the ito clifford integral abstract we present a theory of non commutative stochastic integration analogous to the ito theory it is shown that wick products of fermi fields define martingales and that stochastic integrals with respect to these are defined for adapted operatorvalued square integrable integrands for square integrable martingales associated with an arbitrary probability gage space a stochastic integral is defined and a doob meyer decomposition for supermartingales obtained

proteus a coupled iterative force correction immersed boundary cascaded lattice boltzmann solver for moving and deformable boundary applications abstract many realistic fluid flow problems are characterised by high reynolds numbers and complex moving or deformable geometries in our previous study we presented a novel coupling between an iterative force correction immersed boundary and a multi domain cascaded lattice boltzmann method falagkaris et al and investigated flows around rigid bodies at reynolds numbers up to 1 0 5 here we extend its application to flows around moving and deformable bodies with prescribed motions emphasis is given on the influence of the internal mass on the computation of the aerodynamic forces including deforming boundary applications where the rigid body approximation is no longer valid both the rigid body and the internal lagrangian points approximations are examined the resulting solver has been applied to viscous flows around an in line oscillating cylinder a pitching foil a plunging sd7003 airfoil and a plunging and flapping naca 0014 airfoil good agreement with experimental results and other numerical schemes has been obtained it is shown that the internal lagrangian points approximation accurately captures the internal mass effects in linear and angular motions as well as in deforming motions at reynolds numbers up to 4 1 0 4 in all cases the aerodynamic loads are significantly affected by the internal fluid forces

the immersed boundary lattice boltzmann method for solving fluid particles interaction problems a new computational method the immersed boundary lattice boltzmann method is presented this method is a combination and utilizes the most desirable features of the lattice boltzmann and the immersed boundary methods the method uses a regular eulerian grid for the flow domain and a lagrangian grid to follow particles that are contained in the flow field the rigid body conditions for the fluid and the particles are enforced by a penalty method which assumes that the particle boundary is deformable with a high stiffness constant the velocity field of the fluid and particles is solved by adding a force density term into the lattice boltzmann equation this novel method preserves the advantages of lbm in tracking a group of particles and at the same time provides an alternative and better approach to treating the solid fluid boundary conditions the method also solves the problems of fluctuation of the forces and velocities on the particles when the bounce back boundary conditions are applied this method enables one to simulate problems with particle deformation and fluid structure deformation its results are validated by comparison with results from other methods

comparison of boundary slip for two variants of immersed boundary method in lattice boltzmann framework abstract in this paper the immersed moving boundary lattice boltzmann imb lb method is compared with the single relaxation time and multiple relaxation time versions of the immersed boundary lattice boltzmann ib lb method in terms of the amount of numerical velocity slip produced on solid boundaries the comparisons are performed for both straight and curved boundaries based on the effects of thickness of virtual domain used in the ib method for the first time and relaxation time parameter s of the lb method for the straight boundaries a shear flow problem is studied while for the curved boundaries a falling circular cylinder in an infinite channel is investigated first sensitivities of the ib lb and imb lb methods to the relaxation time parameter s and thickness of virtual domain are presented then the interaction of two solid particles in a shear flow is studied to assess the validity of arguments presented in solving a typical practical problem

stochastic integration on the cuntz algebra o abstract we develop a theory of non commutative stochastic integration with respect to the creation and annihilation process on the full fock space over l2 r our theory largely parallels the theories of non commutative stochastic ito integration on boson and fermion fock space as developed by r hudson and k r parthasarathy it provides the first example of a non commutative stochastic calculus which does not depend on the quantum mechanical commutation or anticommutation relations but it is based on the theory of reduced free products of c algebras by d voiculescu this theory shows that the creation and annihilation processes on the full fock space over l2 r which generate the cuntz algebra o can be interpreted as a generalized brownian motion we should stress the fact that in contrast to the other theories of stochastic integration our integrals converge in the c norm on o i e uniformly rather than in some state dependent strong operator topology or non commutative l2 norm

a geometrical characterization of banach spaces in which martingale difference sequences are unconditional we study banach space valued martingale transforms and in particular characterize those banach spaces for which the classical theorems of the real valued case carry over for example if b is a banach space and 1 0 and zeta x y leq x y text if x leq 1 leq y

