convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping we consider a model of hyperbolic conservation laws with damping and show that the solutions tend to those of a nonlinear parabolic equation time asymptotically the hyperbolic model may be viewed as isentropic euler equations with friction term added to the momentum equation to model gas flow through a porous media in this case our result justifies darcy s law time asymptotically our model may also be viewed as an elastic model with damping

on the estimation of optical flow relations between different approaches and some new results abstract a common basis is suggested for the optical flow estimation approaches of nagel 1983 haralick and lee 1983 and tretiak and pastor 1984 based on a discussion of these approaches an exact solution for the system of partial differential equations proposed by horn and schunck 1981 is given at gray value corners and extrema the insight gained by this solution results in a modification of the oriented smoothness approach of nagel 1983 which thereby becomes considerably simpler in addition the optical flow estimation approach of hildreth 1983 1984 can be shown to represent a kind of special case of this modified oriented smoothness approach in a more direct manner than discussed in nagel 1984

on the superlinear convergence of interior point algorithms for a general class of problems in this paper the authors extend the q superlinear convergence theory recently developed by zhang tapia and dennis for a class of interior point linear programming algorithms to similar interior point algorithms for quadratic programming and for linear complementarily problems this unified approach consists of viewing all these algorithms as a damped newton method applied to perturbations of a general problem a set of sufficient conditions for these algorithms to achieve q superlinear convergence is established the key ingredients consist of asymptotically taking the step to the boundary of the positive orthant and letting the centering parameter approach zero at a specific rate the construction of algorithms that have both the global property of polynomiality and the local property of superlinear convergence will be the subject of further research

asymptotic behavior of solution to nonlinear evolution equations with damping in this paper we consider the global existence and the asymptotic behavior of solutions to the cauchy problem for the following nonlinear evolution equations with ellipticity and dissipative effects e t 1 x xx t x 0 r t 1 x 2 x xx with initial data i x 0 0 x 0 x as x where and are positive constants such that 1 4 1 under the assumption that is sufficiently small we show that if the initial data is a small perturbation of the parabolic system defined by 2 4 which are obtained by the convection diffusion equations 2 1 and solutions to cauchy problem e and i tend asymptotically to the convection diffusion system with exponential rates precisely speaking we derive the asymptotic profile of e by gauss kernel g t x as follows 2 2e 1 4 t xg y t 1 1 sin 2 y 0 cos 2 y 0 dy e 1 t 2e 1 4 t rg t y cos 2 y 1 sin 2 y sin 2 y cos 2 y u0 x y v0 x y dy lp rx e 1 4 to 1 t 1 1p the same problem was studied by tang and zhao s q tang h j zhao nonlinear stability for dissipative nonlinear evolution equations with ellipticity j math anal appl 233 1999 336 358 nishihara k nishihara asymptotic profile of solutions to nonlinear dissipative evolution system with ellipticity z angew math phys 57 4 2006 604 614 for the case of 0 0

the global attractor of a dissipative nonlinear evolution system we study the global smooth solution and the global attractor for a dissipative nonlinear evolution system given by strongly coupled parabolic equations

a superlinear infeasible interior point affine scaling algorithm for lcp we present an infeasible interior point algorithm for monotone linear complementarity problems in which the search directions are affine scaling directions and the step lengths are obtained from simple formulae that ensure both global and superlinear convergence by choosing the value of a parameter in appropriate ways polynomial complexity and convergence with q order up to but not including two can be achieved the only assumption made to obtain the superlinear convergence is the existence of a solution satisfying strict complementarily

on the superlinear and quadratic convergence of primal dual interior point linear programming algorithms this paper presents a convergence rate analysis for interior point primal dual linear programming algorithms conditions that guarantee q superlinear convergence are identified in two distinct theories both state that under appropriate assumptions q superlinear convergence is achieved by asymptotically taking the step to the boundary of the positive orthant and letting the barrier parameter approach zero at a rate that is superlinearly faster than the convergence of the duality gap to zero the first theory makes no nondegeneracy assumption and explains why in recent numerical experimentation q superlinear convergence was always observed the second theory requires the restrictive assumption of primal nondegeneracy however it gives the surprising result that q superlinear convergence can still be attained even if centering is not phased out provided the iterates asymptotically approach the central path the latter theory is extended to produce a satisfactory q quadratic convergence theory it requir

