<Emphasis Type="Italic">m</Emphasis>-Dissipativity of Kolmogorov Operators Corresponding to Burgers Equations with Space-time White Noise

In this article, we prove new a priori estimates on the solutions of the Burgers equation driven by a space-time white noise and on the associated invariant measure. We also prove smoothing properties for the transition semi-group. This is obtained thanks to the introduction of a modified Kolmogorov operator. These results are then used to prove that the Kolmogorov operator associated to the Burgers equation is m-dissipative. This implies several properties on the Kolmogorov equation.      

MORPHOLOGY EVOLUTION IN PTFE AS A FUNCTION OF MELT TIME AND TEMPERATURE——Ⅱ. LOW MOLECULAR WEIGHT FOLDED CHAIN SINGLE CRYSTALS AND BAND STRUCTURES

The effect of sintering dispersed and bulk, low molecular weight (Mn = 50,000 Da), nano-emulsion polytetrafluoroethylene (PTFE) particles near their melting point is described. With the nascent particles consisting of ca.75 nm diameter, hexagonal, single crystals, sintering at, e.g., 350℃, results, initially, in merger of neighboring particles,followed by individual molecular motion on the substrate and the formation of folded chain, lamellar single crystals and spherulites, and on-edge ribbons. It is suggested these structures develop, with time, in the mesomorphic “melt”. Sintering of the bulk resin yields extended chain, band structures, as well as folded chain lamellae; end-surface to end-surface merger,possibly by end-to-end polymerization, occurs with increasing time.     

Graph splicing systems

In this paper, extended graph splicing systems are defined. It is shown that when strings are represented as linear graphs, any recursively enumerable set can be generated by an extended graph splicing system. It is also shown that the computational completeness of extended graph splicing systems can be proved under some constraints too.     

Scaling Laws for the Multidimensional Burgers Equation with Quadratic External Potential

The reordering of the multidimensional exponential quadratic operator in coordinate-momentum space (see X. Wang, C.H. Oh and L.C. Kwek (1998). J. Phys. A.: Math. Gen. 31:4329–4336) is applied to derive an explicit formulation of the solution to the multidimensional heat equation with quadratic external potential and random initial conditions. The solution to the multidimensional Burgers equation with quadratic external potential under Gaussian strongly dependent scenarios is also obtained via the Hopf-Cole transformation. The limiting distributions of scaling solutions to the multidimensional heat and Burgers equations with quadratic external potential are then obtained under such scenarios. AMS Subject Classifications: 60G60, 60G15, 62M15, 60H15     

Steady-state solutions for some simple flows of generalized Burgers fluids

The steady-state solutions for three types of unsteady oscillating flows of generalized Burgers fluids are determined by means of the Fourier sine transforms. These solutions are also presented in equivalent forms in terms of elementary functions exp, sine, cosine, hyperbolic sine and hyperbolic cosine. The similar solutions for Burgers, Oldroyd-B, Maxwell, Second grade and Navier-Stokes fluids can be also obtained as limiting cases of our solutions.     

On the Hardness of Computing Intersection,Union and Minkowski Sum of Polytopes

For polytopes P ₁,P ₂⊂ℝ ^d , we consider the intersection P ₁∩P ₂, the convex hull of the union CH(P ₁∪P ₂), and the Minkowski sum P ₁+P ₂. For the Minkowski sum, we prove that enumerating the facets of P ₁+P ₂ is NP-hard if P ₁ and P ₂ are specified by facets, or if P ₁ is specified by vertices and P ₂ is a polyhedral cone specified by facets. For the intersection, we prove that computing the facets or the vertices of the intersection of two polytopes is NP-hard if one of them is given by vertices and the other by facets. Also, computing the vertices of the intersection of two polytopes given by vertices is shown to be NP-hard. Analogous results for computing the convex hull of the union of two polytopes follow from polar duality. All of the hardness results are established by showing that the appropriate decision version, for each of these problems, is NP-complete.     

Energy properties preserving schemes for Burgers' equation

The Burgers' equation, a simplification of the Navier–Stokes equations, is one of the fundamental model equations in gas dynamics, hydrodynamics, and acoustics that illustrates the coupling between convection/advection and diffusion. The kinetic energy enjoys boundedness and monotone decreasing properties that are useful in the study of the asymptotic behavior of the solution. We construct a family of non‐standard finite difference schemes, which replicate the energy equality and the properties of the kinetic energy. Our approach is based on Mickens' rule [Nonstandard Finite Difference Models of Differential Equations, World Scientific, Singapore, 1994.] of nonlocal approximation of nonlinear terms. More precisely, we propose a systematic nonlocal way of generating approximations that ensure that the trilinear form is identically zero for repeated arguments. We provide numerical experiments that support the theory and demonstrate the power of the non‐standard schemes over the classical ones. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Period-doubling bifurcation in an extended van der Pol system with bounded random parameter

An extended van der Pol system with bounded random parameter subjected to harmonic excitation is investigated by Chebyshev polynomial approximation. Firstly the stochastic extended van der Pol system is reduced into its equivalent deterministic one, solvable by suitable numerical methods. Then we explored nonlinear dynamical behavior about period-doubling bifurcation in stochastic system. Numerical simulations show that similar to the conventional period-doubling phenomenon in deterministic extended van der Pol system, stochastic period-doubling bifurcation may also occur in the stochastic extended van der Pol system. Besides, different from the deterministic case, in addition to the conventional bifurcation parameters, i.e. the amplitude and frequency of harmonic excitation, in the stochastic case the intensity of random parameter should also be taken as a new bifurcation parameter.