accurate three dimensional lid driven cavity flow a chebyshev collocation method in space is introduced which allows an accurate calculation of three dimensional lid driven cavity flows the time integration is carried out by an adams bashforth backward euler scheme the accuracy of the method relies on the representation of the solution as a superposition of stationary local asymptotic solutions and a residual flow field this way the most severe discontinuities in the boundary conditions which arise along the lines where moving and stationary walls meet are taken care of analytically and thus do not spoil the numerical part of the solution calculations are carried out for no slip boundary conditions at the cavity end walls as well as for periodic end wall conditions in general the spatial accuracy is better than fifth order for rigid end wall conditions the accuracy is reduced near the end walls to o n 3 2 but recovers in the bulk tabulated data are provided for the most interesting flow properties

power law distributions for the areas of the basins of attraction on a potential energy landscape energy landscape approaches have become increasingly popular for analyzing a wide variety of chemical physics phenomena basic to many of these applications has been the inherent structure mapping which divides up the potential energy landscape into basins of attraction surrounding the minima here we probe the nature of this division by introducing a method to compute the basin area distribution and applying it to some archetypal supercooled liquids we find that this probability distribution is a power law over a large number of decades with the lower energy minima having larger basins of attraction interestingly the exponent for this power law is approximately the same as that for a high dimensional apollonian packing providing further support for the suggestion that there is a strong analogy between the way the energy landscape is divided into basins and the way that space is packed in self similar space filling hypersphere packings such as the apollonian packing these results suggest that the basins of attraction provide a fractal like tiling of the energy landscape and that a scale free pattern of connections between the minima is a general property of energy landscapes

heteroclinic orbits in a spherically invariant system the existence and stability of structurally stable heteroclinic cycles are discussed in a codimension 2 bifurcation problem with o 3 symmetry when the critical spherical modes l 1 and l 2 occur simultaneously several types of heteroclinic cycles are found which may explain aperiodic attractors found in numerical simulations for the onset of convection in a self gravitating fluid in a spherical shell

invariant helical subspaces for the navier stokes equations three dimensional solutions with helical symmetry are shown to form an invariant subspace for the navier stokes equations uniqueness of weak helical solutions in the sense of leray is proved and these weak solutions are shown to be regular strong solutions existing for arbitrary time t the global universal attractor for the infinite dimensional dynamical system generated by the corresponding semi group of helical flows is shown to be compact and finite dimensional the hausdorff and fractal dimensions of the global attractors are estimated in terms of the governing physical parameters and in terms of the helical parameters for several problems in the class with the most detailed results obtained for rotating hagen poiseuille pipe flow in this case the dimension either hausdorff or fractal up to an absolute constant is bounded from above by frac operatorname re sqrt alpha 2 n 2 where is the axial wavenumber n is the azimuthal wavenumber and re is the reynolds number based on the radius of the pipe these upper bounds are independent of the rotation rate

a staggered semi implicit discontinuous galerkin method for the two dimensional incompressible navier stokes equations in this paper we propose a new spatially high order accurate semi implicit discontinuous galerkin dg method for the solution of the two dimensional incompressible navier stokes equations on staggered unstructured curved meshes while the discrete pressure is defined on the primal grid the discrete velocity vector field is defined on an edge based dual grid the flexibility of high order dg methods on curved unstructured meshes allows to discretize even complex physical domains on rather coarse grids formal substitution of the discrete momentum equation into the discrete continuity equation yields one sparse linear equation system with four non zero blocks per element for only one scalar unknown namely the pressure the method is computationally efficient since the resulting system is not only very sparse but also symmetric and positive definite for appropriate boundary conditions furthermore all the volume and surface integrals needed by the scheme presented in this paper depend only on the geometry and the polynomial degree of the basis and test functions and can therefore be precomputed and stored in a preprocessor stage which leads to savings in terms of computational effort for the time evolution part in this way also the extension to a fully curved isoparametric approach becomes natural and affects only the preprocessing step the method is validated for polynomial degrees up to p 3 by solving some typical numerical test problems and comparing the numerical results with available analytical solutions or other numerical and experimental reference data

