adaptive mesh and algorithm refinement using direct simulation monte carlo adaptive mesh and algorithm refinement amar embeds a particle method within a continuum method at the finest level of an adaptive mesh refinement amr hierarchy the coupling between the particle region and the overlaying continuum grid is algorithmically equivalent to that between the fine and coarse levels of amr direct simulation monte carlo dsmc is used as the particle algorithm embedded within a godunov type compressible navier stokes solver several examples are presented and compared with purely continuum calculations

a solution algorithm for the fluid dynamic equations based on a stochastic model for molecular motion in this paper a stochastic model is presented to simulate the flow of gases which are not in thermodynamic equilibrium like in rarefied or micro situations for the interaction of a particle with others statistical moments of the local ensemble have to be evaluated but unlike in molecular dynamics simulations or dsmc no collisions between computational particles are considered in addition a novel integration technique allows for time steps independent of the stochastic time scale the stochastic model represents a fokker planck equation in the kinetic description which can be viewed as an approximation to the boltzmann equation this allows for a rigorous investigation of the relation between the new model and classical fluid and kinetic equations the fluid dynamic equations of navier stokes and fourier are fully recovered for small relaxation times while for larger values the new model extents into the kinetic regime numerical studies demonstrate that the stochastic model is consistent with navier stokes in that limit but also that the results become significantly different if the conditions for equilibrium are invalid the application to the knudsen paradox demonstrates the correctness and relevance of this development and comparisons with existing kinetic equations and standard solution algorithms reveal its advantages moreover results of a test case with geometrically complex boundaries are presented

on dsmc calculations of rarefied gas flows with small number of particles in cells the direct simulation monte carlo dsmc analysis of two and three dimensional rarefied gas flows requires computational resources of very large proportions one of the major causes for this is that along with the multidimensional computational mesh the standard dsmc approach also requires a large number of particles in each cell of the mesh in order to obtain sufficiently accurate results this paper presents the development and validation of a modified simulation procedure which allows more accurate calculations with a smaller mean number of particles langle n rangle sim1 in the grid cells in the new algorithm the standard dsmc collision scheme is replaced by a two step collision procedure based on bernoulli trials scheme or its simplified version proposed by the author which is applied twice to the cells or subcells of a dual grid within a time step the modified algorithm uses a symmetric strang splitting scheme that improves the accuracy of the splitting scheme to o tau 2 with respect to the time step tau making the modified dsmc method an effective numerical tool for both steady and unsteady gas flow calculations on fine multidimensional grids the latter is particularly important for simulation of vortical and unstable rarefied gas flows the modified simulation scheme might also be useful for dsmc calculations within the subcell areas of a multilevel computational grid

an efficient particle fokker planck algorithm for rarefied gas flows this paper is devoted to the algorithmic improvement and careful analysis of the fokker planck kinetic model derived by jenny et al 1 and gorji et al 2 the motivation behind the fokker planck based particle methods is to gain efficiency in low knudsen rarefied gas flow simulations where conventional direct simulation monte carlo dsmc becomes expensive this can be achieved due to the fact that the resulting model equations are continuous stochastic differential equations in velocity space accordingly the computational particles evolve along independent stochastic paths and thus no collision needs to be calculated therefore the computational cost of the solution algorithm becomes independent of the knudsen number in the present study different computational improvements were persuaded in order to augment the method including an accurate time integration scheme local time stepping and noise reduction for assessment of the performance gas flow around a cylinder and lid driven cavity flow were studied convergence rates accuracy and computational costs were compared with respect to dsmc for a range of knudsen numbers from hydrodynamic regime up to above one in all the considered cases the model together with the proposed scheme give rise to very efficient yet accurate solution algorithms

real groups and sylow 2 subgroups abstract if g is a finite real group and p syl 2 g then p p is elementary abelian this confirms a conjecture of roderick gow in fact we prove a much stronger result that implies gow s conjecture

