co volume methods for degenerate parabolic problems a complementary volume co volume technique is used to develop a physically appealing algorithm for the solution of degenerate parabolic problems such as the stefan problem it is shown that these algorithms give rise to a discrete semigroup theory that parallels the continuous problem in particular the discrete stefan problem gives rise to nonlinear semigroups in both the discretel 1 andh 1 spaces

high lewis number combustion wavefronts a perturbative melnikov analysis the wavefronts associated with a one dimensional combustion model with arrhenius kinetics and no heat loss are analyzed within the high lewis number perturbative limit this situation in which fuel diffusivity is small in comparison to that of heat is appropriate for highly dense fluids a formula for the wavespeed is established by a nonstandard application of melnikov s method and slow manifold theory from dynamical systems and compared to numerical results a simple characterization of the wavespeed correction is obtained it is proportional to the ratio between the exothermicity parameter and the lewis number the perturbation method developed herein is also applicable to more general coupled reaction diffusion equations with strongly differing diffusivities the stability of the wavefronts is also tested using a numerical evans function method

a bundle view of boundary value problems generalizing the gardner jones bundle abstract holomorphic families of linear ordinary differential equations on a finite interval with prescribed parameter dependent boundary conditions are considered from a geometrical viewpoint the gardner jones bundle which was introduced for linearized reaction diffusion equations is generalized and applied to this abstract class of dependent boundary value problems where is a complex eigenvalue parameter the fundamental analytical object of such boundary value problems bvps is the characteristic determinant and it is proved that any characteristic determinant on a jordan curve can be characterized geometrically as the determinant of a transition function associated with the gardner jones bundle the topology of the bundle represented by the chern number then yields precise information about the number of eigenvalues in a prescribed subset of the complex plane this result shows that the gardner jones bundle is an intrinsic geometric property of such dependent bvps the bundle framework is applied to examples from hydrodynamic stability theory and the linearized complex ginzburg landau equation

a nonlinear mixed finite eelement method for a degenerate parabolic equation arising in flow in porous media we study a model nonlinear degenerate advection diffusion equation having application in petroleum reservoir and groundwater aquifer simulation the main difficulty is that the true solution is typically lacking in regularity therefore we consider the problem from the point of view of optimal approximation through time integration we develop a mixed variational form that respects the known minimal regularity and then we develop and analyze two versions of a mixed finite element approximation a simpler semidiscrete time continuous version and a fully discrete version our error bounds are optimal in the sense that all but one of the bounding terms reduce to standard approximation error the exceptional term is a nonstandard approximation error term we also consider our new formulation for the nondegenerate problem showing the usual optimal l 2 error bounds moreover superconvergence is obtained under special circumstances

the gap lemma and geometric criteria for instability of viscous shock profiles an obstacle in the use of evans function theory for stability analysis of traveling waves occurs when the spectrum of the linearized operator about the wave accumulates at the imaginary axis since the evans function has in general been constructed only away from the essential spectrum a notable case in which this difficulty occurs is in the stability analysis of viscous shock profiles here we prove a general theorem the gap lemma concerning the analytic continuation of the evans function associated with the point spectrum of a traveling wave into the essential spectrum of the wave this allows geometric stability theory to be applied in many cases where it could not be applied previously we demonstrate the power of this method by analyzing the stability of certain undercompressive viscous shock waves a necessary geometric condition for stability is determined in terms of the sign of a certain melnikov integral of the associated viscous profile this sign can easily be evaluated numerically we also compute it analytically for solutions of several important classes of systems in particular we show for a wide class of systems that homoclinic solitary waves are linearly unstable confirming these as the first known examples of unstable viscous shock waves we also show that strong heteroclinic undercompressive waves are sometimes unstable similar stability conditions are also derived for lax and overcompressive shocks and for nn

an upper bound on the weight distribution of some systematic codes corresp for a class of binary systematic codes of length n and cardinality 2 k including the class of linear n k binary cyclic codes it is shown that the number of codewords of weight lfloor omega n rfloor cannot be much larger than 2 kh omega

