g 1 approximation of conic sections by quartic bezier curves this paper proposes a new approximation method for conic sections by quartic bezier curves with g 1 end point continuity we give an upper bound on the hausdorff distance between the conic section and the quartic bezier approximation curve and also show that the approximation order is eight moreover using the subdivision scheme at the shoulder point of the conic section the resulting approximation curve has g 2 curvature continuity with the conic section at the endpoints finally we prove that our approximation has a smaller error bound than previous quartic bezier approximations and also present some numerical examples to demonstrate the validity and effectiveness of our method

an approximation method to circular arcs abstract modern cad cam systems are not able to dispose the circles represented by the implicit equations or the parametric equations with trigonometric function parametric rational polynomials are used to approximate the circular arcs a new approximation method by means of quartic bezier curves is presented to effectively approximate the circular arcs it is shown that our method provides the approximation of order eight and yields a smaller error than previous methods by equally spaced subdivision of circular arcs the curvature continuous spline approximation algorithm of the circular arcs is given

optimal parameter values for approximating conic sections by the quartic bzier curves the previous approximation curves of conic section by quartic bzier curves interpolate the conic section at the specified parameter values in this paper by solving the minimax problem we present the optimal parameter values for approximating conic sections by the quartic bzier curves the upper bound on the hausdorff distance between the conic section and the approximation curves is minimized the method of solving the minimax problem is changed to solve a quartic algebraic equation by delicate reasoning

classical solutions to the boltzmann equation for molecules with an angular cutoff an important class of collision kernels in the boltzmann theory are governed by the inverse power law in which the intermolecular potential between two particles is an inverse power of their distance under the grad angular cutoff assumption global in time classical solutions near maxwellians are constructed in a periodic box for all soft potentials with 3 0

the landau equation in a periodic box the landau equation which was proposed by landau in 1936 is a fundamental equation to describe collisions among charged particles interacting with their coulombic force in this article global in time classical solutions near maxwellians are constructed for the landau equation in a periodic box our result also covers a class of generalized landau equations which describes grazing collisions in a dilute gas

a general framework for the optimal approximation of circular arcs by parametric polynomial curves abstract we propose a general framework for a geometric approximation of circular arcs by parametric polynomial curves the approach is based on a constrained uniform approximation of an error function by scalar polynomials the system of nonlinear equations for the unknown control points of the approximating polynomial given in the bezier form is derived and a detailed analysis provided for some low degree cases which were not studied yet at least for these cases the solutions can be in principal written in a closed form and provide the best known approximants according to the simplified radial distance a general conjecture on the optimality of the solution is stated and several numerical examples conforming theoretical results are given

the vlasov poisson landau system in mathbb r 3 _ x for the landau poisson system with coulomb interaction in mathbb r 3 x we prove the global existence uniqueness and large time convergence rates to the maxwellian equilibrium for solutions which start out sufficiently close

exponential decay for soft potentials near maxwellian we consider both soft potentials with angular cutoff and landau collision kernels in the boltzmann theory inside a periodic box we prove that any smooth perturbation near a given maxwellian approaches zero at the rate of e lambda t p for some 0 and 0 p 1 our method is based on an unified energy estimate with appropriate exponential velocity weight our results extend the classical result caflisch of 2 to the case of very soft potential and coulomb interactions and also improve the recent almost exponential decay results by 5 14

on the trend to equilibrium for the vlasov poisson boltzmann equation abstract the dynamics of dilute electrons and plasma can be modeled by vlasov poisson boltzmann equation for which the equilibrium state can be a global maxwellian in this paper we show that the rate of convergence to equilibrium is o t by using a method developed for the boltzmann equation without external force in l desvillettes c villani on the trend to global equilibrium for spatially inhomogeneous kinetic systems the boltzmann equation invent math 159 2005 245 316 in particular the idea of this method is to show that the solution f cannot stay near any local maxwellians for long the improvement in this paper is to handle the effect from the external force governed by the poisson equation moreover by using the macro micro decomposition we simplify the estimation on the time derivatives of the deviation of the solution from the local maxwellian with same macroscopic components