the mckay conjecture and galois automorphisms the main problem of representation theory of finite groups is to find proofs of several conjectures stating that certain global invariants of a finite group g can be computed locally the simplest of these conjectures is the mckay conjecture which asserts that the number of irreducible complex characters of g of degree not divisible by p is the same if computed in a p sylow normalizer of g in this paper we propose a much stronger version of this conjecture which deals with galois automorphisms in fact the same idea can be applied to the celebrated alperin and dade conjectures

hybrid continuum molecular simulations of transient gas flows with rarefaction an adaptively coupled continuum molecular approach for compressible viscous flows in transient calculations is presented the continuum domain is described by the unsteady compressible navier stokes equations and the molecular domain is solved by direct simulation monte carlo a strategy is described to extend the overlapped schwarz method with dirichlet dirichlet boundary condition coupling procedure to transient simulations the method has been successfully validated against full direct simulation monte carlo results for transient simulations of a one dimensional shock tube and a two dimensional pressure driven slit flow a sensitivity analysis showed that the used overlapped schwarz coupling method with dirichlet dirichlet boundary conditions is only weakly sensitive to various parameters e g the exact position of the continuum molecular interface the size of the overlap region and the scatter in the molecular solution which is a clear advantage over the more commonly used flux based coupling technique another advantage of the chosen coupling method is that the continuum and molecular time steps can be decoupled and that in general a coupling time step can be used that is much larger than the molecular time step the study performed also highlighted a limit of the method it was found to be necessary to keep the courant number based on the coupling time step the cell size in the continuum region and the molecular most probable velocity below 1 to avoid instabilities

a reduction theorem for the mckay conjecture the mckay conjecture asserts that for every finite group g and every prime p the number of irreducible characters of g having p degree is equal to the number of such characters of the normalizer of a sylow p subgroup of g although this has been confirmed for large numbers of groups including for example all solvable groups and all symmetric groups no general proof has yet been found in this paper we reduce the mckay conjecture to a question about simple groups we give a list of conditions that we hope all simple groups will satisfy and we show that the mckay conjecture will hold for a finite group g if every simple group involved in g satisfies these conditions also we establish that our conditions are satisfied for the simple groups psl2 q for all prime powers q 4 and for the suzuki groups sz q and ree groups r q where q 2 e or q 3 e respectively and e 1 is odd since our conditions are also satisfied by the sporadic simple group j 1 it follows that the mckay conjecture holds for all primes p for every finite group having an abelian sylow 2 subgroup

shake and bake algorithms for generating uniform points on the boundary of bounded polyhedra we present a class of shake and bake algorithms for generating asymptotically uniform points on the boundary of full dimensional bounded polyhedra we also report results of simulations for some elementary test problems

an asymptotic preserving automatic domain decomposition method for the vlasov poisson bgk system with applications to plasmas in this work we present an efficient strategy to deal with plasma physics simulations in which localized departures from thermodynamical equilibrium are present the method is based on the introduction of intermediate regions which allows smooth transitions between kinetic and fluid zones in this paper we extend domain decomposition techniques obtained through dynamic coupling and buffer zones to the study of plasmas and moreover we combine them with asymptotic preserving and asymptotically accurate strategies for the time integration we use a hybrid scheme in which both kinetic and fluid descriptions are considered and coupled together while the kinetic model is solved by asymptotic preserving and accurate methods in order to guarantee high efficiency and accuracy in all regimes the numerical scheme is validated and its performances are analyzed by numerical simulations

the mckay conjecture is true for the sporadic simple groups abstract the mckay conjecture states that the number of irreducible complex characters of a group g that have degree prime to p is equal to the same number for the sylow p normalizer in g we verify this conjecture for the 26 sporadic simple groups

