high order difference schemes for unsteady one dimensional diffusion convection problems abstract for unsteady 1d diffusion convection problems this paper develops an extensive analysis of two level three point finite difference schemes of order 2 in time and 4 in space this general class of fds includes several schemes independently proposed by different authors one main objective is the identification of difference schemes yielding satisfactory numerical results for strongly convective problems i e when the cell reynolds number b h 2 is greater than unity the stability and the oscillatory behaviour of the schemes are carefully studied and the analyses are completed by some numerical experiments we outline some key points i the great difficulty to obtain accurate numerical results for large values of ii the possibility of virtually optimum schemes is essentially theoretical and requires in practice careful experiments iii for strongly convective problems some second order explicit schemes are almost as efficient and less costly than implicit fourth order schemes

one and a half quantum de finetti theorems when n k systems of an n partite permutation invariant state are traced out the resulting state can be approximated by a convex combination of tensor product states this is the quantum de finetti theorem in this paper we show that an upper bound on the trace distance of this approximation is given by 2 frac kd 2 n where d is the dimension of the individual system thereby improving previously known bounds our result follows from a more general approximation theorem for representations of the unitary group consider a pure state that lies in the irreducible representation u mu nu subset u mu otimes u nu of the unitary group u d for highest weights and let be the state obtained by tracing out u then is close to a convex combination of the coherent states u mu g v mu rangle where g in u d and v mu rangle is the highest weight vector in u

squashed entanglement for multipartite states and entanglement measures based on the mixed convex roof new measures of multipartite entanglement are constructed based on two definitions of multipartite information and different methods of optimizing over extensions of the states one is a generalization of the squashed entanglement where one takes the mutual information of parties conditioned on the state s extension and takes the infimum over such extensions additivity of the multipartite squashed entanglement is proved for both versions of the multipartite information which turn out to be related the second one is based on taking classical extensions this scheme is generalized which enables to construct measures of entanglement based on the mixed convex roof of a quantity which in contrast to the standard convex roof method involves optimization over all decompositions of a density matrix rather than just the decompositions into pure states as one of the possible applications of these results we prove that any multipartite monotone is an upper bound on the amount of multipartite distillable key the findings are finally related to analogous results in classical key agreement

two level compact implicit schemes for three dimensional parabolic problems we derive a class of two level high order implicit finite difference schemes for solving three dimensional parabolic problems with mixed derivatives the schemes are fourth order accurate in space and second or lower order accurate in time depending on the choice of a weighted average parameter m numerical results with m 0 5 are presented to confirm the high accuracy of the derived scheme and to compare it with the standard second order central difference scheme it is shown that the improvement in accuracy does not come at a higher cost of computation and storage since it is possible to choose the grid parameters so that the present scheme requires less work and memory and gives more accuracy than the standard central difference scheme

stability and error estimates of a new high order compact adi method for the unsteady 3d convection diffusion equation abstract in this paper a new high order compact adi method for the unsteady convection diffusion equation in three dimension 3d is considered collecting the truncation error of the finite difference operator by the recursion method we derive a new high order compact finite difference scheme for the unsteady 1d convection diffusion equation firstly then based on the adi method applying a correction technique to reduce the error of the splitting term a high order compact adi method for the unsteady 3d convection diffusion equation is proposed in which we solve a series of 1d problems with strictly diagonal dominant tri diagonal structures instead of the high dimensional ones the scheme is proved to be unconditional stable moreover the regularity and error estimate of the numerical solution are derived finally some numerical examples are performed which confirm the theoretical prediction and show that much better computational accuracy results can be got by applying the new scheme

higher order adi method with completed richardson extrapolation for solving unsteady convection diffusion equations a higher order alternating direction implicit adi method with completed richardson extrapolation is proposed for solving unsteady two dimensional convection diffusion equations the method has sixth order accuracy in space and fourth order accuracy in time completed richardson extrapolation is used to improve the accuracy of the solution obtained from the high order adi method in spatial and temporal domains simultaneously stability analysis is given to show the influence of richardson extrapolation on the unconditionally stable adi method numerical experiments are conducted to test its high accuracy and to compare it with karaa zhang s high order adi method

