coupled lattice Boltzmann method for simulating electrokinetic flows: A localized scheme for the Nernst–Plank model

We present a coupled lattice Boltzmann method (LBM) to solve a set of model equations for electrokinetic flows in micro-/nano-channels. The model consists of the Poisson equation for the electrical potential, the Nernst–Planck equation for the ion concentration, and the Navier–Stokes equation for the flows of the electrolyte solution. In the proposed LBM, the electrochemical migration and the convection of the electrolyte solution contributing to the ion flux are incorporated into the collision operator, which maintains the locality of the algorithm inherent to the original LBM. Furthermore, the Neumann-type boundary condition at the solid/liquid interface is then correctly imposed. In order to validate the present LBM, we consider an electro-osmotic flow in a slit between two charged infinite parallel plates, and the results of LBM computation are compared to the analytical solutions. Good agreement is obtained in the parameter range considered herein, including the case in which the nonlinearity of the Poisson equation due to the large potential variation manifests itself. We also apply the method to a two-dimensional problem of a finite-length microchannel with an entry and an exit. The steady state, as well as the transient behavior, of the electro-osmotic flow induced in the microchannel is investigated. It is shown that, although no external pressure difference is imposed, the presence of the entry and exit results in the occurrence of the local pressure gradient that causes a flow resistance reducing the magnitude of the electro-osmotic flow.     

Integrability and exact solutions of the nonautonomous mixed mKdV–sinh–Gordon equation

In this paper, a nonautonomous mixed mKdV–sinh–Gordon equation with one arbitrary time-dependent variable coefficient is discussed in detail. It is proved that the equation passes the Painlevé test in the case of positive and negative resonances, respectively. Furthermore, a dependent variable transformation is introduced to get its bilinear form. Then, soliton, negaton, positon and interaction solutions are introduced by means of the Wronskian representation. Velocities are found to depend on the time-dependent variable coefficient appearing in the equation and this leads to a wide range of interesting behaviours. The singularities and asymptotic estimate of these solutions are discussed. At last, the superposition formulae for these solutions are also constructed.     

Singleton sets random attractor for stochastic FitzHugh–Nagumo lattice equations driven by fractional Brownian motions

The paper is devoted to the study of the dynamical behavior of the solutions of stochastic FitzHugh–Nagumo lattice equations, driven by fractional Brownian motions, with Hurst parameter greater than 1/2. Under some usual dissipativity conditions, the system considered here features different dynamics from the same one perturbed by Brownian motion. In our case, the random dynamical system has a unique random equilibrium, which constitutes a singleton sets random attractor.     

Low-cost addition–subtraction sequences for the final exponentiation in pairings

In this paper, we address the problem of finding low cost addition–subtraction sequences for situations where a doubling step is significantly cheaper than a non-doubling one. One application of this setting appears in the computation of the final exponentiation step of the reduced Tate pairing defined on ordinary elliptic curves. In particular, we report efficient addition–subtraction sequences for the Kachisa–Schaefer–Scott family of pairing-friendly elliptic curves, whose parameters involve computing the multi-exponentiation of relatively large sequences of exponents with a size of up to 26 bits.     

A combinatorial proof of the Kronecker–Weber Theorem in positive characteristic

In this paper we present a combinatorial proof of the Kronecker–Weber Theorem for global fields of positive characteristic. The main tools are the use of Witt vectors and their arithmetic developed by H.L. Schmid. The key result is to obtain, using counting arguments, how many p-cyclic extensions exist of fixed degree and bounded conductor where only one prime ramifies. We then compare this number with the number of subextensions of cyclotomic function fields of the same type and verify that these two numbers are the same.     

The Boundedness of Composition Operators on Triebel–Lizorkin and Besov Spaces with Different Homogeneities

In this paper, we introduce new Triebel–Lizorkin and Besov Spaces associated with the different homogeneities of two singular integral operators. We then establish the boundedness of composition of two Calder′on–Zygmund singular integral operators with different homogeneities on these Triebel–Lizorkin and Besov spaces.     

Sub-exponential graph coloring algorithm for stencil-based Jacobian computations

Partial differential equations can be discretized using a regular Cartesian grid and a stencil-based method to approximate the partial derivatives. The computational effort for determining the associated Jacobian matrix can be reduced. This reduction can be modeled as a (grid) coloring problem. Currently, this problem is solved by using a heuristic approach for general graphs or by developing a formula for every single stencil. We introduce a sub-exponential algorithm using the Lipton–Tarjan separator in a divide-and-conquer approach to compute an optimal coloring. The practical relevance of the algorithm is evaluated when compared with an exponential algorithm and a greedy heuristic.