development and verification of a coupled dsmc ns scheme using unstructured mesh an efficient and accurate parallel coupled dsmc ns method using three dimensional unstructured grid topology is proposed and verified for the simulation of high speed gas flows involving continuum and rarefied regimes a domain overlapping strategy taking advantage of unstructured data format with dirichlet dirichlet type boundary conditions based on two breakdown parameters is used iteratively to determine the choice of solvers in the spatial domain the selected breakdown parameters for this study include 1 a local maximum knudsen number defined as the ratio of the local mean free path and local characteristic length based on property gradient and 2 a thermal non equilibrium indicator defined as the ratio of the difference between translational and rotational temperatures to the translational temperature a supersonic flow m 4 over a quasi 2 d 25 wedge is employed as the first step in verifying the present coupled method the results of simulation using the coupled method are in excellent agreement with those of the pure dsmc method which is taken as the benchmark solution effects of the size of overlapping regions and the choice of breakdown parameters on the convergence history are discussed results show that the proposed iteratively coupled method predicts the results more accurately as compared to the one shot coupled method which has been often used in practice further a realistic 3 d nitrogen flow which two near continuum parallel orifice jets underexpand into a near vacuum environment is simulated using the present coupled method to demonstrate its capability finally developments in extending the present method are also outlined in this paper

hit and run mixes fast it is shown that the hit and run algorithm for sampling from a convex body k introduced by r l smith mixes in time o n2r2 r2 where r and r are the radii of the inscribed and circumscribed balls of k thus after appropriate preprocessing hit and run produces an approximately uniformly distributed sample point in time o n3 which matches the best known bound for other sampling algorithms we show that the bound is best possible in terms of r r and n

hit and run from a corner we show that the hit and run random walk mixes rapidly starting from any interior point of a convex body this is the first random walk known to have this property in contrast the ball walk can take exponentially many steps from some starting points the proof extends to sampling an exponential density over a convex body

strengthening the mckay conjecture to include local fields and local schur indices abstract a central problem in the representation theory of finite groups is given a prime p the study of relationships between the representations of g and those of p local subgroups of g the simplest of these is perhaps the mckay conjecture which concerns the number of irreducible characters of degree prime to p even though this conjecture remains open in an effort to understand it a number of strengthenings of it have been proposed recently isaacs and navarro and later navarro have proposed interesting stronger conjectures indicating that congruences modulo p of the degree of the characters and the invariance under certain type of fixed galois automorphism could be taken into account as well we propose an even further strengthening of mckay s conjecture which takes into account in addition the p local field of definition of the irreducible characters as well as their p local schur indices

a particle particle hybrid method for kinetic and continuum equations we present a coupling procedure for two different types of particle methods for the boltzmann and the navier stokes equations a variant of the dsmc method is applied to simulate the boltzmann equation whereas a meshfree lagrangian particle method similar to the sph method is used for simulations of the navier stokes equations an automatic domain decomposition approach is used with the help of a continuum breakdown criterion we apply adaptive spatial and time meshes the classical sod s 1d shock tube problem is solved for a large range of knudsen numbers results from boltzmann navier stokes and hybrid solvers are compared the cpu time for the hybrid solver is 3 4 times faster than for the boltzmann solver

verification of dade s conjecture for janko groupj3 abstract in 3 dade made a conjecture expressing the number k b d of characters of a given defect d in a given p block b of a finite group g in terms of the corresponding numbers k b d for blocks b of certain p local subgroups of g several different forms of this conjecture are given in 5 dade claims that the most complicated form of this conjecture called the inductive conjecture 5 8 in 5 will hold for all finite groups if it holds for all covering groups of finite simple groups in this paper we verify the inductive conjecture for all covering groups of the third janko group j 3 in the notation of the atlas 1 this is one step in the inductive proof of the conjecture for all finite groups certain properties of j 3 simplify our task the schur multiplier of j 3 is cyclic of order 3 see 1 p 82 hence there are just two covering groups of j 3 namely j 3 itself and a central extension 3 j 3 of j 3 by a cyclic group z of order 3 we treat these two covering groups separately the outer automorphism group out j 3 of j 3 is cyclic of order 2 see 1 p 82 in this case dade affirms in 5 section 6 that the inductive conjecture for j 3 is equivalent to the much weaker invariant conjecture 2 5 in 5 furthermore dade has proved in 6 that this invariant conjecture holds for all blocks with cyclic defect groups the sylow p subgroups of j 3 are cyclic of order p for all primes dividing j 3 except 2 and 3 so we only need to verify the invariant conjecture for the two primes p 2 and p 3 we do that in theorem 2 10 1 and theorem 3 6 1 below the group out 3 j 3 z of outer automorphisms of 3 j 3 centralizing z is trivial see 1 p 82 in this case dade affirms that the inductive conjecture is equivalent to the projective conjecture 4 4 in 5 again dade has shown in 6 that this projective conjecture holds for all blocks with cyclic defect groups so we only need to verify the projective conjecture for p 2 and p 3 we do that in theorem 4 4 2 and theorem 4 3 1 for the above reasons the four theorems 2 10 1 3 6 1 4 4 2 and 4 3 1 below are sufficient to prove the inductive form of dade s conjecture for all covering groups of j 3 this paper is my ph d thesis it would not have been possible without the help and infinite patience of my advisor everett dade