strong convergence for split step methods in stochastic jump kinetics mesoscopic models in the reaction diffusion framework have gained recognition as a viable approach to describing chemical processes in cell biology the resulting computational problem is a continuous time markov chain on a discrete and typically very large state space due to the many temporal and spatial scales involved many different types of computationally more effective multiscale models have been proposed typically coupling different types of descriptions within the markov chain framework in this work we look at the strong convergence properties of the basic first order strang or lie trotter split step method which is formed by decoupling the dynamics in finite time steps thanks to its simplicity and flexibility this approach has been tried in many different combinations we develop explicit sufficient conditions for pathwise well posedness and convergence of the method including error estimates and we illustrate our findings with numerical examples in doing so we also suggest a certain

slow scale split step tau leap method for stiff stochastic chemical systems abstract tau leaping is a family of algorithms for the approximate simulation of the discrete state continuous time markov chains a motivation for the development of such methods can be found for instance in the fields of chemical kinetics and systems biology it is known that the dynamical behavior of biochemical systems is often intrinsically stiff representing a serious challenge for their numerical approximation the naive extension of stiff deterministic solvers to stochastic integration often yields numerical solutions with either impractically large relaxation times or incorrectly resolved covariance in this paper we propose a splitting heuristic which helps to resolve some of these issues the proposed integrator contains a number of unknown parameters which are estimated for each particular problem from the moment equations of the corresponding linearized system we show that this method is able to reproduce the exact mean and variance of the linear scalar test equation and demonstrates a good accuracy for the arbitrarily stiff systems at least in the linear case the numerical examples for both linear and nonlinear systems are also provided and the obtained results confirm the efficiency of the considered splitting approach

quantum rate distortion coding with auxiliary resources we extend quantum rate distortion theory by considering auxiliary resources that might be available to a sender and receiver performing lossy quantum data compression the first setting we consider is that of quantum rate distortion coding with the help of a classical side channel our result here is that the regularized entanglement of formation characterizes the quantum rate distortion function extending earlier work of devetak and berger we also combine this bound with the entanglement assisted bound from our prior work to obtain the best known bounds on the quantum rate distortion function for an isotropic qubit source the second setting we consider is that of quantum rate distortion coding with quantum side information qsi available to the receiver in order to prove results in this setting we first state and prove a quantum reverse shannon theorem with qsi for tensor power states which extends the known tensor power quantum reverse shannon theorem the achievability part of this theorem relies on the quantum state redistribution protocol while the converse relies on the fact that the protocol can cause only a negligible disturbance to the joint state of the reference and the receiver s qsi this quantum reverse shannon theorem with qsi naturally leads to quantum rate distortion theorems with qsi with or without entanglement assistance

faithful squashed entanglement squashed entanglement is a measure for the entanglement of bipartite quantum states in this paper we present a lower bound for squashed entanglement in terms of a distance to the set of separable states this implies that squashed entanglement is faithful that is it is strictly positive if and only if the state is entangled we derive the lower bound on squashed entanglement from a lower bound on the quantum conditional mutual information which is used to define squashed entanglement the quantum conditional mutual information corresponds to the amount by which strong subadditivity of von neumann entropy fails to be saturated our result therefore sheds light on the structure of states that almost satisfy strong subadditivity with equality the proof is based on two recent results from quantum information theory the operational interpretation of the quantum mutual information as the optimal rate for state redistribution and the interpretation of the regularised relative entropy of entanglement as an error exponent in hypothesis testing the distance to the set of separable states is measured in terms of the locc norm an operationally motivated norm giving the optimal probability of distinguishing two bipartite quantum states each shared by two parties using any protocol formed by local quantum operations and classical communication locc between the parties a similar result for the frobenius or euclidean norm follows as an immediate consequence the result has two applications in complexity theory the first application is a quasipolynomial time algorithm solving the weak membership problem for the set of separable states in locc or euclidean norm the second application concerns quantum merlin arthur games here we show that multiple provers are not more powerful than a single prover when the verifier is restricted to locc operations thereby providing a new characterisation of the complexity class qma

