the ternary commutator obstruction for internal crossed modules abstract in finitely cocomplete homological categories co smash products give rise to possibly higher order commutators of subobjects we use binary and ternary co smash products and the associated commutators to give characterisations of internal crossed modules and internal categories respectively the ternary terms are redundant if the category has the smith is huq property which means that two equivalence relations on a given object commute precisely when their normalisations do in fact we show that the difference between the smith commutator of such relations and the huq commutator of their normalisations is measured by a ternary commutator so that the smith is huq property itself can be characterised by the relation between the latter two commutators this allows us to show that the category of loops does not have the smith is huq property which also implies that ternary commutators are generally not decomposable into nested binary ones thus in contexts where smith is huq need not hold we obtain a new description of internal categories beck modules and double central extensions as well as a decomposition formula for the smith commutator the ternary commutator now also appears in the hopf formula for the third homology with coefficients in the abelianisation functor

the average distances in random graphs with given expected degrees abstract random graph theory is used to examine the small world phenomenon any two strangers are connected through a short chain of mutual acquaintances we will show that for certain families of random graphs with given expected degrees the average distance is almost surely of order log n log d where d is the weighted average of the sum of squares of the expected degrees of particular interest are power law random graphs in which the number of vertices of degree k is proportional to 1 k for some fixed exponent for the case of 3 we prove that the average distance of the power law graphs is almost surely of order log n log d however many internet social and citation networks are power law graphs with exponents in the range 2 3 for which the power law random graphs have average distance almost surely of order log log n but have diameter of order log n provided having some mild constraints for the average distance and maximum degree in particular these graphs contain a dense subgraph which we call the core having nc log log n vertices almost all vertices are within distance log log n of the core although there are vertices at distance log n from the core

contact processes on random graphs with power law degree distributions have critical value 0 if we consider the contact process with infection rate on a random graph on n vertices with power law degree distributions mean field calculations suggest that the critical value c of the infection rate is positive if the power 3 physicists seem to regard this as an established fact since the result has recently been generalized to bipartite graphs by gomez gardenes et al proc natl acad sci usa 105 2008 1399 1404 here we show that the critical value c is zero for any value of 3 and the contact process starting from all vertices infected with a probability tending to 1 as n maintains a positive density of infected sites for time at least exp n 1 for any 0 using the last result together with the contact process duality we can establish the existence of a quasi stationary distribution in which a randomly chosen vertex is occupied with probability it is expected that c as 0 here we show that 1 2 3 and so 2 for 3 thus even though the graph is locally tree like does not take the mean field critical value 1

nontrivial solutions for impulsive fractional differential equations via morse theory abstract in this paper we study the existence of nontrivial solutions for an impulsive fractional differential equation with dirichlet boundary conditions by using morse theory coupled with local linking arguments we obtain some new criteria to guarantee that the impulsive fractional differential equations have at least one nontrivial solution

a two stage contact process on scale free networks we study a two stage contact process on scale free networks as a model for the spread of epidemics we show that any virus starting from a single vertex with arbitrarily small infection rates can last for a super polynomial time with positive probability if the power law exponent 2 this is in sharp contrast with the mean field analysis the estimation of the metastable density is also provided

epidemic thresholds for bipartite networks it is well known that sexually transmitted diseases std spread across a network of human sexual contacts this network is most often bipartite as most std are transmitted between men and women even though network models in epidemiology have quite a long history now there are few general results about bipartite networks one of them is the simple dependence predicted using the mean field approximation between the epidemic threshold and the average and variance of the degree distribution of the network here we show that going beyond this approximation can lead to qualitatively different results that are supported by numerical simulations one of the new features that can be relevant for applications is the existence of a critical value for the infectivity of each population below which no epidemics can arise regardless of the value of the infectivity of the other population

mathematical models for the control of a pest population by infected pest in this paper we propose two mathematical models concerning continuous and impulsive pest control strategies respectively therefore our models are the ordinary differential equations and the impulsive differential equations as a result the global asymptotic stability of the equilibria of the ordinary differential equations is studied in the case when an impulsive control is used it is observed that there exists a globally asymptotically stable susceptible pest eradication periodic solution when the amount of infective pests released periodically is larger than some critical value when the amount of infective pests released is less than this critical value the system is shown to be permanent which implies that the trivial susceptible pest eradication solution loses its stability finally by means of numerical simulation we obtain the critical values of the control variable under different methods of release of infected pests