geometric phases reduction and lie poisson structure for the resonant three wave interaction hamiltonian lie poisson structures of the three wave equations associated with the lie algebras su 3 and su 2 1 are derived and shown to be compatible poisson reduction is performed using the method of invariants and geometric phases associated with the reconstruction are calculated these results can be applied to applications of nonlinear waves in for instance nonlinear optics some of the general structures presented in the latter part of this paper are implicit in the literature our purpose is to put the three wave interaction in the modern setting of geometric mechanics and to explore some new things such as explicit geometric phase formulas as well as some old things such as integrability in this context

identifying communities within energy landscapes potential energy landscapes can be represented as a network of minima linked by transition states the community structure of such networks has been obtained for a series of small lennard jones lj clusters this community structure is compared to the concept of funnels in the potential energy landscape two existing algorithms have been used to find community structure one involving removing edges with high betweenness the other involving optimization of the modularity the definition of the modularity has been refined making it more appropriate for networks such as these where multiple edges and self connections are not included the optimization algorithm has also been improved using monte carlo methods with simulated annealing and basin hopping both often used successfully in other optimization problems in addition to the small clusters two examples with known heterogeneous landscapes the 13 atom cluster lj13 with one labeled atom and the 38 atom cluster lj38 were studied with this approach the network methods found communities that are comparable to those expected from landscape analyses this is particularly interesting since the network model does not take any barrier heights or energies of minima into account for comparison the network associated with a two dimensional hexagonal lattice is also studied and is found to have high modularity thus raising some questions about the interpretation of the community structure associated with such partitions

space time discontinuous galerkin finite element method with dynamic grid motion for inviscid compressible flows ii efficient flux quadrature a new and efficient quadrature rule for the flux integrals arising in the space time discontinuous galerkin discretization of the euler equations in a moving and deforming space time domain is presented and analyzed the quadrature rule is a factor three more efficient than the commonly applied quadrature rule and does not affect the local truncation error and stability of the numerical scheme the local truncation error of the resulting numerical discretization is determined and is shown to be the same as when product gauss quadrature rules are used details of the approximation of the dissipation in the numerical flux are presented which render the scheme consistent and stable the method is successfully applied to the simulation of a three dimensional transonic flow over a deforming wing

neighbor network in a polydisperse hard disk fluid degree distribution and assortativity the neighbor network in a two dimensional polydisperse hard disk fluid with diameter distribution p 4 is examined using constant pressure monte carlo simulations graphs are constructed from vertices disks with edges links connecting each vertex to k neighboring vertices defined by a radical tessellation at packing fractions in the range 0 24 0 36 the decay of the network degree distribution is observed to be consistent with the power law k where the exponent lies in the range 5 6 6 0 comparisons with the predictions of a maximum entropy theory suggest that this apparent power law behavior is not the asymptotic one and that p k k 4 in the limit k this is consistent with the simple idea that for large disks the number of neighbors is proportional to the disk diameter a power law decay of the network degree distribution is one of the characteristics of a scale free network the assortativity of the network is measured and is found to be positive meaning that vertices of equal degree are connected more often than in a random network finally the equation of state is determined and compared with the prediction from a scaled particle theory very good agreement between simulation and theory is demonstrated

a simple based discontinuous galerkin solver for steady incompressible flows in this paper we present how the well known simple algorithm can be extended to solve the steady incompressible navier stokes equations discretized by the discontinuous galerkin method the convective part is discretized by the local lax friedrichs fluxes and the viscous part by the symmetric interior penalty method within the simple algorithm the equations are solved in an iterative process the discretized equations are linearized and an equation for the pressure is derived on the discrete level the equations obtained for each velocity component and the pressure are decoupled and therefore can be solved sequentially leading to an efficient solution procedure the extension of the proposed scheme to the unsteady case is straightforward where fully implicit time schemes can be used various test cases are carried out the poiseuille flow the channel flow with constant transpiration the kovasznay flow the flow into a corner and the backward facing step flow using a mixed order formulation i e order k for the velocity and order k 1 for the pressure the scheme is numerically stable for all test cases convergence rates of k 1 and k in the l 2 norm are observed for velocity and pressure respectively a study of the convergence behavior of the simple algorithm shows that no under relaxation for the pressure is needed which is in strong contrast to the application of the simple algorithm in the context of the finite volume method or the continuous finite element method we conclude that the proposed scheme is efficient to solve the steady incompressible navier stokes equations in the context of the discontinuous galerkin method comprising hp accuracy