billiards in a general domain with random reflections we study stochastic billiards on general tables a particle moves according to its constant velocity inside some domain fancyscript d subset mathbb r d until it hits the boundary and bounces randomly inside according to some reflection law we assume that the boundary of the domain is locally lipschitz and almost everywhere continuously differentiable the angle of the outgoing velocity with the inner normal vector has a specified absolutely continuous density we construct the discrete time and the continuous time processes recording the sequence of hitting points on the boundary and the pair location velocity we mainly focus on the case of bounded domains then we prove exponential ergodicity of these two markov processes we study their invariant distribution and their normal gaussian fluctuations of particular interest is the case of the cosine reflection law the stationary distributions for the two processes are uniform in this case the discrete time chain is reversible though the continuous time process is quasi reversible also in this case we give a natural construction of a chord picked at random in fancyscript d and we study the angle of intersection of the process with a d 1 dimensional manifold contained in fancyscript d

a diffusive information preservation method for small knudsen number flows the direct simulation monte carlo dsmc method is a powerful particle based method for modeling gas flows it works well for relatively large knudsen kn numbers typically larger than 0 01 but quickly becomes computationally intensive as kn decreases due to its time step and cell size limitations an alternative approach was proposed to relax or remove these limitations based on replacing pairwise collisions with a stochastic model corresponding to the fokker planck equation j comput phys 229 1077 2010 j fluid mech 680 574 2011 similar to the dsmc method the downside of that approach suffers from computationally statistical noise to solve the problem a diffusion based information preservation d ip method has been developed the main idea is to track the motion of a simulated molecule from the diffusive standpoint and obtain the flow velocity and temperature through sampling and averaging the ip quantities to validate the idea and the corresponding model several benchmark problems with kn 10 3 10 4 have been investigated it is shown that the ip calculations are not only accurate but also efficient because they make possible using a time step and cell size over an order of magnitude larger than the mean collision time and mean free path respectively

a low variance deviational simulation monte carlo for the boltzmann equation we present an efficient particle method for solving the boltzmann equation the key ingredients of this work are the variance reduction ideas presented in baker and hadjiconstantinou l l baker n g hadjiconstantinou variance reduction for monte carlo solutions of the boltzmann equation physics of fluids 17 2005 art no 051703 and a new collision integral formulation which allows the method to retain the algorithmic structure of direct simulation monte carlo dsmc and thus enjoy the numerous advantages associated with particle methods such as a physically intuitive formulation computational efficiency due to importance sampling low memory usage no discretization in velocity space and the ability to naturally and accurately capture discontinuities in the distribution function the variance reduction achieved by simulating only the deviation from equilibrium results in a significant computational efficiency advantage for low signal flows e g low flow speed compared to traditional particle methods such as dsmc in particular the resulting method can capture arbitrarily small deviations from equilibrium at a computational cost that is independent of the magnitude of this deviation the method is validated by comparing its predictions with dsmc solutions for spatially homogeneous and inhomogeneous problems

