oscillation for solutions of nonlinear neutral differential equations with impulses abstract this paper is concerned with nonlinear neutral differential equations with impulses of the form some oscillation criteria for solutions of this equation are established an interesting example is also given which illustrates that impulses play an important role in giving rise to the oscillation of equations

scaling properties of scale free evolving networks continuous approach the scaling behavior of scale free evolving networks arising in areas such as communications scientific citations collaborations etc is studied we derive universal scaling relations describing properties of such networks and indicate the limits of their validity we show that the main properties of scale free evolving networks may be described in the framework of a simple continuous approach the simplest models of networks growing according to a mechanism of preferential attachment of links to nodes are used we consider different forms of this preference and demonstrate that the range of preferential attachments producing scale free networks is wide we also obtain scaling relations for networks with nonlinear accelerating growth and describe the temporal evolution of the arising distributions size effects char22 the cutoffs of these distributions char22 introduce restrictions for the observation of power law dependences mainly we discuss the so called degree distribution i e the distribution of the number of connections of nodes a scaling form of the distribution of links between pairs of individual nodes for a growing network of citations is also studied we describe the effects of differences between nodes the aging of nodes changes the exponents of the distributions the appearance of a single node with high fitness changes the degree distribution of a network dramatically if its fitness exceeds some threshold value this node captures a finite part of all links of the network we show that permanent random damage to a growing scale free network char22 a permanent deletion of some links char22 radically changes the values of the scaling exponents results of other kinds of permanent damage are described

on the oscillation of an n th order nonlinear neutral delay differential equation abstract the authors consider the n th order nonlinear neutral delay differential equation y t p t y g t n q t f y h t 0 where n 1 q t 0 and 1 they give sufficient conditions for solutions to either oscillate or converge to zero

linearized oscillation of first order nonlinear neutral delay differential equations abstract in this paper we establish a equivalence of oscillation for the first order nonlinear neutral delay differential equation x t px t i 1 m q i f i x t i 0 and its corresponding linear equation x t px t i 1 m q i x t i 0 under a weaker condition on fi

on some nonlinear elliptic equations involving derivatives of the nonlinearity the existence of weak solutions of the dirichlet problem for nonlinear elliptic equations of the type x j a ij x u x i b j u a x u f is proved for nonlinearities b j which are only continuous the proof is based on the observation that an l a priori estimate is valid which is independent of b j the second part contains several uniqueness results and counterexamples showing that uniqueness may fail in general

kamenev type oscillation criteria for even order neutral differential equations with deviating arguments abstract sufficient conditions are established for the oscillation of even order neutral differential equations of the form x t i 1 m p i t x i t n j 1 l q j t f j x j t 0 t t 0 where n 2 is even integer

percolation on correlated networks we reconsider the problem of percolation on an equilibrium random network with degree degree correlations between nearest neighboring vertices focusing on critical singularities at a percolation threshold we obtain criteria for degree degree correlations to be irrelevant for critical singularities we present examples of networks in which assortative and disassortative mixing leads to unusual percolation properties and new critical exponents

on the reynolds lubrication equation let be a bounded open set in r 2 with smooth boundary is the region where two solid bodies are in contact these two solids are moving and the average of their velocities is denoted by v v 1 v 2 the pressure p p x which develops in a fluid layer confined between these two bodies satisfies the so called reynolds lubrication equation 1 h 3 p 6 v h in 2 p p a on here is the density of the fluid 0 its dynamic viscosity h h x is the distance between the two bodies and p a 0 is the given ambient pressure we will assume all along that h is a lipschitz continuous function such that 3 0 h 1 h x h 2 a e x x y 4 h x ha e x x y where h 1 h 2 h are positive constants when the fluid is incompressible i e is a positive constant existence and uniqueness for a solution to 1 2 is a trivial matter since the equation is linear in the compressible case when 5 p the problem becomes more challenging the paper is organized as follows in section 1 we investigate the problem of existence of a solution for 1 2 when is given by 5 in section 2 we consider a case where 1 2 reduces to a one dimensional problem and we study the question of uniqueness as well as the shape of the solution finally in a third section we give some extensions of the results of section 2 in higher dimension

oscillation and existence of positive solutions in a class of higher order neutral equations abstract consider the higher order neutral differential equation x t x t n q t x t 0 t t 0 with q t continuous 0 0 and n odd we establish several new sufficient conditions for the oscillation of all solutions and the existence of a positive solution by an associated ordinary differential equation some of these conditions are sharp and improve the known results in the literature which include the newest one by 8

new nonoscillation criteria for delay differential equations abstract this paper is concerned with the nonoscillation of first order delay differential equations of the form x t p t x t 0 t t 0 where p c t 0 r r 0 t is nondecreasing t t for t t 0 and lim t t new nonoscillation conditions are obtained these conditions concern the case when the well known nonoscillation condition t t p s ds 1 e is not satisfied

