In this paper we prove the existence of a global attractor with respect to the weak topology of a suitable Banach space for a parabolic scalar differential equation describing a non-Newtonian flow. More precisely, we study a model proposed by Hébraud and Lequeux for concentrated suspensions.

We propose a Lax equation for the non-linear sigma model which leads directly to the conserved local charges of the system. We show that the system has two infinite sets of such conserved charges following from the Lax equation, much like dispersionless systems. We show that the system has two Hamiltonian structures which are compatible so that it is truly a bi-Hamiltonian system. However, the ...

This is the second part of a series of papers where we develop rigorous decay estimates for breather solutions of an averaged version of the non-linear Schrödinger equation. In this part we study the diffraction managed discrete nonlinear Schrödinger equation, an equation which describes coupled waveguide arrays with periodic diffraction management geometries. We show that, for vanishing averag...

Special thanks to Aapo Hyvärinen, Patrik O. Hoyer and Takashi Washio. 2 Abstract • Linear structural equation models (linear SEMs) can be used to model data generating processes of variables. • We review a new approach to learn or estimate linear structural equation models. • The new estimation approach utilizes non-Gaussianity of data for model identification and uniquely estimates much wider...

Three exact solutions say φ0 of massless scalar theories on Euclidean space, i.e. D = 6 φ, D = 4 φ and D = 3 φ models are obtained which share similar properties. The information geometry of their moduli spaces coincide with the Euclidean AdS7, AdS5 and AdS4 respectively on which φ0 can be described as a stable tachyon. In D = 4 we recognize that the SU(2) instanton density is proportional to φ...

Solution of the diffusion equation is usually performed with the finite element discretization for the spatial elliptic part of the equation and the time dependency is integrated via some difference scheme, often the trapezoidal rule (Crank-Nicolson) or the unconditionally stable semi-implicit two-step algorithm of Lees. A common procedure is to use Picard’s iteration with the trapezoidal rule....

We give an elementary introduction to some recent polyhedral techniques for understanding and solving systems of multivariate polynomial equations. We provide numerous concrete examples and assume no background in algebraic geometry. Highlights include the following: (1) A completely self-contained proof of an extension of Bernstein’s Theorem. Our extension relates volumes of polytopes with the...

This paper focuses on applying the GDTM-Padé technique to solve the non-linear differential-difference equation. The bell-shaped solitary wave solution of Belov-Chaltikian lattice equation is considered. Comparison between the approximate solutions and the exact ones shows that this technique is an efficient and attractive method for solving the differential-difference equations.

This paper deals with application of the maximum principle for differential equations to the finite difference method for determining upper and lower approximate solutions of the non-linear Burgers’ equation and their error range. In term of mathematical architecture, the paper is based on the maximum principle for parabolic differential equations to establish monotonic residual relations of th...

We study the existence and stability of space-periodic standing waves for the space-periodic cubic nonlinear Schrödinger equation with a point defect determined by a space-periodic Dirac distribution at the origin. This equation admits a smooth curve of positive space-periodic solutions with a profile given by the Jacobi elliptic function of dnoidal type. Via a perturbation method and continuat...