an example of a generalized brownian motion we present an example of a generalized brownian motion it is given by creation and annihilation operators on a twisted fock space ofl2 these operators fulfill for a fixed 1 1 the relationsc f c g c g c f f g 1 f g l2 we show that the distribution of these operators with respect to the vacuum expectation is a generalized gaussian distribution in the sense that all moments can be calculated from the second moments with the help of a combinatorial formula we also indicate that our brownian motion is one component of ann dimensional brownian motion which is invariant under the quantum groups u n of woronowicz with v2

theory and applications of an alternative lattice boltzmann grid refinement algorithm this contribution proposes an alternative lattice boltzmann grid refinement algorithm that overcomes the drawbacks that plague existing approaches we demonstrate that this algorithm is accurate and applicable for all values of the relaxation time we also show that this algorithm can significantly speed up the flow settlement process by using a hierarchy of grid levels the stationary regime can be approached up to a thousand times faster than with a single grid resolution

immersed boundary lattice boltzmann model based on multiple relaxation times as an alterative version of the lattice boltzmann models the multiple relaxation time mrt lattice boltzmann model introduces much less numerical boundary slip than the single relaxation time srt lattice boltzmann model if some special relationship between the relaxation time parameters is chosen on the other hand most current versions of the immersed boundary lattice boltzmann method which was first introduced by feng and improved by many other authors suffer from numerical boundary slip as has been investigated by le and zhang to reduce such a numerical boundary slip an immerse boundary lattice boltzmann model based on multiple relaxation times is proposed in this paper a special formula is given between two relaxation time parameters in the model a rigorous analysis and the numerical experiments carried out show that the numerical boundary slip reduces dramatically by using the present model compared to the single relaxation time based model

on martingale inequalities in non commutative stochastic analysis we develop a non commutativelpstochastic calculus for the clifford stochastic integral anl2theory of which has been developed by barnett streater and wilde the main results are certain non commutativelpinequalities relating clifford integrals and their integrands these results are applied to extend the domain of the clifford integral froml2tol1integrands and we give applications to optional stopping of clifford martingales proving an analog of a theorem of burkholder the stopped clifford processfthas zero expectation providedet in proving these results we establish a number of results relating the clifford integral to the differential calculus in the clifford algebra in particular we show that the clifford integral is given by the divergence operator and we prove an explicit martingale representation theorem both of these results correspond closely to basic results for stochastic analysis on wiener space thus furthering the analogy between the clifford process and brownian motion

stochastic integration with respect to q brownian motion we develop a stochastic integration with respect to a q brownian motion for i e a non commutative process where the operator at and its adjoint fulfill the q commutation relation under the vacuum state expectation we show that this process enjoys a predictable representation type property

stochastic calculus with respect to free brownian motion and analysis on wigner space we define stochastic integrals with respect to free brownian motion and show that they satisfy burkholder gundy type inequalities in operator norm we prove also a version of ito s predictable representation theorem as well as product form and functional form of ito s formula finally we develop stochastic analysis on the free fock space in analogy with stochastic analysis on the wiener space

discrete lattice effects on the forcing term in the lattice boltzmann method we show that discrete lattice effects must be considered in the introduction of a force into the lattice boltzmann equation a representation of the forcing term is then proposed with the representation the navier stokes equation is derived from the lattice boltzmann equation through the chapman enskog expansion several other existing force treatments are also examined

on the greatest prime factor of p 1 with effective constants let p denote a prime in this article we provide the first published lower bounds for the greatest prime factor of p i exceeding p 1 1 2 in which the constants are effectively computable as a result we prove that it is possible to calculate a value x 0 such that for every x x 0 there is a p x with the greatest prime factor of p 1 exceeding x 3 5 the novelty of our approach is the avoidance of any appeal to siegel s theorem on primes in arithmetic progression

vinogradov s mean value theorem via efficient congruencing we obtain estimates for vinogradov s integral that for the rst time approach those conjectured to be the best possible several applications of these new bounds are provided in particular the conjectured asymptotic formula in waring s problem holds for sums of s kth powers of natural numbers whenever s 2k 2 2k 3

on integer solutions of parsell vinogradov systems we prove a sharp upper bound on the number of integer solutions of the parsell vinogradov system in every dimension d ge 2

an immersed boundary lattice boltzmann method for the simulation of the flow past an impulsively started cylinder we present a lattice boltzmann method coupled with an immersed boundary technique for the simulation of bluff body flows the lattice boltzmann method for the modeling of the navier stokes equations is enhanced by a forcing term to account for the no slip boundary condition on a non grid conforming boundary we investigate two alternatives of coupling the boundary forcing term with the grid nodes namely the direct and the interpolated forcing techniques the present lb ib methods are validated in simulations of the incompressible flow past an impulsively started cylinder at low and moderate reynolds numbers we present diagnostics such as the near wall vorticity field and the drag coefficient and comparisons with previous computational and experimental works and assess the advantages and drawbacks of the two techniques