global decaying solution to dissipative nonlinear evolution equations with ellipticity abstract we establish the global existence and decaying results for the cauchy problem of nonlinear evolution equations e t 1 x x xx t 1 x 2 x xx forinitial data with different end states i x 0 x 0 0 x 0 x as x which displays the complexity in between ellipticity and dissipation although the nonlinear term x appears in equation e 1 which makes calculations more complicated due to smoothing effect of the parabolic operator we detail its regularity property and decay estimates when t 0 for the higher order spatial derivatives despite its relatively lower regularity of the initial data and we also discuss the decay estimates furthermore we do not restrict l1 bound on the initial data 0 x 0 x as in 2

nonlinear stability for dissipative nonlinear evolution equations with ellipticity we study the cauchy problem for a set of nonlinear evolution equations which has been proposed for the investigation of nonlinear interaction between ellipticity and dissipation the zero equilibrium is linearly stable for coefficients within certain range we further establish the global existence nonlinear stability and optimal decay rate of the solution for coefficients within the same range though with suitable restriction to initial data

a fast scalable algorithm for discontinuous optical flow estimation multiple moving objects partially occluded objects or even a single object moving against the background gives rise to discontinuities in the optical flow field in corresponding image sequences while uniform global regularization based moderately fast techniques cannot provide accurate estimates of the discontinuous flow field statistical optimization based accurate techniques suffer from excessive solution time a weighted anisotropic smoothness based numerically robust algorithm is proposed that can generate discontinuous optical flow field with high speed and linear computational complexity weighted sum of the first order spatial derivatives of the flow field is used for regularization less regularization is performed where strong gradient information is available the flow field at any point is interpolated more from those at neighboring points along the weaker intensity gradient component such intensity gradient weighted regularization leads to euler lagrange equations with strong anisotropies coupled with discontinuities in their coefficients a robust multilevel iterative technique that recursively generates coarse level problems based on intensity gradient weighted smoothing weights is employed to estimate discontinuous optical flow field experimental results are presented to demonstrate the efficacy of the proposed technique

a morphological affine and galilean invariant scale space for movies we study a model of multiscale analysis or scale space applied to movies this model comes from a thorough formalization that has been done in the theory of scale space of static image this formulation has led one to associate with each multiscale analysis a partial differential equation pde we examine the case of movies and insist on the motion aspects more precisely it has been proved by alvarez guichard lions and morel 1993 that there exists a unique affine and morphological and galilean invariant scale space for movies the amg model this model is described by a partial differential equation we focus on terms appearing in that equation we show that this model provides a reliable definition of an optical multiscale acceleration at the practical level scale is interpreted as a way of characterizing reliable trajectories as we prove by experiments the amg model is a riddle for decimating spurious trajectories due to any kinds of nonadditive impurities and noise simple discrete formulae are given to implement the model

constant potential primal dual algorithms a framework we start with a study of the primal dual affine scaling algorithms for linear programs using ideas from kojima et al mizuno and nagasawa and new potential functions we establish a framework for primal dual algorithms that keep a potential function value fixed we show that if the potential function used in the algorithm is compatible with a corresponding neighborhood of the central path then the convergence proofs simplify greatly our algorithms have the property that all the iterates can be kept in a neighborhood of the central path without using any centering in the search directions

large step interior point algorithms for linear complementarity problems recently kojima megiddo and mizuno showed theoretical convergence of primal dual interior point algorithms with the use of new step length rules for linear programs their rules which only rely on the lengths of steps from the current iterates in the primal and dual spaces to the respective boundaries of the primal and dual feasible regions allow taking large step lengths without performing any line search this paper extends and modifies their analysis to interior point algorithms for positive semidefinite linear complementarily problems global convergence and polynomial time convergence are presented under similar step length rules

a theoretical framework for convex regularizers in pde based computation of image motion many differential methods for the recovery of the optic flow field from an image sequence can be expressed in terms of a variational problem where the optic flow minimizes some energy typically these energy functionals consist of two terms a data term which requires e g that a brightness constancy assumption holds and a regularizer that encourages global or piecewise smoothness of the flow field in this paper we present a systematic classification of rotation invariant convex regularizers by exploring their connection to diffusion filters for multichannel images this taxonomy provides a unifying framework for data driven and flow driven isotropic and anisotropic as well as spatial and spatio temporal regularizers while some of these techniques are classic methods from the literature others are derived here for the first time we prove that all these methods are well posed they posses a unique solution that depends in a continuous way on the initial data an interesting structural relation between isotropic and anisotropic flow driven regularizers is identified and a design criterion is proposed for constructing anisotropic flow driven regularizers in a simple and direct way from isotropic ones its use is illustrated by several examples

