local spectral gap in simple lie groups and applications we introduce a novel notion of local spectral gap for general possibly infinite measure preserving actions we establish local spectral gap for the left translation action gamma curvearrowright g whenever gamma is a dense subgroup generated by algebraic elements of an arbitrary connected simple lie group g this extends to the non compact setting works of bourgain and gamburd invent math 171 83 121 2008 j eur math soc jems 14 1455 1511 2012 and benoist and de saxce invent math 205 337 361 2016 we present several applications to the banach ruziewicz problem orbit equivalence rigidity continuous and monotone expanders and bounded random walks on g in particular we prove that up to a multiplicative constant the haar measure is the unique gamma invariant finitely additive measure defined on all bounded measurable subsets of g

constructing spectral schemes of the immersed interface method via a global description of discontinuous functions a global description of discontinuous functions is introduced in this paper by expressing a discontinuous function as the sum of a smooth function and a correction term determined by jump conditions we turn the unknown function from a discontinuous one into a sufficiently smooth one when solving a differential equation spectral schemes are developed based on this concept with the intention of eliminating or reducing the gibbs oscillation finite difference schemes are also constructed as an alternative of the current immersed interface methods both spectral and finite difference schemes are tested on one and two dimensional cases

simulating the dynamics of inextensible vesicles by the penalty immersed boundary method in this paper we develop an immersed boundary ib method to simulate the dynamics of inextensible vesicles interacting with an incompressible fluid in order to take into account the inextensibility constraint of the vesicle the penalty immersed boundary pib method is used to virtually decouple the fluid and vesicle dynamics as numerical tests of our current pib method the dynamics of single and multiple inextensible vesicles under shear flows have been extensively explored and compared with the previous literature the method is also validated by a series of convergence study which confirms its consistent first order accuracy on the velocity field the vesicle configuration the vesicle area and the perimeter errors in addition the method is also applied to study a binary component vesicle problem

an immersed boundary method with direct forcing for the simulation of particulate flows we present an improved method for computing incompressible viscous flow around suspended rigid particles using a fixed and uniform computational grid the main idea is to incorporate peskin s regularized delta function approach acta numerica 11 2002 1 into a direct formulation of the fluid solid interaction force in order to allow for a smooth transfer between eulerian and lagrangian representations while at the same time avoiding strong restrictions of the time step this technique was implemented in a finite difference and fractional step context a variety of two and three dimensional simulations are presented ranging from the flow around a single cylinder to the sedimentation of 1000 spherical particles the accuracy and efficiency of the current method are clearly demonstrated

computations of multi fluid flows abstract full numerical simulations of three dimensional flows of two or more immiscible fluids of different densities and viscosities separated by a sharp interface with finite surface tension are discussed the method used is based on a finite difference approximation of the full navier stokes equations and explicit tracking of the interface between the fluids preliminary simulations of the rayleigh taylor instability and the motion of bubbles are shown

a fast numerical method for solving the three dimensional stokes equations in the presence of suspended particles abstract a new fast numerical method for solving the three dimensional stokes equations in the presence of suspended particles is presented the fluid dynamics equations are solved on a lattice a particle is represented by a set of points each of which moves at the local fluid velocity and is not constrained to lie on the lattice these points are coupled by forces which resist deformation of the particle these forces contribute to the force density in the stokes equations as a result a single set of fluid dynamics equations holds at all points of the domain and there are no internal boundaries particle size shape and deformability may be prescribed computational work increases only linearly with the number of particles so large numbers 500 1000 of particles may be studied efficiently the numerical method involves implicit calculation of the particle forces by minimizing an energy function and solution of a finite difference approximation to the stokes equations using the fourier toeplitz method the numerical method has been implemented to run on all cray computers the implementation exploits the cray s vectorized arithmetic and on machines with insufficient central memory it performs efficient disk 1 o while storing most of the data on disk applications of the method to sedimentation of one two and many particle systems are described trajectories and settling speeds for two particle sedimentation and settling speed for multiparticle sedimentation from initial distributions on a cubic lattice or at random give good quantitative agreement with existing theories

