a new stream function formulation for the steady euler equations a new stream function formulation is presented for the solution of euler equations in the transonic flow region in this method the stream function and the density are the dependent variables the governing equations for adiabatic flow are then the momentum equations which are solved in the strong conservation law form the method differs from the recent formulation of hafez and lovell which is based on crocco s equation in that it does not require knowledge of the vorticity to study arbitrary geometries the algorithm is combined with the automatic grid solver grape of steger and sorenson results are presented for an naca 0012 airfoil at various mach numbers angles of attack and cylinders comparisons with other solutions of the euler equations show good agreement

development of an iterative boundary layer type solver for axisymmetric separated flows on presente un schema de couplage totalement implicite comparaison de la vitesse de convergence de ce schema avec d autres methodes de couplage resultats numeriques pour des ecoulements laminaires incompressibles decolles sur differentes geometries

graph operations and synchronization of complex networks the effects of graph operations on the synchronization of coupled dynamical systems are studied the operations range from addition or deletion of links to various ways of combining networks and generating larger networks from simpler ones methods from graph theory are used to calculate or estimate the eigenvalues of the laplacian operator which determine the synchronizability of continuous or discrete time dynamics evolving on the network results are applied to explain numerical observations on random scale free and small world networks an interesting feature is that when two networks are combined by adding links between them the synchronizability of the resulting network may worsen as the synchronizability of the individual networks is improved similarly adding links to a network may worsen its synchronizability although it decreases the average distance in the graph

spectra of random graphs with given expected degrees in the study of the spectra of power law graphs there are basically two competing approaches one is to prove analogues of wigner s semicircle law whereas the other predicts that the eigenvalues follow a power law distribution although the semicircle law and the power law have nothing in common we will show that both approaches are essentially correct if one considers the appropriate matrices we will prove that under certain mild conditions the eigenvalues of the normalized laplacian of a random power law graph follow the semicircle law whereas the spectrum of the adjacency matrix of a power law graph obeys the power law our results are based on the analysis of random graphs with given expected degrees and their relations to several key invariants of interest are a number of new values for the exponent where phase transitions for eigenvalue distributions occur the spectrum distributions have direct implications to numerous graph algorithms such as for example randomized algorithms that involve rapidly mixing markov chains

synchrony and complexity in state related eeg networks an application of spectral graph theory the brain may be considered as a synchronized dynamic network with several coherent dynamical units however concerns remain whether synchronizability is a stable state in the brain networks if s

network synchronization diffusion and the paradox of heterogeneity many complex networks display strong heterogeneity in the degree connectivity distribution heterogeneity in the degree distribution often reduces the average distance between nodes but paradoxically may suppress synchronization in networks of oscillators coupled symmetrically with uniform coupling strength here we offer a solution to this apparent paradox our analysis is partially based on the identification of a diffusive process underlying the communication between oscillators and reveals a striking relation between this process and the condition for the linear stability of the synchronized states we show that for a given degree distribution the maximum synchronizability is achieved when the network of couplings is weighted and directed and the overall cost involved in the couplings is minimum this enhanced synchronizability is solely determined by the mean degree and does not depend on the degree distribution and system size numerical verification of the main results is provided for representative classes of small world and scale free networks

numerical solution of transonic stream function equation the stream function equation in conservation form looks similar to the full potential equation and existing methods e g artificial compressibility can be readily applied rotational flows can be calculated once the vorticity due to shocks or nonuniformity is evaluated there are however two main difficulties first the density is not uniquely determined in terms of the flux there are two solutions the subsonic and the supersonic branch with a square root singularity at the sonic point methods to overcome this difficulty are studied and results are presented with some remarks on inviscid separation and closed stream lines second the need of two stream functions for three dimensional calculations is briefly discussed

global synchronization for discrete time stochastic complex networks with randomly occurred nonlinearities and mixed time delays in this paper the problem of stochastic synchronization analysis is investigated for a new array of coupled discrete time stochastic complex networks with randomly occurred nonlinearities rons and time delays the discrete time complex networks under consideration are subject to 1 stochastic nonlinearities that occur according to the bernoulli distributed white noise sequences 2 stochastic disturbances that enter the coupling term the delayed coupling term as well as the overall network and 3 time delays that include both the discrete and distributed ones note that the newly introduced rons and the multiple stochastic disturbances can better reflect the dynamical behaviors of coupled complex networks whose information transmission process is affected by a noisy environment e g internet based control systems by constructing a novel lyapunov like matrix functional the idea of delay fractioning is applied to deal with the addressed synchronization analysis problem by employing a combination of the linear matrix inequality lmi techniques the free weighting matrix method and stochastic analysis theories several delay dependent sufficient conditions are obtained which ensure the asymptotic synchronization in the mean square sense for the discrete time stochastic complex networks with time delays the criteria derived are characterized in terms of lmis whose solution can be solved by utilizing the standard numerical software a simulation example is presented to show the effectiveness and applicability of the proposed results