a colimit of classifying spaces abstract we recall a description of the first non vanishing homotopy group of a certain n 1 ad of spaces and show how it yields group theoretic formulae for homotopy and homology groups of several specific spaces

existence of at least three solutions of a second order three point boundary value problem abstract we study existence of at least three solutions in the presence of two lower and two upper solutions of some second order nonlinear three point boundary value problem of the type x f t x x t i 0 1 x 0 0 x 1 x 0 1 0 1 the growth of f with respect to x is allowed to be quadratic we use some degree theory arguments to get the multiplicity result

multiplicity of solutions for integral boundary value problems of fractional differential equations with upper and lower solutions abstract in this paper we study the existence of multiple solutions for the integral boundary value problems of fractional differential equations by the method of upper and lower solutions and leray schauder degree theory the sufficient conditions about the existence of at least three solutions are obtained moreover it is proved that the integral boundary value problem has at least three positive solutions under the conditions of m 0 and f is nonnegative by given two upper and lower solutions which can be easily obtained through our methods we can present the existence theorem of at least three solutions two examples are also included to illustrate the effectiveness of the proposed results

epidemic dynamics in finite size scale free networks many real networks present a bounded scale free behavior with a connectivity cutoff due to physical constraints or a finite network size we study epidemic dynamics in bounded scale free networks with soft and hard connectivity cutoffs the finite size effects introduced by the cutoff induce an epidemic threshold that approaches zero at increasing sizes the induced epidemic threshold is very small even at a relatively small cutoff showing that the neglection of connectivity fluctuations in bounded scale free networks leads to a strong overestimation of the epidemic threshold we provide the expression for the infection prevalence and discuss its finite size corrections the present paper shows that the highly heterogeneous nature of scale free networks does not allow the use of homogeneous approximations even for systems of a relatively small number of nodes

pure galois theory in categories abstract galois theory j pure appl algebra 19 1980 21 42 2 m barr abstract galois theory ii j pure appl algebra 25 1982 227 247 3 m barr and r diaconescu on locally simply connected toposes and their fundamen tal groups cahiers topologie gt om diffprentielle catigoriques 22 i98 i 301 3 14 4 s u chase d k harrison and a rosenberg galois theory and cohomology of commutative rings mem amer mafh sot 52 1965 15 33 5 s u chase and m e sweedler hopf algebras and galois theory in lecture notes in math vol 97 springer verlag new york berlin 1969 6 i diers categories of boolean sheaves of simple algebras in lecture notes in math vol 1187 springer verlag new york berlin 1986 7 a grothendieck revetements itales et groupe fondamental sga 1 in lecture notes in math vol 269 springer verlag new york berlin 1972 8 j f kennison galois theory and theaters of action in a topos j pure appl algebra 18 1980 149 164 9

solvability of three point boundary value problems for second order differential equations with deviating arguments the monotone iterative technique is used to boundary problems for second order ordinary differential equations with deviating arguments corresponding results are formulated when the problem has extremal solutions or weakly coupled extremal quasi solutions

ising models on power law random graphs we study a ferromagnetic ising model on random graphs with a power law degree distribution and compute the thermodynamic limit of the pressure when the mean degree is finite degree exponent 2 for which the random graph has a tree like structure for this we closely follow the analysis by dembo and montanari ann appl probab 20 2 565 592 2010 which assumes finite variance degrees 3 adapting it when necessary and also simplifying it when possible our results also apply in cases where the degree distribution does not obey a power law

monotone iterative method for the second order three point boundary value problem with upper and lower solutions in the reversed order abstract this article introduces a new concept of upper and lower solutions and studies the existence and uniqueness of solutions of second order three point boundary value problems with upper and lower solutions in the reversed order the sufficient conditions for the existence and uniqueness of solutions are obtained by using the monotone iterative method meantime the iterative sequence for solving a solution and its error estimate formula under the condition of unique solution are given some results of previous literature are extended and improved a numerical example is also included to illustrate the effectiveness of the proposed results

existence and uniqueness of solutions of second order three point boundary value problems with upper and lower solutions in the reversed order abstract this paper studies the existence and uniqueness of solutions of second order three point boundary value problems with lower and upper solutions in the reversed order obtains the sufficient conditions for the existence and uniqueness of solutions by use of the monotone iterative method and gives the iterative sequence for solving a solution and its error estimate formula under the condition of unique solution