self similar disk packings as model spatial scale free networks the network of contacts in space filling disk packings such as the apollonian packing is examined these networks provide an interesting example of spatial scale free networks where the topology reflects the broad distribution of disk areas a wide variety of topological and spatial properties of these systems is characterized their potential as models for networks of connected minima on energy landscapes is discussed

stabilized hp finite element methods for first order hyperbolic problems we analyze the hp version of the streamline diffusion finite element method sdfem and of the discontinuous galerkin finite element method dgfem for first order linear hyperbolic problems for both methods we derive new error estimates on general finite element meshes which are sharp in the mesh width h and in the spectral order p of the method assuming that the stabilization parameter is o h p for piecewise analytic solutions exponential convergence is established on quadrilateral meshes for the dgfem we admit very general irregular meshes and for the sdfem we allow meshes which contain hanging nodes numerical experiments confirm the theoretical results

a projection hybrid finite volume element method for low mach number flows the purpose of this article is to introduce a projection hybrid finite volume element method for low mach number flows of viscous or inviscid fluids starting with a 3d tetrahedral finite element mesh of the computational domain the equation of the transport diffusion stage is discretized by a finite volume method associated with a dual mesh where the nodes of the volumes are the barycenters of the faces of the initial tetrahedra the transport diffusion stage is explicit upwinding of convective terms is done by classical riemann solvers as the q scheme of van leer or the rusanov scheme concerning the projection stage the pressure correction is computed by a piecewise linear finite element method associated with the initial tetrahedral mesh passing the information from one stage to the other is carefully made in order to get a stable global scheme numerical results for several test examples aiming at evaluating the convergence properties of the method are shown

double three wave interaction of four waves lax representations and exact solutions the nonlinear resonant interaction of coherent waves is a fundamental process in the study of wave phenomena which has received a great deal of attention in its many aspects in the present article a system of four interacting waves which constitute two resonant triplets is considered the system is described in a simplified model by a hamiltonian system of eight autonomous ordinary differential equations with time as the independent variable both positive and negative energy waves are allowed in the interaction two distinct lax representations for this system two new classes of exact solutions in terms of elliptic functions a solution in the form of a convergent generic laurent series expansion around a movable pole in the independent variable and a stabilization criterion for the explosive instability that may occur when waves of different energy sign interact are obtained herein

low dimensional chaotic dynamics versus intrinsic stochastic noise a paradigm model several prototype models are introduced here which are designed to elucidate the interaction between heteroclinic low dimensional chaos in the projected nonlinear dynamics and intrinsic stochasticity induced by energy exchange with a bath of fast variables these models are built by coupling a four dimensional ode with known analytical properties including heteroclinic cycles with a suitable deterministic bath of fast variables a systematic strategy for stochastic mode reduction is applied to these models with 104 degrees of freedom to derive four dimensional reduced stochastic equations for the slow variables due to the internal chaotic dynamics of the slow variables the stochastic mode reduction strategy is very robust in this case and yields reduced models which accurately capture the statistical behavior of the original deterministic system furthermore it is also shown here that even in the regime of a weak coupling between the slow variables and the fast heat bath the detailed structure of the stochastic terms derived through the mode elimination procedure is essential for reproducing the statistical behavior of the slow dynamics

the hausdorf dimension of the apollonian packing of circles we formulate the problem of determining the hausdorf dimension df of the apollonian packing of circles as an eigenvalue problem of a linear integral equation we show that solving a finite dimensional approximation to this infinite order matrix equation and extrapolating the results provides a fast algorithm for obtaining high precision numerical estimates for df we find that df 1 305 686 729 10 this is consistent with the rigorously known bounds on df and improves the precision of the existing estimate by three orders of magnitude

stability accuracy and efficiency of a semi implicit method for three dimensional shallow water flow abstract the stability analysis the accuracy and the efficiency of a semi implicit finite difference scheme for the numerical solution of a three dimensional shallow water model are presented and discussed the governing equations are the three dimensional reynolds equations in which pressure is assumed to be hydrostatic the pressure gradient in the momentum equations and the velocities in the vertically integrated continuity equation are discretized with the method with being an implicitness parameter it is shown that the method is stable for 1 2 1 unstable for 1 2 and highest accuracy and efficiency is achieved when 1 2 the resulting algorithm is mass conservative and naturally allows for the simulation of flooding and drying of tidal flats

