Three-dimensional Laguerre tessellation models became quite popular in many areas of physics and biology. They are generated by locally finite configurations of marked points. Randomness is included by assuming that the set of generators is formed by a marked point process. The present paper focuses on 3D marked Gibbs point processes of generators which enable us to specify the desired geometry of the Laguerre tessellation. In order to prove the existence of a stationary Gibbs measure using a general approach of Dereudre, Drouilhet and Georgii \cite{DDG12}, the geometry of Laguerre tessellations and their duals Laguerre Delaunay tetrahedrizations is examined in detail. Since it is difficult to treat the models analytically, their simulations are carried out by Markov chain Monte Carlo techniques.
Consider the forced higher-order nonlinear neutral functional differential equation \[ \frac{{\mathrm d}^n}{{\mathrm d}t^n}[x(t)+C(t) x(t-\tau )]+\sum ^m_{i=1} Q_i(t)f_i(x(t-\sigma _i))=g(t), \quad t\ge t_0, \] where $n, m \ge 1$ are integers, $\tau , \sigma _i\in {\mathbb{R}}^+ =[0, \infty )$, $C, Q_i, g\in C([t_0, \infty ), {\mathbb{R}})$, $f_i\in C(\mathbb{R}, \mathbb{R})$, $(i=1,2,\dots ,m)$. Some sufficient conditions for the existence of a nonoscillatory solution of above equation are obtained for general $Q_i(t)$ $(i=1,2,\dots ,m)$ and $g(t)$ which means that we allow oscillatory $Q_i(t)$ $(i=1,2,\dots ,m)$ and $g(t)$. Our results improve essentially some known results in the references.
We consider the problem of influencing the motion of an electrically conducting fluid with an applied steady magnetic field. Since the flow is originating from buoyancy, heat transfer has to be included in the model. The stationary system of magnetohydrodynamics is considered, and an approximation of Boussinesq type is used to describe the buoyancy. The heat sources given by the dissipation of current and the viscous friction are not neglected in the fluid. The vessel containing the fluid is embedded in a larger domain, relevant for the global temperature- and magnetic field- distributions. Material inhomogeneities in this larger region lead to transmission relations for the electromagnetic fields and the heat flux on inner boundaries. In the presence of transparent materials, the radiative heat transfer is important and leads to a nonlocal and nonlinear jump relation for the heat flux. We prove the existence of weak solutions, under the assumption that the imposed velocity at the boundary of the fluid remains sufficiently small.
The paper presents an existence result for global solutions to the finite dimensional differential inclusion y ′ ∈ F(y), F being defined on a closed set K. A priori bounds for such solutions are provided.
We consider the following quasilinear Neumann boundary-value problem of the type − ∑ N i=1 ∂ ⁄ ∂xi ai ( x, ∂u ⁄ ∂xi ) + b(x)|u| p0(x)−2 u = f(x, u) + g(x, u) in Ω, ∂u ⁄ ∂γ = 0 on ∂Ω. We prove the existence of infinitely many weak solutions for our equation in the anisotropic variable exponent Sobolev spaces and we give some examples.
In this paper we investigate the existence of mild solutions to second order initial value problems for a class of delay integrodifferential inclusions with nonlocal conditions. We rely on a fixed point theorem for condensing maps due to Martelli.
The paper deals with the existence of multiple positive solutions for the boundary value problem (ϕ(p(t)u (n−1))(t))′ + a(t)f(t, u(t), u′ (t), . . . , u(n−2)(t)) = 0, 0 < t < 1, u (i) (0) = 0, i = 0, 1, . . . , n − 3, u (n−2)(0) = mP−2 i=1 αiu (n−2)(ξi), u(n−1)(1) = 0, where ϕ: R → R is an increasing homeomorphism and a positive homomorphism with ϕ(0) = 0. Using a fixed-point theorem for operators on a cone, we provide sufficient conditions for the existence of multiple positive solutions to the above boundary value problem.
In this paper, necessary and sufficient conditions for the existence of nonoscillatory solutions of the forced nonlinear difference equation ∆(xn = pnxτ(n)) + f(n, xσ(n)) = qn are obtained. Examples are included to illustrate the results.