application of a finite element algorithm to the solution of steady transonic euler equations a finite element algorithm for the solution of two dimensional steady euler equations is presented which through a clebsch type transformation for the velocity vector solves the conservation of mass equation with one primary variable and two additional equations for the convection of two new variables the accuracy and efficiency of this scheme is discussed and the second order accuracy attained in the analysis of the convection of the vorticity is demonstrated together with the efficient treatment of rotational and irrotational flow subregions a sample problem is used to show the accuracy of the numerical scheme and its convergence characteristics

global bifurcation results for a fractional equation in rn abstract in this paper we study the existence of positive solution for the following class of fractional equation p s u g x f u in r n 0 u x 1 x r n lim x u x 0 where n 2 s s 0 1 0 f r r is a continuous function g r n r is a bounded continuous function that changes sign and s u is the fractional laplacian the main tools used are a theorem due to crandall and rabinowitz 14 and the global bifurcation theorem due to rabinowitz 23

network synchronization spectral versus statistical properties we consider synchronization of weighted networks possibly with asymmetrical connections focusing on causal relations rather than the observed correlations we show that the synchronizability of networks cannot be directly inferred from their statistical properties small local changes in the network structure can sensitively affect the eigenvalues relevant for synchronization while the gross statistical network properties remain essentially unchanged consequently commonly used statistical properties including the degree distribution degree homogeneity average degree average distance degree correlation and clustering coefficient can fail to characterize the synchronizability of networks in terms of causal relations despite the observed correlations c 2006 elsevier b v all rights reserved

factors that predict better synchronizability on complex networks while shorter characteristic path length has in general been believed to enhance synchronizability of a coupled oscillator system on a complex network the suppressing tendency of the heterogeneity of the degree distribution even for shorter characteristic path length has also been reported to see this we investigate the effects of various factors such as the degree characteristic path length heterogeneity and betweenness centrality on synchronization and find a consistent trend between the synchronization and the betweenness centrality the betweenness centrality is thus proposed as a good indicator for synchronizability

a mathematical analysis of the effects of hebbian learning rules on the dynamics and structure of discrete time random recurrent neural networks we present a mathematical analysis of the effects of hebbian learning in random recurrent neural networks with a generic hebbian learning rule including passive forgetting and different timescales for neuronal activity and learning dynamics previous numerical work has reported that hebbian learning drives the system from chaos to a steady state through a sequence of bifurcations here we interpret these results mathematically and show that these effects involving a complex coupling between neuronal dynamics and synaptic graph structure can be analyzed using jacobian matrices which introduce both a structural and a dynamical point of view on neural network evolution furthermore we show that sensitivity to a learned pattern is maximal when the largest lyapunov exponent is close to 0 we discuss how neural networks may take advantage of this regime of high functional interest

transonic flow calculations using the euler equations an implicit finite difference method with implicit boundary conditions is employed to solve the steady euler equations for flows past arbitrary geometries the resulting code is used to investigate in a systematic way various aspects of flow past airfoils at transonic speeds such as method of solution boundary conditions grid stretching and generation shock and sonic point operators the kiitta condition and smoothing results ob tained are in good agreement with results of other codes moreover it appears that the method of solution employed is such that kutta s condition need not be invoked this statement appears to be valid for other existing schemes employed in the solution of euler equations he primitive variable form of the system of partial differential equations governing the steady motion of an inviscid fluid euler equations are first order and of mixed elliptic and hyperbolic type because of this two approaches are used in obtaining steady state solutions for subcritical and supercritical flows past bodies the first seeks temporally asymptotic solutions of the time dependent equations1 7 while the second exploits relaxation methods developed for the solution of second order equations by embedding euler equations in a second order system8 or by using a stream function formulation 9 the purpose of this investigation is to develop an efficient algorithm for the steady euler equations in primitive variable form and use it to carry out numerical experiments designed to investigate in a systematic way the various aspects of flows past airfoils at transonic speeds these include method of solution i e peaceman rachford10 pr vs douglas gunn1 1 dg boundary conditions grid generation and stretching shock and sonic point operators smoothing and the kutta condition the resulting algorithm solves the strong con servation form of the euler equations with the energy equation replaced by the statement that the total enthalpy is constant using an adi finite difference scheme and implicit boundary conditions to study arbitrary geometries the algorithm is combined with the automatic grid solver grape of steger and sorenson 12 with this capability the code provides a general and efficient method for the solution of flowfields past arbitrary geometries for all speed ranges the achieved efficiency of this code can be traced to two main reasons the resulting block tridiagonal matrix for two dimensional flows is 3 x 3 and not 4 x 4 moreover the code incorporates a procedure that selects an optimum relaxation factor for each iteration formulation of the problem