sampling methods for the quasistationary regime of epidemic processes on regular and complex networks a major hurdle in the simulation of the steady state of epidemic processes is that the system will unavoidably visit an absorbing disease free state at sufficiently long times due to the finite size of the networks where epidemics evolves in the present work we compare different quasistationary qs simulation methods where the absorbing states are suitably handled and the thermodynamical limit of the original dynamics can be achieved we analyze the standard qs sqs method where the sampling is constrained to active configurations the reflecting boundary condition rbc where the dynamics returns to the pre absorbing configuration and hub reactivation hr where the most connected vertex of the network is reactivated after a visit to an absorbing state we apply the methods to the contact process cp and susceptible infected susceptible sis models on regular and scale free networks the investigated methods yield the same epidemic threshold for both models for cp that undergoes a standard collective phase transition the methods are equivalent for sis whose phase transition is ruled by the hub mutual reactivation the sqs and hr methods are able to capture localized epidemic phases while rbc is not we also apply the autocorrelation time as a tool to characterize the phase transition and observe that this analysis provides the same finite size scaling exponents for the critical relaxation time for the investigated methods finally we verify the equivalence between rbc method and a weak external field for epidemics on networks

subordination and superordination for univalent solutions for fractional differential equations in this article we establish the existence and uniqueness of univalent solution for fractional differential equation moreover we illustrate some properties of this solution containing differential and integral subordination properties

the contact process on trees the contact process on an infinite homogeneous tree is shown to exhibit at least two phase transitions as the infection parameter is varied for small values of a single infection eventually dies out for larger the infection lives forever with positive probability but eventually leaves any finite set the survival probability is a continuous function of and the proof of this is much easier than it is for the contact process on d dimensional integer lattices for still larger the infection converges in distribution to a nontrivial invariant measure for an n ary tree with n large the first of these transitions occurs when 1 n and the second occurs when 1 2 n e n nonhomogeneous trees whose vertices have degrees varying between 1 and n behave essentially as homogeneous n ary trees provided that vertices of degree n are not too rare in particular letting n go to galton watson trees whose vertices have degree n with probability that does not decrease exponentially with n may have both phase transitions occur together at 0 the nature of the second phase transition is not yet clear and several problems are mentioned in this regard

existence results for fractional differential systems through a local minimization principle in this paper we investigate the existence of one weak solution for a class of fractional differential systems the approach is based on variational methods for smooth functionals defined on reflexive banach spaces the main result is also demonstrated with examples

higher central extensions and hopf formulae abstract higher extensions and higher central extensions which are of importance to non abelian homological algebra are studied and some fundamental properties are proven as an application a direct proof of the invariance of the higher hopf formulae is obtained

existence and approximation of solutions for nonlinear functional differential equations with periodic boundary value conditions abstract this paper is concerned with the existence and approximation of solutions for a class of first order functional differential equations with periodic boundary conditions we present a new comparison result and extend previous results

high prevalence regimes in the pair quenched mean field theory for the susceptible infected susceptible model on networks reckoning of pairwise dynamical correlations significantly improves the accuracy of mean field theories and plays an important role in the investigation of dynamical processes in complex networks in this work we perform a nonperturbative numerical analysis of the quenched mean field theory qmf and the inclusion of dynamical correlations by means of the pair quenched mean field pqmf theory for the susceptible infected susceptible model on synthetic and real networks we show that the pqmf considerably outperforms the standard qmf theory on synthetic networks of distinct levels of heterogeneity and degree correlations providing extremely accurate predictions when the system is not too close to the epidemic threshold while the qmf theory deviates substantially from simulations for networks with a degree exponent 2 5 the scenario for real networks is more complicated still with pqmf significantly outperforming the qmf theory however despite its high accuracy for most investigated networks in a few cases pqmf deviations from simulations are not negligible we found correlations between accuracy and average shortest path while other basic network metrics seem to be uncorrelated with the theory accuracy our results show the viability of the pqmf theory to investigate the high prevalence regimes of recurrent state epidemic processes in networks a regime of high applicability

boundary value problems involving upper and lower solutions in reverse order in this paper we discuss the existence of extreme solutions of the boundary value problem for a class of first order functional equations with a nonlinear boundary condition in the presence of a lower solution a and an upper solution b with email protected email protected we establish existence results of extreme solutions by using the method of upper and lower solutions and a monotone iterative technique