reversible low density parity check codes low density parity check ldpc codes may be decoded using a circuit implementation of the sum product algorithm which maps the factor graph of the code by reusing the decoder for encoding both tasks can be performed using the same circuit thus reducing area and verification requirements motivated by this iterative encoding techniques based upon the graphical representation of the code are proposed code design constraints which ensure encoder convergence are presented and then used to design iteratively encodable codes while also preventing 4 cycle creation we show how the jacobi method for iterative matrix inversion can be applied to finite field matrices viewed as message passing and employed as the core of an iterative encoder we present an algebraic construction of 4 cycle free iteratively encodable codes using circulant matrices analysis of these codes identifies a weakness in their structure due to a repetitive pattern in the factor graph the graph supports pseudo codewords of low pseudo weight in order to remove the repetitive pattern in the graph we propose a recursive technique for generating iteratively encodable codes the new codes offer flexibility in the choice of code length and rate and performance that compares well to randomly generated quasi cyclic and extended euclidean geometry codes

a recursive approach to low complexity codes a method is described for constructing long error correcting codes from one or more shorter error correcting codes referred to as subcodes and a bipartite graph a graph is shown which specifies carefully chosen subsets of the digits of the new codes that must be codewords in one of the shorter subcodes lower bounds to the rate and the minimum distance of the new code are derived in terms of the parameters of the graph and the subeodes both the encoders and decoders proposed are shown to take advantage of the code s explicit decomposition into subcodes to decompose and simplify the associated computational processes bounds on the performance of two specific decoding algorithms are established and the asymptotic growth of the complexity of decoding for two types of codes and decoders is analyzed the proposed decoders are able to make effective use of probabilistic information supplied by the channel receiver e g reliability information without greatly increasing the number of computations required it is shown that choosing a transmission order for the digits that is appropriate for the graph and the subcodes can give the code excellent burst error correction abilities the construction principles

expander codes using expander graphs we construct a new family of asymptotically good linear error correcting codes these codes have linear time sequential decoding algorithms and logarithmic time parallel decoding algorithms that use a linear number of processors we present both randomized and explicit constructions of these codes experimental results demonstrate the good performance of the randomly chosen codes

the compound matrix method for ordinary differential systems abstract a generalization of the compound matrix method is presented to deal with eigenvalue and boundary value problems involving unstable systems of ordinary differential equations details are given for fourth and sixth order problems next is shown that a simple equivalence relation exists between the compounds of the solution matrices of the eigenvalue problems and their adjoints and how this relation can be exploited to simplify the calculation of the adjoint eigenfunctions is discussed using the orr sommerfeld problem as an example it is also shown how the techniques used in the derivation of certain auxiliary systems which play a crucial role in the generalization of the compound matrix method can also provide an alternate method for the direct computation of certain quantities defined in terms of the eigenfunctions finally there is a brief discussion of the relationship between the compound matrix method anf the riccati method

stability and instability of solitary waves of the fifth order kdv equation a numerical framework abstract the spectral problem associated with the linearization about solitary waves of the generalized fifth order kdv equation is formulated in terms of the evans function a complex analytic function whose zeros correspond to eigenvalues a numerical framework based on a fast robust shooting algorithm on exterior algebra spaces is introduced the complete algorithm has several new features including a rigorous numerical algorithm for choosing starting values a new method for numerical analytic continuation of starting vectors the role of the grassmannian g 2 c 5 in choosing the numerical integrator and the role of the hodge star operator for relating 2 c 5 and 3 c 5 and deducing a range of numerically computable forms for the evans function the algorithm is illustrated by computing the stability and instability of solitary waves of the fifth order kdv equation with polynomial nonlinearity

