postprocessing the galerkin method the finite element case a postprocessing technique developed earlier for spectral methods is extended here to galerkin finite element methods for dissipative evolution partial differential equations the postprocessing amounts to solving a linear elliptic problem on a finer grid or higher order space once the time integration on the coarser mesh is completed this technique increases the convergence rate of the finite element method to which it is applied and this is done at almost no additional computational cost the numerical experiments presented here show that the resulting postprocessed method is computationally more efficient than the method to which it is applied say quadratic finite elements as well as standard methods of similar order of convergence as the postprocessed one say cubic finite elements the error analysis of the new method is performed in l2 and in l infty norms

optimal error bounds for two grid schemes applied to the navier stokes equations abstract we consider two grid mixed finite element schemes for the spatial discretization of the incompressible navier stokes equations a standard mixed finite element method is applied over the coarse grid to approximate the nonlinear navier stokes equations while a linear evolutionary problem is solved over the fine grid the previously computed galerkin approximation to the velocity is used to linearize the convective term for the analysis we take into account the lack of regularity of the solutions of the navier stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data optimal error bounds are obtained

two level method based on finite element and crank nicolson extrapolation for the time dependent navier stokes equations a fully discrete two level finite element method the two level method is presented for solving the two dimensional time dependent navier stokes problem the method requires a crank nicolson extrapolation solution u h tau 0 p h tau 0 on a spatial time coarse grid j h tau 0 and a backward euler solution u h tau p h tau on a space time fine grid j h tau the error estimates of optimal order of the discrete solution for the two level method are derived compared with the standard crank nicolson extrapolation method the one level method based on a space time fine grid j h tau the two level method is of the error estimates of the same order as the one level method in the h1 norm for velocity and the l2 norm for pressure however the two level method involves much less work than the one level method

element wise a posteriori estimates based on hierarchical bases for non linear parabolic problems we show that the issue of a posteriori estimate the errors in the numerical simulation of non linear parabolic equations can be reduced to a posteriori estimate the errors in the approximation of an elliptic problem with the right hand side depending on known data of the problem and the computed numerical solution a procedure to obtain local error estimates for the p version of the finite element method by solving small discrete elliptic problems with right hand side the residual of the p fem solution is introduced the boundary conditions are inherited by those of the space of hierarchical bases to which the error estimator belongs we prove that the error in the numerical solution can be reduced by adding the estimators that behave as a locally defined correction to the computed approximation when the error being estimated is that of a elliptic problem constant free local lower bounds are obtained the local error estimation procedure is applied to non linear parabolic differential equations in several space dimensions some numerical experiments for both the elliptic and the non linear parabolic cases are provided copyright 2005 john wiley sons ltd

a spectral element method for the navier stokes equations with improved accuracy we present a procedure to enhance the accuracy of spectral element methods for the evolutionary navier stokes equations the approximations to the velocity and the pressure obtained using a spectral element method are postprocessed by solving a discrete stokes problem only once when the time evolution is completed the analysis shows that this procedure improves the order of convergence for the approximations to the velocity and the pressure the choice of a discrete pressure space endowed with an inf sup condition independent of the discretization parameter leads to optimal error bounds for the pressure

two level method and some a priori estimates in unsteady navier stokes calculations a two level method proposed for quasielliptic problems is adapted in this paper to the simulation of unsteady incompressible navier stokes flows the method requires a solution of a nonlinear problem on a coarse grid and a solution of linear symmetric problem on a fine grid the scaling between these two grids is superlinear approximation stability and convergence aspects of a fully discrete scheme are considered stability properties of the two level scheme are compared with those for a commonly used semi implicit scheme some new estimates are also proved for the latter

multi level spectral galerkin method for the navier stokes problem i spatial discretization a multi level spectral galerkin method for the two dimensional non stationary navier stokes equations is presented the method proposed here is a multiscale method in which the fully nonlinear navier stokes equations are solved only on a low dimensional space subsequent approximations are generated on a succession of higher dimensional spaces j 2 j by solving a linearized navier stokes problem around the solution on the previous level error estimates depending on the kinematic viscosity 0 1 are also presented for the j level spectral galerkin method the optimal accuracy is achieved when we demonstrate theoretically that the j level spectral galerkin method is much more efficient than the standard one level spectral galerkin method on the highest dimensional space

