design of capacity approaching irregular low density parity check codes we design low density parity check ldpc codes that perform at rates extremely close to the shannon capacity the codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides our theoretical analysis of the codes is based on the work of richardson and urbanke see ibid vol 47 no 2 p 599 618 2000 assuming that the underlying communication channel is symmetric we prove that the probability densities at the message nodes of the graph possess a certain symmetry using this symmetry property we then show that under the assumption of no cycles the message densities always converge as the number of iterations tends to infinity furthermore we prove a stability condition which implies an upper bound on the fraction of errors that a belief propagation decoder can correct when applied to a code induced from a bipartite graph with a given degree distribution our codes are found by optimizing the degree structure of the underlying graphs we develop several strategies to perform this optimization we also present some simulation results for the codes found which show that the performance of the codes is very close to the asymptotic theoretical bounds

a remark on the two dimensional water wave problem with surface tension abstract we consider the motion of a two dimensional interface between air above and an irrotational incompressible inviscid infinitely deep water below with surface tension present we propose a new way to reduce the original problem into an equivalent quasilinear system which is related to the interface s tangent angle and a quantity related to the difference of tangential velocities of the interface in the lagrangian and the arc length coordinates the new way is relatively simple because it involves only taking differentiation and the real and the imaginary parts then if assuming that waves are periodic we establish a priori energy inequality

an implicit immersed boundary method for three dimensional fluid membrane interactions we present an implicit immersed boundary method for the incompressible navier stokes equations capable of handling three dimensional membrane fluid flow interactions the goal of our approach is to greatly improve the time step by using the jacobian free newton krylov method jfnk to advance the location of the elastic membrane implicitly the most attractive feature of this jacobian free approach is newton like nonlinear convergence without the cost of forming and storing the true jacobian the generalized minimal residual method gmres which is a widely used krylov subspace iterative method is used to update the search direction required for each newton iteration each gmres iteration only requires the action of the jacobian in the form of matrix vector products and therefore avoids the need of forming and storing the jacobian matrix explicitly once the location of the boundary is obtained the elastic forces acting at the discrete nodes of the membrane are computed using a finite element model we then use the immersed boundary method to calculate the hydrodynamic effects and fluid structure interaction effects such as membrane deformation the present scheme has been validated by several examples including an oscillatory membrane initially placed in a still fluid capsule membranes in shear flows and large deformation of red blood cells subjected to stretching force

a boundary integral method for simulating the dynamics of inextensible vesicles suspended in a viscous fluid in 2d we present a new method for the evolution of inextensible vesicles immersed in a stokesian fluid we use a boundary integral formulation for the fluid that results in a set of nonlinear integro differential equations for the vesicle dynamics the motion of the vesicles is determined by balancing the non local hydrodynamic forces with the elastic forces due to bending and tension numerical simulations of such vesicle motions are quite challenging on one hand explicit time stepping schemes suffer from a severe stability constraint due to the stiffness related to high order spatial derivatives and a milder constraint due to a transport like stability condition on the other hand an implicit scheme can be expensive because it requires the solution of a set of nonlinear equations at each time step we present two semi implicit schemes that circumvent the severe stability constraints on the time step and whose computational cost per time step is comparable to that of an explicit scheme we discretize the equations by using a spectral method in space and a multistep third order accurate scheme in time we use the fast multipole method fmm to efficiently compute vesicle vesicle interaction forces in a suspension with a large number of vesicles we report results from numerical experiments that demonstrate the convergence and algorithmic complexity properties of our scheme

a numerical study of the self similar solutions of the schrodinger map we present a numerical study of the self similar solutions of the localized induction approximation of a vortex filament these self similar solutions which are explicitly known constitute a one parameter family and develop a singularity in finite time we study a number of boundary conditions that allow us to reproduce the mechanism of singularity formation some related questions are also considered

radial fingering in a hele shaw cell a weakly nonlinear analysis abstract the saffman taylor viscous fingering instability occurs when a less viscous fluid displaces a more viscous one between narrowly spaced parallel plates in a hele shaw cell experiments in radial flow geometry form fan like patterns in which fingers of different lengths compete spread and split our weakly nonlinear analysis of the instability predicts these phenomena which are beyond the scope of linear stability theory finger competition arises through enhanced growth of sub harmonic perturbations while spreading and splitting occur through the growth of harmonic modes nonlinear mode coupling enhances the growth of these specific perturbations with appropriate relative phases as we demonstrate through a symmetry analysis of the mode coupling equations we contrast mode coupling in radial flow with rectangular flow geometry