a high order exponential scheme for solving 1d unsteady convection diffusion equations in this paper a high order exponential hoe scheme is developed for the solution of the unsteady one dimensional convection diffusion equation the present scheme uses the fourth order compact exponential difference formula for the spatial discretization and the 2 2 pade approximation for the temporal discretization the proposed scheme achieves fourth order accuracy in temporal and spatial variables and is unconditionally stable numerical experiments are carried out to demonstrate its accuracy and to compare it with analytic solutions and numerical results established by other methods in the literature the results show that the present scheme gives highly accurate solutions for all test examples and can get excellent solutions for convection dominated problems

an implicit high accuracy variable mesh scheme for 1 d non linear singular parabolic partial differential equations abstract in this paper we propose a new two level implicit difference scheme of o k 2 h l 1 kh l h l 3 for the solution of non linear parabolic equation eu xx x t u u x u t 0 x t 0 subject to appropriate initial and dirichlet boundary conditions prescribed where k 0 h l 0 are mesh sizes in t and x coordinates respectively and e 0 is a small parameter in addition we also discuss a new explicit variable mesh difference scheme of o kh l h l 3 for the estimates of u x in all cases we require only three spatial variable grid points the proposed schemes require less algebra and three evaluations of function a special technique is required to solve singular parabolic equations the proposed variable mesh scheme when applied to a linear diffusion equation is shown to be stable for all h l 0 and k 0 computational results are provided to support our derived schemes and analysis

on the numerical solution of nonlinear black scholes equations nonlinear black scholes equations have been increasingly attracting interest over the last two decades since they provide more accurate values by taking into account more realistic assumptions such as transaction costs risks from an unprotected portfolio large investor s preferences or illiquid markets which may have an impact on the stock price the volatility the drift and the option price itself in this paper we will focus on several models from the most relevant class of nonlinear black scholes equations for european and american options with a volatility depending on different factors such as the stock price the time the option price and its derivatives due to transaction costs we will analytically approach the option price by transforming the problem for a european call option into a convection diffusion equation with a nonlinear term and the free boundary problem for an american call option into a fully nonlinear nonlocal parabolic equation defined on a fixed domain following sevcovic s idea finally we will present the results of different numerical discretization schemes for european options for various volatility models including the leland model the barles and soner model and the risk adjusted pricing methodology model

continuity of quantum conditional information we prove continuity of quantum conditional information s 12 2 with respect to the uniform convergence of states and obtain a bound which is independent of the dimension of the second party this can e g be used to prove the continuity of squashed entanglement

quantum discord a measure of the quantumness of correlations two classically identical expressions for the mutual information generally differ when the systems involved are quantum this difference defines the quantum discord it can be used as a measure of the quantumness of correlations separability of the density matrix describing a pair of systems does not guarantee vanishing of the discord thus showing that absence of entanglement does not imply classicality we relate this to the quantum superposition principle and consider the vanishing of discord as a criterion for the preferred effectively classical states of a system i e the pointer states

fast computational approach to the delta greek of non linear black scholes equations abstract in this paper we consider a class of non linear option pricing models the focus is on the numerical investigation of delta equation where the unknown solution is the first spatial derivative of the option value we construct and analyze monotone and sign preserving finite difference schemes for the problems newton s and picard s iterative procedures for solving the non linear systems of algebraic equations are proposed on this base in order to improve the computational efficiency we develop fast two grid algorithms numerical experiments using also richardson extrapolation in time are discussed in terms of accuracy convergence and efficiency

parallelization processor communication and error analysis in lattice kinetic monte carlo in this paper we study from a numerical analysis perspective the fractional step kinetic monte carlo fs kmc algorithms proposed in 1 for the parallel simulation of spatially distributed particle systems on a lattice fs kmc are fractional step algorithms with a time stepping windowt and as such they are inherently partially asynchronous since there is no processor com munication during the periodt in this contribution we primarily focus on the error analysis of fs kmc algorithms as approximations of conventional serial kinetic monte carlo kmc a key as pect of our analysis relies on emphasizing a goal oriented approach for suitably defined macroscopic observables e g density energy correlations surface roughness rather than focusing on strong topology estimates for individual trajectories one of the key implications of our error analysis is that it allows us to address systematically the processor communication of different parallelization strategies for kmc by comparing their partial asynchrony which in turn is measured by their respective fractional time stept for a prescribed error tolerance