cumulants of partitions we utilize the formal equivalence between the number partitioning problem and a harmonically trapped ideal bose gas within the microcanonical ensemble for characterizing the probability distribution which governs the number of addends occurring in an unrestricted partition of a natural number n by deriving accurate asymptotic formulae for its coefficients of skewness and excess it is shown that this distribution remains non gaussian even when n is made arbitrarily large both skewness and excess vary substantially before settling to their constant limiting values for n 1010

an implicit boundary approach for viscous compressible high reynolds flows using a hybrid remeshed particle hydrodynamics method abstract we present an implicit boundary particle method with background mesh adaptation we use a brinkman penalisation to represent the boundary of the domain and a remeshed particle method to simulate viscous flow with high reynolds numbers a penalty term is added to the navier stokes equations to impose the boundary conditions the boundary conditions are enforced to a specific precision with no need to modify the numerical method or change the grid achieving an implicit approach for flow around complex boundaries geometries simulation the main idea of the brinkman penalisation method is to model the solid obstacle as a porous medium the governing equations for the compressible fluid and penalised navier stokes for the porous medium are solved simultaneously without need for interface conditions the accuracy of the method is tested for a number of benchmark problems starting with simple cases such as a periodic laminar flow in a channel poiseuille flow to more complex problems such as the lid driven cavity with high reynolds numbers the reflection and propagation of a compressible shock wave and finally swirling flow in a model two stroke diesel engine the remeshed particle mesh method coupled with brinkman penalisation provides a good quality simulation and the results are in agreement with analytical or reference solutions

proteus a direct forcing method in the simulations of particulate flows a new and efficient direct numerical method for the simulation of particulate flows is introduced the method combines desired elements of the immersed boundary method the direct forcing method and the lattice boltzmann method adding a forcing term in the momentum equation enforces the no slip condition on the boundary of a moving particle by applying the direct forcing scheme proteus eliminates the need for the determination of free parameters such as the stiffness coefficient in the penalty scheme or the two relaxation parameters in the adaptive forcing scheme the method presents a significant improvement over the previously introduced immersed boundary lattice boltzmann method ib lbm where the forcing term was computed using a penalty method and a user defined parameter the method allows the enforcement of the rigid body motion of a particle in a more efficient way compared to the bounce back scheme used in the conventional lbm the direct forcing method provides a smoother computational boundary for particles and is capable of achieving results at higher reynolds number flows by using a set of lagrangian points to track the boundary of a particle proteus eliminates any need for the determination of the boundary nodes that are prescribed by the bounce back scheme at every time step it also makes computations for particles of irregular shapes simpler and more efficient proteus has been developed in two as well as three dimensions this new method has been validated by comparing its results with those from experimental measurements for a single sphere settling in an enclosure under gravity as a demonstration of the efficiency and capabilities of the present method the settling of a large number 1232 of spherical particles is simulated in a narrow box under two different boundary conditions it is found that when the no slip boundary condition is imposed at the front and rear sides of the box the particles motion is significantly hindered under the periodic boundary conditions the particles move faster the simulations show that the sedimentation characteristics in a box with periodic boundary conditions at the two sides are very close to those found in the sedimentation of two dimensional circular particles

a spectral element discontinuous galerkin lattice boltzmann method for nearly incompressible flows we present a spectral element discontinuous galerkin lattice boltzmann method for solving nearly incompressible flows decoupling the collision step from the streaming step offers numerical stability at high reynolds numbers in the streaming step we employ high order spectral element discontinuous galerkin discretizations using a tensor product basis of one dimensional lagrange interpolation polynomials based on gauss lobatto legendre grids our scheme is cost effective with a fully diagonal mass matrix advancing time integration with the fourth order runge kutta method we present a consistent treatment for imposing boundary conditions with a numerical flux in the discontinuous galerkin approach we show convergence studies for couette flows and demonstrate two benchmark cases with lid driven cavity flows for re 400 5000 and flows around an impulsively started cylinder for re 550 9500 computational results are compared with those of other theoretical and computational work that used a multigrid method a vortex method and a spectral element model