an efficient numerical method for studying interfacial motion in two dimensional creeping flows we present new methods for computing the motion of two dimensional closed interfaces in a slow viscous flow the interfacial velocity is found through the solution to an integral equation whose analytic formulation is based on complex variable theory for the biharmonic equation the numerical methods for solving the integral equations are spectrally accurate and employ a fast multipole based iterative solution procedure which requires only o n operations where n is the number of nodes in the discretization of the interface the interface is described spectrally and we use evolution equations that preserve equal spacing in arclength of the marker points a small scale decomposition is performed to extract the dominant term in the evolution of the interface and we show that this dominant term leads to a cfl type stability constraint when in an equal arclength frame this term is linear and we show that implicit time integration schemes that are explicit in fourier space can be formulated we verify this analysis through several numerical examples

contour dynamics of incompressible 3 d fluids in a porous medium with different densities we consider the problem of the evolution of the interface given by two incompressible fluids through a porous medium which is known as the muskat problem and in two dimensions it is mathematically analogous to the two phase hele shaw cell we focus on a fluid interface given by a jump of densities being the equation of the evolution obtained using darcy s law we prove local well posedness when the smaller density is above stable case and in the unstable case we show ill posedness

the zero surface tension limit of three dimensional interfacial darcy flow abstract we study the zero surface tension limit of three dimensional interfacial darcy flow we start with a proof of well posedness of three dimensional interfacial darcy flow for any positive value of the surface tension coefficient the primary tool for this well posedness proof is an energy estimate the time of existence for these solutions will in general go to zero with the surface tension parameter however in the case that a stability condition is satisfied by the initial data we prove an additional energy estimate establishing that the time of existence can be made uniform in the surface tension parameter then an additional estimate allows the limit to be taken as surface tension vanishes demonstrating that three dimensional interfacial darcy flow without surface tension is the limit of three dimensional interfacial darcy flow with surface tension as surface tension vanishes this provides a new proof of existence of solutions for the problem without surface tension

on the utility of laguerre series for the envelope pdf in multipath fading channels it is well known that multipath fading significantly affects the performance of communication systems in order to incorporate the impact of this random phenomenon on system analysis and design in many cases we need to calculate the probability density function pdf of the received signal envelope in multipath fading channels in this paper we consider a general multipath fading channel with arbitrary number of paths where the amplitudes of multipath components are arbitrary correlated positive random variables independent of phases whereas the phases are independent and identically distributed random variables with uniform distributions since the integral form of the envelope pdf for such a general channel model is too complicated to be used for analytic calculations we propose two infinite expansions for the pdf a laguerre series and a power series based on the tight uniform upper bounds on the truncation error of these two infinite series we show that the laguerre series is superior to the power series due to the fact that for a fixed number of terms it yields a smaller truncation error this laguerre series with a finite number of terms which expresses the envelope pdf just in terms of simple polynomial exponential kernels is particularly useful for mathematical performance prediction of communication systems in those indoor and outdoor multipath propagation environments where the number of strong multipath components is small

self similar solutions for the 1 d schrodinger map on the hyperbolic plane in this paper we will study self similar solutions for x t x x ss the equivalent in the minkowski 3 space mathbb r 2 1 to the localized induction approximation flow trying to adapt some results given by gutierrez rivas and vega we will show the existence of a one parameter family of smooth solutions developing a corner in finite time the main difference with respect to the euclidean case studied by those authors will be the proof of the boundedness of t e 1 and e 2 the equivalents of t b and n in mathbb r 2 1

joint physical layer coding and network coding for bidirectional relaying we consider a communication system where two transmitters wish to exchange information through a central relay the transmitter and relay nodes exchange data over synchronized average power constrained additive white gaussian noise channels with a real input with signal to noise ratio snr of snr an upper bound on the capacity is 1 2 log 1 snr bits per transmitter per use of the multiple access phase and broadcast phase of the bidirectional relay channel we show that using lattice codes and lattice decoding we can obtain a rate of 1 2 log 1 2 snr bits per transmitter which is essentially optimal at high snr the main idea is to decode the sum of the codewords modulo a lattice at the relay followed by a broadcast phase which performs slepian wolf coding we also show that if the two transmitters use identical lattices with minimum angle decoding we can achieve the same rate of 1 2 log 1 2 snr the proposed scheme can be thought of as a joint physical layer network layer code which outperforms other recently proposed analog network coding schemes

short note moore s law and the saffman taylor instability ten years ago hou lowengrub and shelley 4 published a state of the art boundary integral simulation of a classical viscous fingering problem the saffman taylor instability 12 in terms of complexity and level of detail those computations 4 are still among the most ramified and accurately computed interfacial instability patterns that have appeared in the literature since 1994 the computational power of a standard workstation has increased a hundredfold as predicted by moore s law 7 the purpose of this note is to consider moore s law and its consequences in computational science and in particular its impact on studying the saffman taylor instability we illustrate moore s law and fast algorithms in action by presenting the worlds largest viscous fingering simulation to date viscous fingering is one of the fundamental interfacial instabilities in fluid dynamics perturbations to an expanding circular air bubble displacing a viscous fluid in a thin gap flow device become unstable resulting in intricate densely branched interfacial patterns the viscous fingering problem is governed by the three dimensional incompressible navier stokes equations with moving free boundaries at the fluid air interface without simplification this problem is computationally intractable and will remain so for the foreseeable future the simulations in this note build upon decades of advances in mathematical modeling and numerical methods i the fluid dynamics in a thin gap is reduced from the threedimensional navier stokes equations by asymptotic analysis to a darcy s law 12