some randomized code constructions from group actions we study in this paper randomized constructions of binary linear codes that are invariant under the action of some group on the bits of the codewords we study a non abelian randomized construction corresponding to the action of the dihedral group on a single copy of itself as well as a randomized abelian construction based on the action of an abelian group on a number of disjoint copies of itself cyclic codes have been extensively studied over the last 40 years however it is still an open question as to whether there exist asymptotically good binary cyclic codes we argue that by using a slightly more complex group than a cyclic group namely the dihedral group the existence of asymptotically good codes that are invariant under the action of the group on itself can be guaranteed in particular we show that for infinitely many block lengths a random ideal in the binary group algebra of the dihedral group is an asymptotically good rate half code with a high probability we argue also that a random code that is invariant under the action of an abelian group g of odd order on k disjoint copies of itself satisfies the binary gilbert varshamov gv bound with a high probability for rate 1 k under a condition on the family of groups the underlying condition is in terms of the growth of the smallest dimension of a nontrivial f sub 2 representation of the group and is satisfied by roughly most abelian groups of odd order and specifically by almost all cyclic groups of prime order

asymptotic enumeration methods for analyzing ldpc codes we show how asymptotic estimates of powers of polynomials with nonnegative coefficients can be used in the analysis of low density parity check ldpc codes in particular we show how these estimates can be used to derive the asymptotic distance spectrum of both regular and irregular ldpc code ensembles we then consider the binary erasure channel bec using these estimates we derive lower bounds on the error exponent under iterative decoding of ldpc codes used over the bec both regular and irregular code structures are considered these bounds are compared to the corresponding bounds when optimal maximum likelihood ml decoding is applied

self dual doubly even 2 quasi cyclic transitive codes are asymptotically good in this correspondence we prove that the class of binary self dual doubly even 2 quasi cyclic transitive codes is asymptotically good this improves a recent result of bazzi and mitter ieee trans inf theory vol 52 pp 3210 3219 2006 the proof is based on the study of a particular class of codes invariant under dihedral groups using a blend of representation theory and probabilistic arguments the methods are closely related to those used in bazzi and mitter in order to complete the proof a number theoretical result of hasse is needed

quasi cyclic codes of index 1 frac 1 3 we introduce quasi cyclic codes of index 1 1 3 and construct a class of such codes generated by pairs of polynomials by investigating the pair of circulant matrices associated with the generator pair of polynomials we obtain the generator matrix of any code of the class using a probabilistic method we prove that for any positive real number delta such that the asymptotic gv bound at 2 delta is greater than 1 2 the probability that the relative minimal distance of the code in the class is greater than delta is almost 1 and the probability that the rate of the code equals to 1 4 is also almost 1 an obvious consequence is that the quasi cyclic codes of index 1 1 3 are asymptotically good

convergence of the finite element method for the porous media equation with variable exponent in this work we study the convergence of the finite element method when applied to the following parabolic equation u t div u gamma mathbf x nabla u f mathbf x t mathbf x in omega subset r m t in 0 t since the problem may be of degenerate type we utilize an approximate problem regularized by introducing a parameter varepsilon we prove under certain conditions on gamma and f that the weak solution of the approximate problem converges to the weak solution of the initial problem when the parameter varepsilon tends to zero discrete solutions are built using the finite element method and the convergence of these for the weak solution of the approximate problem is proved finally we present some numerical results of a matlab implementation of the method

evaluating the evans function order reduction in numerical methods we consider the numerical evaluation of the evans function a wronskian like determinant that arises in the study of the stability of travel ling waves constructing the evans function involves matching the solutions of a linear ordinary differential equation depending on the spectral parameter the problem becomes stiff as the spectral parameter grows consequently the gauss legendre method has previously been used for such problems however more recently methods based on the magnus expansion have been proposed here we extensively examine the stiff regime for a general scalar schrodinger operator we show that although the fourth order magnus method suffers from order reduction a fortunate cancellation when computing the evans matching function means that fourth order convergence in the end result is preserved the gauss legendre method does not suffer from order reduction but it does not experience the cancellation either and thus it has the same order of conver gence in the end result finally we discuss the relative merits of both methods as spectral tools