postprocessing finite element methods for the navier stokes equations the fully discrete case an accuracy enhancing postprocessing technique for finite element discretizations of the navier stokes equations is analyzed the technique had been previously analyzed only for semidiscretizations and fully discrete methods are addressed in the present paper we show that the increased spatial accuracy of the postprocessing procedure is not affected by the errors arising from any convergent time stepping procedure further refined bounds are obtained when the time stepping procedure is either the backward euler method or the two step backward differentiation formula

the postprocessed mixed finite element method for the navier stokes equations a postprocessing technique for mixed finite element methods for the incompressible navier stokes equations is studied the technique was earlier developed for spectral and standard finite element methods for dissipative partial differential equations the postprocessing amounts to solving a stokes problem on a finer grid or higher order space once the time integration on the coarser mesh is completed the analysis presented here shows that this technique increases the convergence rate of both the velocity and the pressure approximations numerical experiments are presented that confirm both this increase in the convergence rate and the corresponding improvement in computational efficiency

enhancing network robustness against malicious attacks in a recent work schneider et al proc natl acad sci usa 108 3838 2011 the authors proposed a simple measure for network robustness under malicious attacks on nodes using a greedy algorithm they found that the optimal structure with respect to this quantity is an onion structure in which high degree nodes form a core surrounded by rings of nodes with decreasing degree however in real networks the failure can also occur in links such as dysfunctional power cables and blocked airlines accordingly complementary to the node robustness measurement r n we propose a link robustness index r l we show that solely enhancing r n cannot guarantee the improvement of r l moreover the structure of an r l optimized network is found to be entirely different from that of an onion network in order to design robust networks that are resistant to a more realistic attack condition we propose a hybrid greedy algorithm that takes both the r n and r l into account we validate the robustness of our generated networks against malicious attacks mixed with both nodes and links failure finally some economical constraints for swapping the links in real networks are considered and significant improvement in both aspects of robustness is still achieved

cascading failures on correlated interdependent networks with dependency groups abstract the topologies of interdependent networks markedly affect the cascading failures and phase transition however the effects of interlayer degree correlations on cascading failures on interdependent networks with dependency groups have not been investigated in this paper we propose a cascading failure model on correlated interdependent networks with dependency groups and use a percolation theory to describe the final state of the dynamics through extensive numerical simulations and theoretical analysis we find that interlayer degree correlations enhance the robustness of the system on networks with homogeneous degree distribution the system always exhibits a discontinuous phase transition however on networks with heterogeneous degree distributions the phase transition is continuous discontinuous for strong weak interlayer degree correlations the theoretical and numerical predictions agree well with each other

the postprocessed mixed finite element method for the navier stokes equations refined error bounds a postprocessing technique for mixed finite element methods for the incompressible navier stokes equations is analyzed the postprocess which amounts to solving a linear stokes problem is shown to increase the order of convergence of the method to which it is applied by one unit times the logarithm of the mesh diameter in proving the error bounds some superconvergence results are also obtained contrary to previous analysis of the postprocessing technique in the present paper we take into account the loss of regularity suffered by the solutions of the navier stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data as in h g heywood and r rannacher siam j numer anal 25 1988 pp 489 512 where the same hypothesis is assumed no better than fifth order convergence is achieved

structure of shells in complex networks we define shell l in a network as the set of nodes at distance l with respect to a given node and define rl as the fraction of nodes outside shell l in a transport process information or disease usually diffuses from a random node and reach nodes shell after shell thus understanding the shell structure is crucial for the study of the transport property of networks we study the statistical properties of the shells of a randomly chosen node for a randomly connected network with given degree distribution we derive analytically the degree distribution and average degree of the nodes residing outside shell l as a function of rl further we find that rl follows an iterative functional form rl phi rl 1 where phi is expressed in terms of the generating function of the original degree distribution of the network our results can explain the power law distribution of the number of nodes bl found in shells with l larger than the network diameter d which is the average distance between all pairs of nodes for real world networks the theoretical prediction of rl deviates from the empirical rl we introduce a network correlation function c rl identical with rl phi rl 1 to characterize the correlations in the network where rl is the empirical value and phi rl 1 is the theoretical prediction c rl 1 indicates perfect agreement between empirical results and theory we apply c rl to several model and real world networks we find that the networks fall into two distinct classes i a class of poorly connected networks with c rl 1 where a larger smaller fraction of nodes resides outside inside distance l from a given node than in randomly connected networks with the same degree distributions examples include the watts strogatz model and networks characterizing human collaborations such as citation networks and the actor collaboration network ii a class of well connected networks with c rl 1 examples include the barabasi albert model and the autonomous system internet network