symmetrical dense optical flow estimation with occlusions detection traditional techniques of dense optical flow estimation do not generally yield symmetrical solutions the results will differ if they are applied between images i 1 and i 2 or between images i 2 and i 1 in this work we present a method to recover a dense optical flow field map from two images while explicitely taking into account the symmetry across the images as well as possible occlusions in the flow field the idea is to consider both displacements vectors from i 1 to i 2 and i 2 to i 1 and to minimise an energy functional that explicitely encodes all those properties this variational problem is then solved using the gradient flow defined by the euler lagrange equations associated to the energy to prove the importance of the concepts of symmetry and occlusions for optical flow computation we have extended a classical approach to handle those experiments clearly show the added value of these properties to improve the accuracy of the computed flows figures appear in color in the online version of this paper

reliable estimation of dense optical flow fields with large displacements in this paper we show that a classic optical flow technique by nagel and enkelmann 1986 ieee trans pattern anal mach intell vol 8 pp 565 593 can be regarded as an early anisotropic diffusion method with a diffusion tensor we introduce three improvements into the model formulation that i avoid inconsistencies caused by centering the brightness term and the smoothness term in different images ii use a linear scale space focusing strategy from coarse to fine scales for avoiding convergence to physically irrelevant local minima and iii create an energy functional that is invariant under linear brightness changes applying a gradient descent method to the resulting energy functional leads to a system of diffusion reaction equations we prove that this system has a unique solution under realistic assumptions on the initial data and we present an efficient linear implicit numerical scheme in detail our method creates flow fields with 100 density over the entire image domain it is robust under a large range of parameter variations and it can recover displacement fields that are far beyond the typical one pixel limits which are characteristic for many differential methods for determining optical flow we show that it performs better than the optical flow methods with 100 density that are evaluated by barron et al 1994 int j comput vision vol 12 pp 43 47 our software is available from the internet

superlinear primal dual affine scaling algorithms for lcp we describe an interior point algorithm for monotone linear complementarity problems in which primal dual affine scaling is used to generate the search directions the algorithm is shown to have global and superlinear convergence with q order up to but not including two the technique is shown to be consistent with a potential reduction algorithm yielding the first potential reduction algorithm that is both globally and superlinearly convergent

asymptotics in nonlinear evolution system with dissipation and ellipticity on quadrant abstract in this paper we consider an initial boundary value problem for some nonlinear evolution system with dissipation and ellipticity we establish the global existence and furthermore obtain the l p p 2 decay rates of solutions corresponding to diffusion waves the analysis is based on the energy method and pointwise estimates

convergence to diffusion waves for nonlinear evolution equations with different end states in this paper we consider the global existence and the asymptotic decay of solutions to the cauchy problem for the following nonlinear evolution equations with ellipticity and dissipative effects e t 1 x f x xx t 1 x x xx with initial data i x 0 0 x 0 x asx where and are positive constants such that 1 s 4 1 s is defined in 1 14 under the assumption that is sufficiently small we show that if the initial data is a small perturbation of the diffusion waves defined by 2 5 which are obtained by the diffusion equations 2 1 solutions to cauchy problem e and i tend asymptotically to those diffusion waves with exponential rates the analysis is based on the energy method the similar problem was studied by tang and zhao s q tang h j zhao nonlinear stability for dissipative nonlinear evolution equations with ellipticity j math anal appl 233 1999 336 358 for the case of 0 0

on the mathematical foundations of smoothness constraints for the determination of optical flow and for surface reconstruction gradient based approaches to the computation of optical flow often use a minimization technique incorporating a smoothness constraint on the optical flow field the author derives the most general form of such a smoothness constraint that is quadratic in first derivatives of the grey level image intensity function based on three simple assumptions about the smoothness constraint 1 it must be expressed in a form that is independent of the choice of cartesian coordinate system in the image 2 it must be positive definite and 3 it must not couple different component of the optical flow it is shown that there are essentially only four such constraints any smoothness constraint satisfying 1 2 or 3 must be a linear combination of these four possibly multiplied by certain quantities invariant under a change in the cartesian coordinate system beginning with the three assumptions mentioned above the author mathematically demonstrates that all best known smoothness constraints appearing in the literature are special cases of this general form and in particular that the weight matrix introduced by h h nagel is essentially modulo invariant quantities the only physically plausible such constraint