remarks on periodic boundary value problems for functional differential equations we extend some results on existence and approximation of solution for a class of first order functional differential equations with periodic boundary conditions we show the validity of the monotone iterative technique under weaker hypotheses and present some examples

higher hopf formulae for homology via galois theory we use janelidze s categorical galois theory to extend brown and ellis s higher hopf formulae for homology of groups to arbitrary semi abelian monadic categories given such a category alpha and a chosen birkhoff subcategory beta of alpha thus we describe the barr beck derived functors of the reflector of alpha onto beta in terms of centralization of higher extensions in case alpha is the category gp of all groups and beta is the category ab of all abelian groups this yields a new proof for brown and ellis s formulae we also give explicit formulae in the cases of groups vs k nilpotent groups groups vs k solvable groups and precrossed modules vs crossed modules

dynamical correlations and pairwise theory for the symbiotic contact process on networks the two species symbiotic contact process 2scp is a stochastic process in which each vertex of a graph may be vacant or host at most one individual of each species vertices with both species have a reduced death rate representing a symbiotic interaction while the dynamics evolves according to the standard single species contact process rules otherwise we investigate the role of dynamical correlations on the 2scp on homogeneous and heterogeneous networks using pairwise mean field theory this approach is compared with the ordinary one site theory and stochastic simulations we show that our approach significantly outperforms the one site theory in particular the stationary state of the 2scp model on random regular networks is very accurately reproduced by the pairwise mean field even for relatively small values of vertex degree where expressive deviations of the standard mean field are observed the pairwise approach is also able to capture the transition points accurately for heterogeneous networks and provides rich phase diagrams with transitions not predicted by the one site method our theoretical results are corroborated by extensive numerical simulations

on the three subobjects lemma and its higher order generalisations abstract we solve a problem mentioned in the article 1 of berger and bourn we prove that in the context of an algebraically coherent semi abelian category two natural definitions of the lower central series coincide in a first standard approach nilpotency is defined as in group theory via nested binary commutators of the form x x x in a second approach higher higgins commutators of the form x x x are used to define nilpotent objects 18 16 20 the two are known to be different in general for instance in the context of loops the definition of bruck 6 is of the former kind while the commutator associator filtration of mostovoy 27 26 and his co authors is of the latter type another example in the context of moufang loops is given in 1 in this article we show that the two streams of development agree in any algebraically coherent semi abelian category such are for instance all orzech categories of interest 31 our proof of this result is based on a higher order version of the three subobjects lemma of 8 which extends the classical three subgroups lemma from group theory to categorical algebra it says that any n fold higgins commutator k 1 k n of normal subobjects k i x may be decomposed into a join of nested binary commutators

monotone iterative technique for impulsive integro differential equations with periodic boundary conditions this paper investigates the existence of minimal and maximal solutions of impulsive integro differential equations with periodic boundary conditions by establishing a comparison result and using the method of upper and lower solutions and the monotone iterative technique

generalized viscoelastic models their fractional equations with solutions recently fractional calculus fc has encountered much success in the description of complex dynamics in particular fc has proved to be a valuable tool to handle viscoelastic aspects in this paper we construct fractional rheological constitutive equations on the basis of well known mechanical models especially the maxwell the kelvin voigt the zener and the poynting thomson model to this end we introduce a fractional element in addition to the standard purely elastic and purely viscous elements as we proceed to show many of the fractional differential equations which we obtain by this construction method admit closed form analytical solutions in terms of fox h functions of the minag leffler type

extinction time for the contact process on general graphs we consider the contact process on finite and connected graphs and study the behavior of the extinction time that is the amount of time that it takes for the infection to disappear in the process started from full occupancy we prove without any restriction on the graph g that if the infection rate lambda is larger than the critical rate of the one dimensional process then the extinction time grows faster than exp g log g kappa for any constant kappa 1 where g denotes the number of vertices of g also for general graphs we show that the extinction time divided by its expectation converges in distribution as the number of vertices tends to infinity to the exponential distribution with parameter 1 these results complement earlier work of mountford mourrat valesin and yao in which only graphs of bounded degrees were considered and the extinction time was shown to grow exponentially in n here we also provide a simpler proof of this fact