cauchy problem for the vlasov poisson boltzmann system the dynamics of dilute electrons can be modelled by the fundamental vlasov poisson boltzmann system which describes mutual interactions of the electrons through collisions in the self consistent electric field in this paper it is shown that any smooth perturbation of a given global maxwellian leads to a unique global in time classical solution when either the mean free path is small or the background charge density is large moreover the solution converges to the global maxwellian when time tends to infinity the analysis combines the techniques used in the study of conservation laws with the decomposition of the boltzmann equation introduced in 17 19 by obtaining new entropy estimates for this physical model

the vlasov maxwell boltzmann system near maxwellians perhaps the most fundamental model for dynamics of dilute charged particles is described by the vlasov maxwell boltzmann system in which particles interact with themselves through collisions and with their self consistent electromagnetic field despite its importance no global in time solutions weak or strong have been constructed so far it is shown in this article that any initially smooth periodic small perturbation of a given global maxwellian which preserves the same mass total momentum and reduced total energy 22 leads to a unique global in time classical solution for such a master system the construction is based on a recent nonlinear energy method with a new a priori estimate for the dissipation the linear collision operator l not its time integration is positive definite for any solutionf t x v with small amplitude to the vlasov maxwell boltzmann system 8 and 12 as a by product such an estimate also yields an exponential decay for the simpler vlasov poisson boltzmann system 24

some new g1 quartic parametric approximants of circular arcs abstract in this paper a method for approximation of circular arc by quartic bezier curve is presented interpolation by geometrically continuous g 1 parametric polynomials is considered and necessary and sufficient conditions for the existence of the solution are established optimal asymptotic approximation order eight is also confirmed for the obtained approximant application of our new solution to approximation of conic sections is presented and some numerical examples provided

on polynomial approximation of circular arcs and helices we present a simple method for polynomial approximation of circular arcs and helices by expressing the trigonometric functions using the two point taylor expansion we obtain the degree 2n 1 polynomial for the approximation problem in an efficient way which is very convenient to increase the degree of polynomial by adding new terms an upper bound on the approximation error is available so that we can obtain the lowest degree polynomial curve that can approximate a circular arc or helix segment within any user prescribed error tolerance

the best g 1 cubic and g 2 quartic bezier approximations of circular arcs we obtain cubic and quartic bezier approximations of circular arcs that respectively satisfy g 1 and g 2 end point interpolation conditions we identify the necessary and sufficient conditions for such approximations to be the best in the sense that they have the minimum hausdorff distance to the circular arc we then establish the existence and uniqueness of these best approximations and present practical methods to calculate them which are verified by examples

a note on quintic polynomial approximation of generalized cornu spiral segments in two recent papers cripps et al 2010 3 and cross and cripps 2012 2 quintic polynomial approximations of the generalized cornu spirals have been studied by considering g 2 continuity and g 3 continuity at the end points respectively the quintic curve is constructed so that the maximum curvature error is within the specified tolerance in this paper we provide corrections to the typing errors in cross and cripps 2012 2 and propose a simpler and more efficient method for the g 2 constrained quintic polynomial approximation where the g 2 conditions are always satisfied by four free variables also we introduce a new error measure of the maximum curvature error which can greatly reduce the computational time when looking for the solution numerical experiments demonstrate the effectiveness of the new measure

the vlasov poisson boltzmann system for the whole range of cutoff soft potentials abstract the dynamics of dilute electrons can be modeled by the fundamental one species vlasov poisson boltzmann system which describes mutual interactions of the electrons through collisions in the self consistent electrostatic field for cutoff intermolecular interactions although there is some progress on the construction of global smooth solutions to its cauchy problem near maxwellians recently the problem for the case of very soft potentials remains unsolved by introducing a new time velocity weighted energy method and based on some new optimal temporal decay estimates on the solution itself and some of its derivatives with respect to both the spatial and the velocity variables it is shown in this manuscript that the cauchy problem of the one species vlasov poisson boltzmann system for all cutoff soft potentials does exist a unique global smooth solution for general initial perturbation which is unnecessary to satisfy the neutral condition imposed in 13 for the case of cutoff moderately soft potentials but is assumed to be small in certain weighted sobolev spaces our approach applies also to the case of cutoff hard potentials and thus provides a satisfactory global well posedness theory to the one species vlasov poisson boltzmann system near maxwellians for the whole range of cutoff intermolecular interactions in the perturbative framework