space time discontinuous galerkin finite element method with dynamic grid motion for inviscid compressible flows i general formulation a new space time discontinuous galerkin finite element method for the solution of the euler equations of gas dynamics in time dependent flow domains is presented the discontinuous galerkin discretization results in an efficient elementwise conservative upwind finite element method which is particularly well suited for local mesh refinement the upwind scheme uses a formulation of the hllc flux applicable to moving meshes and several formulations for the stabilization operator to ensure that monotone solutions around discontinuities are investigated the non linear equations of the space time discretization are solved using a multigrid accelerated pseudo time integration technique with an optimized runge kutta method the linear stability of the pseudo time integration method is investigated for the linear advection equation the numerical scheme is demonstrated with simulations of the flow field in a shock tube a channel with a bump and an oscillating naca 0012 airfoil these simulations show that using the data at the superconvergence points the accuracy of the numerical discretization is o h5 2 in space for smooth subsonic flows both on structured and on locally refined meshes and that the space time adaptation can significantly improve the accuracy and efficiency of the numerical method

non periodic driving of coupled oscillators a spherical swing abstract nonlinearly coupled damped oscillators at 1 1 frequency ratio one oscillator being driven coherently for efficient excitation are exemplified by a spherical swing with some phase mismatch between drive and response for certain damping range excitation is found to succeed if it lags behind but to produce a chaotic attractor if it leads the response although a period doubling sequence for damping increasing leads to the attractor this is actually born as a hard as regards amplitude bifurcation at a zero growth rate parametric line as damping decreases an unstable fixed point crosses an invariant plane to enter as saddle focus a phase space domain of physical solutions a second hard bifurcation occurs at the zero mismatch line the saddle focus leaving that domain times on the attractor diverge when approaching either line leading to exactly one dimensional and noninvertible limit maps which are analytically determined

heteroclinic cycles and modulated travelling waves in a system with d 4 symmetry we study a system of two complex ordinary differential equations with d4 symmetry describing the effects of the symmetry breaking o 2 to d4 on an interaction between fourier modes with wavenumbers in the ratio 1 2 this change from continuous to discrete symmetry is relevant to the effect of introducing riblets on a wall to reduce boundary layer drag numerical simulations planar analysis and melnikov s method reveal the persistence under this symmetry breaking of just two of the continuous group orbit of heteroclinic cycles of the o 2 system it is further shown that one of these cycles destabilizes to regular modulated travellin waves while the other loses stability to a new type of solution modulated travelling waves which quasi periodically reverse their direction of propagation a bifurcation diagram for this behavior and the subsequent destabilization of modulated travelling waves to travelling waves is established the method of averaging is used to show that some of the quasiperiodic solutions of the o 2 system persist under the symmetry breaking while others break into sets of periodic orbits

cluster algorithm for hard spheres and related systems in this paper we present a cluster algorithm for the simulation of hard spheres and related systems in this algorithm a copy of the configuration is rotated with respect to a randomly chosen pivot point the two systems are then superposed and clusters of overlapping spheres in the joint system are isolated each of these clusters can be flipped independently a process which generates non local moves in the original configuration a generalization of this algorithm which works perfectly well at small density can be made to work successfully at densities around the solid liquid transition point in the two dimensional hard sphere system

complex network study of brazilian soccer players departamento de f isica e inform atica instituto de f isica de s ao carlos universidade de s ao paulo c p 369 13560 970 s ao carlos sp brazilalthough being a very popular sport in many countries soccer has not received much attentionfrom the scienti c community in this paper we study soccer from a complex network point of view first we consider a bipartite network with two kinds of vertices or nodes the soccer players andthe clubs real data were gathered from the 32 editions of the brazilian soccer championship in atotal of 13 411 soccer players and 127 clubs we nd a lot of interesting and perhaps unsuspectedresults the probability that a brazilian soccer player has worked at n clubs or played m gamesshows an exponential decay while the probability that he has scored g goals is power law now iftwo soccer players who have worked at the same club at the same time are connected by an edge then a new type of network arises composed exclusively by soccer players nodes our analysisshows that for this network the degree distribution decays exponentially we determine the exactvalues of the clustering coe cient the assortativity coe cient and the average shortest path lengthand compare them with those of the erd os r enyi and con guration model the time evolution ofthese quantities are calculated and the corresponding results discussed