a moving overset grid method for interface dynamics applied to non newtonian hele shaw flow we present a novel moving overset grid scheme for the accurate and efficient long time simulation of an air bubble displacing a non newtonian fluid in the prototypical thin film device the hele shaw cell we use a two dimensional generalization of darcy s law that accounts for shear thinning of a non newtonian fluid in the limit of weak shear thinning the pressure is found from a ladder of two linear elliptic boundary value problems each to be solved in the whole fluid domain a moving body fitted grid is used to resolve the flow near the interface while most of the fluid domain is covered with a fixed cartesian grid our use of body conforming grids reduces grid anisotropy effects and allows the accurate modeling of boundary conditions

interface evolution the hele shaw and muskat problems we study the dynamics of the interface between two incompressible 2 d ows where the evolution equation is obtained from darcy s law the free boundary is given by the discontinuity among the densities and viscosities of the uids this physical scenario is known as the two dimensional muskat problem or the two phase hele shaw ow we prove local existence in sobolev spaces when initially the dierence of the gradients of the pressure in the normal direction has the proper sign an assumption which is also known as the rayleigh taylor condition

distribution of the sum of two sine waves and gaussian noise the cumulative distribution of the envelope of the sum of two sine waves and narrowband gaussian noise the difference of the phases of the sine waves being uniformly distributed over 0 2 pi is expressed as an integral that is evaluated by numerical quadrature approximation for the distribution valid in the limit of large signal to noise ratio are derived the probability density function of the envelope is similarly treated

a method to compute statistics of large noise induced perturbations of nonlinear schrodinger solitons we describe in detail the application of importance sampling to numerical simulations of large noise induced perturbations in soliton based optical transmission systems governed by the nonlinear schrodinger equation the method allows one to concentrate the samples in monte carlo simulations around those noise realizations that are most likely to produce the large pulse deformations connected with errors and it yields computational speedups of several orders of magnitude over standard monte carlo simulations we demonstrate the method by using it to calculate the probability density functions associated with pulse amplitude frequency timing and phase fluctuations in a prototypical soliton based communication system

density evolution for nonbinary ldpc codes under gaussian approximation this paper extends the work on density evolution for binary low density parity check ldpc codes with gaussian approximation to ldpc codes over gf q we first generalize the definition of channel symmetry for nonbinary inputs to include q ary phase shift keying psk modulated channels for prime q and binary modulated channels for q that is a power of 2 for the well defined q ary input symmetric output channel we prove that under the gaussian assumption the density distribution for messages undergoing decoding is fully characterized by q 1 quantities assuming uniform edge weights we further show that the density of messages computed by the check node decoder cnd is fully defined by a single number we then present the approximate density evolution for regular and irregular ldpc codes and show that the q 1 dimensional integration involved can be simplified using a dimensionality reduction algorithm for the important case of q 2p through application of approximate density evolution and linear programming we optimize the degree distribution of ldpc codes over gf 3 and gf 4 the optimized irregular ldpc codes demonstrate performance close to the shannon capacity for long codewords we also design gf q codes for high order modulation by using the idea of a channel adapter we find that codes designed in this fashion outperform those optimized specifically for the binary additive white gaussian noise awgn channel for a short codewords and a spectral efficiency of 2 bits per channel use b cu

a maximum principle for the muskat problem for fluids with different densities we study the fluid interface problem given by two incompressible fluids with different densities evolving by darcy s law this scenario is known as the muskat problem for fluids with the same viscosities being in two dimensions mathematically analogous to the two phase hele shaw cell we prove in the stable case the denser fluid is below a maximum principle for the l norm of the free boundary

new cartesian grid methods for interface problems using the finite element formulation new finite element methods based on cartesian triangulations are presented for two dimensional elliptic interface problems involving discontinuities in the coefficients the triangulations in these methods do not need to fit the interfaces the basis functions in these methods are constructed to satisfy the interface jump conditions either exactly or approximately both non conforming and conforming finite element spaces are considered corresponding interpolation functions are proved to be second order accurate in the maximum norm the conforming finite element method has been shown to be convergent with cartesian triangulations these new methods can be used as finite difference methods numerical examples are provided to support the methods and the theoretical analysis

semi implicit surface tension formulation with a lagrangian surface mesh on an eulerian simulation grid we present a method for applying semi implicit forces on a lagrangian mesh to an eulerian discretization of the navier stokes equations in a way that produces a sparse symmetric positive definite system the resulting method has semi implicit and fully coupled viscosity pressure and lagrangian forces we apply our new framework for forces on a lagrangian mesh to the case of a surface tension force which when treated explicitly leads to a tight time step restriction by applying surface tension as a semi implicit lagrangian force the resulting method benefits from improved stability and the ability to take larger time steps the resulting discretization is also able to maintain parasitic currents at low levels