numerical methods for flows through porous media i the degenerate parabolic equation aut v oulvvu v 1 has been used to model the flow of gas through a porous medium error estimates for continuous and discrete time finite element procedures to approximate the solution of this equation are proved and several new regularity results are given 1 a porous medium equation introduction we shall study the porous medium equation 1 1 au at v i u i vu on 2 x 0 t 1 2 au an 0 on aa x o t 1 3 u x 0 uo x on 2 where v 1 is a parameter and q is a bounded domain in rn n 3 with a smooth boundary the initial function u0 is assumed to be nonnegative and four times continuously differentiable on u notice that the compatibility condition auo an 0 holds on asa our main result is the derivation of error estimates for numerical approximations to the problem 1 1 1 3 which we shall refer to as the porous medium equation or ipme the pme does not in general admit classical solutions existence and uniqueness of weak solutions was proved in one space dimension by oleinik kalashnikov and czou 15 16 and in several space dimensions by lions 12 these proofs concern the pme with different boundary conditions but the arguments carry over to the pme 1 1 1 3 the maximum principle implies that since u0 is nonnegative on 92 u x t is nonnegative for all x t e q x 0 t see 15 16 if u0 is nonzero the neumann boundary condition implies that u will eventually become strictly positive and 1 1 will become nondegenerate for all time t to to sufficiently large we can rewrite 1 1 in the form 1 4 au at ak u on qx 0 t received may 11 1979 revised july 24 1981 1980 mathematics subject classification primary 35k65 65n30 65n15 current address reservoir engineering branch petroleum engineering division bp exploration dos britannic house moore lane london ec2y 9bu united kingdom d1983 american mathematical society 0025 5718 82 0000 1 063 08 25 435 this content downloaded from 157 55 39 55 on tue 23 aug 2016 04 12 54 utc all use subject to http about jstor org terms

instability of the hocking stewartson pulse and its implications for three dimensional poiseuille flow the linear stability problem for the hocking stewartson pulse obtained by linearizing the complex ginzburg landau cgl equation is formulated in terms of the evans function a complex analytic function whose zeros correspond to stability exponents a numerical algorithm based on the compound matrix method is developed for computing the evans function using values in the cgl equation associated with spanwise modulation of plane poiseuille flow we show that the hocking stewartson pulse associated with points along the neutral curve is always linearly unstable due to a real positive eigenvalue implications for the spanwise structure of nonlinear poiseuille problem between parallel plates are also discussed

reduced complexity iterative decoding of low density parity check codes based on belief propagation two simplified versions of the belief propagation algorithm for fast iterative decoding of low density parity check codes on the additive white gaussian noise channel are proposed both versions are implemented with real additions only which greatly simplifies the decoding complexity of belief propagation in which products of probabilities have to be computed also these two algorithms do not require any knowledge about the channel characteristics both algorithms yield a good performance complexity trade off and can be efficiently implemented in software as well as in hardware with possibly quantized received values

a combined finite volume nonconforming mixed hybrid finite element scheme for degenerate parabolic problems we propose and analyze a numerical scheme for nonlinear degenerate parabolic convection diffusion reaction equations in two or three space dimensions we discretize the diffusion term which generally involves an inhomogeneous and anisotropic diffusion tensor over an unstructured simplicial mesh of the space domain by means of the piecewise linear nonconforming crouzeix raviart finite element method or using the stiffness matrix of the hybridization of the lowest order raviart thomas mixed finite element method the other terms are discretized by means of a cell centered finite volume scheme on a dual mesh where the dual volumes are constructed around the sides of the original mesh checking the local peclet number we set up the exact necessary amount of upstream weighting to avoid spurious oscillations in the convection dominated case this technique also ensures the validity of the discrete maximum principle under some conditions on the mesh and the diffusion tensor we prove the convergence of the scheme only supposing the shape regularity condition for the original mesh we use a priori estimates and the kolmogorov relative compactness theorem for this purpose the proposed scheme is robust only 5 point 7 point in space dimension three locally conservative efficient and stable which is confirmed by numerical experiments