numerical simulation of axisymmetric free surface flows this paper describes an extension of the gensmac code for solving two dimen sional free surface flows to axisymmetric flows like gensmac the technique is finite difference based and embodies but considerably extends the smac simplified marker and cell ideas it incorporates adaptive time stepping and an accurate representation of the free surfaces while at the same time only uses surface particles to define the free surfaces greatly increasing the computational speed in addition it employs a graphic interface with solid modeling techniques to provide enhanced three dimensional visualization various simulations are undertaken to illustrate and validate typical flows both g i taylor s viscous jet plunging into a fluid and a liquid drop splashing onto a fluid are simulated also the important industrial application of container filling is illustrated finally a comparison is made with the linear theory of standing waves and the code is validated by a numerical convergence study

systematic derivation of jump conditions for the immersed interface method in three dimensional flow simulation in this paper we systematically derive jump conditions for the immersed interface method siam j numer anal 31 1994 pp 1019 1044 siam j sci comput 18 1997 pp 709 735 to simulate three dimensional incompressible viscous flows subject to moving surfaces the surfaces are represented as singular forces in the navier stokes equations which give rise to discontinuities of flow quantities the principal jump conditions across a closed surface of the velocity the pressure and their normal derivatives have been derived by lai and li appl math lett 14 2001 pp 149 154 in this paper we first extend their derivation to generalized surface parametrization starting from the principal jump conditions we then derive the jump conditions of all first second and third order spatial derivatives of the velocity and the pressure we also derive the jump conditions of first and second order temporal derivatives of the velocity using these jump conditions the immersed interface method is applicable to the simulation of three dimensional incompressible viscous flows subject to moving surfaces where near the surfaces the first and second order spatial derivatives of the velocity and the pressure can be discretized with respectively third and second order accuracy and the first order temporal derivatives of the velocity can be discretized with second order accuracy

an aschbacher oʼnan scott theorem for countable linear groups abstract the purpose of this note is to extend the classical aschbacher o nan scott theorem on finite groups to the class of countable linear groups this relies on the analysis of primitive actions carried out in gelander and glasner 2008 gg08 unlike the situation for finite groups we show here that the number of primitive actions depends on the type linear groups of almost simple type admit infinitely and in fact unaccountably many primitive actions while affine and diagonal groups admit only one the abundance of primitive permutation representations is particularly interesting for rigid groups such as simple and arithmetic ones

groups where free subgroups are abundant given an infinite topological group g and a cardinal 0 we say that g is almost free if the set of tuples gi i g which freely generate free subgroups of g is dense in g in this note we examine groups having this property and construct examples for instance we show that if g is a non discrete hausdorff topological group that contains a dense free subgroup of rank 0 then g is almost free a consequence of this is that for any infinite set the group of all permutations of is almost 2 free we also show that an infinite topological group is almost 0 free if and only if it is almost n free for each positive integer n this generalizes the work of dixon and gartside and knight

implicit second order immersed boundary methods with boundary mass abstract the immersed boundary method is a computational framework for problems involving the interaction of a fluid and immersed elastic structures immersed boundary computations typically evaluate the elastic forces explicitly in the configuration of the immersed elastic structure in many applications this results in a severe restriction on the time step we present a semi implicit and a fully implicit second order accurate immersed boundary method the methods provide a natural way to handle mass on the immersed elastic structures we demonstrate their performance for a prototypical fluid structure interaction problem the methods are shown to possess superior stability properties that significantly alleviate the typically severe time step restriction of explicit computations

an immersed boundary method with formal second order accuracy and reduced numerical viscosity a formally second order accurate immersed boundary method is presented and tested in this paper we apply this new scheme to simulate the flow past a circular cylinder and study the effect of numerical viscosity on the accuracy of the computation by comparing the numerical results with those of a first order method the numerical evidence shows that the new scheme has less numerical viscosity and is therefore a better choice for the simulation of high reynolds number flows with immersed boundaries

generating semisimple groups by tori abstract let g be a simple real lie group with finite center we prove that given a non central element g in g there is an elliptic element h in g such that h and g h g 1 generate a dense subsemigroup of g

