N-Capacity, N-harmonic radius and N-harmonic transplantation

For the N-Laplacian ΔN on a regular domain the N-Green's function exists. This allows us to define the N-Robin's function and the N-harmonic radius. We show their basic properties and extend the method of the harmonic transplantation to that of the N-harmonic transplantation. The method is used to estimate the first eigenvalue of the N-Laplacian ΔN. We also give another proof of the isoperimetric inequality for the N-capacity and give the isoperimetric inequality for the N-harmonic radius.     

Potential comparison and asymptotics in scalar conservation laws without convexity

Two kinds of optimal convergence orders in L¹-norm to a self-similar solution are proved or conjectured for various evolutionary problems so far. The first convergence order is of the magnitude of the similarity solution itself and the second one is of order 1/t. Employing a potential comparison technique to scalar conservation laws we may easily see that these asymptotic convergence orders are related to space and time translation of potentials. We present the technique clearly in the simple setting of scalar conservation laws in one space dimension.     

Asymptotic stabilization of entropy solutions to scalar conservation laws through a stationary feedback law

In this paper, we study the problem of asymptotic stabilization by closed loop feedback for a scalar conservation law with a convex flux and in the context of entropy solutions. Besides the boundary data, we use an additional control which is a source term acting uniformly in space.     

Boundary singularities of solutions of N-harmonic equations with absorption

We study the boundary behaviour of solutions u of −ΔNu+|u|^q−1u=0 in a bounded smooth domain Ω⊂RN subject to the boundary condition u=0 except at one point, in the range q>N−1. We prove that if q?2N−1 such an u is identically zero, while, if N−1<q<2N−1, u inherits a boundary behaviour which either corresponds to a weak singularity, or to a strong singularity. Such singularities are effectively constructed.     

Asymptotic behavior of solutions to viscous conservation laws with slowly varying external forces

This paper considers the existence and large time behavior of solutions to the convection-diffusion equation u _t −Δu+b(x)·∇(u|u| ^{q −1})=f(x, t) in ℝ ⁿ ×[0,∞), where f(x, t) is slowly decaying and q≥1+1/n (or in some particular cases q≥1). The initial condition u ₀ is supposed to be in an appropriate L ^p space. Uniform and nonuniform decay of the solutions will be established depending on the data and the forcing term.This work is partially supported by an AMO Grant     

Regularity of solutions to scalar conservation laws with a force

We prove regularity estimates for entropy solutions to scalar conservation laws with a force. Based on the kinetic form of a scalar conservation law, a new decomposition of entropy solutions is introduced, by means of a decomposition in the velocity variable, adapted to the non-degeneracy properties of the flux function. This allows a finer control of the degeneracy behavior of the flux. In addition, this decomposition allows to make use of the fact that the entropy dissipation measure has locally finite singular moments. Based on these observations, improved regularity estimates for entropy solutions to (forced) scalar conservation laws are obtained.     

New modifications of Potra-Pták’s method with optimal fourth and eighth orders of convergence

In this paper, we present two new iterative methods for solving nonlinear equations by using suitable Taylor and divided difference approximations. Both methods are obtained by modifying Potra-Pták’s method trying to get optimal order. We prove that the new methods reach orders of convergence four and eight with three and four functional evaluations, respectively. So, Kung and Traub’s conjecture Kung and Traub (1974) [2], that establishes for an iterative method based on n evaluations an optimal order p=2ⁿ⁻¹ is fulfilled, getting the highest efficiency indices for orders p=4 and p=8, which are 1.587 and 1.682.We also perform different numerical tests that confirm the theoretical results and allow us to compare these methods with Potra-Pták’s method from which they have been derived, and with other recently published eighth-order methods.     

N-soliton solution of a lattice equation related to the discrete MKdV equation

We present an N-soliton solution of a lattice equation related to the discrete MKdV equation under an arbitrary boundary value at infinity.     

The N threshold policy for the GI/M/1 queue

We study a GI/M/1 queue with an N threshold policy. In this system, the server stops attending the queue when the system becomes empty and resumes serving the queue when the number of customers reaches a threshold value N. Using the embeded Markov chain method, we obtain the stationary distributions of queue length and waiting time and prove the stochastic decomposition properties.     

On invertibility and zero-augmentability ofN-gon orders

A class of finite partially ordered sets isinvertible if the inverse (dual) of every poset in the class is in the class, and iszero-augmentable if the addition of a new element below all others yields a poset in the class for each member. This paper demonstrates that certain classes of posets that have representations byN-gons in the plane ordered by proper inclusion are neither invertible nor zero-augmentable. AMS subject classification (1980). 06A10.     