bifurcations of poiseuille flow between parallel plates three dimensional solutions with large spanwise wavelength the spatial dynamics approach is applied to the analysis of bifurcations of the three dimensional poiseuille flow between parallel plates in contrast to the classical studies we impose time periodicity as well as spatial periodicity with period 2 in the streamwise direction however we make no assumptions on the behavior in the spanwise direction except the uniform closeness of the bifurcating solution to the basic flow in an abstract setting it is shown how the dimension of the critical eigenspace of the spatial dynamics analysis can be uniquely determined from the classical linear stability problem for the three dimensional poiseuille problem we are able to find all relevant coefficients from the analysis of the purely two dimensional problem moreover we are able to analyze precisely the influence of a spanwise pressure gradient and the associated spanwise mass flux the study of the reduced problem shows that there are two different kinds of solutions spirals and ribbons which are 2 p periodic in the spanwise direction as in the couette taylor problem and both of them bifurcate in the same direction

numerical study of the porous medium equation with absorption variable exponents of nonlinearity and free boundary abstract in this paper we study an application of the moving mesh method to the porous medium equation with absorption and variable exponents of nonlinearity in 2d domains with moving boundaries the boundary s movement is governed by an equation prompted by the darcy law and the spatial discretization is defined by a triangulation of the domain at each finite element the solution is approximated by piecewise polynomial functions of degree r 1 using lagrange interpolating polynomials in area coordinates the vertices of the triangles move according to a system of differential equations which is added to the equations of the problem the resulting system is converted into a system of ordinary differential equations in time variable which is solved using a suitable integrator the integrals that arise in the system of ordinary differential equations are calculated using the gaussian quadrature finally we present some numerical results of application of this technique

error analysis for characteristics based methods for degenerate parabolic problems we consider characteristics based finite element methods for solving nonlinear degenerate advection diffusion equations these equations have applications in the simulation of petroleum reservoirs and groundwater aquifers and in the modeling of free boundary problems standard finite element galerkin methods have been studied for these equations in this paper we analyze the characteristics based finite element methods for them the main difficulty in the analysis is that the equations are degenerate and the solution lacks regularity here we develop a technique that respects the degeneracy and the known minimal regularity this technique is based on the green operator for standard elliptic equations and is developed directly for the degenerate advection diffusion equations we concentrate our analysis on the modified method of characteristics mmoc and one of its variants the modified method of characteristics with adjusted advection mmocaa which conserves mass we derive error estimates in various norms the extension to other variants is discussed the present technique is also applied to nondegenerate problems error estimates previously obtained for the mmoc are derived under much weaker regularity assumptions on the solution and the error estimates for the mmocaa appear new even in the nondegenerate case finally numerical results are presented to show the sharpness of the error estimates derived

error estimates for a class of degenerate parabolic equations the degenerate parabolic equation u t delta beta u is considered in the cylinder omega times 0 t where omega subset r n is bounded beta 0 beta 0 0 and beta geq0 e g beta u u u m 1 m 1 a fully discrete scheme is given using c0 piecewise linear finite elements in space and a backward difference scheme in time l2 error estimates for the approximation of u and of beta u are proven for that purpose we employ the fact that the free boundary has a finite speed of propagation

thresholds of random quasi abelian codes for a q ary random quasi abelian code with fixed coindex and constant rate r it is shown that the gilbert varshamov gv bound is a threshold point if r is less than the gv bound at 0 1 q 1 then the probability of the relative distance of the random code being greater than approaches 1 as the index goes to infinity whereas if r is bigger than the gv bound at then the probability approaches 0 as a corollary there exist numerous asymptotically good quasi abelian codes attaining the gv bound

linear time encodable and decodable error correcting codes we present a new class of asymptotically good linear error correcting codes these codes can be both encoded and decoded in linear time they can also be encoded by logarithmic depth circuits of linear size and decoded by logarithmic depth circuits of size o nlogn we present both randomized and explicit constructions of these codes