observability transitions in correlated networks yang wang and motter phys rev lett 109 258701 2012 analyzed a model for network observability transitions in which a sensor placed on a node makes the node and the adjacent nodes observable the size of the connected components comprising the observable nodes is a major concern of the model we analyze this model in random heterogeneous networks with degree correlation with numerical simulations and analytical arguments based on generating functions we find that negative degree correlation makes networks more observable this result holds true both when the sensors are placed on nodes one by one in a random order and when hubs preferentially receive the sensors finally we numerically optimize networks with a fixed degree sequence with respect to the size of the largest observable component optimized networks have negative degree correlation induced by the resulting hub repulsive structure the largest hubs are rarely connected to each other in contrast to the rich club phenomenon of networks

oscillation criteria for certain nth order differential equations with deviating arguments abstract oscillation criteria for nth order differential equations with deviating arguments of the form x n 1 t 1 x n 1 t f t x g t 0 n even are established where g c t0 r f c t0 r r and 0 is a constant

uncovering disassortativity in large scale free networks mixing patterns in large self organizing networks such as the internet the world wide web and social and biological networks are often characterized by degree degree dependencies between neighboring nodes in this paper we propose a new way of measuring degree degree dependencies one of the problems with the commonly used assortativity coefficient is that in disassortative networks its magnitude decreases with the network size we mathematically explain this phenomenon and validate the results on synthetic graphs and real world network data as an alternative we suggest to use rank correlation measures such as spearman s rho our experiments convincingly show that spearman s rho produces consistent values in graphs of different sizes but similar structure and it is able to reveal strong positive or negative dependencies in large graphs in particular we discover much stronger negative degree degree dependencies in web graphs than was previously thought rank correlations allow us to compare the assortativity of networks of different sizes which is impossible with the assortativity coefficient due to its genuine dependence on the network size we conclude that rank correlations provide a suitable and informative method for uncovering network mixing patterns

measuring mixing patterns in complex networks by spearman rank correlation coefficient abstract in this paper we utilize spearman rank correlation coefficient to measure mixing patterns in complex networks compared with the widely used pearson coefficient spearman coefficient is rank based nonparametric and size independent thus it is more effective to assess linking patterns of diverse networks especially for large size networks we demonstrate this point by testing a variety of empirical and artificial networks moreover we show that normalized spearman ranks of stubs are subject to an interesting linear rule where the correlation coefficient is just the spearman coefficient this compelling linear relationship allows us to directly produce networks with any prescribed spearman coefficient our method apparently has an edge over the well known uncorrelated configuration model

existence and uniqueness for several non linear elliptic problems arising in lubrication theory we consider the non linear elliptic equation a x p p b x p with positive dirichlet boundary data the coefficients a and b are taken from models used in lubrication theory andin particular are not d negative values of p we prove some general existence anduniqueness results for a family of mod els which extendrelatedresults in the literature these results allow us to prove existence uniqueness andpositivity of the solution to advanced compressible lubrication models such as the kinetic based fukui kaneko model and the second order slip model we also consider a spring like model of compliant foil compressible bearing andweaken some hypotheses of previous results on more classical mod els such as the standard reynolds model and the first order slip model

organization of growing random networks the organizational development of growing random networks is investigated these growing networks are built by adding nodes successively and linking each to an earlier node of degree k with an attachment probability a k when a k grows more slowly than linearly with k the number of nodes with k links n k t decays faster than a power law in k while for a k growing faster than linearly in k a single node emerges which connects to nearly all other nodes when a k is asymptotically linear n k t ensuremath sim mathrm tk ensuremath ensuremath nu with ensuremath nu dependent on details of the attachment probability but in the range 2l ensuremath nu l ensuremath infty the combined age and degree distribution of nodes shows that old nodes typically have a large degree there is also a significant correlation in the degrees of neighboring nodes so that nodes of similar degree are more likely to be connected the size distributions of the in and out components of the network with respect to a given node namely its descendants and ancestors are also determined the in component exhibits a robust s ensuremath 2 power law tail where s is the component size the out component has a typical size of order mathrm ln t and it provides basic insights into the genealogy of the network

large scale topological and dynamical properties of the internet we study the large scale topological and dynamical properties of real internet maps at the autonomous system level collected in a 3 yr time interval we find that the connectivity structure of the internet presents statistical distributions settled in a well defined stationary state the large scale properties are characterized by a scale free topology consistent with previous observations correlation functions and clustering coefficients exhibit a remarkable structure due to the underlying hierarchical organization of the internet the study of the internet time evolution shows a growth dynamics with aging features typical of recently proposed growing network models we compare the properties of growing network models with the present real internet data analysis

measuring degree degree association in networks the pearson correlation coefficient is commonly used for quantifying the global level of degree degree association in complex networks here we use a probabilistic representation of the underlying network structure for assessing the applicability of different association measures to heavy tailed degree distributions theoretical arguments together with our numerical study indicate that pearson s coefficient often depends on the size of networks with equal association structure impeding a systematic comparison of real world networks in contrast kendall gibbons ensuremath tau b is a considerably more robust measure of the degree degree association

existence of equilibria in articulated bearings the existence of equilibrium solutions for a lubricated system consisting of an articulated body sliding over a flat plate is considered though this configuration is very common it corresponds to the popular tilting pad thrust bearings the existence problem has only been addressed in extremely simplified cases such as planar sliders of infinite width our results show the existence of at least one equilibrium for a quite general class of nonplanar slider shapes we also extend previous results concerning planar sliders