a sharp continuity estimate for the von neumann entropy we derive an inequality relating the entropy difference between two quantum states to their trace norm distance sharpening a well known inequality due to fannes in our inequality equality can be attained for every prescribed value of the trace norm distance

high order finite volume schemes for the advection diffusion equation a high order finite volume method based on piecewise interpolant polynomials is proposed to discretize spatially the one dimensional and two dimensional advection diffusion equation evolution equations for the mean values of each control volume are integrated in time by a classical fourth order runge kutta since our work focuses on the behaviour of the spatial discretization the time step is chosen small enough to neglect the time integration error two dimensional interpolants are built by means of one dimensional interpolants it is shown that when the degree of the one dimensional interpolant q is odd the proper selection of a fixed stencil gives rise to centred schemes of order q 1 in order not to lose precision due to the change of stencil near boundaries the degree of the interpolants close to boundaries is raised to q 1 four test cases with small values of diffusion are integrated with high order methods it is shown that the spatial discretization of the advection diffusion equation with periodic boundary conditions leads to normal discretization matrices and asymptotic stability must be assured to bound the spatial discretization error once the asymptotic stability is assured by means of the spectra of the discretization matrix the spatial error is of the order of the truncation error however it is shown that the discretization of the advection diffusion equation with arbitrary boundary conditions gives rise to non normal matrices if asymptotic stability is assured the spatial order of steady solutions is of the order of the truncation error but for transient processes the order of the spatial error is determined by both the truncation error and the norm of the exponential matrix of the spatial discretization the use of the pseudospectra of the discretization matrix is proposed as a valuable tool to analyse the transient error of the numerical solution copyright 2001 john wiley sons ltd

a fourth order compact adi method for solving two dimensional unsteady convection diffusion problems n this article an exponential high order compact ehoc alternating direction implicit adi method in which the crank nicolson scheme is used for the time discretization and an exponential fourth order compact difference formula for the steady state 1d convection diffusion problem is used for the spatial discretization is presented for the solution of the unsteady 2d convection diffusion problems the method is temporally second order accurate and spatially fourth order accurate which requires only a regular five point 2d stencil similar to that in the standard second order methods the resulting ehoc adi scheme in each adi solution step corresponds to a strictly diagonally dominant tridiagonal matrix equation which can be inverted by simple tridiagonal gaussian decomposition and may also be solved by application of the one dimensional tridiagonal thomas algorithm with a considerable saving in computing time the unconditionally stable character of the method was verified by means of the discrete fourier or von neumann analysis numerical examples are given to demonstrate the performance of the method proposed and to compare mostly it with the high order adi method of karaa and zhang and the spatial third order compact scheme of note and tan

multipartite quantum correlations and local recoverability characterizing genuine multipartite quantum correlations in quantum physical systems has historically been a challenging problem in quantum information theory more recently however the total correlation or multipartite information measure has been helpful in accomplishing this goal especially with the multipartite symmetric quantum msq discord piani et al 2008 phys rev lett 100 090502 doi 10 1103 physrevlett 100 090502 1 and the conditional entanglement of multipartite information cemi yang et al 2008 phys rev lett 101 140501 doi 10 1103 physrevlett 101 140501 2 here we apply a recent and significant improvement of strong subadditivity of quantum entropy fawzi renner 2014 in order to develop these quantities further in particular we prove that the msq discord is nearly equal to zero if and only if the multipartite state for which it is evaluated is approximately locally recoverable after performing measurements on each of its systems furthermore we prove that the cemi is a faithful entanglement measure i e it vanishes if and only if the multipartite state for which it is evaluated is a fully separable state along the way we provide an operational interpretation of the msq discord in terms of the partial state distribution protocol which in turn as a special case gives an interpretation for the original discord quantity finally we prove an inequality that could potentially improve upon the fawzi renner inequality in the multipartite context but it remains an open question to determine whether this is so 1 http dx doi org 10 1103 physrevlett 100 090502 2 http dx doi org 10 1103 physrevlett 101 140501