an efficient semi implicit immersed boundary method for the navier stokes equations the immersed boundary method is one of the most useful computational methods in studying fluid structure interaction on the other hand the immersed boundary method is also known to require small time steps to maintain stability when solved with an explicit method many implicit or approximately implicit methods have been proposed in the literature to remove this severe time step stability constraint but none of them give satisfactory performance in this paper we propose an efficient semi implicit scheme to remove this stiffness from the immersed boundary method for the navier stokes equations the construction of our semi implicit scheme consists of two steps first we obtain a semi implicit discretization which is proved to be unconditionally stable this unconditionally stable semi implicit scheme is still quite expensive to implement in practice next we apply the small scale decomposition to the unconditionally stable semi implicit scheme to construct our efficient semi implicit scheme unlike other implicit or semi implicit schemes proposed in the literature our semi implicit scheme can be solved explicitly in the spectral space thus the computational cost of our semi implicit schemes is comparable to that of an explicit scheme our extensive numerical experiments show that our semi implicit scheme has much better stability property than an explicit scheme this offers a substantial computational saving in using the immersed boundary method

sobolev s spaces on time scales and its applications to a class of second order hamiltonian systems on time scales abstract in this paper we study the sobolev s spaces on time scales and their properties as applications we present a recent approach via variational methods and the critical point theory to obtain the existence of solutions for a class of second order hamiltonian systems on time scales by establishing a proper variational setting three existence results for systems under consideration are obtained finally three examples are presented to illustrate the feasibility and effectiveness of the existence results

modeling elastic shells immersed in fluid we describe a numerical method to simulate an elastic shell immersed in a viscous incompressible fluid the method is developed as an extension of the immersed boundary method using shell equations based on the kirchhoff love and the planar stress hypotheses a detailed derivation of the shell equations used in the numerical method is presented this derivation as well as the numerical method uses techniques of differential geometry our main motivation for developing this method is its use in constructing a comprehensive threedimensional computational model of the cochlea the inner ear the central object of study within the cochlea is the basilar membrane which is immersed in fluid and whose elastic properties rather resemble those of a shell we apply the method to a specific example which is a prototype of a piece of the basilar membrane and study the convergence of the method in this case some typical features of cochlear mechanics are already captured in this simple model in particular numerical experiments have shown a traveling wave propagating from the base to the apex of the model shell in response to external excitation in the fluid c 2004 wiley periodicals inc

existence results for dynamic sturm liouville boundary value problems via variational methods abstract several conditions ensuring existence of solutions of a dynamic sturm liouville boundary value problem are derived variational methods are utilized in the proofs an example illustrating the main results is given

validation of a simple method for representing spheres and slender bodies in an immersed boundary method for stokes flow on an unbounded domain we test the efficacy of using a single lagrangian point to represent a sphere and a one dimensional array of such points to represent a slender body in a new immersed boundary method for stokes flow a numerical parameter the spacing of the eulerian grid is used to determine the effective radius of the immersed sphere or slender body such representations are much less expensive computationally than those with two or three dimensional meshes of lagrangian points to perform this test we develop a numerical method to solve the discretized stokes equations on an unbounded eulerian grid which contains an arbitrary configuration of lagrangian points that apply force to the fluid and that move with the fluid we compare results computed with this new immersed boundary method to known results for spheres and rigid cylinders in stokes flow in r 3 we find that for certain choices of parameters the interactions with the fluid of a single lagrangian point accurately replicate those of a sphere of some particular radius independent of the location of the point with respect to the eulerian grid the interactions of a linear array of lagrangian points for certain choices of parameters accurately replicate those of a cylinder of some particular radius independent of the position and orientation of the array with respect to the eulerian grid the effective radius of the sphere and the effective radius of the cylinder turn out to be related in a simple and natural way our results suggest recipes for choosing parameters that should be useful to practitioners one surprising result is that one must not use too many lagrangian points in an array another is that the approximate delta functions traditionally used in the immersed boundary method perform much better than higher order delta functions with the same support

a fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies application to particulate flow abstract in this article we discuss a methodology that allows the direct numerical simulation of incompressible viscous fluid flow past moving rigid bodies the simulation methods rest essentially on the combination of a lagrange multiplier based fictitious domain methods which allow the fluid flow computations to be done in a fixed flow region b finite element approximations of the navier stokes equations occurring in the global model c time discretizations by operator splitting schemes in order to treat optimally the various operators present in the model the above methodology is particularly well suited to the direct numerical simulation of particulate flow such as the flow of mixtures of rigid solid particles and incompressible viscous fluids possibly non newtonian we conclude this article with the presentation of the results of various numerical experiments including the simulation of store separation for rigid airfoils and of sedimentation and fluidization phenomena in two and three dimensions

maximally transitive semigroups of n n matrices we prove that for every n 1 there exists a pair of n n matrices that generates a topologically n transitive semigroup action on kn where k r or c equivalently we construct dense 2 generator subsemigroups of gl n k for all n 1

numerical simulation of a cylinder in uniform flow in this study a virtual boundary technique is applied to the numerical simulation of stationary and moving cylinders in uniform flow this approach readily allows the imposition of a no slip boundary within the flow field by a feedback forcing term added to the momentum equations in the present work this technique is used with a high order finite difference method effectively eliminating spurious oscillations caused by the feedback forcing when used with spectrally discretized flow solvers very good agreement is found between the present calculations and previous computational and experimental results for steady and time dependent flow at low reynolds numbers

a numerical method for suspension flow abstract peskin s j comput phys 25 220 1977 immersed boundary technique is modified to give a new numerical method for studying a fluid with suspended elastic particles as before the presence of the suspended particles is transmitted to the fluid through a force density term in the fluid equations as a result one set of equations holds in the entire computational domain eliminating the need to apply boundary conditions on the surface of suspended objects the new method computes the force density by discretizing the stress strain constitutive equations for an elastic solid on a grid using data provided by clusters of lagrangian points this approach clearly specifies the material properties of the suspended objects a simple data structure for the lagrangian points makes it easy to model suspended solids with arbitrary shape and size the method is validated by comparing numerical results for elastic vibrations and particle settling in viscous fluids with theory and analysis the capability of the method to do a wide range of problems is illustrated by qualitative results for lubrication and cavity flow problems

a topological tits alternative let k be a local eld and gln k a linear group over k we prove that either contains a relatively open solvable subgroup or it contains a relatively dense free subgroup this result has applications in dynamics riemannian foliations and pronite groups

tether force constraints in stokes flow by the immersed boundary method on a periodic domain the immersed boundary method is an algorithm for simulating the interaction of immersed elastic bodies or boundaries with a viscous incompressible fluid the immersed elastic material is represented in the fluid equations by a system or field of applied forces the particular case of stokes flow with applied forces on a periodic domain involves two related mathematical complications one of these is that an arbitrary constant vector may be added to the fluid velocity and the other is the constraint that the integral of the applied force must be zero typically forces defined on a freely floating elastic immersed boundary or body satisfy this constraint but there are many important classes of forces that do not for example the so called tether forces that are used to prescribe the simulated configuration of an immersed boundary possibly in a time dependent manner typically do not sum to zero another type of force that does not have zero integral is a uniform force density that may be used to simulate an overall pressure gradient driving flow through a system we present a method for periodic stokes flow that when used with tether points admits the use of all forces irrespective of their integral over the domain a byproduct of this method is that the additive constant velocity associated with periodic stokes flow is uniquely determined indeed the additive constant is chosen at each time step so that the sum of the tether forces balances the sum of any other forces that may be applied