finite element approximation of the nonstationary navier stokes problem part iii smoothing property and higher order error estimates for spatial discretization this paper continues our error analysis of finite element galerkin approximation of the nonstationary navier stokes equations optimal order error estimates both local and global are derived for higher order finite elements under appropriate assumptions about the smoothness and stability of the solution these assumptions take into account the loss of regularity at t 0 that one generally has to expect in the absence of higher order nonlocal compatibility conditions for the data of the problem

stability of higher order hood taylor methods the stability of a higher order hood taylor method for the approximation of the stationary stokes equations using continuous piecewise polynomials of degree 3 to approximate velocities and continuo

the postprocessing galerkin and nonlinear galerkin methods a truncation analysis point of view we revisit the postprocessing algorithm and give a justification from a classical truncation analysis point of view we assume a perturbation expansion for the high frequency mode component of solutions to the underlying equation keeping terms to certain orders we then generate approximate systems which correspond to numerical schemes we show that the first two leading order methods are in fact the postprocessed galerkin and postprocessed nonlinear galerkin methods respectively hence postprocessed galerkin is a natural leading order method more natural than the standard galerkin method for approximating solutions of parabolic dissipative pdes the analysis is presented in the framework of the two dimensional navier stokes equation nse however similar analysis may be done for any parabolic dissipative nonlinear pde the truncation analysis is based on asymptotic estimates in time for the low and high mode components we also introduce and investigate an alternative postprocessing scheme whic

from m clusters to m noncrossing partitions via exceptional sequences let w be a finite crystallographic reflection group the generalized catalan number of w coincides both with the number of clusters in the cluster algebra associated to w and with the number of noncrossing partitions for w natural bijections between these two sets are known for any positive integer m both m clusters and m noncrossing partitions have been defined and the cardinality of both these sets is the fuss catalan number cm w we give a natural bijection between these two sets by first establishing a bijection between two particular sets of exceptional sequences in the bounded derived category db h for any finite dimensional hereditary algebra h

partial tilting modules over m replicated algebras abstract let a be a hereditary algebra over an algebraically closed field k and a m be the m replicated algebra of a given an a m module t we denote by t the number of non isomorphic indecomposable summands of t in this paper we prove that a partial tilting a m module t is a tilting a m module if and only if t a m and that every partial tilting a m module has complements as an application we deduce that the tilting quiver k a m of a m is connected moreover we investigate the number of complements to almost tilting modules over duplicated algebras

spectral properties of supra laplacian for partially interdependent networks abstract the spectrum of the laplacian matrices of complex networks is a key factor in network functionality in this paper the spectral properties of supra laplacian for partially interdependent networks are investigated based on matrix perturbation theory refined results of the eigenvalue properties of laplacian matrices are provided which shows that the size relationship of the first order approximate solutions of the eigenvalues remains unchanged even if there is perturbation using these results the theoretical approximate formulae of the minimum non zero eigenvalue and the maximum eigenvalue of supra laplacian for partially interdependent networks are derived respectively the outcomes in this paper are more general and need fewer calculations than that in the literature 13 14 finally the simulations and applications of synchronizability and diffusion process verify the feasibility and effectiveness of the proposed solutions the findings can be instructive for networks when linking to the spectral properties of the supra laplacian

effects of group size distribution on cascading failure in partially interdependent networks abstract cascading failures on interdependent networks has attracted much attention in the field of network science however the effects of the distribution of group size on cascading failures still puzzled us in this paper we propose a cascading failure model on two layered partially interdependent networks with groups in which all nodes in the same group are functional dependent on each other a percolation theory is developed to investigate the final giant component and the theory well agrees with numerical simulations through extensive numerical simulations and theoretical analyses we find a discontinuous phase transition in the system in addition the final giant component increases with the heterogeneity of group size distribution and the percolation threshold decreases with the heterogeneity of group size distribution furthermore the phase transition is discontinuous for relatively strong dependency while the phase transition is continuous for relatively weak dependency the degree distribution of the interdependent networks does not qualitatively alter the above presented phenomena