stability of the self similar dynamics of a vortex filament in this paper we continue our investigation of self similar solutions of the vortex filament equation also known as the binormal flow or the localized induction equation our main result is the stability of the self similar dynamics of small perturbations of a given self similar solution the proof relies on finding precise asymptotics in space and time for the tangent and the normal vectors of the perturbations a main ingredient in the proof is the control of the evolution of weighted norms for a cubic one dimensional schrodinger equation connected to the binormal flow by hasimoto s transform

stable methods for vortex sheet motion in the presence of surface tension boundary integral techniques provide a convenient way to study the evolution of an interface between inviscid liquids several studies have revealed that standard numerical approximations tend to lead to unstable methods and various remedies have been introduced and tested in this paper we conduct a stability analysis of the linearized equations with a particular objective in mind the determination of how the discrete system fails to capture the physical dispersion relation precisely for the available discrete modes we discover two reasons for the typical failure in numerical discretizations one is the inability of the mesh to represent the vorticity created by surface tension effects on the finest scale and the other is the inaccuracies in the evaluation of the boundary integral for the velocity with the insight gained from our linear analysis we propose a new method that is spectrally accurate and linearly stable further the exact dispersion relation is obtained for all the available discrete modes numerical tests suggest that the method is also stable in the nonlinear regime however our method runs into difficulties generic to methods based on lagrangian motion the markers accumulate near a stagnation point on the interface forcing us to use an ever decreasing time step in our explicit method we introduce a redistribution of markers to overcome this difficulty when we redistribute according to equal arclength we find excellent agreement with a method based on preserving equal spacing in arclength

design of regular 2 d sub c ldpc codes over gf q using their binary images in this paper a method to design regular 2 dc ldpc codes over gf q with both good waterfall and error floor properties is presented based on the algebraic properties of their binary image first the algebraic properties of rows of the parity check matrix h associated with a code are characterized and optimized to improve the waterfall then the algebraic properties of cycles and stopping sets associated with the underlying tanner graph are studied and linked to the global binary minimum distance of the code finally simulations are presented to illustrate the excellent performance of the designed codes

a well posed numerical method to track isolated conformal map singularities in hele shaw flow we present a new numerical method for calculating an evolving 2d hele shaw interface when surface tension effects are neglected in the case where the flow is directed from the less viscous fluid into the more viscous fluid the motion of the interface is ill posed small deviations in the initial condition will produce significant changes in the ensuing motion this situation is disastrous for numerical computation as small roundoff errors can quickly lead to large inaccuracies in the computed solution our method of computation is most easily formulated using a conformal map from the fluid domain into a unit disk the method relies on analytically continuing the initial data and equations of motion into the region exterior to the disk where the evolution problem becomes well posed the equations are then numerically solved in the extended domain the presence of singularities in the conformal map outside of the disk introduces specific structures along the fluid interface our method can explicitly track the location of isolated pole and branch point singularities allowing us to draw connections between the development of interfacial patterns and the motion of singularities as they approach the unit disk in particular we are able to relate physical features such as finger shape side branch formation and competition between fingers to the nature and location of the singularities the usefulness of this method in studying the formation of topological singularities self intersections of the interface is also pointed out

on the pdf of the sum of random vectors there are various cases in physics and engineering sciences especially communications where one requires the envelope probability density function pdf of the sum of several random sinusoidal signals according to the correspondence between a random sinusoidal signal and a random vector the sum of random vectors can be considered as an abstract mathematical model for the above sum now it is desired to obtain the pdf of the length of the resulting vector considering the common and reasonable assumption of uniform distributions for the angles of the vectors many researchers have obtained the pdf of the length of the resulting vector only for special cases however in this paper the pdf is obtained for the most general case in which the lengths of vectors are arbitrary dependent random variables this pdf is in the form of a definite integral which may be inappropriate for analytic manipulations and numerical computations so an appropriate infinite laguerre expansion is also derived finally the results are applied to solve a typical example in computing the scattering cross section of random scatterers

channel coding with multilevel phase signals a coding technique is described which improves error performance of synchronous data links without sacrificing data rate or requiring more bandwidth this is achieved by channel coding with expanded sets of multilevel phase signals in a manner which increases free euclidean distance soft maximum likelihood ml decoding using the viterbi algorithm is assumed following a discussion of channel capacity simple hand designed trellis codes are presented for 8 phase shift keying psk and 16 quadrature amplitude shift keying qask modulation these simple codes achieve coding gains in the order of 3 4 db it is then shown that the codes can be interpreted as binary convolutional codes with a mapping of coded bits into channel signals which we call mapping by set partitioning based on a new distance measure between binary code sequences which efficiently lower bounds the euclidean distance between the corresponding channel signal sequences a search procedure for more powerful codes is developed codes with coding gains up to 6 db are obtained for a variety of multilevel phase modulation schemes simulation results are presented and an example of carrier phase tracking is discussed