finite cloud method a true meshless technique based on a fixed reproducing kernel approximation we introduce fixed moving and multiple fixed kernel techniques for the construction of interpolation functions over a scattered set of points we show that for a particular choice of nodal volumes the fixed moving and multiple fixed kernel approaches are identical to the fixed moving and multiple fixed least squares approaches a finite cloud method which combines collocation with a fixed kernel technique for the construction of interpolation functions is presented as a true meshless technique for the numerical solution of partial differential equations numerical results are presented for several one and two dimensional problems including examples from elasticity heat conduction thermoelasticity stokes flow and piezoelectricity copyright 2001 john wiley sons ltd

fokker planck dsmc algorithm for simulations of rarefied gas flows a fokker planck based particle monte carlo algorithm was devised recently for simulations of rarefied gas flows by the authors 1 3 the main motivation behind the fokker planck fp model is computational efficiency which could be gained due to the fact that the resulting stochastic processes are continuous in velocity space this property of the model leads to simulations where the computational cost becomes independent of the knudsen number kn 3 however the fokker planck model which can be seen as a diffusion approximation of the boltzmann equation becomes less accurate as kn increases in this study we propose a hybrid fokker planck direct simulation monte carlo fp dsmc solution method which is applicable for the whole range of kn the objective of this algorithm is to retain the efficiency of the fp scheme at low kn kn 1 and to employ conventional dsmc at high kn kn 1 since the computational particles employed by the fp model represent the same data as in dsmc the coupling between the two methods is straightforward the new ingredient is a switching criterion which would ideally result in a hybrid scheme with the efficiency of the fp method and the accuracy of dsmc for the whole kn range here we adopt the number of collisions in a given computational cell and for a given time step size as a decision criterion in order to switch between the fp model and dsmc for assessment of the hybrid algorithm different test cases including flow impingement and flow expansion through a slit were studied both accuracy and efficiency of the model are shown to be excellent for the presented test cases

time relaxed monte carlo methods for the boltzmann equation a new family of monte carlo schemes is introduced for the numerical solution of the boltzmann equation of rarefied gas dynamics the schemes are inspired by the wild sum expansion of the solution of the boltzmann equation for maxwellian molecules and consist of a novel time discretization of the equation in particular high order terms in the expansion are replaced by the equilibrium maxwellian distribution the two main features of the schemes are high order accuracy in time and asymptotic preservation the first property allows to recover accurate solutions with time steps larger than those required by direct simulation monte carlo dsmc while the latter guarantees that for the vanishing knudsen number the numerical solution relaxes to the local maxwellian conservation of mass momentum and energy are preserved by the scheme numerical results on several space homogeneous problems show the improvement of the new schemes over standard dsmc applications to a one dimensional shock wave problem are also presented

a random polynomial time algorithm for approximating the volume of convex bodies a randomized polynomial time algorithm for approximating the volume of a convex body k in n dimensional euclidean space is presented the proof of correctness of the algorithm relies on recent theory of rapidly mixing markov chains and isoperimetric inequalities to show that a certain random walk can be used to sample nearly uniformly from within k

efficient monte carlo procedures for generating points uniformly distributed over bounded regions we consider the monte carlo problem of generating points uniformly distributed within an arbitrary bounded measurable region the class of markovian methods considered generate points asymptotically uniformly distributed within the region computational experience suggests the methods are potentially superior to conventional rejection techniques for large dimensional regions

stochastic billiards for sampling from the boundary of a convex set stochastic billiards can be used for approximate sampling from the boundary of a bounded convex set through the markov chain monte carlo paradigm this paper studies how many steps of the underlying markov chain are required to get samples approximately from the uniform distribution on the boundary of the set for sets with an upper bound on the curvature of the boundary our main theorem implies a polynomial time algorithm for sampling from the boundary of such sets

new refinements of the mckay conjecture for arbitrary finite groups let g be an arbitrary finite group and fix a prime number p the mckay conjecture asserts that g and the normalizer in g of a sylow p subgroup have equal numbers of irreducible characters with degrees not divisible by p the alperin mckay conjecture is version of this as applied to individual brauer p blocks of g we offer evidence that perhaps much stronger forms of both of these conjectures are true