critical value for the contact process with random recovery rates and edge weights on regular tree abstract in this paper we are concerned with contact processes with random recovery rates and edge weights on rooted regular trees t n let and be two nonnegative random variables such that p m 1 for some m 0 for each vertex x on t n x is an independent copy of while for each edge e on t n e is an independent copy of an infected vertex x becomes healthy at rate x while an infected vertex y infects an healthy neighbor z at rate proportional to y z for this model we prove that the critical value under the annealed measure approximately equals n e e 1 1 as n grows to infinity furthermore we show that the critical value under the quenched measure equals that under the annealed measure when the cluster containing the root formed with edges with positive weights is infinite

robustness and fragility of the susceptible infected susceptible epidemic models on complex networks we analyze two alterations of the standard susceptible infected susceptible sis dynamics that preserve the central properties of spontaneous healing and infection capacity of a vertex increasing unlimitedly with its degree all models have the same epidemic thresholds in mean field theories but depending on the network properties simulations yield a dual scenario in which the epidemic thresholds of the modified sis models can be either dramatically altered or remain unchanged in comparison with the standard dynamics for uncorrelated synthetic networks having a power law degree distribution with exponent 5 2 differences are more remarkable for 3 where a finite threshold is found in the modified models in contrast with the vanishing threshold of the original one this duality is elucidated in terms of epidemic lifespan on star graphs we verify that the activation of the modified sis models is triggered in the innermost component of the network given by a k core decomposition for 3 the activation in the modified dynamics is collective involving essentially the whole network while it is triggered by hubs in the standard sis the duality also appears in the finite size scaling of the critical quantities where mean field behaviors are observed for the modified but not for the original dynamics our results feed the discussions about the most proper conceptions of epidemic models to describe real systems and the choices of the most suitable theoretical approaches to deal with these models

nature of the epidemic threshold for the susceptible infected susceptible dynamics in networks we develop an analytical approach to the susceptible infected susceptible epidemic model that allows us to unravel the true origin of the absence of an epidemic threshold in heterogeneous networks we find that a delicate balance between the number of high degree nodes in the network and the topological distance between them dictates the existence or absence of such a threshold in particular small world random networks with a degree distribution decaying slower than an exponential have a vanishing epidemic threshold in the thermodynamic limit

higher central extensions and cohomology abstract we establish a galois theoretic interpretation of cohomology in semi abelian categories cohomology with trivial coefficients classifies central extensions also in arbitrarily high degrees this allows us to obtain a duality in a certain sense between internal homology and external cohomology in semi abelian categories these results depend on a geometric viewpoint of the concept of a higher central extension as well as the algebraic one in terms of commutators

epidemic dynamics and endemic states in complex networks we study by analytical methods and large scale simulations a dynamical model for the spreading of epidemics in complex networks in networks with exponentially bounded connectivity we recover the usual epidemic behavior with a threshold defining a critical point below that the infection prevalence is null on the contrary on a wide range of scale free networks we observe the absence of an epidemic threshold and its associated critical behavior this implies that scale free networks are prone to the spreading and the persistence of infections whatever spreading rate the epidemic agents might possess these results can help understanding computer virus epidemics and other spreading phenomena on communication and social networks

fundamental group functors in descent exact homological categories abstract we study the notion of a fundamental group in the framework of descent exact homological categories this setting is sufficiently large to include several categories of algebraic nature such as almost abelian categories semi abelian categories and categories of topological semi abelian algebras for many adjunctions in this context we describe the fundamental groups by generalised brown ellis hopf formulae for the integral homology of groups

on monotone method for first and second order periodic boundary value problems and periodic solutions of functional differential equations abstract in this paper we show that the method of monotone iterative technique is valid to obtain two monotone sequences that converge uniformly to extremal solutions of first and second order periodic boundary value problems and periodic solutions of functional differential equations we obtain some new results relative to the lower solution and upper solution with

monotone iterative method for first order functional difference equations with nonlinear boundary value conditions in this paper we show that the method of monotone iterative coupled with the upper and lower solutions is valid to obtain constructive proofs to the existence of solutions for first order functional difference equations with nonlinear boundary value conditions an example is given to illustrate the results obtained