decay rates of solutions to the cauchy problem for dissipative nonlinear evolution equations abstract in this paper we study the global existence and the asymptotic behavior of the solutions to the cauchy problem for the following nonlinear evolution equations with ellipticity and dissipative effects e t 1 x x x t 1 x x x x with initial data i x 0 0 x 0 x as x where and are positive constants such that 1 1 through constructing a correct function x t defined by 2 13 and using the energy method we show sup x r x t x x x t 0 as t and the solutions decay with exponential rates the same problem was studied by tang and zhao s q tang h j zhao nonlinear stability for dissipative nonlinear evolution equations with ellipticity j math anal appl 233 1999 336 358 for the case of 0 0

a note on boundary layer of a nonlinear evolution system with damping and diffusions abstract in this paper we consider an initial boundary value problem for a nonlinear evolution system with damping and diffusions on the strip our main purpose is to investigate zero diffusion limits which formally yield parabolic hyperbolic coupled equations resulting from parabolic parabolic coupled equations it is shown that the boundary layer thickness is of the order o with 0 3 4 as the diffusion parameter goes to zero

on functionals with greyvalue controlled smoothness terms for determining optical flow the modification by h h nagel 1987 of the approach developed by b k p horn and b g schunck 1981 for determining optical flow is generalized to the case where local motion information is given by more than one constraint equation applying this scheme to three constraint equations reported in the literature as a special case a generalization of nagel s approach is obtained an existence and uniqueness result of solutions under very general conditions that in turn ensures the applicability of standard techniques to compute an approximate solution is presented

large time profile of solutions for a dissipative nonlinear evolution system with conservational form we examine the cauchy problem for a nonlinear dissipative evolution system with conservation form with initial condition where and are positive constants such that and 1 we establish the global existence and decay rate of the solution subject to the parameter restriction the optimal decay rate is obtained if 0 0 l1 r r2 furthermore the global existence of the solution to the same problem has been studied in jian and chen 1998 acta math sin 14 17 34 without giving the decay rate and optimal decay order of the solution in which 0 0 h1 r r2 l1 r r2

global smooth solutions to the spatially periodic cauchy problem for dissipative nonlinear evolution equations abstract the existence and uniqueness are proved for global classical solutions of the spatially periodic cauchy problem to the following system of parabolic equations formula which was proposed as a substitute for the rayleigh benard equation and can lead to lorenz equations

on the convergence of the affine scaling algorithm the affine scaling algorithm first proposed by dikin is presently enjoying great popularity as a potentially effective means of solving linear programs an outstanding question about this algorithm concerns its convergence in the presence of degeneracy in this paper we give new convergence results for this algorithm that do not require any non degeneracy assumption on the problem in particular we show that if the stepsize choice of either dikin or barnes or vanderbei et al is used then the algorithm generates iterates that converge at least linearly with a convergence ratio of 1 beta sqrt n wheren is the number of variables and 0 1 is a certain stepsize ratio for one particular stepsize choice which is an extension of that of barnes we show that the cost of the limit point is within o 1 of the optimal cost and for sufficiently small roughly proportional to how close the cost of the nonoptimal vertices are to the optimal cost is exactly optimal we prove the latter result by using an unusual proof technique that of analyzing the ergodic convergence of the corresponding dual vectors for the special case of network flow problems with integer data we show that it suffices to take 1 6mc wherem is the number of constraints andc is the sum of the cost coefficients to attain exact optimality

direct methods for recovering motion we have developed direct methods for recovering the motion of an observer in a static environment in the case of pure rotation pure translation and arbitrary motion when the rotation is known some of these methods are based on the minimization of the difference between the observed time derivative of brightness and that predicted from the spatial brightness gradient given the estimated motion we minimize the square of the integral of this difference taken over the image region of interest other methods presented here exploit the fact that surfaces have to be in front of the observer in order to be seen

an investigation of smoothness constraints for the estimation of displacement vector fields from image sequences a mapping between one frame from an image sequence and the preceding or following frame can be represented as a displacement vector field in most situations the mere gray value variations do not provide sufficient information in order to estimate such a displacement vector field supplementary constraints are necessary for example the postulate that a displacement vector field varies smoothly as a function of the image position taken as a general requirement this creates difficulties at gray value transitions which correspond to occluding contours nagel therefore introduced the oriented smoothness requirement which restricts variations of the displacement vector field only in directions with small or no variation of gray values this contribution reports results of an investigation about how such an oriented smoothness constraint may be formulated and evaluated