parallel fokker planck dsmc algorithm for rarefied gas flow simulation in complex domains at all knudsen numbers a major challenge for the conventional direct simulation monte carlo dsmc technique lies in the fact that its computational cost becomes prohibitive in the near continuum regime where the knudsen number kn characterizing the degree of rarefaction becomes small in contrast the fokker planck fp based particle monte carlo scheme allows for computationally efficient simulations of rarefied gas flows in the low and intermediate kn regime the fokker planck collision operator instead of performing binary collisions employed by the dsmc method integrates continuous stochastic processes for the phase space evolution in time this allows for time step and grid cell sizes larger than the respective collisional scales required by dsmc dynamically switching between the fp and the dsmc collision operators in each computational cell is the basis of the combined fp dsmc method which has been proven successful in simulating flows covering the whole kn range until recently this algorithm had only been applied to two dimensional test cases in this contribution we present the first general purpose implementation of the combined fp dsmc method utilizing both shared and distributed memory parallelization this implementation provides the capability for simulations involving many particles and complex geometries by exploiting state of the art computer cluster technologies

algorithm refinement for stochastic partial differential equations ii correlated systems we analyze a hybrid particle continuum algorithm for a hydrodynamic system with long ranged correlations specifically we consider the so called train model for viscous transport in gases which is based on a generalization of the random walk process for the diffusion of momentum this discrete model is coupled with its continuous counterpart given by a pair of stochastic partial differential equations at the interface between the particle and continuum computations the coupling is by flux matching giving exact mass and momentum conservation this methodology is an extension of our stochastic algorithm refinement ar hybrid for simple diffusion f alexander a garcia d tartakovsky algorithm refinement for stochastic partial differential equations i linear diffusion j comput phys 182 2002 47 66 results from a variety of numerical experiments are presented for steady state scenarios in all cases the mean and variance of density and velocity are captured correctly by the stochastic hybrid algorithm for a non stochastic version i e using only deterministic continuum fluxes the long range correlations of velocity fluctuations are qualitatively preserved but at reduced magnitude

extension of a multiscale particle scheme to near equilibrium viscous flows a recently proposed low diffusion ld equilibrium particle method based on the direct simulation monte carlo dsmc method is modified for use with near equilibrium viscous flows a finite volume discretization of the viscous terms in the compressible navier stokes equations is used to incorporate diffusive transport effects into the ld particle method and a velocity and temperature slip wall boundary condition is employed for improved accuracy in the slip flow knudsen number regime the modified method is compared with both dsmc and theory for a series of unsteady boundary layer problems and excellent agreement is observed the computational cost of the modified ld particle method is shown for a representative near equilibrium case to be roughly four orders of magnitude lower than that of dsmc due to reduced scatter as well as less stringent cell size and time step requirements in the ld method simulation procedures are outlined for a strongly coupled hybrid algorithm where the ld particle method is used in continuum flowfield regions and dsmc is employed in nonequilibrium regions the hybrid scheme is evaluated through a comparison with numerical and experimental data for a flow of n2 through a small convergent divergent nozzle into a near vacuum and hybrid simulation results are generally found to agree very well with other available data i introduction as flows involving a wide range of characteristic length scales appear in a number of different engineering applications including those related to atmospheric flow around reentry or hypersonic vehicles high altitude rocket plumes flows within and around micro electro mechanical systems and any supersonic flow where internal shock structures are of interest in these types of flows a near equilibrium gas velocity distribution may exist through much of the flowfield as the equilibrating effect of intermolecular collisions dominates over other processes such as inhomogeneous diffusive transport or gas surface interaction which tend to pull the velocity distribution away from equilibrium however some flowfield regions may have characteristic length scales comparable to or smaller than the local mean free path so that the influence of collisions does not dominate and the velocity distribution diverges considerably from the equilibrium limit while simulation of near equilibrium flowfield regions may be efficiently performed using computational fluid dynamics cfd techniques based on the navier stokes equations nonequilibrium regions must be simulated using more expensive techniques based on the boltzmann equation the boltzmann equation is the governing equation for dilute gas flows at arbitrary knudsen numbers and its derivation follows from assumptions of molecular chaos and binary intermolecular collisions with no approximations regarding the shape of the velocity distribution the most mature and commonly used simulation method for the boltzmann equation is the direct simulation monte carlo dsmc method first introduced by bird 1 in a dsmc simulation a large number of representative particles are tracked through a computational grid and move and collide in a manner consistent with physical arguments underlying the boltzmann equation while this method may be applied to both nonequilibrium and continuum flowfield regions cell size and time step limitations often make it prohibitively expensive for simulating continuum flows where global characteristic length scales are far larger than the mean free path in practice such multiscale flows are usually simulated using both dsmc and cfd methods by applying cfd techniques in near equilibrium regions where the navier stokes equations are valid and using dsmc elsewhere in the flow in the simplest type of hybrid cfd dsmc approach a cfd simulation is performed on a domain which 1