a model porous medium equation with variable exponent of nonlinearity existence uniqueness and localization properties of solutions abstract we study the localization properties of weak solutions to the dirichlet problem for the degenerate parabolic equation u t div u x t u f with variable exponent of nonlinearity we prove the existence and uniqueness of weak solutions and establish conditions on the problem data and the exponent x t sufficient for the existence of such properties as finite speed of propagation of disturbances the waiting time effect finite time vanishing of the solution it is shown that the solution may instinct in a finite time even if x 0 in the problem domain but max 0

stability of bright solitary wave solutions to perturbed nonlinear schrodinger equations abstract the propagation of pulses in ideal nonlinear optical fibers without loss is governed by the nonlinear schrodinger equation nls when considering realistic fibers one must examine perturbed nls equations with the particular perturbation depending on the physical situation that is being modeled a common example is the complex ginzburg landau equation cgl which is a dissipative perturbation it is known that some of the stable bright solitons of the nls survive a dissipative perturbation such as the cgl given that a wave persists it is then important to determine its stability with respect to the perturbed nls a major difficulty in analyzing the stability of solitary waves upon adding dissipative terms is that eigenvalues may bifurcate out of the essential spectrum since the essential spectrum of the nls is located on the imaginary axis such eigenvalues may lead to an unstable wave in fact eigenvalues can pop out of the essential spectrum even if the unperturbed problem has no eigenvalue embedded in the essential spectrum here we present a technique which can be used to track these bifurcating eigenvalues as a consequence we are able to locate the spectrum for bright solitary wave solutions to various perturbed nonlinear schrodinger equations and determine precise conditions on parameters for which the waves are stable in addition we show that a particular perturbation the parametrically forced nls equation supports stable multi bump solitary waves the technique for tracking eigenvalues which bifurcate from the essential spectrum is very general and should therefore be applicable to a larger class of problems than those presented here

stability criterion for bright solitary waves of the perturbed cubic quintic schro dinger equation abstract the stability of the bright solitary wave solution to the perturbed cubic quintic schrodinger equation is considered it is shown that in a certain region of parameter space these solutions are unstable with the instability being manifested as a small positive eignevalue furthermore it is shown that in the complimentary region of parameter space there are no small unstable eigenvalues the proof involves a novel calculation of the evans function which is of interest in its own right as a consequence of the eigenvalue calculation it is additionally shown that n bump bright solitary waves bifurcate from the primary wave

anisotropic mesh adaptation for finite element solution of anisotropic porous medium equation abstract anisotropic porous medium equation apme is developed as an extension of the porous medium equation pme for anisotropic porous media a special analytical solution is derived for apme for time independent diffusion anisotropic mesh adaptation for linear finite element solution of apme is discussed and numerical results for two dimensional examples are presented the solution errors using anisotropic adaptive meshes show second order convergence

finite element approximation of the fast diffusion and the porous medium equations 1the fast diffusion equation u t delta u m 1 u 0 m 1 and the porous medium equation 1 m infty are studied in a parabolic cylinder omega times 0 t a fully discrete galerkin approximation is considered using c 0 piecewise linear finite elements in space and the backward euler time discretization a priori error estimates in quasi norms and rates of convergence are proved

the instability of the hocking stewartson pulse and its geometric phase in the hopf bundle this work demonstrates an innovative numerical method for counting and locating eigenvalues with the evans function utilizing the geometric phase in the hopf bundle the technique calculates the winding of the evans function about a contour in the spectral plane describing the eigenvalues enclosed by the contour for the hocking stewartson pulse of the complex ginzburg landau equation locating eigenvalues with the geometric phase in the hopf bundle was proposed by way 2009 and proven by grudzien et al 2016 way demonstrated his proposed method for the hocking stewartson pulse and this manuscript redevelops this example as in the proof of the method in grudzien et al 2016 modifying his numerical shooting argument and introduces new numerical results concerning the phase transition