nonlinear stability of strong planar rarefaction waves for the relaxation approximation of conservation laws in several space dimensions abstract in this paper we show that a strong planar rarefaction wave is nonlinear stable namely it is an attractor for the relaxation approximation of the scalar conservation laws in several space dimensions compared with former results obtained by t p liu 1987 comm math phys 108 153 175 and t luo 1997 j differential equations 133 255 279 our main novelty lies in the fact that the planar rarefaction waves do not need to be small and in the one dimensional case the initial disturbance can also be chosen arbitrarily large

a superlinearly convergent polynomial primal dual interior point algorithm for linear programming the choices of the centering parameter and the step length parameter are the fundamental issues in primal dual interior point algorithms for linear programming various choices for these two parameters that lead to polynomial algorithms have been proposed recently zhang tapia and dennis derived conditions for the choices of the two parameters that were sufficient for superlinear or quadratic convergence however prior to this work it had not been shown that these conditions for fast convergence are compatible with the choices that lead to polynomiality none of the existing polynomial primal dual interior point algorithms satisfy these fast convergence requirements this paper gives an affirmative answer to the question can a primal dual algorithm be both polynomial and superlinearly convergent for general problems a large step algorithm that possesses both polynomiality and under the assumption of the convergence of the iteration sequence q superlinear convergence is constructed and analyzed

quadratic convergence in a primal dual method we show that the mizuno todd ye o nl iteration predictor corrector primal dual interior point algorithm for linear programming is quadratically convergent our proof does not assume that the problems be nondegenerate we do not assume that the iterate generated by the algorithm be convergent an assumption common to all previous asymptotic convergence analysis

potential reduction methods in mathematical programming we provide a survey of interior point methods for linear programming and its extensions that are based on reducing a suitable potential function at each iteration we give a fairly complete overview of potential reduction methods for linear programming focusing on the possibility of taking long steps and the properties of the barrier function that are necessary for the analysis we then describe briefly how the methods and results can be extended to certain convex programming problems following the approach of nesterov and todd we conclude with some open problems

asymptotics of dissipative nonlinear evolution equations with ellipticity different end states in this paper we consider the global existence and asymptotic behaviors of solutions to the cauchy problem for the following nonlinear evolution equations with ellipticity and dissipative effects e t 1 x xx t 1 x 2 x xx with initial data i x 0 0 x 0 x as x where and are positive constants such that 1 4 1 under the assumption that is sufficiently small we show that if the initial data is a small perturbation of the diffusion waves defined by 2 5 which are obtained by the diffusion equations 2 1 solutions to cauchy problem e and i tend asymptotically to those diffusion waves with exponential rates the analysis is based on the energy method the same problem was studied by tang and zhao j math anal appl 233 1999 336 358 for the case of 0 0

optimal decay rates to diffusion wave for nonlinear evolution equations with ellipticity abstract we derive the optimal convergence rates to diffusion wave for the cauchy problem of a set of nonlinear evolution equations with ellipticity and dissipative effects t 1 x x x t 1 x x x x subject to the initial data with end states x 0 0 x 0 x as x where and are positive constants such that 1 4 1 introducing the auxiliary function to avoid the difference of the end states we show that the solutions to the reformulated problem decay as t with the optimal decay order the decay properties of the solution in the l 2 sense which are not optimal were already established in paper c j zhu z y zhang h yin convergence to diffusion waves for nonlinear evolution equations with ellipticity and damping and with different end states acta math sinica english ed in press the main element of this paper is to obtain the optimal decay order in the sense of l p space for 1 p which is based on the application of fourier analysis and interpolation inequality under some suitable restrictions on coefficients and moreover we discuss the asymptotic behavior of the solution to general system 1 1 at the end however the optimal decay rates of the solution to general system 1 1 remains unknown

a primal dual variant of the iri imai algorithm for linear programming a local acceleration method for primal dual potential reduction algorithms is introduced the method developed here uses modified newton search directions to minimize the tanabe todd ye tty potential function and can be regarded as a primal dual variant of the iri imai algorithm based on the multiplicative analogue of karmarkar s potential function when iterates are close to an optimal solution the tty potential function has negative curvature along the generated search directions therefore large reductions in the potential function can be obtained guaranteeing polynomial and quadratic convergence to nondegenerate solutions

