This work investigates mosaic patterns for the one-dimensional cellular neural networks with various boundary conditions. These patterns can be formed by combining the basic patterns. The parameter space is partitioned so that the existence of basic patterns can be determined for each parameter region. The mosaic patterns can then be completely characterized through formulating suitable transit...

A convective Cahn-Hilliard type equation of sixth order that describes the faceting of a growing surface is considered with periodic boundary conditions. By using a Galerkin approach the existence of weak solutions to this sixth order partial differential equation is established in L2(0, T ; Ḣ3 per). Furthermore stronger regularity results have been derived and these are used to prove uniquenes...

We prove existence results for the solutions of the periodic boundary value problem concerning the n-th order functional differential equation with impulses effects ⎩ ⎨ ⎧ Δ ∈ α α − , p , 1, = k)), t (x ,), t (x (I =) t (x ], T [0, t. e. a))), t ((x ,)) t ((x), t (x , t (f =) t (x k 1) n (k k , i k) i (m 1) n (L L L and the periodic boundary conditions 1. , 0, =), (= (0)) () (− n i T x x i i L O...

We prove that when a countable group admits a nontrivial Floyd-type boundary, then every nonelementary and metrically proper subgroup contains a noncommutative free subgroup. This generalizes the corresponding well-known results for hyperbolic groups and groups with infinitely many ends. It also shows that no finitely generated amenable group admits a nontrivial boundary of this type. This impr...

We consider stability properties of discrete-time bilinear lters. Simple su cient conditions are given for bounded-input bounded-output stability (with not necessarily zero initial conditions), lp stability, and three other important types of stability. In particular, conditions are given under which asymptotically periodic inputs produce asymptotically periodic outputs with the same period. Re...

We study solutions of Type IIB supergravity, which describe the geometries dual to supersymmetric Wilson lines in N = 4 super–Yang–Mills. We show that the solutions are uniquely specified by one function which satisfies a Laplace equation in two dimensions. We show that if this function obeys a certain Dirichlet boundary condition, the corresponding geometry is regular, and we find a simple int...

Settings for the Stokes Problem As a first step to considering the Navier-Stokes equations, we will consider various settings for the Stokes problem of finding u⃗,p which satisfy −ν∇2u⃗ + ∇p = f⃗ in U ⊂ Rn 1 div u⃗ = 0. We could consider this system of equations on: 1. a. U = Rn : in this case we would have to impose some ”behavior at ∞" conditions such as u⃗ → ψ⃗ as | x⃗| → ∞ b. U ⊂ Rn, open bounde...

Ventcel boundary conditions are second order differential conditions that appear in asymptotic models. Like Robin boundary conditions, they lead to wellposed variational problems under a sign condition of a coefficient. Nevertheless situations where this condition is violated appeared in several works. The wellposedness of such problems was still open. This manuscript establishes, in the generi...

It has recently become clear that simulations under amorphous boundary conditions (ABCs) can provide valuable information on the dynamics and thermodynamics of disordered systems with no obvious ordered parameter. In particular, they allow to detect a correlation length that is not measurable with standard correlation functions. Here we explain what exactly is meant by ABCs, discuss their relat...

We prove that a certain finite difference scheme converges to the weak solution of the Cauchy problem on a finite interval with periodic boundary conditions for the Camassa– Holm equation ut−uxxt +3uux−2uxuxx−uuxxx = 0 with initial data u|t=0 = u0 ∈ H1([0, 1]). Here it is assumed that u0 − u′′ 0 ≥ 0 and in this case, the solution is unique, globally defined, and energy preserving.