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

analysis of adaptive mesh refinement for imex discontinuous galerkin solutions of the compressible euler equations with application to atmospheric simulations the resolutions of interests in atmospheric simulations require prohibitively large computational resources adaptive mesh refinement amr tries to mitigate this problem by putting high resolution in crucial areas of the domain we investigate the performance of a tree based amr algorithm for the high order discontinuous galerkin method on quadrilateral grids with non conforming elements we perform a detailed analysis of the cost of amr by comparing this to uniform reference simulations of two standard atmospheric test cases density current and rising thermal bubble the analysis shows up to 15 times speed up of the amr simulations with the cost of mesh adaptation below 1 of the total runtime we pay particular attention to the implicit explicit imex time integration methods and show that the ark2 method is more robust with respect to dynamically adapting meshes than bdf2 preliminary analysis of preconditioning reveals that it can be an important factor in the amr overhead the compiler optimizations provide significant runtime reduction and positively affect the effectiveness of amr allowing for speed ups greater than it would follow from the simple performance model

a discontinuous hp finite element method for convection diffusion problems this paper presents a new method which exhibits the best features of both finite volume and finite element techniques special attention is given to the issues of conservation flexible accuracy and stability the method is elementwise conservative the order of polynomial approximation can be adjusted element by element and the stability is not based on the introduction of artificial diffusion but on the use of a very particular finite element formulation with discontinuous basis functions the method supports h p and hp approximations and can be applied to any type of meshes including non matching grids a priori error estimates and numerical experiments on representative model problems indicate that the method is robust and capable of delivering high accuracy

semi implicit discontinuous galerkin methods for the incompressible navier stokes equations on adaptive staggered cartesian grids in this paper a new high order semi implicit discontinuous galerkin method si dg is presented for the solution of the incompressible navier stokes equations on staggered space time adaptive cartesian grids amr in two and three space dimensions the pressure is written in the form of piecewise polynomials on the main grid which is dynamically adapted within a cell by cell amr framework according to the time dependent main grid different face based spatially staggered dual grids are defined for the piece wise polynomials of the respective velocity components although the resulting adaptive staggered grids are more complex than classical uniform cartesian meshes the numerical scheme can still be written in a rather compact form thanks to the use of a tensor product formulation for the definition of the nodal basis in the d dimensional space d 2 3 all the discrete operators can be efficiently written as a combination of linear one dimensional operators acting in the d space directions separately arbitrary high order of accuracy is achieved in space while a very simple semi implicit time discretization is obtained via an explicit discretization of the nonlinear convective terms and an implicit discretization of the pressure gradient in the momentum equation and of the divergence of the velocity field in the continuity equation the real advantages of the staggered grid arise in the solution of the schur complement associated with the saddle point problem of the discretized incompressible navier stokes equations i e after substituting the discrete momentum equations into the discrete continuity equation this leads to a linear system for only one unknown the scalar pressure indeed the resulting linear pressure system is shown to be symmetric and positive definite in order to avoid a quadratic stability condition for the parabolic terms given by the viscous stress tensor an implicit discretization is also used for the diffusive terms in the momentum equation a particular feature of our staggered dg approach is that the viscous stress tensor is discretized on the dual mesh this corresponds to the use of a lifting operator but on the staggered grid both linear systems for pressure and velocity are very efficiently solved by means of a classical matrix free conjugate gradient method for which fast convergence is observed moreover it should be noticed that all test cases shown in this paper have been performed without the use of any preconditioner due to the explicit discretization of the nonlinear convective terms the final algorithm is stable within the classical cfl type time step restriction for explicit dg methods based on the fluid velocity moreover by using a suitable local time stepping technique based on the ader dg framework each element can use its optimal local time step also within the amr context the new space time adaptive staggered dg scheme has been thoroughly verified for polynomial degrees up to n 9 for a large set of non trivial test problems in two and three space dimensions for which analytical numerical or experimental reference solutions exist the high order of accuracy of the scheme is shown by means of a numerical convergence table obtained by comparing the numerical solution against a smooth analytical solution to the knowledge of the authors this is the first staggered semi implicit dg scheme for the incompressible navier stokes equations on space time adaptive meshes in two and three space dimensions

the helmholtzian equation stabilizing mechanisms for quadratic nonlinear fields hexagonal resonant triad patterns are shown to exist as stable solutions of a particular type of nonlinear field where no cubic field nonlinearity is present the zero dc fourier mode is shown to stabilize these patterns produced by a pure quadratic field nonlinearity closed form solutions and stability results are obtained near the critical point complimented by numerical studies far from the critical point these results are obtained using a neural field based on the helmholtzian operator constraints on structure and parameters for a general pure quadratic neural field which supports hexagonal patterns are obtained