distributed lagrange multiplier fictitious domain method in the framework of lattice boltzmann method for fluid structure interactions a new implementation of the lattice boltzmann method lbm for fluid structure interactions is presented the idea of the distributed lagrange multiplier fictitious domain method dlm fd is introduced in the framework of the lattice boltzmann algorithm this implementation employs a fixed mesh for the solution of the fluid problem and a lagrangian formulation for the solid problem the main advantage of the method is that the re meshing procedure normally required in the ale method is circumvented numerical examples are provided to verify the algorithm and illustrate the capacity of the method to deal with the fluid elastic solid interactions

the lattice boltzmann equation a new tool for computational fluid dynamics abstract we present a series of applications which demonstrate that the lattice boltzmann equation is an adequate computational tool to address problems spanning a wide spectrum of fluid regimes ranging from laminar to fully turbulent flows in two and three dimensions

asymptotic expansion of mathieu power series and trigonometric mathieu series abstract we consider a generalized mathieu series where the summands of the classical mathieu series are multiplied by powers of a complex number the mellin transform of this series can be expressed by the polylogarithm or the hurwitz zeta function from this we derive a full asymptotic expansion generalizing known expansions for alternating mathieu series another asymptotic regime for trigonometric mathieu series is also considered to first order by applying known results on the asymptotic behavior of trigonometric series

proteus a coupled iterative force correction immersed boundary multi domain cascaded lattice boltzmann solver abstract most realistic fluid flow problems are characterized by high reynolds numbers and complex boundaries over the last ten years immersed boundary methods that are able to cope with realistic geometries have been applied to lattice boltzmann lb methods these methods however have normally been applied to low reynolds number problems here we present a novel coupling between an iterative force correction immersed boundary zhang et al 2016 and a multi domain cascaded lb method the iterative force correction immersed boundary method has been selected due to the improved accuracy of the computation while the cascaded lb formulation is used due to its superior stability at high reynolds numbers the coupling is shown to improve both the stability and numerical accuracy of the solution the resulting solver has been applied to viscous flow up to a reynolds number of 100000 passed a naca 0012 airfoil at a 10 degree angle of attack good agreement with results obtained using a body fitted navier stokes solver has been obtained the formulation provides a straight forward and efficient method for modeling realistic geometries and could easily be extended to problems with moving boundaries

application of multi block approach in the immersed boundary lattice boltzmann method for viscous fluid flows the immersed boundary lattice boltzmann method was presented recently to simulate the rigid particle motion it combines the desirable features of the lattice boltzmann and immersed boundary methods it uses a regular eulerian grid for the flow domain and a lagrangian grid for the boundary for the lattice boltzmann method as compared with the single relaxation time collision scheme the multi relaxation time collision scheme has better computational stability due to separation of the relaxations of various kinetic models especially near the geometric singularity so the multi relaxation time collision scheme is used to replace the single relaxation time collision scheme in the original immersed boundary lattice boltzmann method in order to obtain an accurate result very fine lattice grid is needed near the solid boundary to reduce the computational effort local grid refinement is adopted to offer high resolution near a solid body and to place the outer boundary far away from the body so the multi block scheme with the multi relaxation time collision model is used in the immersed boundary lattice boltzmann method in each block uniform lattice spacing can still be used in order to validate the multi block approach for the immersed boundary lattice boltzmann method with multi relaxation time collision scheme the numerical simulations of steady and unsteady flows past a circular cylinder and airfoil are carried out and good results are obtained

an adaptive wavelet collocation method for fluid structure interaction at high reynolds numbers two mathematical approaches are combined to calculate high rey nolds number incompressible fluid structure interaction a wavelet method to dynamically adapt the computational grid to flow intermittency and obstacle motion and brinkman penalization to enforce solid boundaries of arbitrary complexity we also implement a wavelet based multilevel solver for the poisson problem for the pressure at each time step the method is applied to two dimensional flow around fixed and moving cylinders for reynolds numbers in the range 3 times 10 1 le re le 10 5 the compression ratios of up to 1000 are achieved for the first time it is demonstrated in actual dynamic simulations that the compression scales like re 1 2 over five orders of magnitude while computational complexity scales like re this represents a significant improvement over the classical complexity estimate of re 9 4 for two dimensional turbulence

noncommutative stochastic integration through decoupling abstract we present an abstract approach to noncommutative stochastic integration in the context of a finite von neumann algebra equipped with a normal faithful tracial state with respect to processes with tensor or freely independent increments satisfying a stationarity condition using a decoupling technique we obtain necessary and sufficient conditions for stochastic integrability of l p processes with respect to such integrators we apply the theory to stochastic integration with respect to boson and free brownian motion

