Partial-limit Solutions and Rational Solutions with Parameter for the Fokas-Lenells Equation

A partial-limit procedure is applied to soliton solutions of the Fokas-Lenells equation. Multiple-pole solutions related to real repeated eigenvalues are obtained. For the envelop | u | 2 , the simplest solution corresponds to a real double eigenvalue, showing a solitary wave with algebraic decay. Two such solitons allow elastic scattering but asymptotically with zero phase shift. Single eigenvalue with higher multiplicity gives rise to rational solutions which contain an intrinsic parameter, live on a zero background, and have slowly-changing amplitudes.

Nonlocal and local models for taxis in cell migration: a rigorous limit procedure

A rigorous limit procedure is presented which links nonlocal models involving adhesion or nonlocal chemotaxis to their local counterparts featuring haptotaxis and classical chemotaxis, respectively. It relies on a novel reformulation of the involved nonlocalities in terms of integral operators applied directly to the gradients of signal-dependent quantities. The proposed approach handles both model types in a unified way and extends the previous mathematical framework to settings that allow for general solution-dependent coefficient functions. The previous forms of nonlocal operators are compared with the new ones introduced in this paper and the advantages of the latter are highlighted by concrete examples. Numerical simulations in 1D provide an illustration of some of the theoretical findings.

Propagators, BCFW recursion and new scattering equations at one loop

We investigate how loop-level propagators arise from tree level via a forward-limit procedure in two modern approaches to scattering amplitudes, namely the BCFW recursion relations and the scattering equations formalism. In the first part of the paper, we revisit the BCFW construction of one-loop integrands in momentum space, using a convenient parametrisation of the D-dimensional loop momentum. We work out explicit examples with and without supersymmetry, and discuss the non-planar case in both gauge theory and gravity. In the second part of the paper, we study an alternative approach to one-loop integrands, where these are written as worldsheet formulas based on new one-loop scattering equations. These equations, which are inspired by BCFW, lead to standard Feynman-type propagators, instead of the ‘linear’-type loop-level propagators that first arose from the formalism of ambitwistor strings. We exploit the analogies between the two approaches, and present a proof of an all-multiplicity worldsheet formula using the BCFW recursion.

Nonlinear self-dual network equations: modulation instability, interactions of higher-order discrete vector rational solitons and dynamical behaviours

The nonlinear self-dual network equations that describe the propagations of electrical signals in nonlinear LC self-dual circuits are explored. We firstly analyse the modulation instability of the constant amplitude waves. Secondly, a novel generalized perturbation ( M , N − M )-fold Darboux transform (DT) is proposed for the lattice system by means of the Taylor expansion and a parameter limit procedure. Thirdly, the obtained perturbation (1, N − 1)-fold DT is used to find its new higher-order rational solitons (RSs) in terms of determinants. These higher-order RSs differ from those known results in terms of hyperbolic functions. The abundant wave structures of the first-, second-, third- and fourth-order RSs are exhibited in detail. Their dynamical behaviours and stabilities are numerically simulated. These results may be useful for understanding the wave propagations of electrical signals.

Asymptotic Behavior of Magnetostrictive Elasticity for Microstructured Materials

According to the classical theory of Weiss, Landau, and Lifshitz, in a ferromagnetic body there is a spontaneous magnetization field m, such that ∥m∥ = τ0 = const in all points of this material Ω. In any stationary configuration, this ferromagnetic body consists of areas (Weiss domains) in which the magnetization is uniform (i.e. m = const) separated by thin transition layers (Bloch walls). Such stationary configuration corresponds to the minimum point of the magnetostrictive free energy E. We are considering an elastic magnetostrictive body in our paper. The elastic magnetostrictive free energy Eδ depends on a small parameter δ such that δ → 0. As usual, the displacement field is denoted by u. We will show that each sequence of minimizers (ui, mi) contains a subsequence that converges to a couple of fields (u0, m0). By means of a Γ-limit procedure we will show that this couple (u0, m0) is a minimizer of the new functional E0. This new functional E0 describes the magnetic-elastic properties of the body with microstructure.

On weak regularity requirements of the relaxation modulus in viscoelasticity

The existence and uniqueness of solution to a one-dimensional hyperbolic integro-differential problem arising in viscoelasticity is here considered. The kernel, in the linear viscoelasticity equation, represents the relaxation function which is characteristic of the considered material. Specifically, the case of a kernel, which does not satisfy the classical regularity requirements is analysed. This choice is suggested by applications according to the literature to model a wider variety of materials. A notable example of kernel, not satisfying the classical regularity requirements, is represented by a wedge continuous function. Indeed, the linear integro-differential viscoelasticity equation, characterised by a suitable wedge continuous relaxation function, is shown to give the classical linear wave equation via a limit procedure.

Strain gradient visco-plasticity with dislocation densities contributing to the energy

We consider the energetic description of a visco-plastic evolution and derive an existence result. The energies are convex, but not necessarily quadratic. Our model is a strain gradient model in which the curl of the plastic strain contributes to the energy. Our existence results are based on a time-discretization, the limit procedure relies on Helmholtz decompositions and compensated compactness.

Qualitative Analysis of Modified Hand Test

Qualitative analysis of modified Hand Test was carried out on 500 participants further bifurcated into 350 normal, 50 maladjusted, 50 neurotic and 50 psychotic participants. Their ages ranged from 11 years to 90 years with mean age of 34.44 SD (17.34). The qualitative analysis based up on seventeen categories which includes ambivalent, automatic phrase, cylindrical, denial, emotion, gross, hiding, immature, impotent, inanimate, movement, oral, and perplexity, sensual, sexual and original. Original Purposive sampling technique was used. Modified Hand test with four new adapted stimulus was administered in accordance with described instructions by its author. Testing the limit procedure was applied only for psychotic group. Post-test inquiry was held to clarify certain responses. The results of the study depicted interesting features which differentiates four groups. E.g. normal group did not produce any sexual, hiding, repetition responses. More number of repetitive responses were found in mal adjective and neurotic groups. Certain new areas were also explored like introjections produced by psychotic group. Need for altruism by doctors sub group of normal population. Direction by teachers. Interesting results are expected with other different sample and are likely to provide insight in order to understand human behavior in tradition of idiosyncratic approach.

Fokker–Planck equations in the modeling of socio-economic phenomena

We present and discuss various one-dimensional linear Fokker–Planck-type equations that have been recently considered in connection with the study of interacting multi-agent systems. In general, these Fokker–Planck equations describe the evolution in time of some probability density of the population of agents, typically the distribution of the personal wealth or of the personal opinion, and are mostly obtained by linear or bilinear kinetic models of Boltzmann type via some limit procedure. The main feature of these equations is the presence of variable diffusion, drift coefficients and boundaries, which introduce new challenging mathematical problems in the study of their long-time behavior.

REMARKS ON THE FORMULA FOR THE MOMENTS OF THE PÓLYA PROBABILITY DISTRIBUTION

ABSTRACT The probability distribution of a random variable can be characterized by some numbers called parameters of the distribution. The moments belong to the most frequently used parameters. We focus on the Pólya distribution because it is easy to obtain from it as special cases some very important in the statistics distributions, as binomial, negative binomial and Poisson (the last one in the limit procedure). In 1972 G. Mühlbach gave very interesting formulae for the moments of the Pólya distribution. The author did not investigate the evaluation of the numerical efficacy of the formula for the moments. We will show that it is possible to demonstrate this formula in a simpler form, which has a practical significance and importance.