a boundary integral method for computing the dynamics of an epitaxial island in this paper we present a boundary integral method for computing the quasi steady evolution of an epitaxial island the problem consists of an adatom diffusion equation with desorption on terrace and a kinetic boundary condition at the step island boundary the normal velocity for step motion is determined by a two sided flux the integral formulation of the problem involves both double and single layer potentials due to the kinetic boundary condition numerical tests on a growing shrinking circular or a slightly perturbed circular island are in excellent agreement with the linear analysis demonstrating that the method is stable efficient and spectrally accurate in space nonlinear simulations for the growth of perturbed circular islands show that sharp tips and flat edges will form during growth instead of the usual dense branching morphology seen throughout physical and biological systems driven out of equilibrium in particular bales zangwill instability is manifested in the form of wave like fronts meandering instability around the tip regions the numerical techniques presented here can be applied generally to a class of free moving boundary problems in physical and biological science

boundary integral techniques for multi connected domains abstract several boundary integral techniques are available for the computation of the solution to laplace s equation in multi connected domains however for cases where the domain is changing such as incompressible inviscid fluid flow with free surfaces iterative methods are highly attractive the paper describes one such formulation and tests it on circular and elliptic annuli it is necessary to use interpolated quadrature points to maintain accuracy when regions of the annuli are thin

pairwise check decoding for ldpc coded two way relay block fading channels partial decoding has the potential to achieve a larger capacity region than full decoding in two way relay twr channels existing partial decoding realizations are however designed for gaussian channels and with a static physical layer network coding plnc in this paper we propose a new solution for joint network coding and channel decoding at the relay called pairwise check decoding pcd for low density parity check ldpc coded twr system over block fading channels the main idea is to form a check relationship table check relation tab for the superimposed ldpc coded packet pair in the multiple access ma phase in conjunction with an adaptive plnc mapping in the broadcast bc phase using pcd we then present a partial decoding method two stage closest neighbor clustering with pcd ts cnc pcd with the aim of minimizing the worst pairwise error probability moreover we propose the minimum correlation optimization mco for selecting the better check relation tabs simulation results confirm that the proposed ts cnc pcd offers a sizable gain over the conventional xor with belief propagation bp in fading channels

stability analysis of finite difference pseudospectral and fourier galerkin approximations for time dependent problems we consider finite difference pseudospectral and fourier galerkin methods for the approximate solution of time dependent problems the paper provides a unified framework for the stability analysis of all three discrete methods in particular the problem of stability for highly accurate stencils is studied in some detail

nonlinear simulations of elastic fingering in a hele shaw cell this work is motivated by the recent experiments of two reacting fluids in a hele shaw cell podgorski et al 2007 and associated linear stability analysis of a curvature weakening model he et al 2012 unlike the classical hele shaw problem posed for moving interfaces with surface tension the curvature weakening model is concerned with a newly produced gel like phase that stiffens the interface thus the interface is modeled as an elastic membrane with curvature dependent rigidity that reflects geometrically induced breaking of intermolecular bonds here we are interested in exploring the long time interface dynamics in the nonlinear regime we perform simulations using a spectrally accurate boundary integral method together with a rescaling scheme to dramatically speed up the intrinsically slow evolution of the interface we find curvature weakening inhibits tip splitting and promotes side branching morphology at long times numerical results reveal that there exist nonlinear stable self similarly evolving morphologies

a boundary integral approach to unstable solidification we consider the supercooled stefan problem with a general anisotropic curvature and velocity dependent boundary condition on the moving interface we present numerical methods based on an integral equation formulation and including a new algorithm for moving curves with curvature dependent velocity these methods compute a periodic interface with o t accuracy where t is the time step previous work has been limited to short time spans and achieved slightly less than o t12 accuracy accurate numerical results are seen to agree with the predictions of linear stability theory this agreement has eluded previous authors because their numerical methods suffered from grid effects and their linear stability theory was incorrect we study the long time evolution of an unstable interface our computations exhibit the beginnings of a sidebranching instability when the boundary condition includes anisotropy and tip splitting in the isotropic case

a boundary integral formulation of quasi steady fluid wetting this paper considers the motion of a liquid droplet on a solid surface when capillary relaxation is much faster than the motion of the contact line the fluid geometry and its dynamical evolution can be characterized in terms of the contact line alone this problem can be cast in terms of boundary integral equations involving a dirichlet neumann map coupled to a volume conservation constraint a computational method for this formulation is described which has two principal advantages over approaches which track the entire free surface 1 only the curve which describes the contact line is computed and 2 the resulting method exhibits only mild numerical stiffness obviating the need for implicit timestepping effects of both capillary and body forces are considered computational examples include surface inhomogeneities topological transitions and cusp formation