asymptotic behavior of solutions for p system with relaxation in this paper we consider the asymptotic behavior of solutions for the cauchy problem for p system with relaxationvt ux 0 ut p v x 1e f v u e with initial data v u x 0 v 0 x u 0 x v u v 0 asx i we are interested to show the solutions of e i tend also to the equilibrium rarefaction waves and the traveling waves even if the limits v u of the initial data at x do not satisfy the equilibrium equation i e u f v when the limits of the initial data at infinity satisfy equilibrium states liu 9 studied the stability of rarefaction waves and traveling waves for the general 2 2 hyperbolic conservation laws with relaxation

computation and analysis of image motion a synopsis of current problems and methods the goal of this paper is to offer a structured synopsis of the problems in image motion computation and analysis and of the methods proposed exposing the underlying models and supporting assumptions a sufficient number of pointers to the literature will be given concentrating mostly on recent contributions emphasis will be on the detection measurement and segmentation of image motion tracking and deformable motion issues will be also addressed finally a number of related questions which could require more investigations will be presented

primal dual affine scaling interior point methods for linear complementarity problems a first order affine scaling method and two m th order affine scaling methods for solving monotone linear complementarity problems lcps are presented all three methods produce iterates in a wide neighborhood of the central path the first order method has o nl 2 lognl 2 log lognl 2 iteration complexity if the lcp admits a strict complementary solution then both the duality gap and the iteration sequence converge superlinearly with q order two if m omega log sqrt n l then both higher order methods have o sqrt n l iteration complexity the q order of convergence of one of the methods is m 1 for problems that admit a strict complementarity solution while the q order of convergence of the other method is m 1 2 for general monotone lcps

superlinearly convergent o sqrt n l iteration interior point algorithms for linear programming and the monotone linear complementarity problem zhang and tapia have recently developed an interior point algorithm for the linear programming problem that has the property that the duality gap converges to zero q quadratically in the nondegenerate case and if the iterates converge q superlinearly in the degenerate case their algorithm also solves the integer model for lp in o nl iterations where l is the input length of the lp this paper presents an algorithm which maintains the quadratic superlinear convergence property but solves lp in o sqrt n l iterations in the integer model also the algorithm is generalized into an algorithm which solves lcp the lcp algorithm which is similar to an adaptive path following algorithm proposed by mizuno yoshise and kikuchi has similar local convergence properties if we assume that the iterates converge to a strictly complementary solution

a numerical method for systems of conservation laws of mixed type admitting hyperbolic flux splitting the present treatment of elliptic regions via hyperbolic flux splitting and high order methods proposes a flux splitting in which the corresponding jacobians have real and positive negative eigenvalues while resembling the flux splitting used in hyperbolic systems the present generalization of such splitting to elliptic regions allows the handling of mixed type systems in a unified and heuristically stable fashion the van der waals fluid dynamics equation is used convergence with good resolution to weak solutions for various riemann problems are observed

convergence rates of vanishing diffusion limit on nonlinear hyperbolic system with damping and diffusion the aim of this paper is to study the global unique solvability on c solution to the cauchy problem of a special 2 2 nonlinear hyperbolic system with damping and diffusion furthermore we also investigate the convergence rates as the diffusion parameter goes to zero it is shown that the convergence rates in c norm is of the order o the novelty of analysis taken in the present article is to obtain the uniform in a priori estimates on hs norm for any s n which are implemented by introducing induction we focus on two cases in this paper both and since the uniform in a priori estimates obtained are distinctly different for different cases

on partial differential equations related to lorenz system partial differential equations are constructed such that a truncation scheme as adopted by lorenz j atmos sci 20 130 1963 will lead to lorenz equations the partial differential equations are much simpler than those of the rayleigh bernard problem and are essentially of the mixed type various aspects of the partial differential equations are explored it is suggested that the switching back and forth between the elliptic and hyperbolic regimes represents the chaotic behavior of the system in the context of partial differential equations

a self adjusting interior point algorithm for linear complementarity problems abstract based on the min max principle the standard centering equation in the interior point method is replaced by the optimality condition of a new proximity measure function thus a self adjusting mechanism is constructed in the new perturbed system the newton direction can be adjusted self adaptively according to the information of last iterates a self adjusting interior point method is given based on the new perturbed system numerical comparison is made between this algorithm and a primal dual interior point algorithm using standard perturbed system results demonstrate the efficiency and some advantages of the proposed algorithm