A family of Boussinesq systems has been proposed in [8] to describe the bi-directional propagation of small amplitude long waves on the surface of shallow water. In this paper, we investigate the boundary stabilization of the generalized higher order Boussinesq systems of KdV–type posed on a interval. We design a two-parameter family of feedback laws for which the solutions are globally defined...

We present two solutions to the firing synchronization problem on the ring, an 8-state minimal-time solution and a 6-state non-minimal-time solution. Both solutions use fewer states than the previous best-known minimal-time automaton, a 16-state solution due to Culik. We also give the first lower bounds on the number of states needed for solutions to the ring firing synchronization problem. We ...

In this paper, we investigate the following quasilinear and singular problem : { −∆pu = λ uδ + u q in Ω u|∂Ω = 0 , u > 0 in Ω (1) where Ω is an open bounded domain with smooth boundary, 1 < p, p− 1 < q and λ, δ > 0. We first prove that there exist weak solutions for λ > 0 small in W 1,p 0 (Ω)∩C(Ω̄) if and only if δ < 2+ 1 p−1 . Investigating the radial symmetric case (Ω = BR(0)), we prove by a s...

We study stochastic equations Xt = x0 + ∫ t 0 b(u,Xu−) dZu, where Z is an one-dimensional symmetric stable process of index α with 0 < α ≤ 2, b : [0,∞) × IR → IR is a measurable diffusion coefficient, and x0 ∈ IR is the initial value. We give sufficient conditions for the existence of weak solutions. Our main results generalize results of P. A. Zanzotto [18] who dealt with homogeneous diffusion...

We consider the long time limit theorems for the solutions of a discrete wave equation with a weak stochastic forcing. The multiplicative noise conserves the energy and the momentum. We obtain a time-inhomogeneous Ornstein-Uhlenbeck equation for the limit wave function that holds both for square integrable and statistically homogeneous initial data. The limit is understood in the point-wise sen...

The existence and regularity of Young measure-valued solutions and weak solutions to non-Newtonian flows are considered. Galerkin approximation and an L2 compactness theorem are main ingredients for the proof of the existence of Young measurevalued solutions. Under a certain convexity condition for the energy, we prove that Young measure-valued solutions are weak solutions. Also, for the limite...

In this paper, we verify that a general p(x)-Laplacian Neumann problem has at least three weak solutions, which generalizes the corresponding result of the reference [R. A. Mashiyev, Three Solutions to a Neumann Problem for Elliptic Equations with Variable Exponent, Arab. J. Sci. Eng. 36 (2011) 1559-1567].

We investigate refinements of two solutions, the saddle and the weak saddle, defined by Shapley (1964) for two-player zero-sum games. Applied to weak tournaments, the first refinement, the mixed saddle, is unique and gives us a new solution, generally lying between the GETCHA and GOTCHA sets of Schwartz (1972, 1986). In the absence of ties, all three solutions reduce to the usual top cycle set....

A precise definition of “weak [quantum] measurements” and “weak value” (of a quantum observable) is offered, and simple finite dimensional examples are given showing that weak values are not unique and therefore probably do not correspond to any physical attribute of the system being “weakly” measured, contrary to impressions given by most of the literature on weak measurements. A possible math...

We study a class of semi-linear mean curvature equations −Mu = H + λf(u) where M is the mean curvature operator. We show that there exists an extremal parameter λ∗ such that this equation admits a minimal weak solutions for all λ ∈ [0, λ∗], while no weak solutions exists for λ > λ∗. In the radial case, we then show that minimal solutions are classical solutions for all λ ∈ [0, λ∗] and that anot...