removing the stiffness of elastic force from the immersed boundary method for the 2d stokes equations the immersed boundary method has evolved into one of the most useful computational methods in studying fluid structure interaction on the other hand the immersed boundary method is also known to suffer from a severe timestep stability restriction when using an explicit time discretization in this paper we propose several efficient semi implicit schemes to remove this stiffness from the immersed boundary method for the two dimensional stokes flow first we obtain a novel unconditionally stable semi implicit discretization for the immersed boundary problem using this unconditionally stable discretization as a building block we derive several efficient semi implicit schemes for the immersed boundary problem by applying the small scale decomposition to this unconditionally stable discretization our stability analysis and extensive numerical experiments show that our semi implicit schemes offer much better stability property than the explicit scheme unlike other implicit or semi implicit schemes proposed in the literature our semi implicit schemes 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 but with a much better stability property

on the probability density function of the squared envelope of a sum of random phase vectors a recursive solution is presented for the probability density function of the sum of n independent random phase vectors the recursion parameter is n the number of vectors in the sum this approach allows one to rapidly compute a complete set of these density functions for values of n 2 3 n max where n max typically corresponds to the total number of system users in a multiuser fhma mfsk application or one plus the total number of jamming tones in an fh mfsk spread spectrum application such evaluations are necessary for exact calculation of the average error probability performance of such systems

rayleigh taylor breakdown for the muskat problem with applications to water waves the muskat problem models the evolution of the interface between two different fluids in porous media the rayleigh taylor condition is natural to reach linear stability of the muskat problem we show that the rayleigh taylor condition may hold initially but break down in finite time as a consequence of the method used we prove the existence of water waves turning

the zero surface tension limit of two dimensional water waves we consider two dimensional water waves of infinite depth periodic in the horizontal direction it has been proven by wu in the slightly different nonperiodic setting that solutions to this initial value problem exist in the absence of surface tension recently ambrose has proven that solutions exist when surface tension is taken into account in this paper we provide a shorter more elementary proof of existence of solutions to the water wave initial value problem both with and without surface tension our proof requires estimating the growth of geometric quantities using a renormalized arc length parametrization of the free surface and using physical quantities related to the tangential velocity of the free surface using this formulation we find that as surface tension goes to 0 the water wave without surface tension is the limit of the water wave with surface tension far from being a simple adaptation of previous works our method requires a very original choice of variables these variables turn out to be physical and well adapted to both cases c

computation of axisymmetric suction flow through porous media in the presence of surface tension abstract the effect of small surface tension on a class of axisymmetric flows with suction is studied numerically the dynamic evolution of a blob of incompressible viscous fluid surrounded by air and drawn into an interior sink is considered the velocity field of the viscous fluid is assumed to satisfy darcy s law and thus the motion is that of a flow through porous media the fluid interface motion is computed using a highly accurate boundary integral method this method combines recent numerical techniques to achieve efficient and high order space and time discretizations for this type of flow through accurate computations it is shown that in the presence of small surface tension the dynamic behavior of the axisymmetric flows is very similar to that of the two dimensional hele shaw counterparts a long finger develops in the fluid interface and forms a cone singularity when it reaches the sink before all the fluid is sucked out the finger bulges and develops a neck for sufficiently small surface tension the bulge neck formation is enhanced by the additional component of the mean curvature in the three dimensional flow however its effect is not strong enough to cause the interface to pinch off at the neck for the data considered here

decoding algorithms for nonbinary ldpc codes over gf q in this letter we address the problem of decoding nonbinary low density parity check ldpc codes over finite fields gf q with reasonable complexity and good performance in the first part of the letter we recall the original belief propagation bp decoding algorithm and its fourier domain implementation we show that the use of tensor notations for the messages is very convenient for the algorithm description and understanding in the second part of the letter we introduce a simplified decoder which is inspired by the min sum decoder for binary ldpc codes we called this decoder extended min sum ems we show that it is possible to greatly reduce the computational complexity of the check node processing by computing approximate reliability measures with a limited number of values in a message by choosing appropriate correction factors or offsets we show that the ems decoder performance is quite good and in some cases better than the regular bp decoder the optimal values of the factor and offset correction are obtained asymptotically with simulated density evolution our simulations on ultra sparse codes over very high order fields show that nonbinary ldpc codes are promising for applications which require low frame error rates for small or moderate codeword lengths the ems decoder is a good candidate for practical hardware implementations of such codes