a full analysis of a new second order finite volume approximation based on a low order scheme using general admissible spatial meshes for the unsteady one dimensional heat equation abstract the present work is an extension of our previous works 3 4 5 which dealt with first order both in time and space and second order time accurate second order in time and first order in space implicit finite volume schemes for parabolic equations we aim in this work and some forthcoming studies at getting higher order both in time and space finite volume approximations for the exact solution of parabolic equations using the class of spatial generic meshes introduced recently in 13 we focus in the present contribution on the one dimensional heat equation and its implicit finite volume scheme described in 3 the implicit finite volume scheme approximating the one dimensional heat equation we consider hereafter referred to as the basic finite volume scheme yields linear systems to be solved successively the matrices involved in these linear systems are tridiagonal the finite volume approximate solution is of order h k where h resp k is the mesh size of the spatial resp time discretization we construct a new finite volume approximation of order h k 2 in several discrete norms which allows us to get approximations of order two for the exact solution and its first derivatives this new high order approximation can be computed using the same linear systems involved in the basic finite volume scheme while the right hand sides are corrected the construction of these right hand sides includes the approximations of the second third and fourth spatial derivatives of the exact solution the computation of the approximation of these high order derivatives can be performed using the same matrices stated above with another two tridiagonal matrices the manner by which this new high order approximation is constructed can be repeated to compute successively finite volume approximations of arbitrary order using the same matrices stated above these high order approximations can be obtained on any one dimensional admissible finite volume mesh in the sense of 12 without any restrictive condition on the spatial mesh a full analysis for the stated theoretical results as well as some numerical examples supporting the theory is presented the results obtained in the present study are based essentially on two facts the first fact is the use of the results provided in 3 which state the convergence order of the finite volume approximate solution in several norms the second fact is the comparison between the stated new higher order approximations and suitable auxiliary finite volume approximations

a rigorous derivation of the chemical master equation it is widely believed that the chemical master equation has no rigorous microphysical basis and hence no a priori claim to validity this view is challenged here through arguments purporting to show that the chemical master equation is exact for any gas phase chemical system that is kept well stirred and thermally equilibrated

an upwind finite difference method for a nonlinear black scholes equation governing european option valuation under transaction costs in this paper we develop a numerical method for a nonlinear parabolic partial differential equation arising from pricing european options under transaction costs the method is based on an upwind finite difference scheme for the spatial discretization and a fully implicit time stepping scheme we prove that the system matrix from this scheme is an m matrix and that the approximate solution converges unconditionally to the viscosity solution to the equation by showing that the scheme is consistent monotone and unconditionally stable a newton iterative algorithm is proposed for solving the discretized nonlinear system of which the jacobian matrix is shown to be also an m matrix numerical experiments are performed to demonstrate the accuracy and robustness of the method

optimal quantum source coding with quantum side information at the encoder and decoder consider many instances of an arbitrary quadripartite pure state of four quantum systems abcd alice holds the ac part of each state bob holds b while r represents all other parties correlated with abc alice is required to redistribute the c systems to bob while asymptotically preserving the overall purity we prove that this is possible using q qubits of communication and e ebits of shared entanglement between alice and bob provided that q ges 1 2i c d b and q e ges h c b proving the optimality of the luo devetak outer bound the optimal qubit rate provides the first known operational interpretation of quantum conditional mutual information we also show how our protocol leads to a fully operational proof of strong subaddivity and uncover a general organizing principle in analogy to thermodynamics that underlies the optimal rates