a posteriori error estimations for mixed finite element approximations to the navier stokes equations a posteriori estimates for mixed finite element discretizations of the navier stokes equations are derived we show that the task of estimating the error in the evolutionary navier stokes equations can be reduced to the estimation of the error in a steady stokes problem as a consequence any available procedure to estimate the error in a stokes problem can be used to estimate the error in the nonlinear evolutionary problem a practical procedure to estimate the error based on the so called postprocessed approximation is also considered both the semidiscrete in space and the fully discrete cases are analyzed some numerical experiments are provided

postprocessing the linear finite element method we extend the idea of the postprocessing galerkin methods for dissipative evolution equations to the case of the linear finite element method the postprocessing technique has been developed earlier for spectral methods and for higher order finite element methods the analysis shows that this procedure improves the order of convergence of the piecewise linear galerkin finite element approximation in the h1 norm we show by means of numerical experiments that there is no improvement in the order of convergence in the l2 norm

bootstrap percolation on complex networks we consider bootstrap percolation on uncorrelated complex networks we obtain the phase diagram for this process with respect to two parameters f the fraction of vertices initially activated and p the fraction of undamaged vertices in the graph we observe two transitions the giant active component appears continuously at a first threshold there may also be a second discontinuous hybrid transition at a higher threshold avalanches of activations increase in size as this second critical point is approached finally diverging at this threshold we describe the existence of a special critical point at which this second transition first appears in networks with degree distributions whose second moment diverges but whose first moment does not we find a qualitatively different behavior in this case the giant active component appears for any fg0 and pg0 and the discontinuous transition is absent this means that the giant active component is robust to damage and also is very easily activated we also formulate a generalized bootstrap process in which each vertex can have an arbitrary threshold

finite element approximation of the nonstationary navier stokes problem i regularity of solutions and second order error estimates for spatial discretization this is the first part of a work dealing with the rigorous error analysis of finite element solutions of the nonstationary navier stokes equations second order error estimates are proven for spatial discretization using conforming or nonconforming elements the results indicate a fluid like behavior of the approximations even in the case of large data so long as the solution remains regular the analysis is based on sharp a priori estimates for the solution particularly reflecting its behavior as t to 0 and as t to infty it is shown that the regularity customarily assumed in the error analysis for corresponding parabolic problems cannot be realistically assumed in the case of the navier stokes equations as it depends on nonlocal compatibility conditions for the data the results which are presented here are independent of such compatibility conditions which cannot be verified in practice

rigid objects triangulated subfactors and abelian localizations we show that the abelian category mathsf mod text mathcal x of coherent functors over a contravariantly finite rigid subcategory mathcal x in a triangulated category mathcal t is equivalent to the gabriel zisman localization at the class of regular maps of a certain factor category of mathcal t and moreover it can be calculated by left and right fractions thus we generalize recent results of buan and marsh we also extend recent results of iyama yoshino concerning subfactor triangulated categories arising from mutation pairs in mathcal t in fact we give a classification of thick triangulated subcategories of a natural pretriangulated factor category of mathcal t and a classification of functorially finite rigid subcategories of mathcal t if the latter has serre duality in addition we characterize 2 cluster tilting subcategories along these lines finally we extend basic results of keller reiten concerning the gorenstein and the calabi yau property for categories arising from certain rigid not necessarily cluster tilting subcategories as well as several results of the literature concerning the connections between 2 cluster tilting subcategories of triangulated categories and tilting subcategories of the associated abelian category of coherent functors

nonlinear galerkin methods and mixed finite elements two grid algorithms for the navier stokes equations a nonlinear galerkin method using mixed finite elements is presented for the two dimensional incompressible navier stokes equations the scheme is based on two finite element spaces x h and x h for the approximation of the velocity defined respectively on one coarse grid with grid size h and one fine grid with grid size h ll h and one finite element space m h for the approximation of the pressure nonlinearity and time dependence are both treated on the coarse space we prove that the difference between the new nonlinear galerkin method and the standard galerkin solution is of the order of h 2 both in velocity h 1 omega and pressure l 2 omega norm we also discuss a penalized version of our algorithm which enjoys similar properties