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

truncated newton methods and the modeling of complex immersed elastic structures truncated newton minimization methods are combined with peskin s immersed boundary method to facilitate investigation of the dynamic interaction between a viscous incompressible fluid and immersed elastic objects of complex structure applications to aquatic animal locomotion and platelet aggregation during blood clotting are presented 1993 john wiley sons inc

numerical simulation of two dimensional flows over a circular cylinder using the immersed boundary method in this work a virtual boundary method is applied to the numerical simulation of a uniform flow over a cylinder the force source term added to the two dimensional navier stokes equations guarantees the imposition of the no slip boundary condition over the body fluid interface these equations are discretized using the finite differences method the immersed boundary is represented with a finite number of lagrangian points distributed over the solid fluid interface a cartesian grid is used to solve the fluid flow equations the key idea is to propose a method to calculate the interfacial force without ad hoc constants that should usually be adjusted for the type of flow and the type of the numerical method when this kind of model is used in the present work this force is calculated using the navier stokes equations applied to the lagrangian points and then distributed over the eulerian grid the main advantage of this approach is that it enables calculation of this force field even if the interface is moving or deforming it is unnecessary to locate the eulerian grid points near this immersed boundary the lift and drag coefficients and the strouhal number calculated for an immersed cylinder are compared with previous experimental and numerical results for different reynolds numbers

weak maximum principle and accessory problem for control problems on time scales abstract in this paper we derive the first and second variations for a nonlinear time scale optimal control problem with control and state endpoints equality constraints using the first variation a first order necessary condition for weak local optimality is obtained under the form of a weak maximum principle generalizing the dubois reymond lemma to the optimal control setting and time scales a second order necessary condition in terms of the accessory problem is derived by using the nonnegativity of the second variation at all admissible directions the control problem is studied under a controllability assumption and with or without the shift in the state variable these two forms of the problem are shown to be equivalent

recurrent orbits of subgroups of local complex analytic diffeomorphisms we show recurrent phenomena for orbits of groups of local complex analytic diffeomorphisms that have a certain subgroup or image by a morphism of groups that is non virtually solvable in particular we prove that a non virtually solvable subgroup of local biholomorphisms has always recurrent orbits i e there exists an orbit contained in its set of limit points

physalis a new o n method for the numerical simulation of disperse systems potential flow of spheres this paper presents a new approach to the direct numerical simulation of potential problems with many spherical internal boundaries e g many spheres in potential flow the basic idea is to use a local analytic representation valid near the particle and to match it to an external field calculated by a standard finite difference or finite element method in this way the geometric complexity arising from the irregular relation between the particle boundary and the underlying mesh is avoided and fast solvers can be used the results suggest that the computational effort increases less than proportionally to the number of particles and additionally that meshes that would be excessively coarse as measured in terms of particle radius in a conventional calculation can be used without significant loss of accuracy in separate if preliminary work the same approach has been extended to the simulation of viscous flow about spheres and cylinders at finite reynolds numbers

a spectral element formulation of the immersed boundary method for newtonian fluids a spectral element formulation of the immersed boundary method ibm is presented the spectral element formulation se ibm is a generalisation of the finite element immersed boundary method fe ibm based on high order approximations of the fluid variables several schemes for tracking the movement of the immersed boundary are considered and a semi implicit euler scheme is shown to offer advantages in terms of accuracy and efficiency high order spectral element approximations provide improved area conservation properties of the ibm due to the incompressibility constraint being more accurately satisfied superior orders of convergence are obtained for se ibm compared with fe ibm in both l2 and h1 norms the area conservation and convergence properties of the scheme are demonstrated on a series of benchmark problems

a boundary value problem for fourth order elastic beam equations multiplicity results for a fourth order nonlinear boundary value problem are presented the proof of the main result is based on the critical point theory

2 generation of simple lie algebras and free dense subgroups of algebraic groups abstract we construct generating pairs of simple lie algebras in characteristic zero we apply this construction to exhibit infinite series of 2 generator zariski dense subgroups that are free of rank 2 of the simple algebraic groups sl n c sp n c g 2 c

product set phenomena for countable groups we develop in this paper novel techniques to analyze local combinatorial structures in product sets of two subsets of a countable group which are large with respect to certain classes of not necessarily invariant means on the group our methods heavily utilize the theory of c algebras and random walks on groups as applications of our methods we extend and quantify a series of recent results by jin bergelson furstenberg weiss beiglbock bergelson fish griesmer and di nasso lupini to general countable groups