fourth order compact scheme for space fractional advection diffusion reaction equations with variable coefficients abstract in this work a new fourth order compact approximation is derived for riemann liouville space fractional derivatives modified wave numbers are obtained for various approximations of fractional derivatives considering these modified wave numbers fourier analysis of differencing errors is presented to quantify the resolution characteristics of various approximations a novel fourth order compact scheme is developed for space fractional advection diffusion reaction equations with variable coefficients in one and two dimensions in the proposed compact scheme the second derivative approximation of unknowns is approximated using the value of these unknowns and their first derivative approximations this splitting of second derivative approximations allows us to obtain a similar system of linear equations as in zhao and tian 2017 which presents only second order accurate finite difference method for advection diffusion reaction equations in one dimension the stability of the proposed compact scheme is demonstrated numerically furthermore the proposed compact scheme is employed to space fractional black scholes equation for pricing european options as an application of fractional derivatives in mathematical finance numerical illustrations are presented to validate the theoretical claims

short note a high order pade adi method for unsteady convection diffusion equations a high order alternating direction implicit adi method for computations of unsteady convection diffusion equations is proposed by using fourth order pade schemes for spatial derivatives the present scheme is fourth order accurate in space and second order accurate in time the solution procedure consists of a number of tridiagonal matrix operations which make the computation cost effective the method is unconditionally stable and shows higher accuracy and better phase and amplitude error characteristics than the standard second order adi method d w peaceman h h rachford jr the numerical solution of parabolic and elliptic differential equations journal of the society of industrial and applied mathematics 3 1959 28 41 and the fourth order adi scheme of karaa and zhang high order adi method for solving unsteady convection diffusion problem journal of computational physics 198 2004 1 9

squashed entanglement an additive entanglement measure in this paper we present a new entanglement monotone for bipartite quantum states its definition is inspired by the so called intrinsic information of classical cryptography and is given by the halved minimum quantum conditional mutual information over all tripartite state extensions we derive certain properties of the new measure which we call squashed entanglement it is a lower bound on entanglement of formation and an upper bound on distillable entanglement furthermore it is convex additive on tensor products and superadditive in general continuity in the state is the only property of our entanglement measure which we cannot provide a proof for we present some evidence however that our quantity has this property the strongest indication being a conjectured fannes type inequality for the conditional von neumann entropy this inequality is proved in the classical case

an optimal time stepping algorithm for unsteady advection diffusion problems we consider the numerical solution of a linear time dependent advection diffusion problem by an implicit two weight three point finite difference scheme we extend the scheme proposed by chadha and madden 2011 to incorporate an optimal time step selection algorithm for the method the optimal values of the weights involved in the scheme have been obtained by using the notion of an equivalent differential equation warming and hyett 1974 and by enforcing certain constraints due to von neumann stability and a discrete maximum principle we obtain sharp bounds for optimal time stepping in terms of mesh width and problem data furthermore we formulate the time step selection as a root finding problem and use this to show the optimal time step exists and is unique the resulting method based on optimal values of weights and optimal time stepping is of fifth order in space and third order in time the results of extensive numerical experiments are presented for several classic test problems for various values of the advection and diffusion parameters these show that our proposed method is superior quantitatively and qualitatively to other schemes that can be represented within the framework of two weighted schemes we also present comparisons with other high order methods and an application of the method to a problem with non smooth data

numerical solution of unsteady advection dispersion equation arising in contaminant transport through porous media using neural networks a soft computing approach based on artificial neural network ann and optimization is presented for the numerical solution of the unsteady one dimensional advection dispersion equation ade arising in contaminant transport through porous media a length factor ann method based on automatic satisfaction of arbitrary boundary conditions bcs was chosen for the numerical solution of ade the strength of ann is exploited to construct a trial approximate solution tas for ade in a way that it satisfies the initial or bcs exactly an unsupervised error is constructed in approximating the solution of ade which is minimized by training ann using gradient descent algorithm gda two challenging test problems of ade are considered in this paper in which the first problem has steep boundary layers near x 1 and many numerical methods create non physical oscillation near steep boundaries also for the second problem many numerical schemes suffer from computational noise and instability issues the proposed method is advantageous as it does not require temporal discretization for the solution of the ades as well as it does not suffer from numerical instability the reliability and effectiveness of the presented algorithm is investigated by sufficient large number of independent runs and comparison of results with other existing numerical methods the results show that the present method removes the difficulties arising in the solution of the ades and provides solution with good accuracy