postprocessing the finite element method for semilinear parabolic problems in this paper we consider postprocessing of the finite element method for semilinear parabolic problems the postprocessing amounts to solving a linear elliptic problem on a finer grid or higher order space once the time integration on the coarser mesh is completed the convergence rate is increased at almost no additional computational cost this procedure was introduced and analyzed in garcia archilla and titi siam j numer anal 37 2000 pp 470 499 we extend the analysis to the fully discrete case and prove error estimates for both space and time discretization the analysis is based on error estimates for the approximation of time derivatives by difference quotients

robustness of a network formed by n interdependent networks with a one to one correspondence of dependent nodes many real world networks interact with and depend upon other networks we develop an analytical framework for studying a network formed by n fully interdependent randomly connected networks each composed of the same number of nodes n the dependency links connecting nodes from different networks establish a unique one to one correspondence between the nodes of one network and the nodes of the other network we study the dynamics of the cascades of failures in such a network of networks non caused by a random initial attack on one of the networks after which a fraction p of its nodes survives we find for the fully interdependent loopless non that the final state of the non does not depend on the dynamics of the cascades but is determined by a uniquely defined mutual giant component of the non which generalizes both the giant component of regular percolation of a single network n 1 and the recently studied case of the mutual giant component of two interdependent networks n 2 we also find that the mutual giant component does not depend on the topology of the non and express it in terms of generating functions of the degree distributions of the network our results show that for any n 2 there exists a critical p p c 0 below which the mutual giant component abruptly collapses from a finite nonzero value for p p c to zero for p 2 a rr non is stable for any n with p c 1 this results arises from the critical role played by singly connected nodes which exist in an er non and enhance the cascading failures but do not exist in a rr non

finite element approximations of the nonstationary navier stokes problem part iv error estimates for second order time discretization this paper provides an error analysis for the crank nicolson method of time discretization applied to spatially discrete galerkin approximations of the nonstationary navier stokes equations second order error estimates are proven locally in time under realistic assumptions about the regularity of the solution for approximations of an exponentially stable solution these local error estimates are extended uniformly in time as text t to infty

breakdown of interdependent directed networks increasing evidence shows that real world systems interact with one another via dependency connectivities failing connectivities are the mechanism behind the breakdown of interacting complex systems e g blackouts caused by the interdependence of power grids and communication networks previous research analyzing the robustness of interdependent networks has been limited to undirected networks however most real world networks are directed their in degrees and out degrees may be correlated and they are often coupled to one another as interdependent directed networks to understand the breakdown and robustness of interdependent directed networks we develop a theoretical framework based on generating functions and percolation theory we find that for interdependent erd s renyi networks the directionality within each network increases their vulnerability and exhibits hybrid phase transitions we also find that the percolation behavior of interdependent directed scale free networks with and without degree correlations is so complex that two criteria are needed to quantify and compare their robustness the percolation threshold and the integrated size of the giant component during an entire attack process interestingly we find that the in degree and out degree correlations in each network layer increase the robustness of interdependent degree heterogeneous networks that most real networks are but decrease the robustness of interdependent networks with homogeneous degree distribution and with strong coupling strengths moreover by applying our theoretical analysis to real interdependent international trade networks we find that the robustness of these real world systems increases with the in degree and out degree correlations confirming our theoretical analysis

a posteriori error estimations for mixed finite element approximations to the navier stokes equations based on newton type linearization abstract in this paper we derive a posteriori error estimates for inf sup stable mixed finite element approximations to the evolutionary navier stokes equations we reduce the problem of getting a posteriori error estimations of a non linear evolutionary problem to that of getting a posteriori estimations of a linear steady problem the main idea is based on the fact that the solutions of some newton type linearized navier stokes equations around the plain galerkin approximations approach the solution of the original navier stokes equations with a bigger rate of convergence than the plain galerkin method as a consequence the difference between the galerkin approximations and the solution of the linearized problem is an estimator of the galerkin error moreover since the galerkin approximation of the evolutionary navier stokes equations is also the galerkin approximation of the linearized equations any a posteriori estimator of the error in the newton type linearized navier stokes equations is also an a posteriori error estimator of the full navier stokes equations