2 d parachute simulation by the immersed boundary method parachute aerodynamics involves an interaction between the flexible elastic porous parachute canopy and the high speed airflow relative to the parachute through which the parachute falls computer simulation of parachute dynamics typically simplifies the problem in various ways e g by considering the parachute as a rigid bluff body here we avoid such simplification by using the immersed boundary ib method to study the full fluid structure interaction the ib method is generalized to handle porous immersed boundaries and the generalized method is used to study the influence of porosity on parachute stability

a cartesian grid method for solving the two dimensional streamfunction vorticity equations in irregular regions we describe a method for solving the two dimensional navier stokes equations in irregular physical domains our method is based on an underlying uniform cartesian grid and second order finite difference finite volume discretizations of the streamfunction vorticity equations geometry representing stationary solid obstacles in the flow domain is embedded in the cartesian grid and special discretizations near the embedded boundary ensure the accuracy of the solution in the cut cells along the embedded boundary we determine a distribution of vorticity sources needed to impose the no slip flow conditions this distribution appears as a right hand side term in the discretized fluid equations and so we can use fast solvers to solve the linear systems that arise to handle the advective terms we use the high resolution algorithms in clawpack we show that our stokes solver is second order accurate for steady state solutions and that our full navier stokes solver is between first and second order accurate and reproduces results from well studied benchmark problems in viscous fluid flow finally we demonstrate the robustness of our code on flow in a complex domain

uniform independence in linear groups we show that for any non virtually solvable finitely generated group of matrices over any field there is an integer m such that given an arbitrary finite generating set for the group one may find two elements a and b that are both products of at most m generators such that a and b are free generators of a free subgroup this uniformity result improves the original statement of the tits alternative

simulations of valveless pumping in an open elastic tube mathematical models and numerical simulations of flows driven by pumping without valves valveless pumping are presented this work has been originally motivated by a biomedical objective to explain the complicated valveless blood flow mechanism in the circulation for instance the fetal circulation before the formation of valves and blood circulation during cardiopulmonary resuscitation a mathematical model of valveless pumping either in a closed loop system or in an open system consists of a couple of tubes with different elasticities or radii in this work we develop new models which are not necessarily connected by two different materials only one open elastic tube and an open elastic tube contained in a closed almost rigid tank although only one soft material is used or the soft and rigid materials are not connected we have observed the existence of a net flow driven by the periodic compress and release action and the important features of valveless pumping that have been observed in earlier models or experiments we have also shown that there exist mostly one directional net flows inside the elastic tube if a long and thin tube is considered and the pumping is applied to the short portion near the edge of the tube this might explain why most earlier reports on the physical experiments could not easily observe the existence of both directional net flows depending on the parameters such as frequency another important result is that the direction and the magnitude of a net flow can be explained by the sign and the amount of power which is work done on the fluid by the fluid pressure and the elastic wall over one period respectively a new feature in this work is that only one elastic material is sufficient to show the existence of a net flow in a valveless pump system because of a simple structure in our new model it is much easier to construct a valveless pump system applicable to real world applications such as a microelectromechanical system device

a critical point approach for a second order dynamic sturm liouville boundary value problem with p laplacian abstract in this paper we give conditions guaranteeing the existence of at least three solutions for a second order dynamic sturm liouville boundary value problem involving two parameters in the proofs of the results we utilize critical point theory and variational methods in addition an example is given in order to illustrate our results

cocycle superrigidity for translation actions of product groups let g be either a profinite or a connected compact group and be finitely generated dense subgroups assuming that the left translation action of on g is strongly ergodic we prove that any cocycle for the left right translation action of on g with values in a countable group is virtually cohomologous to a group homomorphism moreover we prove that the same holds if g is a not necessarily compact connected simple lie group provided that contains an infinite cyclic subgroup with compact closure we derive several applications to oe and w superrigidity in particular we obtain the first examples of compact actions of f2 f2 which are w superrigid