no local broadcasting theorem for multipartite quantum correlations we prove that the correlations present in a multipartite quantum state have an operational quantum character even if the state is unentangled as long as it does not simply encode a multipartite classical probability distribution said quantumness is revealed by the new task of local broadcasting i e of locally sharing preestablished correlations which is feasible if and only if correlations are stricly classical our operational approach leads to natural definitions of measures for quantumness of correlations it also reproduces the standard no broadcasting theorem as a special case

numerical modelling of linear and nonlinear diffusion equations by compact finite difference method in this work accurate solutions to linear and nonlinear diffusion equations were introduced a combination of a sixth order compact finite difference scheme in space and a low storage third order total variation diminishing runge kutta scheme in time have been used for treatment of these equations the computed results with the use of this technique have been compared with the exact solution to show the accuracy of it here the approximate solution to the diffusion equations has been obtained easily and elegantly with neither transforming nor linearizing the equation the present method is seen to be a very good alternative method to some existing techniques for realistic problems

pathwise error bounds in multiscale variable splitting methods for spatial stochastic kinetics stochastic computational models in the form of pure jump processes occur frequently in the description of chemical reactive processes of ion channel dynamics and of the spread of infections in po

point processes and queues martingale dynamics i martingales 1 histories and stopping times 2 martingales 3 predictability 4 square integrable martingales references solutions to exercises chapter i ii point processes queues and intensities 1 counting processes and queues 2 watanabe s characterization 3 stochastic intensity general case 4 predictable intensities 5 representation of queues 6 random changes of time 7 cryptographic point processes references solutions to exercises chapter ii iii integral representation of point process martingales 1 the structure of internal histories 2 regenerative form of the intensity 3 the representation theorem 4 hilbert space theory of poissonian martingales 5 useful extensions references solutions to exercises chapter iii iv filtering 1 the theory of innovations 2 state estimates for queues and markov chains 3 continuous states and nontrivial prehistory references solutions to exercises chapter iv v flows in markovian networks of queues 1 single station the historical results and the filtering method 2 jackson s networks 3 burke s output theorem for networks 4 cascades and loops in jackson s networks 5 independence and poissonian flows in markov chains references solutions to exercises chapter v vi likelihood ratios 1 radon nikodym derivatives and tests of hypotheses 2 changes of intensities a la girsanov 3 filtering by the method of the probability of reference 4 applications 5 the capacity of a point process channel 6 detection formula187 references189 solutions to exercises chapter vi vii optimal control 1 modeling intensity controls 2 dynamic programming for intensity controls complete observation case 3 input regulation a case study in impulsive control 4 attraction controls 5 existence via likelihood ratio references solutions to exercises chapter vii viii marked point processes 1 counting measure and intensity kernels 2 martingale representation and filtering 3 radon nikodym derivatives 4 towards a general theory of intensity references solutions to exercises chapter viii a1 background in probability and stochastic processes 1 introduction 2 monotone class theorem 3 random variables 4 expectations 5 conditioning and independence 6 convergence 7 stochastic processes 8 markov processes references a2 stopping times and point process histories 1 stopping times 2 changes of time and meyer dellacherie s integration formula 3 point process histories references a3 wiener driven dynamical systems 1 ito s stochastic integral 2 square integrable brownian martingales 3 girsanov s theorem references a4 stieltjes lebesgue calculus 1 the stieltjes lebesgue integral 2 the product and exponential formulas references general bibliography

a sixth order compact finite difference scheme to the numerical solutions of burgers equation abstract a numerical solution of the one dimensional burgers equation is obtained using a sixth order compact finite difference method to achieve this a tridiagonal sixth order compact finite difference scheme in space and a low storage third order total variation diminishing runge kutta scheme in time have been combined the scheme is implemented to solve two test problems with known exact solutions comparisons of the computed results with exact solutions showed that the method is capable of achieving high accuracy and efficiency with minimal computational effort the present results are also seen to be more accurate than some available results given in the literature