relative homology and maximal l orthogonal modules abstract let be an artin algebra and let m 1 and m 2 be generator cogenerators for mod such that the global dimension of their endomorphism ring is finite we characterize tilting modules of the form hom m 2 m 1 with respect to the relative theories induced by m 1 and m 2 on mod and apply our results to maximal l orthogonal modules

universality and scaling laws in the cascading failure model with healing cascading failures may lead to dramatic collapse in interdependent networks where the breakdown takes place as a discontinuity of the order parameter in the cascading failure cf model with healing there is a control parameter which at some value suppresses the discontinuity of the order parameter however up to this value of the healing parameter the breakdown is a hybrid transition meaning that besides this first order character the transition shows scaling too in this paper we investigate the question of universality related to the scaling behavior recently we showed that the hybrid phase transition in the original cf model has two sets of exponents describing respectively the order parameter and the cascade statistics which are connected by a scaling law in the cf model with healing we measure these exponents as a function of the healing parameter we find two universality classes in the wide range below the critical healing value the exponents agree with those of the original model while above this value the model displays trivial scaling meaning that fluctuations follow the central limit theorem

a postprocessed galerkin method with chebyshev or legendre polynomials we present an approximate inertial manifold based postprocess to enhance chebyshev or legendre spectral galerkin methods we prove that the postprocess improves the order of convergence of the galerkin solution yielding the same accuracy as the nonlinear galerkin method numerical experiments show that the new method is computationally more efficient than galerkin and nonlinear galerkin methods new approximation results for chebyshev polynomials are presented

a small eddy correction method for nonlinear dissipative evolutionary equations considering the interaction between the large and small eddy components of solutions and using the idea of the newton iteration a small eddy correction method is proposed for approximating and numerically solving nonlinear dissipative pdes of parabolic type in particular the navier stokes equations nse we assume that the large eddy approximation to the solution is known formally applying the newton iterative procedure to the small eddy equation we then generate approximate systems it is shown that the first two steps in fact lead to the standard galerkin method sgm and the so called optimum nonlinear galerkin method ong and therefore the small eddy correction method is actually a certain generalization of sgm and ong the boundedness and convergence analysis are presented in the framework of the two dimensional nse the results show that the small eddy correction method can greatly improve the accuracy of sgm approximate solutions

on modules with trivial self extensions throughout this note all rings will be self basic connected artin algebras over a fixed commutative local artin ring k and all modules will be finitely generated homomorphisms will be written on the opposite side of the scalars by mod a resp a mod we denote the category of all right resp left a modules for an artin algebra a the ordinary duality functor is denoted by d mod a 2 a mod i e d hom f with the minimal injective cogenerator i over k we are interested in the bimodules bta with the following properties

generalized postprocessed approximations to the navier stokes equations based on two grids abstract in this paper we generalize previous postprocessed approximations to the navier stokes equations the idea of postprocessing in the context of mixed finite element approximations is the following once a standard galerkin mixed finite element approximation is computed using a mesh we will call coarse of size h at a fixed time t we postprocess the approximation using a finer mesh of size h h by solving a steady problem with data based on the galerkin approximation in the literature one can find different ways to postprocess the galerkin approximation depending on the problem that is solved over the fine mesh at the final time t basically one can solve a stokes problem an oseen type problem or a newton type problem in the present paper we present a method based on several parameters that generalize the above postprocessing techniques depending on the values chosen for the different parameters one can recover one of the old known postprocessing procedures but also produce a new different method we get error bounds for the generalized method valid for any of the values of the different parameters in all the cases the postprocessed method has a rate of convergence one unit bigger than the rate of convergence of the plain galerkin method in terms of the coarse mesh h and optimal in terms of the fine mesh h the computational added cost of postprocessing is however negligible for the error analysis we do not assume nonlocal compatibility conditions for the true solutions of the navier stokes equations some numerical experiments show the performance of the method for different values of the parameters

a posteriori error estimates for fully discrete nonlinear parabolic problems abstract we present estimates for the spatial error in fully discrete approximations to nonlinear parabolic problems that extend the a posteriori estimates for the continuous time semi discretization introduced in de frutos and novo j de frutos j novo a posteriori error estimation with the p version of the finite element method for nonlinear parabolic differential equations comput methods appl mech engrg 191 2002 4893 4904 the error indicator is defined as the difference between the fully discrete galerkin and the fully discrete postprocessed approximations we prove that the main term of the temporal error in both the fully discrete galerkin and the postprocessed approximation is the same as a consequence the spatial approximation error can be independently estimated with regard to the time approximation error some numerical experiments are provided