a staggered semi implicit spectral discontinuous galerkin scheme for the shallow water equations a spatially arbitrary high order semi implicit spectral discontinuous galerkin dg scheme for the numerical solution of the shallow water equations on staggered control volumes is derived and discussed the free surface elevation and the momentum are expressed in terms of the same polynomials that are used as basis and as test functions each unknown is however defined on a different set of control volumes that are spatially staggered with respect to each other a semi implicit time integration yields a stable and efficient mass conservative algorithm the use of a staggered mesh has the advantage that after substitution of the momentum equations into the mass conservation equation only one single block penta diagonal system must be solved for the new free surface location which is a scalar quantity subsequently the new momentum components are directly obtained the staggered semi implicit approach makes the present dg scheme different from other published dg schemes the sparse linear system of the resulting discrete wave equation for the free surface can be conveniently solved by a matrix free gmres algorithm furthermore for the shallow water equations the proposed scheme can be written as a quadrature free method where all surface and volume integrals can be precomputed once and for all on the reference element and assembled into universal matrices and tensors for the special case n 0 the proposed method reduces to a classical semi implicit finite difference scheme the proposed semi implicit scheme is particularly well suited for low froude number flows the method is validated on some typical academic benchmark problems using polynomial degrees of up to n 20

resonant triad dynamics in weakly damped faraday waves with two frequency forcing abstract many of the interesting patterns seen in recent multi frequency faraday experiments can be understood on the basis of three wave interactions resonant triads in this paper we consider two frequency forcing and focus on a resonant triad occurring near the bicritical point where two pattern forming modes with distinct wavenumbers onset simultaneously this triad has been observed directly in the form of rhomboids and has also been implicated in the formation of quasipatterns and superlattices we show how the symmetries of the undamped unforced problem time translation time reversal and hamiltonian structure can be used when the damping is weak to obtain general scaling laws and additional qualitative properties of the normal form coefficients governing the pattern selection process near onset such features help to explain why this particular triad is seen only for certain low forcing ratios and predict the existence of drifting solutions and heteroclinic cycles we confirm the anticipated parameter dependence of the coefficients and investigate its dynamical consequences using coefficients derived numerically from a quasipotential formulation of the faraday problem due to zhang and vinals

generalized locally toeplitz sequences spectral analysis and applications to discretized partial differential equations abstract starting from the finite difference discretization of an elliptic second order pde as i j 1 d x i a i j x x j u x b x over a bounded domain we introduce the notion of generalized locally toeplitz sequence of matrices the singular value distribution the eigenvalue distribution in the hermitian case is studied and characterized for generalized locally toeplitz sequences in terms of weighted multidimensional szeg formulas this extends preceding results attributed to tilli which concern the unilevel case the application of this theoretic analysis to the numerical solution of pdes is finally discussed

hp adaptive discontinuous galerkin finite element methods for first order hyperbolic problems we consider the a posteriori error analysis of hp discontinuous galerkin finite element approximations to first order hyperbolic problems in particular we discuss the question of error estimation for linear functionals such as the outflow flux and the local average of the solution based on our a posteriori error bound we design and implement the corresponding adaptive algorithm to ensure reliable and efficient control of the error in the prescribed functional to within a given tolerance this involves exploiting both local polynomial degree variation and local mesh subdivision the theoretical results are illustrated by a series of numerical experiments

resonant triad interactions in symmetric systems abstract we analyse resonant triad interactions in symmetric systems the case of coupling at quadratic order in amplitudes and the case of coupling at cubic order are considered we study modulated travelling waves and heteroclinic cycles in amplitude equations and we discuss their stability types we describe possible dynamical regimes and spatial patterns arising from triad interactions of long waves in rapidly rotating hagen poiseuille flow