contact processes with random connection weights on regular graphs abstract in this paper we study the asymptotic critical value of contact processes with random connection weights sitting on a degree increasing sequence of r regular graph g r we propose a method to generalize the asymptotics results for c z d and c t d of classical contact processes as well as of recent work for contact processes on complete graphs with random connection weights only the lower bound is rigorously proved it is conjectured however that the lower bound gives the right asymptotic behavior for comparison purposes we also introduce binary contact path processes with random connection weights whose asymptotic behavior of the critical value is obtained

critical value for contact processes on clusters of oriented bond percolation in this paper we are concerned with contact processes on open clusters of oriented bond percolation in zd where the disease spreads along the direction of open edges we show that the two critical values in the quenched and annealed cases are equal with probability one and are asymptotically equal to dp 1 as the dimension d grows to infinity where p is the probability that the edge is open

multiple solutions for a coupled system of nonlinear fractional differential equations via variational methods in this paper a coupled system of nonlinear fractional differential equations is considered the existence of at least three distinct weak solutions is obtained by means of the variational methods and the critical point theory due to bonanno and marano in addition an example is presented to illustrate the feasibility and effectiveness of the main results

the contact process on a finite set ii in this paper we complete the work started in part i we show that if sigma n is the time that the contact process on 1 ldots n first hits the empty set then for lambda lambda c the critical value for the process on z there is a positive constant gamma lambda so that log sigma n n rightarrow gamma lambda in probability as n rightarrow infty we also give a new simple proof that sigma n e sigma n converges to a mean one exponential the keys to the proof of the first result are a planar graph duality for the contact process and an observation of j chayes and l chayes that exponential decay rates for connections in strips approach the decay rates in the plane as the width of the strip goes to infty

phase transition of the susceptible infected susceptible dynamics on time varying configuration model networks we present a degree based theoretical framework to study the susceptible infected susceptible sis dynamics on time varying rewired configuration model networks using this framework on a given degree distribution we provide a detailed analysis of the stationary state using the rewiring rate to explore the whole range of the time variation of the structure relative to that of the sis process this analysis is suitable for the characterization of the phase transition and leads to three main contributions 1 we obtain a self consistent expression for the absorbing state threshold able to capture both collective and hub activation 2 we recover the predictions of a number of existing approaches as limiting cases of our analysis providing thereby a unifying point of view for the sis dynamics on random networks 3 we obtain bounds for the critical exponents of a number of quantities in the stationary state this allows us to reinterpret the concept of hub dominated phase transition within our framework it appears as a heterogeneous critical phenomenon observables for different degree classes have a different scaling with the infection rate this phenomenon is followed by the successive activation of the degree classes beyond the epidemic threshold

higher commutator conditions for extensions in mal tsev categories abstract we define a galois structure on the category of pairs of equivalence relations in an exact mal tsev category and characterize central and double central extensions in terms of higher commutator conditions these results generalize both the ones related to the abelianization functor in exact mal tsev categories and the ones corresponding to the reflection from the category of internal reflexive graphs to the subcategory of internal groupoids some examples and applications are given in the categories of groups precrossed modules precrossed lie algebras and compact groups

random graphs with arbitrary degree distributions and their applications recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the poisson degree distributions that have been widely studied in the past in this paper we develop in detail the theory of random graphs with arbitrary degree distributions in addition to simple undirected unipartite graphs we examine the properties of directed and bipartite graphs among other results we derive exact expressions for the position of the phase transition at which a giant component first forms the mean component size the size of the giant component if there is one the mean number of vertices a certain distance away from a randomly chosen vertex and the average vertex vertex distance within a graph we apply our theory to some real world graphs including the worldwide web and collaboration graphs of scientists and fortune 1000 company directors we demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world while in others there is a measurable discrepancy between theory and reality perhaps indicating the presence of additional social structure in the network that is not captured by the random graph

solvability and numerical simulations for bvps of fractional coupled systems involving left and right fractional derivatives abstract we study the three point boundary value problems for the fractional differential coupled systems with both left and right fractional derivatives in the paper firstly the upper lower solutions method for the systems is presented and some existence results to the systems solutions are also obtained secondly we also give out an iterative approach for finding approximate solutions of the systems finally we consider a concrete example where we obtain the existence result of the solution and also give out its numerical solution and the graphical simulation