postprocessing the galerkin method a novel approach to approximate inertial manifolds a postprocess of the standard galerkin method for the discretization of dissipative equations is presented the postprocessed galerkin method uses the same approximate inertial manifold phi app to approximate the high wave number modes of the solution as in the nonlinear galerkin nlg method however in this postprocessed galerkin method the value of phi app is calculated only once after the time integration of the standard galerkin method is completed contrary to the nlg in which phi app evolves with time and affects the time evolution of the lower wave number modes the postprocessed galerkin method which is much cheaper to implement computationally than the nlg is shown in the case of fourier modes to possess the same rate of convergence accuracy as the simplest version of the nlg which is based on either the foias manley temam approximate inertial manifold or the euler galerkin approximate inertial manifold this is proved for some problems in one and two spatial dimensions including the navier stokes equations under periodic boundary conditions the advantages of postprocessing that we present here apply not only to the standard galerkin method but also to the computationally more efficient pseudospectral method

an approximate inertial manifolds approach to postprocessing the galerkin method for the navier stokes equation in a recent paper we have introduced a postprocessing procedure for the galerkin method for dissipative evolution partial differential equations with periodic boundary conditions the postprocessing technique uses approximate inertial manifolds to approximate the high modes the small scale components in the exact solutions in terms of the galerkin approximations which in this case play the role of the lower modes large scale components this procedure can be seen as a defect correction technique but contrary to standard procedures the correction is computed only when the time evolution is completed here we extend these results to more realistic boundary conditions specifically we study in detail the two dimensional navier stokes equations subject to homogeneous nonslip dirichlet boundary conditions we also discuss other equations such as reaction diffusion systems and the cahn hilliard equations

robustness of interdependent and interconnected clustered networks abstract in real world most systems show significant clustering and it is more practical to investigate the behaviors of clustered network previous studies are mostly focused on single clustered network and coupled clustered networks with dependency links here we study two clustered networks coupled with both interdependent and interconnected links by introducing generating function of the joint degree distribution when the networks are fully dependent we obtain the analytical solution of giant component p we show rich phase transition phenomena and analyze their behaviors we find that as dependency coupling strength increases the system changes from second order phase transition through hybrid transition to first order phase transition for weak dependency coupling strength q a corresponding to second order phase transition we find that clustering has almost no effect on the robustness of network but for strong dependency coupling strength q a corresponding to first order transition the more clustered system is more vulnerable at the same time we notice that when the system is more clustered the hybrid order region is almost unchangeable the first order region becomes smaller and the second order region is larger additionally we can see that the bigger the clustering coefficient c is the bigger the second order region becomes for the same c the density of connectivity links between networks is higher the second order region becomes smaller and the density of connectivity links within each network is higher the second order region becomes bigger

wakamatsu tilting modules abstract we study a generalization of tilting modules to modules of possibly infinite projective dimension introduced by wakamatsu in j algebra 114 1988 106 114 in particular we characterize these modules in terms of suitable subcategories of finitely generated modules and in terms of cotorsion theories

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

percolation of a general network of networks percolation theory is an approach to study the vulnerability of a system we develop an analytical framework and analyze the percolation properties of a network composed of interdependent networks netonet typically percolation of a single network shows that the damage in the network due to a failure is a continuous function of the size of the failure i e the fraction of failed nodes in sharp contrast in netonet due to the cascading failures the percolation transition may be discontinuous and even a single node failure may lead to an abrupt collapse of the system we demonstrate our general framework for a netonet composed of n classic erd ifmmode mbox h o else h o fi s r enyi er networks where each network depends on the same number m of other networks i e for a random regular network rr formed of interdependent er networks the dependency between nodes of different networks is taken as one to one correspondence i e a node in one network can depend only on one node in the other network no feedback condition in contrast to a treelike netonet in which the size of the largest connected cluster mutual component depends on n the loops in the rr netonet cause the largest connected cluster to depend only on m and the topology of each network but not on n we also analyzed the extremely vulnerable feedback condition of coupling where the coupling between nodes of different networks is not one to one correspondence in the case of netonet formed of er networks percolation only exhibits two phases a second order phase transition and collapse and no first order percolation transition regime is found in the case of the no feedback condition in the case of netonet composed of rr networks there exists a first order phase transition when the coupling strength q fraction of interdependency links is large and a second order phase transition when q is small our insight on the resilience of coupled networks might help in designing robust interdependent systems