global bifurcation results for a semilinear elliptic equation on all of mathbb r n we prove the existence of positive solutions for the semilinear elliptic equation u x g x f u x 0 u 1 for x r lim x u x 0 which arises in population genetics under the hypotheses that n 3 4 5 and g lies in l n 2 ir n we establish the existence of a principal eigenvalue 1 for the corresponding linearized problem and making use of the asymptotic properties of solutions and local and global bifurcation theory prove the existence of a continuum of solutions lying in the space d 1 2 extending from 1 to

nonisentropic potential formulation for transonic flows a potential equation for nonisentropic transonic flows is formulated this procedure captures shock waves with rankine hugoniot strengths but retains the simplicity of the traditional potential equation numerical computations are presented that verify the efficiency of the present procedure it is also shown that crocco s theorem is not applicable to potential flows a new crocco s theorem valid for potential flows is derived and it is shown that transonic flows with nonconstant shock strengths can never be irrotational irrespective of whether the flow is isentropic or not

best constants for sobolev inequalities for higher order fractional derivatives we obtain sharp constants for sobolev inequalities for higher order fractional derivatives as an application we give a new proof of a theorem of w beckner concerning conformally invariant higher order differential operators on the sphere

inviscid flows over a cylinder in this paper we simulate steady inviscid flows over a cylinder using potential and stream functions including entropy and vorticity corrections for incompressible subsonic transonic and supersonic flows the present hierarchical formulation is equivalent to euler equations satisfying conservation of mass momentum and energy standard numerical schemes and iterative algorithms are used for the mixed type potential or stream function equation and for the convection equations of entropy and vorticity augmented with the proper artificial dissipation necessary for stability of computations typical results for a cylinder in uniform and shear flows are presented and the merits of the present approach are briefly discussed

synchronization analysis of linearly coupled networks of discrete time systems abstract in this paper we present a framework for analysis of synchronization of linearly coupled map lattices lcmls the dynamics at each node of the coupled map lattices is described by a general difference equation which can be of chaotic or others as for coupling configuration of the linearly coupled map lattices it is not assumed that the coupling matrix is symmetric and irreducible criteria are obtained for both local and global synchronization these criteria reveal that two factors influence synchronization dynamical behaviors at each node and coupling configuration by these criteria one can verify whether the lcml is synchronized which is important and useful for both understanding and interpreting synchronization phenomena and designing coupling configuration a scale d is presented to measure the synchronization capability of a linearly coupled map lattice as we show d can be regarded as a function of the smallest and the largest eigenvalues of the coupling matrix various kinds of coupling schemes including complex networks such as small world sw lattice and scale free network are studied the quantity d is carefully analyzed and calculated for each case

connection graph stability method for synchronized coupled chaotic systems this paper elucidates the relation between network dynamics and graph theory a new general method to determine global stability of total synchronization in networks with different topologies is proposed this method combines the lyapunov function approach with graph theoretical reasoning in this context the main step is to establish a bound on the total length of all paths passing through an edge on the network connection graph in particular the method is applied to the study of synchronization in rings of 2k nearest neighbor coupled oscillators a rigorous bound is given for the minimum coupling strength sufficient for global synchronization of all oscillators this bound is explicitly linked with the average path length of the coupling graph contrary to the master stability function approach developed by pecora and carroll the connection graph stability method leads to global stability of synchronization and it permits not only constant but also time dependent interaction coefficients in a companion paper blinking model and synchronization in small world networks with a time varying coupling see this issue this method is extended to the blinking model of small world networks where in addition to the fixed 2 k nearest neighbor interactions all the remaining links are rapidly switched on and off independently of each other