impact of the heterogeneity of adoption thresholds on behavior spreading in complex networks abstract individuals always exhibit distinct personal profiles ranging from education social experiences to risk tolerance which induces a vastly different response in the behavior spreading dynamics to describe this phenomenon a widely used method is to assign a distinct adoption threshold for each individual which so far has eluded theoretical analysis in this paper we propose a behavior spreading model to describe the heterogeneity of individuals in adopting the behavior in which the adoption threshold follows the gaussian distribution based on the edge based compartmental theory the theoretical solutions of the behavior spreading size and the threshold are given for discontinuous and continuous phase transitions through theoretical analysis and extensive numerical simulations the impact of the heterogeneity of adoption thresholds on behavior spreading is demonstrated and we find that the phase transition depends on the mean and standard deviation of the adoption thresholds for small values of the mean adoption threshold the system exhibits a continuous phase transition for large values of the mean adoption threshold the phase transition is discontinuous continuous for the small large standard deviation of the adoption thresholds there is a remarkable agreement between theoretical predictions and numerical simulations

universality for distances in power law random graphs we survey the recent work on phase transition and distances in various random graph models with general degree sequences we focus on inhomogeneous random graphs the configuration model and affine preferential attachment models and pay special attention to the setting where these random graphs have a power law degree sequence this means that the proportion of vertices with degree k in large graphs is approximately proportional to k for some 1 since many real networks have been empirically shown to have power law degree sequences these random graphs can be seen as more realistic models for real complex networks than classical random graphs such as the erd s renyi random graph it is often suggested that the behavior of random graphs should have a large amount of universality meaning in this case that random graphs with similar degree sequences share similar behavior we survey the available results on graph distances in power law random graphs that are consistent with this prediction

discrete stochastic modeling for epidemics in networks in this paper we construct a microscopic mechanism for the epidemic processes in heterogeneous populations which contains two basic assumptions all edges of the network are broken and reconnected at every time step among the epidemic process the probability of randomly chosen two half edges to make a pair is identical we define the stochastic epidemic process and get the epidemic distributions numerically according to the transmission probability two different phases are observed which means the onset of phase transition and the threshold value is very small under the thermodynamic limit the process can be approximated by a deterministic dynamical system about this deterministic system we get the analytical threshold value which is consistent with the numerical results of the epidemic distributions

existence of solutions for sequential fractional integro differential equations and inclusions with nonlocal boundary conditions abstract we investigate the existence of solutions for caputo type sequential fractional integro differential equations and inclusions subject to nonlocal boundary conditions involving riemann liouville and riemann stieltjes integrals for the proofs of our main theorems we use the contraction mapping principle and the krasnosel skii fixed point theorem for the sum of two operators in the case of fractional equations and the nonlinear alternative of leray schauder type for kakutani maps and the covitz nadler fixed point theorem in the case of fractional inclusions some examples are presented to illustrate our results

three nonnegative solutions for fractional differential equations with integral boundary conditions in this paper we study multiple nonnegative solutions for the fractional differential equation with integral boundary conditions cd ax t f t x t 0 email protected 0 1 x 0 email protected 0 1g 0 s x s ds x 1 email protected 0 1g 1 s x s ds x k 0 email protected 0 1g k s x s ds k 2 3 a by means of leggett williams fixed point theorem some new results on the existence of at least three nonnegative solutions are obtained

3 3 lemma and protomodularity abstract the classical 3 3 lemma and snake lemma valid in any abelian category still hold in any quasi pointed the map 0 1 is a mono regular and protomodular category some applications are given in this abstract context concerning the denormalization of kernel maps and the normalization of internals groupoids i e associated crossed modules

existence of multiple solutions for second order boundary value problems we give conditions on f involving pairs of lower and upper solutions which lead to the existence of at least three solutions of the two point boundary value problem y f x y y 0 x epsilon 0 1 y 0 0 y 1 in the special case f x y y f y greater than or equal to 0 we give growth conditions on f and apply our general result to show the existence of three positive solutions we give an example showing this latter result is sharp our results extend those of avery and of lakshmikantham et al

resolutions higher extensions and the relative mal tsev axiom abstract we study how the concept of higher dimensional extension which comes from categorical galois theory relates to simplicial resolutions for instance an augmented simplicial object is a resolution if and only if its truncation in every dimension gives a higher extension in which sense resolutions are infinite dimensional extensions or higher extensions are finite dimensional resolutions we also relate certain stability conditions of extensions to the kan property for simplicial objects this gives a new proof of the fact that a regular category is mal tsev if and only if every simplicial object is kan using a relative setting of extensions