the 2 1 steady hopf mode interaction in the two layer be nard problem abstract the interaction of hopf and steady modes with wavenumber ratio 2 1 is investigated for a critical situation two different immiscible liquids lie in layers between horizontal walls and are heated from below a situation with a pair of complex conjugate eigenvalues at wavenumber and a real eigenvalue at wavenumber 2 is at criticality weakly nonlinear amplitude equations are derived for the interaction of these oscillatory and steady modes the two modes generate a two parameter bifurcation the coefficients involved in the equations are determined numerically based on the physical parameters of the system at criticality three obvious equilibrium solutions of the amplitude equations are the steady solution the traveling waves and the mixed standing waves the eigenvalues governing the stability of these solutions are found explicitly numerical results and bifurcation diagrams are given for the critical situation the steady solution and the traveling wave solution are unstable there is a region of stability for the standing wave solution a new equilibrium solution the asymmetric mixed mode is found to be stable in a parameter range bifurcations from the standing wave solution and the asymmetric mixed mode are described

iterative solution of piecewise linear systems and applications to flows in porous media the correct numerical modeling of free surface hydrodynamic problems often requires to have the solution of special linear systems whose coefficient matrix is a piecewise constant function of the solution itself in doing so one may fulfill relevant physical constraints the existence the uniqueness and two constructive iterative methods to solve a piecewise linear system of the form max boldsymbol l min boldsymbol u mathbf x t mathbf x mathbf b are analyzed the methods are shown to have a finite termination property i e they converge to an exact solution in a finite number of steps and actually they converge very quickly as confirmed by a few numerical tests which are derived from the mathematical modeling of flows in porous media

universality in two dimensional cellular structures evolving by cell division and disappearance the dynamics of two dimensional cellular networks is written in terms of coupled population equations which describe how the population of s sided cells is affected by cell division and disappearance in these equations the effect of the rest of the foam on the disappearing or dividing cell is treated as a local mean field under not too restrictive conditions the equilibrium distribution p s of cells satisfies a linear difference equation of order two or higher the population equations are asymptotically integrable the asymptotic integrability implies a universal distribution p s approximately cs kzs for large values of s which is also the boltzmann distribution associated with the maximum entropy inference asymptotic integrability of the population equations is absent in a global mean field approximation the importance of short range topological information to control the evolution of foams is thus confirmed

a high order semi implicit discontinuous galerkin method for the two dimensional shallow water equations on staggered unstructured meshes a well balanced spatially arbitrary high order accurate semi implicit discontinuous galerkin scheme is presented for the numerical solution of the two dimensional shallow water equations on staggered unstructured non orthogonal grids the semi implicit method is derived in such a fashion that all relevant integrals can be precomputed and stored in a preprocessing stage so that the extension to curved isoparametric elements is natural and does not increase the computational effort of the simulation at runtime for p 0 the resulting scheme becomes a generalization of the classical semi implicit finite volume finite difference scheme of casulli and walters 2000 25 but with less conditions on the grid geometry the method proposed in this paper allows large time steps with respect to the surface wave speed gh and is thus particularly suitable for low froude number flows the approach is validated on some typical academic benchmark problems using polynomial degrees up to p 6

a semi implicit semi lagrangian p adaptive discontinuous galerkin method for the shallow water equations a semi implicit and semi lagrangian discontinuous galerkin method for the shallow water equations is proposed for applications to geophysical scale flows a non conservative formulation of the advection equation is employed in order to achieve a more treatable form of the linear system to be solved at each time step the method is equipped with a simple p adaptivity criterion that allows to adjust dynamically the number of local degrees of freedom employed to the local structure of the solution numerical results show that the method captures well the main features of gravity and inertial gravity waves as well as reproducing correct solutions in nonlinear test cases with analytic solutions the accuracy and effectiveness of the method are also demonstrated by numerical results obtained at high courant numbers and with automatic choice of the local approximation degree

new type of complex dynamics in the 1 2 spatial resonance abstract the amplitude equations describing the 1 2 spatial resonance in o 2 symmetric systems are re examined complex dynamical behavior associated with a new class of global connections is identified and analyzed these connections form further from the mode interaction point where the structurally stable heteroclinic cycles described by armbruster et al physica d 29 1988 257 no longer exist the resulting behavior is characteristic of systems with o 2 z 2 symmetry in which the symmetry z 2 is weakly broken