the vlasov poisson landau system in a periodic box the classical vlasov poisson landau system describes dynamics of a collisional plasma interacting with its own electrostatic eld as well as its grazing collisions such grazing collisions are modeled by the famous landau fokker planck collision kernel proposed by landau in 1936 we construct global unique solutions to such a system for initial data which have small weighted h norms but can have large h k 3 norms with high velocity moments our construction is based on accumulative study on the landau kernel in the past decade g1 sg1 3 with four extra ingredients to overcome the speci c mathematical di culties present in the vlasov poisson landau system a new exponential weight of electric potential to cancel the growth of the velocity a new velocity weight to capture the weak velocity di usion in the landau kernel a decay of the electric eld to close the energy estimate and a new bootstrap argument to control the propagation of the high moments and regularity with large amplitude

the vlasov poisson boltzmann system in the whole space the hard potential case abstract this paper is concerned with the cauchy problem on the vlasov poisson boltzmann system for hard potentials in the whole space when the initial data is a small perturbation of a global maxwellian a satisfactory global existence theory of classical solutions to this problem together with the corresponding temporal decay estimates on the global solutions is established our analysis is based on time decay properties of solutions and a new time velocity weight function which is designed to control the large velocity growth in the nonlinear term for the case of non hard sphere interactions

circular sector area preserving approximation of circular arcs by geometrically smooth parametric polynomials abstract the quality of the approximation of circular arcs by parametric polynomials is usually measured by the hausdorff distance it is sometimes important that a parametric polynomial approximant additionally preserves some particular geometric property in this paper we study the circular sector area preserving parametric polynomial approximants of circular arcs a general approach to this problem is considered and corresponding nonlinear equations are derived for the approximants possessing the maximal order of geometric smoothness a scalar nonlinear equation is analyzed in detail for the parabolic the cubic and the quartic case the existence of the admissible solution is confirmed moreover the uniqueness of the solution with the optimal approximation order with respect to the radial distance is proved theoretical results are confirmed by numerical examples

quintic polynomial approximation of log aesthetic curves by curvature deviation log aesthetic curves lacs possessing monotone curvature and including many classical curves have been widely used to describe fair shapes in geometric modeling however they are generally represented in non polynomial form and are thus not compatible with current cad systems in this paper we present quintic polynomial approximation of lac segments for a given lac segment a quintic g 2 interpolating bezier curve is obtained by minimizing a curvature based error metric with the advantage of being more likely to preserve the monotone curvature property numerical experiments demonstrate that our method can usually generate better results than the previous methods in terms of the deviation in positions and curvatures

global hilbert expansion for the vlasov poisson boltzmann system the dynamics of an electron gas in a constant ion background can be decribed by the vlasov poisson boltzmann system at the kinetic level or by the compressible euler poisson system at the fluid level we prove that any solution of the vlasov poisson boltzmann system near a smooth local maxwellian with a small irrotational velocity converges global in time to the corresponding solution to the euler poisson system as the mean free path e goes to zero we use a recent l2 l framework in the boltzmann theory to control the higher order remainder in the hilbert expansion uniformly in e and globally in time

global existence of classical solutions to the vlasov poisson boltzmann system the time evolution of the distribution function for the charged particles in a dilute gas is governed by the vlasov poisson boltzmann system when the force is self induced and its potential function satisfies the poisson equation in this paper we give a satisfactory global existence theory of classical solutions to this system when the initial data is a small perturbation of a global maxwellian moreover the convergence rate in time to the global maxwellian is also obtained through the energy method the proof is based on the theory of compressible navier stokes equations with forcing and the decomposition of the solutions to the boltzmann equation with respect to the local maxwellian introduced in 23 and elaborated in 31

approximation of circular arcs by bezier curves abstract for the circular arc of angle 0 we present the explicit form of the best gc 3 quartic approximation and the best gc 2 quartic approximations of various types and give the explicit form of the hausdorff distance between the circular arc and the approximate bezier curves for each case we also show the existence of the gc 4 quintic approximations to the arc and find the explicit form of the best gc 3 quintic approximation in certain constraints and their distances from the arc all approximations we construct in this paper have the optimal order of approximation twice of the degree of approximate bezier curves

existence and multiplicity results to neumann problems for elliptic equations involving the p laplacian abstract in this paper existence results of positive solutions to a neumann problem involving the p laplacian are established multiplicity results are also pointed out the approach is based on variational methods