Asymptotic behavior of solutions to a system of viscous conservation laws related to a phase transition problem

We study the asymptotic behavior of solutions for a system of viscous conservation laws with discontinuous initial data. We discuss mainly the case where the system without the viscosity term is of hyperbolic elliptic mixed type. This problem is related to a phase transition problem. We study the initial value problem and show the decay rates of solutions to piecewise constant states where two phases coexist. The modification necessary for the hyperbolic case is also discussed.     

A nonlinear functional for general scalar hyperbolic conservation laws

A generalized entropy functional was introduced in [T.-P. Liu, T. Yang, A new entropy functional for scalar conservation laws, Comm. Pure Appl. Math. 52 (1999) 1427-1442] for the scalar hyperbolic conservation laws with convex flux function. This functional was crucially used in the functional approach to the L¹ stability study on the system of hyperbolic conservation laws when each characteristic field is either genuinely nonlinear or linearly degenerate. However, how to construct the generalized entropy functional for scalar conservation laws with general flux, and then how to apply the functional approach to the L¹ study on general systems are still open. In this paper, we construct a new nonlinear functional which gives some partial answer to this question and we expect the analysis will shed some light on the future investigation in this direction.     

Analysis of the GI/Geo/1 queue with N-policy

We consider a discrete-time single server N  -policy GI/Geo/1$\mathit{GI}/\mathit{Geo}/1$ queueing system. The server stops servicing whenever the system becomes empty, and resumes its service as soon as the number of waiting customers in the queue reaches N. Using an embedded Markov chain and a trial solution approach, the stationary queue length distribution at arrival epochs is obtained. Furthermore, we obtain the stationary queue length distribution at arbitrary epochs by using the preceding result and a semi-Markov process. The sojourn time distribution is also presented.     

Invariant manifolds and the stability of traveling waves in scalar viscous conservation laws

The stability of traveling wave solutions of scalar viscous conservation laws is investigated by decomposing perturbations into three components: two far-field components and one near-field component. The linear operators associated to the far-field components are the constant coefficient operators determined by the asymptotic spatial limits of the original operator. Scaling variables can be applied to study the evolution of these components, allowing for the construction of invariant manifolds and the determination of their temporal decay rate. The large time evolution of the near-field component is shown to be governed by that of the far-field components, thus giving it the same temporal decay rate. We also give a discussion of the relationship between this geometric approach and previous results, which demonstrate that the decay rate of perturbations can be increased by requiring that initial data lie in appropriate algebraically weighted spaces.     

The interaction processes of the N-soliton solutions for an extended generalization of Vakhnenko equation

The N-soliton solutions for an extended generalization of Vakhnenko equation are calculated by applying the improved Hirota method and the variable transformations. The N-soliton solutions can be expressed explicitly. Furthermore, the interaction processes are discussed for the N-soliton solutions.     

Spaces of N-homogeneous polynomials between Fréchet spaces

Properties of the Nachbin-ported topology on spaces of N-homogeneous polynomials between Fréchet spaces are investigated by applying results on derived functors.     

On energy conservation and P-stability

When applied to the classical dynamical equations, generally, conventional time-stepping methods cannot conserve the total energy exactly, but up to their algebraic order. To obtain an energy-conserving method, we add an energy calibrated parameter to some conventional ones. The periodic stability and phase-lag property of this type of energy-conserving methods are investigated in this note. Some numerical results are reported to illustrate our conclusions.     

Convergence of a stochastic particle approximation for fractional scalar conservation laws

We are interested in a probabilistic approximation of the solution to scalar conservation laws with fractional diffusion and nonlinear drift. The probabilistic interpretation of this equation is based on a stochastic differential equation driven by an α-stable Lévy process and involving a nonlinear drift. The approximation is constructed using a system of particles following a time-discretized version of this stochastic differential equation, with nonlinearity replaced by interaction. We prove convergence of the particle approximation to the solution of the conservation law as the number of particles tends to infinity whereas the discretization step tends to 0 in some precise asymptotics.     

The pyramidal configurations of N+1 bodies cannot rotate

For the (N+1)-body problem, we assume that N bodies are at the vertices of a unit regular polygon and the (N+1)st body is along the vertical line normal to the plane formed by the former N bodies. If N bodies rotate at the unit circle and the (N+1)st body oscillates along the vertical line of the plane formed by the former N bodies and passing through the geometrical center, then we prove that the (N+1)st body must locate at the geometrical center of unit regular polygon.