enhancing neural network performance via assortativity the performance of attractor neural networks has been shown to depend crucially on the heterogeneity of the underlying topology we take this analysis a step further by examining the effect of degree degree correlations assortativity on neural network behavior we make use of a method recently put forward for studying correlated networks and dynamics thereon both analytically and computationally which is independent of how the topology may have evolved we show how the robustness to noise is greatly enhanced in assortative positively correlated neural networks especially if it is the hub neurons that store the information

oscillatory behaviour of higher order neutral type nonlinear forced differential equation with oscillating coefficients abstract oscillation criteria are given for higher order neutral type nonlinear forced differential equation with oscillating coefficients of the form y t p t y t n i 1 m q i t f i y i t s t

oscillation of impulsive neutral delay differential equations abstract in this paper effective sufficient conditions for the oscillation of all solutions of impulsive neutral delay differential equations of the form x t p t x t q t x t sgn x t 0 t t 0 x t k b k x t k k 1 2 are established our results reveal the fact that the oscillatory properties of all solutions of eqs and may be caused by the impulsive perturbations though the corresponding neutral delay differential equation without impulses i e eq admits a nonoscillatory solution it is also seen that the oscillatory behavior of all solutions of eq can be inherited by eqs and under certain impulsive perturbations some examples are also given to illustrate the applicability of the results obtained

rate equation approach for correlations in growing network models we propose a rate equation approach to compute two vertex correlations in scale free growing network models based on the preferential attachment mechanism the formalism based on previous work of szabo et al fphys rev e 67 056102 s2002dg for the clustering spectrum measuring three vertex correlations is based on a rate equation in the continuous degree and time approximation for the average degree of the nearest neighbors of vertices of degree k with an appropriate boundary condition we study the properties of both two and three vertex correlations for linear preferential attachment models and also for a model yielding a large clustering coefficient the expressions obtained are checked by means of extensive numerical simulations the rate equation proposed can be generalized to more sophisticated growing network models and also extended to deal with related correlation measures as an example we consider the case of a recently proposed model of weighted networks for which we are able to compute a weighted two vertex correlation function taking into account the strength of the interactions between connected vertices

a oscillation theorem for a class of even order neutral differential equations abstract the even order neutral differential equation 1 d n dt n x t h t x t f t x g t 0 is considered under the following conditions n 2 is even 0 h c r g c t 0 lim t g t f c t 0 r uf t u 0 for t u t 0 r and f t u is nondecreasing in u r for each fixed t t 0 it is shown that 1 is oscillatory if and only if the certain non neutral differential equation is oscillatory for the case where 0 h t h t

oscillations of delay differential equations with variable coefficients consider the delay differential equation t p t x t 0 where p t c t0 r and is a positive constant we show that every solution of this equation oscillates if tt p t dt 1 e for sufficiently large t and t0 p t exp tt p s ds 1 e 1 dt

the oscillation of higher order differential equations with deviating arguments in this paper some new criteria for the oscillation of higher order functional differential equations of the form lnx t f t x g t 0 n is even are established some of our results are obtained via comparing it with second order ordinary linear and first order delay differential equations the oscillation of this equation when n is odd is also considered then we shall use the obtained results to study the oscillatory behavior of the neutral functional differential equations of the form ln x t c t x t f t x g t 0 n is even and the damped functional differential equations of the type lnx t h t x g t ddtx h t 0 n is even the obtained results extend improve and correlate a number of existing criteria

oscillation and existence of positive solutions for odd order neutral equations with integrally small coefficients abstract consider the odd order neutral differential equation formula we establish several new sufficient conditions for the oscillation of all solutions and the existence of a positive solution when p t 1 is allowed to oscillate and the usual divergent condition q s ds is not satisfied these conditions are almost sharp and improve some known results in the literature some examples are given to demonstrate the advantage of our results

zero pearson coefficient for strongly correlated growing trees we obtained pearson s coefficient of strongly correlated recursive networks growing by preferential attachment of every new vertex by m edges we found that the pearson coefficient is exactly zero in the infinite network limit for the recursive trees m 1 if the number of connections of new vertices exceeds one mg1 then the pearson coefficient in the infinite networks equals zero only when the degree distribution exponent ensuremath gamma does not exceed 4 we calculated the pearson coefficient for finite networks and observed a slow power law like approach to an infinite network limit our findings indicate that pearson s coefficient strongly depends on size and details of networks which makes this characteristic virtually useless for quantitative comparison of different networks

oscillation of second order nonlinear neutral differential equations abstract in this paper some sufficient conditions for oscillation and nonoscillation are obtained for the second order nonlinear neutral differential equation x t p t x t q t f x t 0 t 0 where 0 p t 1 q t 0 0

oscillation criteria for nth order nonlinear neutral differential equations abstract sufficient conditions are established for the oscillation of nth order neutral differential equations of the form r t x t x t 1 p t x t n 1 q t f x t 0 t t 0 where n 2 is even integer 1 p q c t 0 r f c r r the results obtained extend some of the known results