quantum stochastic calculus with maximal operator domains quantum stochastic calculus is extended in a new formulation in which its stochastic integrals achieve their natural and maximal domains operator adaptedness conditional expectations and stochastic integrals are all defined simply in terms of the orthogonal projections of the time filtration of fock space together with sections of the adapted gradient operator free from exponential vector domains our stochastic integrals may be satisfactorily composed yielding quantum ito formulas for operator products as sums of stochastic integrals the calculus has seen two reformulations since its discovery one closely related to classical ito calculus the other to noncausal stochastic analysis and malliavin calculus our theory extends both of these approaches and may be viewed as a synthesis of the two the main application given here is existence and uniqueness for the attal meyer equations for implicit definition of quantum stochastic integrals

a curved no slip boundary condition for the lattice boltzmann method we present a generalization of the no slip boundary condition by latt et al j latt b chopard o malaspinas m deville a michler straight velocity boundaries in the lattice boltzmann method physical review e 77 5 2008 056703 from straight to curved geometries for the lattice boltzmann bhatnager gross krook method lbgk the boundary condition is based on a reconstruction of the populations from the density velocity and rate of strain for curved boundaries the reconstruction reduces the question of accuracy to a technical issue of interpolation we present a method of interpolation allowing a very accurate representation of the curved boundary the resulting boundary condition is verified for three different test cases taylor couette flow in between rotating cylinders laminar flow around a cylinder and flow past an impulsively started cylinder demonstrating its second order accuracy and low error constant the present boundary is stable for relaxation frequencies close to two

non commutative martingale inequalities we prove the analogue of the classical burkholder gundy inequalites for non commutative martingales as applications we give a characterization for an ito clifford integral to be an lp martingale via its integrand and then extend the ito clifford integral theory in l2 developed by barnett streater and wilde to lp for all 1 p we include an appendix on the non commutative analogue of the classical fefferman duality between h1 and bmo

combined immersed boundary finite difference methods for three dimensional complex flow simulations a second order accurate highly efficient method is developed for simulating unsteady three dimensional incompressible flows in complex geometries this is achieved by using boundary body forces that allow the imposition of the boundary conditions on a given surface not coinciding with the computational grid the governing equations therefore can be discretized and solved on a regular mesh thus retaining the advantages and the efficiency of the standard solution procedures two different forcings are tested showing that while the quality of the results is essentially the same in both cases the efficiency of the calculation strongly depends on the particular expression a major issue is the interpolation of the forcing over the grid that determines the accuracy of the scheme this ranges from zeroth order for the most commonly used interpolations up to second order for an ad hoc velocity interpolation the present scheme has been used to simulate several flows whose results have been validated by experiments and other results available in the literature finally in the last example we show the flow inside an ic piston cylinder assembly at high reynolds number to our knowledge this is the first example in which the immersed boundary technique is applied to a full three dimensional complex flow with moving boundaries and with a reynolds number high enough to require a subgrid scale turbulence model

a high order explicit method for the computation of flow about a circular cylinder in this paper a finite difference method for computing the solutions of the incompressible navier stokes equations for flow about a circular cylinder in two dimensions is presented a stream function vorticity formulation of the equation is used and the numerical method incorporates recent developments in the computation of vorticity boundary conditions as well as far field boundary conditions three schemes are described one of second order accuracy one of fourth order accuracy and a hybrid method which is second order accurate in the computation of the vorticity transport and fourth order accurate in the determination of the stream function fully resolved solutions for flow past a cylinder have been computed over a range of reynolds numbers from 1000 to 9500 comparisons are made between the results obtained with methods of different orders of accuracy as well as of the effectiveness of the vorticity and far field boundary conditions

vinogradov s integral and bounds for the riemann zeta function the main result is an upper bound for the riemann zeta function in the critical strip zeta sigma it le a t b 1 sigma 3 2 log 2 3 t with a 76 2 and b 4 45 valid for frac12 le sigma le 1 and t ge 3 the previous best constant b was 18 5 tools include a variant of the korobov vinogradov method of bounding exponential sums an explicit version of t d wooley s bounds for vinogradov s integral and explicit bounds for mean values of exponential sums over numbers without small prime factors also using methods of wooley an auxiliary result is the exponential sum bound s n t le 9 463 n 1 1 133 66 lambda 2 where n is a positive integer t is a real number lambda log t log n and s n t max 0 2000 mathematical subject classification primary 11m06 11n05 11l15 secondary 11d72 11m35