a numerical study on parasitic capillary waves using unsteady conformal mapping this paper describes fully nonlinear computation of unsteady motion of parasitic capillary waves that appear on the front face of steep gravity waves progressing on water of infinite depth within the framework of irrotational plane flow as an alternative to the widely used boundary integral method with mixed eulerian lagrangian mel time updating we focus on a numerical method based on unsteady conformal mapping which will be hereafter referred to as the unsteady hodograph transformation uht method in this method we solve the nonlinear evolution equations to find an unsteady conformal map in a complex plane with which the flow domain is mapped onto the unit disk while the free surface is fixed on the unit circle the aim of this work is to compare the uht method with the mel method and find a more efficient method to compute parasitic capillary waves from linear stability analysis it is found that a critical difference between these two methods arises from the kernel of cotangent function in singular integrals and the uht method can avoid some numerical instability due to it numerical examples demonstrate that the uht method is more suitable than the mel method for not only parasitic capillary waves but also capillary dominated waves in particular the uht method requires no artificial techniques such as filtering to control numerical errors in these examples in addition another major difference between the two methods is observed in terms of the clustering property of sample points on the free surface depending on the restoring force of waves gravity or surface tension

numerical methods for the landau lifshitz equation we discuss numerical schemes for various forms of the landau lifshitz equation a new simple projection method is introduced and is shown to be unconditionally stable the advantages over other schemes are also demonstrated numerically

the korteweg de vries hierarchy as dynamics of closed curves in the plane the korteweg de vries modified korteweg de vries and harry dym hierarchies of integrable systems are shown to be equivalent to a hierarchy of chiral shape dynamics of closed curves in the plane these purely local dynamics conserve an infinite number of global geometric properties of the curves such as perimeter and enclosed area several techniques used to study these integrable systems are shown to have simple differential geometric interpretations connections with incompressible inviscid fluid flow in two dimensions are suggested

efficient numerical methods for multiple surfactant coated bubbles in a two dimensional stokes flow we present efficient and highly accurate numerical methods to compute the deformation of surfactant coated two dimensional bubbles in a slow viscous flow surfactant acts to locally alter the surface tension and thereby change the nature of the interface motion in this paper we restrict our attention to the case of a dilute insoluble surfactant the convection diffusion equation for the surfactant concentration on the interface is coupled with the stokes equations in the fluid domain through a boundary condition based on the laplace young condition the stokes equations are first recast as an integral equation and then solved using a fast multipole accelerated iterative procedure the computational cost per time step is only o n log n operations with n being the number of discretization points on the interface the bubble interfaces are described by a spectral mesh and is advected according to the fluid velocity in such a manner so as to preserve equal arc length spacing of marker points this equal arc length framework has the dual advantage of dynamically maintaining the spatial mesh and allowing efficient implicit treatment of the stiffest terms in the dynamics several phenomenologically different examples are presented

validity of the korteweg de vries approximation for the two dimensional water wave problem in the arc length formulation we consider the two dimensional water wave problem in an infinitely long canal of finite depth both with and without surface tension it has been proven by several authors that long wavelength solutions to this problem can be approximated over a physically relevant timespan by solutions of the korteweg de vries equation or for certain values of the surface tension by solutions of the kawahara equation these proofs are formulated either in lagrangian or in eulerian coordinates in this paper we provide a new proof which is simpler more elementary and shorter moreover the rigorous justification of the kdv approximation can be given for the cases with and without surface tension together by one proof in our proof we parametrize the free surface by arc length and use some geometrically and physically motivated variables with good regularity properties this formulation of the water wave problem has already been of great usefulness for ambrose and masmoudi to simplify the proof of the local well posedness of the water wave problem in sobolev spaces 2011 wiley periodicals inc

an efficient boundary integral method for the mullins sekerka problem we use a boundary integral technique to study the two space dimensional mullins sekerka free boundary problem which originates from a study of solidification and liquidation of materials of negligible specific heat this is an area preserving and curve shortening motion evolution equations for the free boundaries are derived in terms of the tangent angle and total arclength which makes a small scale decomposition possible and the fourier transform a powerful tool in numerical calculations with this formulation implicit schemes can be implemented to avoid the difficult numerical stiffness associated with explicit schemes we can compute solutions up to the time when there is a topological change i e when particles touch or break up our numerical results for systems of a single particle or multi particles provide some valuable information in the particle dynamics such as the circularization of each individual particle and the mass transfer between different particles during particle interactions

extending fields in a level set method by solving a biharmonic equation we present an approach for computing extensions of velocities or other fields in level set methods by solving a biharmonic equation the approach differs from other commonly used approaches to velocity extension because it deals with the interface fully implicitly through the level set function no explicit properties of the interface such as its location or the velocity on the interface are required in computing the extension these features lead to a particularly simple implementation using either a sparse direct solver or a matrix free conjugate gradient solver furthermore we propose a fast poisson preconditioner that can be used to accelerate the convergence of the latter we demonstrate the biharmonic extension on a number of test problems that serve to illustrate its effectiveness at producing smooth and accurate extensions near interfaces a further feature of the method is the natural way in which it deals with symmetry and periodicity ensuring through its construction that the extension field also respects these symmetries