on dense free subgroups of lie groups abstract we give a method for constructing dense and free subgroups in real lie groups in particular we show that any dense subgroup of a connected semisimple real lie group g contains a free group on two generators which is still dense in g and that any finitely generated dense subgroup in a connected non solvable lie group h contains a dense free subgroup of rank 2 dim h this answers a question of carriere and ghys and it gives an elementary proof to a conjecture of connes and sullivan on amenable actions which was first proved by zimmer

smoothed profile method for particulate flows error analysis and simulations abstract we re formulate and analyze a new method for particulate flows the so called smoothed profile method spm first proposed in y nakayama r yamamoto simulation method to resolve hydrodynamic interactions in colloidal dispersions phys rev e 71 2005 036707 which uses a fixed computational mesh without conformation to the geometry of the particles the method represents the particles by certain smoothed profiles to construct a body force term added into the navier stokes equations spm imposes accurately and efficiently the proper conditions at the particle fluid interface in particular while the original method employs a fully explicit time integration scheme we develop a high order semi implicit splitting scheme which we implement in the context of spectral hp element discretization first we show that the modeling error of spm has a non monotonic dependence on the time step size t it is a function of t where is the kinematic viscosity of fluid and is the interface thickness of the smoothed profile subsequently we present several steady and unsteady simulations including flow past 3d complex shaped particles and compare against direct numerical simulations and the force coupling method fcm

non differentiable functionals and applications to elliptic problems with discontinuous nonlinearities abstract multiple critical points theorems for non differentiable functionals are established applications both to elliptic variational hemivariational inequalities and eigenvalue problems with discontinuous nonlinearities are then presented

a front tracking method for viscous incompressible multi fluid flows abstract a method to simulate unsteady multi fluid flows in which a sharp interface or a front separates incompressible fluids of different density and viscosity is described the flow field is discretized by a conservative finite difference approximation on a stationary grid and the interface is explicitly represented by a separate unstructured grid that moves through the stationary grid since the interface deforms continuously it is necessary to restructure its grid as the calculations proceed in addition to keeping the density and viscosity stratification sharp the tracked interface provides a natural way to include surface tension effects both two and three dimensional full numerical simulations of bubble motion are presented

analysis of stiffness in the immersed boundary method and implications for time stepping schemes the immersed boundary method is known to exhibit a high degree of numerical stiffness associated with the interaction of immersed elastic fibres with the surrounding fluid we perform a linear analysis of the underlying equations of motion for immersed fibres and identify a discrete set of fibre modes which are associated solely with the presence of the fibre this work generalises our results in a previous paper 1995 siam j appl math 55 1577 by incorporating the effect of spreading the singular fibre force over a finite smoothing radius corresponding to the approximate delta function used in the immersed boundary method we investigate the stability of the fibre modes their stiffness and their dependence on the problem parameters focusing on the influence of smoothing we then extend the analytical results to include the effect of time discretisation and draw conclusions about the time step restrictions on various explicit schemes as well as the convergence rates for an iterative semi implicit method we draw comparisons with computations and show how the results can be applied to help in choosing alternate time stepping schemes that are specially tailored to handle the stiffness in immersed fibres in particular we present numerical results that show how fully explicit runge kutta schemes perform in comparison with the best of the semi implicit schemes currently in use

improved volume conservation in the computation of flows with immersed elastic boundaries this paper introduces a recipe for the construction of a finite difference divergence operator the coefficients of which are completely determined once an interpolation scheme has been chosen substitution of this divergence operator for the previously used divergence based on central differences makes a dramatic improvement in the overall volume conservation that is observed in an immersed boundary computation this improvement is particularly important for computations in which an elastic boundary separates chambers containing fluid at substantially different pressures a situation which is prominent in cardiac fluid dynamics during the contraction of the ventricles