the orr sommerfeld equation on a manifold the most effective and widely used methods for integrating the orr sommerfeld equation by shooting are the continuous orthogonalization method and the compound matrix method in this paper we consider this problem from a differential geometric point of view a new definition of orthogonalization is presented restriction of the orr sommerfeld to a complex stiefel manifold and this definition leads to a new formulation of continuous orthogonalization which differs in a precise and interesting geometric way from existing orthogonalization routines present orthogonalization methods based on davey9s algorithm are shown to have a different differential geometric interpretation restriction of the orr sommerfeld equation to a complex grassmanian manifold this leads us to introduce the concept of a grassmanian integrator which preserves linear independence and not necessarily orthogonality using properties of grassmanian manifolds and their tangent spaces a new grassmanian integrator is introduced that generalizes davey9s algorithm furthermore it is shown that the compound matrix method is a dual grassmanian integrator it uses plucker coordinates for integrating on a grassmanian manifold and this characterization suggests a new algorithm for constructing the compound matrices extension of the differential geometric framework to general systems of linear ordinary differential equations is also discussed

a study on moving mesh finite element solution of the porous medium equation an adaptive moving mesh finite element method is studied for the numerical solution of the porous medium equation with and without variable exponents and absorption the method is based on the moving mesh partial differential equation approach and employs its newly developed implementation the implementation has several improvements over the traditional one including its explicit compact form of the mesh velocities ease to program and less likelihood of producing singular meshes three types of metric tensor that correspond to uniform and arclength based and hessian based adaptive meshes are considered the method shows first order convergence for uniform and arclength based adaptive meshes and second order convergence for hessian based adaptive meshes it is also shown that the method can be used for situations with complex free boundaries emerging and splitting of free boundaries and the porous medium equation with variable exponents and absorption two dimensional numerical results are presented

a polynomial time algorithm for rayleigh ratio on discrete variables replacing spectral techniques for expander ratio normalized cut and cheeger constant a general form of minimizing the rayleigh ratio on discrete variables is shown here for the first time to be polynomial time solvable this is significant because major problems in clustering partitioning and imaging can be presented as the rayleigh ratio minimization on discrete variables and an orthogonality constraint these challenging problems are modeled as the normalized cut problem the graph expander ratio problem the cheeger constant problem or the conductance problem all of which are np hard these problems have traditionally been solved heuristically using the spectral technique a unified framework is provided here whereby all these problems are formulated as a constrained minimization form of a quadratic ratio referred to here as the rayleigh ratio the quadratic ratio is to be minimized on discrete variables and a single sum constraint that we call the balance or orthogonality constraint when the discreteness constraints on the variables are disregarded the resulting continuous relaxation is solved by the spectral method it is shown here that the rayleigh ratio minimization subject to the discreteness constraints requiring each variable to assume one of two values in b 1 is solvable in strongly polynomial time equivalent to a single minimum s t cut algorithm on a graph of same size as the input graph for any nonnegative value of b this discrete form for the rayleigh ratio problem was often assumed to be np hard not only is it shown here that the discrete rayleigh ratio problem is polynomial time solvable but also the algorithm is more efficient than the spectral algorithm furthermore an experimental study demonstrates that the new algorithm provides in practice an improvement often dramatic on the quality of the results of the spectral method both in terms of approximating the true optimum of the rayleigh ratio problem on both the discrete variables and the balance constraint and in terms of the subjective partition quality a further contribution here is the introduction of a problem the quantity normalized cut generalizing all the rayleigh ratio problems the discrete version of that problem is also solved with the efficient algorithm presented this problem is shown in a companion paper to enable the modeling of features essential to clustering that are valuable in practical applications

bifurcation of homoclinic orbits to a saddle focus in reversible systems with so 2 symmetry we study reversible so 2 invariant vector fields in r4 depending on a real parameter e which possess for e 0 a primary family of homoclinic orbits t h0 s1 under a transversality condition with respect to e the existence of homoclinic n pulse solutions is demonstrated for a sequence of parameter values e n k 0 for k the existence of cascades of 2l3m pulse solutions follows by showing their transversality and then using induction the method relies on the construction of an so 2 equivariant poincare map which after factorization is a composition of two involutions a logarithmic twist map and a smooth global map reversible periodic orbits of this map corresponds to reversible periodic or homoclinic solutions of the original problem as an application we treat the steady complex ginzburg landau equation for which a primary homoclinic solution is known explicitly