removing the stiffness from interfacial flows with surface tension a new formulation and new methods are presented for computing the motion of fluid interfaces with surface tension in two dimensional irrotational and incompressible fluids through the laplace young condition at the interface surface tension introduces high order terms both nonlinear and nonlocal into the dynamics this leads to severe stability constraints for explicit time integration methods and makes the application of implicit methods difficult this new formulation has all the nice properties for time integration methods that are associated with having a linear constant coefficient highest order term that is using this formulation we give implicit time integration methods that have no high order time step stability constraint associated with surface tension and are explicit in fourier space the approach is based on a boundary integral formulation and applies more generally even to problems beyond the fluid mechanical context here they are applied to computing with high resolution the motion of interfaces in hele shaw flows and the motion of free surfaces in inviscid flows governed by the euler equations one hele shaw computation shows the behavior of an expanding gas bubble over long time as the interface undergoes successive tip splittings and finger competition a second computation shows the formation of a very ramified interface through the interaction of surface tension with an unstable density stratification in euler flows the computation of a vortex sheet shows its roll up through the kelvin helmholtz instability this motion culminates in the late time self intersection of the interface creating trapped bubbles of fluid this is we believe a type of singularity formation previously unobserved for such flows in 2d finally computations of falling plumes in an unstably stratified boussinesq fluid show a very similar behavior

partially implicit motion of a sharp interface in navier stokes flow we develop a numerical method for the coupled motion of navier stokes flow with an elastic interface of zero thickness which exerts tension and bending forces on the fluid the interface motion is made partially implicit by approximating a backward euler step in the high wavenumbers as in the small scale decomposition method of hou lowengrub and shelley this modified step is combined with the method of beale and layton j t beale a t layton a velocity decomposition approach for moving interfaces in viscous fluids j comput phys 228 2009 3358 67 the fluid velocity is found by computing the stokes velocity and a more regular remainder the resulting scheme is second order in space and first order in time it can be made second order in time by extrapolation the discontinuities in the pressure and velocity gradient are preserved the partially implicit method allows much larger time steps than an explicit method with negligible added effort the formulas in the fourier transform for the implicit approximation in high wavenumbers are similar to those derived in hou and shi t y hou z shi an efficient semi implicit immersed boundary method for the navier stokes equations j comput phys 227 2008 9138 69 in a different context

an orthonormal laguerre expansion yielding rice s envelope density function for two sine waves in noise through an orthonormal laguerre expansion expressions are derived for a lesser known rician probability distribution the probability density function pdf of the envelope of two fixed amplitude randomly phased sine waves in narrowband gaussian noise and for the integral of the density the cumulative distribution function cdf the principal formula derived has been checked analytically numerically and approximately graphically analytically the moment generating function for the pdf of the square of the envelope has been found to be a three term product of elementary functions times an i sub 0 bessel function and thus to be in closed form in confirmation the same result has been secured via another more direct route

numerical methods for multiple inviscid interfaces in creeping flows we present new highly accurate and efficient methods for computing the motion of a large number of two dimensional closed interfaces in a slow viscous flow the interfacial velocity is found through the solution to an integral equation whose analytic formulation is based on complex variable theory for the biharmonic equation the numerical methods for solving the integral equations are spectrally accurate and employ a fast multipole based iterative solution procedure which requires only o n operations where n is the number of nodes in the discretization of the interface the interface is described spectrally and we use evolution equations that preserve equal arclength spacing of the marker points we assume that the fluid on one side of the interface is inviscid and we discuss two different physical phenomena bubble dynamics and interfacial motion driven by surface tension viscous sintering applications from buoyancy driven bubble interactions the motion of polydispersed bubbles in an extensional flow and the removal of void spaces through viscous sintering are considered and we present large scale fully resolved simulations involving o 100 closed interfaces

dynamics near unstable interfacial fluids we study three examples of unstable interfacial fluid motions vortex sheets with surface tension hele shaw flows with surface tension and vortex patches in all three cases the nonlinear dynamics of a large class of smooth perturbations is proven to be characterized by the corresponding fastest linear growing mode s up to the time scale of log frac 1 delta where delta is the magnitude of the initial perturbation in all three cases the analysis is based on an unified analytical framework that includes precise bounds on the growth of the linearized operator given by an explicit solution formula as well as a special sharp nonlinear energy growth estimate our main contribution is establishing this nonlinear energy growth estimate for each interface problem in certain high energy norms

fingering instabilities in adhesive failure we study the fingering instabilities that occur in the debonding of model adhesives we perform a linear analysis of the instability in the lifting hele shaw cell relevant for the failure of soft adhesives the analysis is done for a newtonian fluid but can be extended to shear thinning fluids without any problem when compared to experiment significant differences are found between the linear theory and the experimental results we therefore discuss three dimensional effects that are found to improve the agreement nevertheless significant differences between theory and experiment persist