on the growth of l 2 invariants for sequences of lattices in lie groups we study the asymptotic behaviour of betti numbers twisted torsion and other spectral invariants of sequences of locally symmetric spaces our main results are uniform versions of the degeorge wallach theorem of a theorem of delorme and various other limit multiplicity theorems a basic idea is to adapt the notion of benjamini schramm convergence bs convergence originally introduced for sequences of finite graphs of bounded degree to sequences of riemannian manifolds and analyze the possible limits we show that bs convergence of locally symmetric spaces g k implies convergence in an appropriate sense of the normalized relative plancherel measures associated to l2 g this then yields convergence of normalized multiplicities of unitary representations betti numbers and other spectral invariants on the other hand when the corresponding lie group g is simple and of real rank at least two we prove that there is only one possible bs limit i e when the volume tends to infinity locally symmetric spaces always bs converge to their universal cover g k this leads to various general uniform results when restricting to arbitrary sequences of congruence covers of a fixed arithmetic manifold we prove a strong quantitative version of bs convergence which in turn implies upper estimates on the rate of convergence of normalized betti numbers in the spirit of sarnak xue an important role in our approach is played by the notion of invariant random subgroups for higher rank simple lie groups g we exploit rigidity theory and in particular the nevo stuck zimmer theorem and kazhdan s property t to obtain a complete understanding of the space of irs s of g 2017 department of mathematics princeton university

sturm liouville eigenvalue problems on time scales for sturm liouville eigenvalue problems on time scales with separated boundary conditions we give an oscillation theorem and establish rayleigh s principle our results not only unifly the corresponding theories for differential and difference equations but are also new in the discrete case

an immersed boundary method based on the lattice boltzmann approach in three dimensions with application the immersed boundary ib method originated by peskin has been popular in modeling and simulating problems involving the interaction of a flexible structure and a viscous incompressible fluid the navier stokes n s equations in the ib method are usually solved using numerical methods such as fft and projection methods here in our work the n s equations are solved by an alternative approach the lattice boltzmann method lbm compared to many conventional n s solvers the lbm can be easier to implement and more convenient for modeling additional physics in a problem this alternative approach adds extra versatility to the immersed boundary method in this paper we discuss the use of a 3d lattice boltzmann model d3q19 within the ib method we use this hybrid approach to simulate a viscous flow past a flexible sheet tethered at its middle line in a 3d channel and determine a drag scaling law for the sheet our main conclusions are 1 the hybrid method is convergent with first order accuracy which is consistent with the immersed boundary method in general 2 the drag of the flexible sheet appears to scale with the inflow speed which is in sharp contrast with the square law for a rigid body in a viscous flow

numerical simulation of 3d bubbles rising in viscous liquids using a front tracking method the rise of bubbles in viscous liquids is not only a very common process in many industrial applications but also an important fundamental problem in fluid physics an improved numerical algorithm based on the front tracking method originally proposed by tryggvason and his co workers has been validated against experiments over a wide range of intermediate reynolds and bond numbers using an axisymmetric model j hua j lou numerical simulation of bubble rising in viscous liquid j comput phys 22 2007 769 795 in the current paper this numerical algorithm is further extended to simulate 3d bubbles rising in viscous liquids with high reynolds and bond numbers and with large density and viscosity ratios representative of the common air water two phase flow system to facilitate the 3d front tracking simulation mesh adaptation is implemented for both the front mesh on the bubble surface and the background mesh on the latter mesh the governing navier stokes equations for incompressible newtonian flow are solved in a moving reference frame attached to the rising bubble specifically the equations are solved using a finite volume scheme based on the semi implicit method for pressure linked equations simple algorithm and it appears to be robust even for high reynolds numbers and high density and viscosity ratios the 3d bubble surface is tracked explicitly using an adaptive unstructured triangular mesh the numerical model is integrated with the software package paramesh a block based adaptive mesh refinement amr tool developed for parallel computing paramesh allows background mesh adaptation as well as the solution of the governing equations in parallel on a supercomputer further peskin distribution function is applied to interpolate the variable values between the front and the background meshes detailed sensitivity analysis about the numerical modeling algorithm has been performed the current model has also been applied to simulate a number of cases of 3d gas bubbles rising in viscous liquids e g air bubbles rising in water simulation results are compared with experimental observations both in aspect of terminal bubble shapes and terminal bubble velocities in addition we applied this model to simulate the interaction between two bubbles rising in a liquid which illustrated the model s capability in predicting the interaction dynamics of rising bubbles