percolation of partially interdependent scale free networks we study the percolation behavior of two interdependent scale free sf networks under random failure of 1 p fraction of nodes our results are based on numerical solutions of analytical expressions and simulations we find that as the coupling strength between the two networks q reduces from 1 fully coupled to 0 no coupling there exist two critical coupling strengths q 1 and q 2 which separate three different regions with different behavior of the giant component as a function of p i for q q 1 an abrupt collapse transition occurs at p p c ii for q 2 3 q 1 decreases with increasing here is the scaling exponent of the degree distribution p k proportionality k b in the hybrid transition at the q 2 0 for 3 thus the known theoretical p c 0 for a single network with 3 is expected to be valid also for strictly partial interdependent networks

targeted attack on networks coupled by connectivity and dependency links abstract coupled systems used to increase capacity were shown beneficial as long as it does not open pathways to cascades previous studies on the robustness of coupled networks except for interdependent networks are almost the cases of random attack many challenges remain exist in targeted attack problem of coupled networks since nodes within coupled networks show different functions for each network this paper both analytically and numerically analyzed the robustness of coupled networks under three types of targeted attacking strategies including attacking on nodes by considering internal and external degree internal degree only and external degree only for coupled network with both interdependent and interconnected links all degree distributions of intra and inter connectivity links are poissonian we find the system undergoes from second to first order phase transition as coupling strength q increases the fraction of nodes in the giant component p at stable state the critical phase transition threshold p c first order threshold p c i and second order threshold p c i i and the critical point p c q c separating the first and second order phase transitions are analytically obtained for three types of attacking strategies with attacking probability parameter 0 1 we also find the system becomes more vulnerable as the average degree of intra links k or inter links k decreases therefore the minimum average degrees k m i n and k m i n to maintain the system stable are obtained for the case of 0 1 moreover we discussed three typical cases of coupled networks interdependent networks k 0 interacting networks q 0 and bipartite network k 0 q 0 the analytical expressions of p p c and p c q c only for interdependent are given respectively besides we study the equivalence relations between interdependent networks and coupled networks with connectivity and dependency links for the same p c the results imply that we can adjust the parameters q k and k to make the two systems have the same robustness moreover the results imply that one should not only protect nodes with high degree of intra links or inter links but also defend nodes the sum of whose degree of intra links and inter links is big can avoid the system becoming vulnerable

on faithfully balanced modules f cotilting and f auslander algebras abstract the aim of this paper is to give a unification of the theory of co tilting modules and higher auslander correspondence by using faithfully balanced modules i e faithful modules with the double centralizer property in the language of relative homological algebra following auslander solberg a relative version of faithfully balancedness is given which will be called 1 f faithful the relative versions of higher auslander correspondence and brenner butler theorem are also developed

analysis of percolation behaviors of clustered networks with partial support dependence relations we carry out a study of percolation behaviors of clustered networks with partial support dependence relations by adopting two different attacking strategies attacking only one network and both networks which help to further understand real coupled networks for two different attacking strategies we find that the system changes from a second order phase transition to a first order phase transition as coupling strengthq increases we also notice that the first order region becomes smaller and the secondorder region becomes larger as average degree or clustering coefficient increases and as the average supported degree approaches infinity coupled clustered networks become independent and only the second order transition is observed which is similar toq 0 furthermore we find that clustering coefficient has a significant impact on robustness of the system for strong coupling strength but for weak coupling strength it has little influence especially for attacking both networks the study implies that we can obtain a more robust network by reducing clustering coefficient and increasing average degree for strong coupling strength however for weak coupling strength a more robust network is obtained only by increasing average degree for the same support average degree

a posteriori error estimation with the p version of the finite element method for nonlinear parabolic differential equations we analyze an a posteriori error estimator for nonlinear parabolic differential equations in several space dimensions the spatial discretization is carried out using the p version of the finite element method the error estimates are obtained by solving an elliptic problem at the desired times when the estimation is wanted some numerical experiments prove the efficiency of the error estimation