convergence of a finite volume scheme for nonlinear degenerate parabolic equations one approximates the entropy weak solution u of a nonlinear parabolic degenerate equation u t rm div mathbf q f u delta phi u 0 by a piecewise constant function u mathcal d using a discretization mathcal d in space and time and a finite volume scheme the convergence of u mathcal d to u is shown as the size of the space and time steps tend to zero in a first step estimates on u mathcal d are used to prove the convergence up to a subsequence of u mathcal d to a measure valued entropy solution called here an entropy process solution a result of uniqueness of the entropy process solution is proved yielding the strong convergence of u mathcal d to it u some on a model equation are shown

analysis of expanded mixed finite element methods for a nonlinear parabolic equation modeling flow into variably saturated porous media we present an analysis of expanded mixed finite element methods applied to richards equation a nonlinear parabolic partial differential equation modeling the flow of water into a variably saturated porous medium we consider the full range of saturated to completely unsaturated media in the case of the lowest order raviart thomas spaces and the range of all possible saturations we bound the h 1 norm of the error in capacity in terms of approximation error this estimate uses a time integrated scheme and the kirchhoff transformation to handle a degeneracy in the case of completely unsaturated flow optimal convergence is then shown for a nonlinear form related to the error in the capacity for the case of saturated to partially saturated flow convergence rates depending on the holder continuity of the capacity term are derived last optimal convergence of pressures and fluxes is stated for the case of strictly partially saturated flow

oscillatory instability of traveling waves for a kdv burgers equation the stability of traveling wave solutions of a generalization of the kdv burgers equation o u u p o u 03u a o u is studied as the parameters p and a are varied the eigenvalue problem for the linearized evolution of perturbations is analyzed by numerically computing evans function d a an analytic function whose zeros correspond to discrete eigenvalues in particular the number of unstable eigenvalues in the complex plane is evaluated by computing the winding number of d a analytical and numerical evidence suggests that a hopf bifurcation occurs for oscillatory traveling wave profiles in certain parameter ranges dynamic simulations suggest that the bifurcation is subcritical no stable time periodic solution is found

combustion waves in hydraulically resistant porous media in a special parameter regime abstract in this paper we study the stability of fronts in a reduction of a well known pde system that is used to model the combustion in hydraulically resistant porous media more precisely we consider the original pde system under the assumption that one of the parameters of the model the lewis number is chosen in a specific way and with initial conditions of a specific form for a class of initial conditions then the number of unknown functions is reduced from three to two for the reduced system the existence of combustion fronts follows from the existence results for the original system the stability of these fronts is studied here by a combination of energy estimates and numerical evans function computations and nonlinear analysis when applicable we then lift the restriction on the initial conditions and show that the stability results obtained for the reduced system extend to the fronts in the full system considered for that specific value of the lewis number the fronts that we investigate are proved to be either absolutely unstable or convectively unstable on the nonlinear level

numerical exterior algebra and the compound matrix method the compound matrix method which was first proposed for numerically integrating systems of differential equations in hydrodynamic stability on k 2 3 dimensional subspaces of mathbb c n by using compound matrices as coordinates is reformulated in a coordinate free way using exterior algebra spaces bigwedge k mathbb c n

dynamic scaling laws in two types of extended hamiltonian systems at dissipation onset the transition to dissipation in one dimensional extended hamiltonian systems with saddle node bifurcations of stationary solutions is characterized three different systems are studied i nonlinear schrodinger flow past a localized obstacle ii sine gordon pendulum chains forced by a local torque iii electrically charged nonlinear schrodinger flows in case i no frequency gap is present in the dispersion relation in contrast in cases ii and iii a minimum frequency for propagating waves exists in the gapless case the growth rates of the unstable modes and the frequency of supercritical soliton emission are found to scale as the square root of the bifurcation parameter no subcriticality is observed in contrast when a frequency gap is present subcritical soliton emission takes place logarithmic and one fourth power scaling laws are found respectively at the bottom and top of the subcriticality window