We consider the problem of existence of entire solutions to the Allen-Cahn equation Δu + u - u(3) = 0 in , usually regarded as a prototype for the modeling of phase transition phenomena. In particular, exploiting the link between the Allen-Cahn equation and minimal surface theory in dimensions N ≥ 9, we find a solution, u, with ∂(x(N))u > 0, such that its level sets are close to a nonplanar, mi...

We study a singular limit problem of the Allen-Cahn equation with Neumann boundary conditions and general initial data of uniformly bounded energy. We prove that the time-parametrized family of limit energy measures is Brakke’s mean curvature flow with a generalized right angle condition on the boundary.

This paper presents a new methodology based on AHP and ANP (Saaty, 1980, 1996) to estimate the value and transfer fees of professional football players. The paper extends the AHP multicriteria methodology proposed by Moreno-Jiménez et al. (2010) for estimating the value of these sportsmen. The new approach multiplicatively combines the (intrinsic) value of the player (estimated by means of AHP ...

Electoral systems are complex entities composed of a set of phases that form a process to which performance parameters can be associated. One of the key points of every electoral system is represented by the electoral formula that can be characterized by a wide spectrum of properties that, according to Arrow’s Impossibility Theorem and other theoretical results, cannot be all satisfied at the s...

The lattice Boltzmann (LB) method is used to study the kinetics of domain growth of a binary fluid in a number of geometries modeling porous media. Unlike the traditional methods which solve the Cahn-Hilliard equation, the LB method correctly simulates fluid properties, phase segregation, interface dynamics and wetting. Our results, based on lattice sizes of up to 4096× 4096, do not show eviden...

We derive an exact nonlinear nonlocal macroscopic equation for the time evolution of the conserved order parameter rsr, td of a microscopic model binary alloy undergoing phase segregation: a d-dimensional lattice gas evolving via Kawasaki exchange dynamics, satisfying detailed balance for a Hamiltonian with a long range pair potential gdJsgjxjd. The macroscopic evolution is on the spatial scale...

In this paper, we reformulate the diffuse interface model of the tumor growth (S.M. Wise et al., Three-dimensional multispecies nonlinear tumor growth-I: model and numerical method, J. Theor. Biol. 253 (2008) 524--543). In the new proposed model, we use the conservative second-order Allen--Cahn equation with a space--time dependent Lagrange multiplier instead of using the fourth-order Cahn--Hil...

We study the one-dimensional Cahn-Hilliard equation with an additional driving term representing, say, the effect of gravity. We find that the driving field E has an asymmetric effect on the solution for a single stationary domain wall (or ‘kink’), the direction of the field determining whether the analytic solutions found by Leung [J. Stat. Phys. 61, 345 (1990)] are unique. The dynamics of a k...