on the complexity of computing mixed volumes this paper gives various positive and negative results on the complexity of the problem of computing and approximating mixed volumes of polytopes and more general convex bodies in arbitrary dimension on the negative side we present several p hardness results that focus on the difference of computing mixed volumes versus computing the volume of polytopes we show that computing the volume of zonotopes is p hard while each corresponding mixed volume can be computed easily but also give examples showing that computing mixed volumes is hard even when computing the volume is easy on the positive side we derive a randomized algorithm for computing the mixed volumes v overbrace k 1 ld k 1 m 1 overbrace k 2 dots k 2 m 2 dots overbrace k s dots k s m s of well presented convex bodies k 1 dots k s where m 1 dots m s in n 0 and m 1 geq n psi n with psi n o frac log n log log n the algorithm is an interpolation method based on polynomial time randomized algorithms for computing the volume of convex bodies this paper concludes with applications of our results to various problems in discrete mathematics combinatorics computational convexity algebraic geometry geometry of numbers and operations research

a combined continuum dsmc technique for multiscale analysis of microfluidic filters a multiscale method that combines continuum fluid models with the direct simulation monte carlo dsmc method is presented the continuum regions are treated by stokes equations and a scattered point based finite cloud method is employed to solve the stokes equations the continuum and dsmc regions are combined by an overlapped schwarz alternating method with dirichlet dirichlet type boundary conditions a scattered point interpolation scheme is developed to interpolate the solution between subdomains the convergence characteristics of the multiscale approach are investigated in detail specifically the dependence of convergence on the overlap size the dsmc noise and the number of time steps employed in the dsmc algorithm are studied while the convergence depends weakly on the dsmc noise and the overlap size the number of dsmc time steps simulated in each coupling iteration should be selected so that the total time steps simulated until convergence of the coupled process is close to the time constant of the dsmc subsystem steady state analysis of microfluidic filters is studied in detail using the multiscale approach the multiscale approach is also applied for the simulation of a membrane with an array of microfluidic filters and a dual stage microfluidic device with an array of microfluidic filters for particle trapping and sorting

generation of the maxwellian inflow distribution this paper presents several efficient exact acceptance rejection methods for generating the maxwellian inflow distribution the velocity distribution of gas molecules crossing a plane the new methods are demonstrated to be computationally faster and more accurate than the schemes commonly used for open boundary conditions in particle simulations

random sampling billiard walk algorithm abstract hit and run is known to be one of the best random sampling algorithms its mixing time is polynomial in dimension however in practice the number of steps required to obtain uniformly distributed samples is rather high we propose a new random walk algorithm based on billiard trajectories numerical experiments demonstrate much faster convergence to the uniform distribution

stochastic search in a convex region a stochastic search strategy is proposed for locating a possibility mobile target in a bounded convex region of the plane the strategy is asymptotically minimax as e 0 with respect to the time required to get within e of the target the proof involves the study of first passages to time dependent boundaries by a certain semi markov process