a pseudospectral method for solution of the three dimensional incompressible navier stokes equations abstract a new chebyshev pseudospectral technique based on the projection method that was previously applied by the authors to the solution of two dimensional incompressible navier stokes equations in primitive variables for nonperiodic boundary conditions is extended to solve the three dimensional navier stokes equations the crucial point of the method is the requirement that the continuity equation be satisfied everywhere in the domain on the boundaries as well as in the interior the key feature of the work presented in this paper is that the computer storage requirements of the full matrix inversion resulting from direct solution of the pressure poisson equation in three dimensions is greatly reduced by considering an eigenfunction decomposition the method was tested on a two dimensional driven cavity flow and the results were compared with those of the most accurate finite difference calculation the three dimensional driven cavity flow was then calculated at the same reynolds numbers as the two dimensional cases i e re 100 400 and 1000 in the calculated results three dimensional boundary effects were observed in all cases and became more apparent with increasing reynolds number

are randomly grown graphs really random we analyze a minimal model of a growing network at each time step a new vertex is added then with probability ensuremath delta two vertices are chosen uniformly at random and joined by an undirected edge this process is repeated for t time steps in the limit of large t the resulting graph displays surprisingly rich characteristics in particular a giant component emerges in an infinite order phase transition at ensuremath delta 1 8 at the transition the average component size jumps discontinuously but remains finite in contrast a static random graph with the same degree distribution exhibits a second order phase transition at ensuremath delta 1 4 and the average component size diverges there these dramatic differences between grown and static random graphs stem from a positive correlation between the degrees of connected vertices in the grown graph older vertices tend to have higher degree and to link with other high degree vertices merely by virtue of their age we conclude that grown graphs however randomly they are constructed are fundamentally different from their static random graph counterparts

why social networks are different from other types of networks we argue that social networks differ from most other types of networks including technological and biological networks in two important ways first they have nontrivial clustering or network transitivity and second they show positive correlations also called assortative mixing between the degrees of adjacent vertices social networks are often divided into groups or communities and it has recently been suggested that this division could account for the observed clustering we demonstrate that group structure in networks can also account for degree correlations we show using a simple model that we should expect assortative mixing in such networks whenever there is variation in the sizes of the groups and that the predicted level of assortative mixing compares well with that observed in real world networks

scale free characteristics of random networks the topology of the world wide web the world wide web forms a large directed graph whose vertices are documents and edges are links pointing from one document to another here we demonstrate that despite its apparent random character the topology of this graph has a number of universal scale free characteristics we introduce a model that leads to a scale free network capturing in a minimal fashion the self organization processes governing the world wide web c 2000 elsevier science b v all rights reserved pacs 84 35 i 64 60 fr 87 23 ge

a generic hard transition to chaos abstract a hard in amplitude transition to chaos in a class of dissipative flows of broad applicability is presented for positive values of a parameter no matter how small a fully developed chaotic attractor exists within some domain of additional parameters whereas no chaotic behavior exists for 0 as is made positive an unstable fixed point reaches an invariant plane to enter a phase half space of physical solutions the ghosts of a line of fixed points and a rich heteroclinic structure existing at 0 make the limits t 0 non commuting and allow an exact description of the chaotic flow the formal structure of flows that exhibit the transition is determined a subclass of such flows coupled oscillators in near resonance at any 2 q frequency ratio with representing linear excitation of the first oscillator is fully analysed

percolation in hierarchical scale free nets we study the percolation phase transition in hierarchical scale free nets depending on the method of construction the nets can be fractal or small world the diameter grows either algebraically or logarithmically with the net size assortative or disassortative a measure of the tendency of like degree nodes to be connected to one another or possess various degrees of clustering the percolation phase transition can be analyzed exactly in all these cases due to the self similar structure of the hierarchical nets we find different types of criticality illustrating the crucial effect of other structural properties aside from the scale free degree distribution of the nets

hexagonal patterns in finite domains in many mathematical models for pattern formation a regular hexagonal pattern is stable in an infinite region however laboratory and numerical experiments are carried out in finite domains and this imposes certain constraints on the possible patterns in finite rectangular domains it is shown that a regular hexagonal pattern cannot occur if the aspect ratio is rational in practice it is found experimentally that in a rectangular region patterns of irregular hexagons are often observed this work analyses the geometry and dynamics of irregular hexagonal patterns these patterns occur in two different symmetry types either with a reflection symmetry involving two wavenumbers or without symmetry involving three different wavenumbers the relevant amplitude equations are studied to investigate the detailed bifurcation structure in each case it is shown that hexagonal patterns can bifurcate subcritically either from the trivial solution or from a pattern of rolls numerical simulations of a model partial differential equation are also presented to illustrate the behaviour