existence of positive solutions for various boundary value problems in this paper we afford a concise approach to guarantee the existence of positive solutions for nonlinear second order ordinary differential equations with respect to various boundary conditions

network geometry with flavor from complexity to quantum geometry network geometry is attracting increasing attention because it has a wide range of applications ranging from data mining to routing protocols in the internet at the same time advances in the understanding of the geometrical properties of networks are essential for further progress in quantum gravity in network geometry simplicial complexes describing the interaction between two or more nodes play a special role in fact these structures can be used to discretize a geometrical d dimensional space and for this reason they have already been widely used in quantum gravity here we introduce the network geometry with flavor s 1 0 1 ngf describing simplicial complexes defined in arbitrary dimension d and evolving by a nonequilibrium dynamics the ngf can generate discrete geometries of different natures ranging from chains and higher dimensional manifolds to scale free networks with small world properties scale free degree distribution and nontrivial community structure the ngf admits as limiting cases both the bianconi barabasi models for complex networks the stochastic apollonian network and the recently introduced model for complex quantum network manifolds the thermodynamic properties of ngf reveal that ngf obeys a generalized area law opening a new scenario for formulating its coarse grained limit the structure of ngf is strongly dependent on the dimensionality d in d 1 ngfs grow complex networks for which the preferential attachment mechanism is necessary in order to obtain a scale free degree distribution instead for ngf with dimension d 1 it is not necessary to have an explicit preferential attachment rule to generate scale free topologies we also show that ngf admits a quantum mechanical description in terms of associated quantum network states quantum network states evolve by a markovian dynamics and a quantum network state at time t encodes all possible ngf evolutions up to time t interestingly the ngf remains fully classical but its statistical properties reveal the relation to its quantum mechanical description in fact the dimensional faces of the ngf have generalized degrees that follow either the fermi dirac boltzmann or bose einstein statistics depending on the flavor s and the dimensions d and

critical random hypergraphs the emergence of a giant set of identifiable vertices we consider a model for random hypergraphs with idemifiability an analogue of connectedness this model has a phase transition in the proportion of identifiable vertices when the underlying random graph becomes critical the phase transition takes various forms depending on the values of the parameters controlling the different types of hyperedges it may be continuous as in a random graph in fact when there are no higher order edges it is exactly the emergence of the giant component in this case there is a sequence of possible sizes of components including but not restricted to n 2 3 alternatively the phase transition may be discontinuous we are particularly interested in the nature of the discontinuous phase transition and are able to exhibit precise asymptotics our method extends a result of aldous ann probab 25 1997 812 854 on component sizes in a random graph

existence of positive solutions for a system of generalized lidstone problems in this paper we study the existence of positive solutions for the system of generalized lidstone problems for ordinary differential equations 1 mu 2 m f 1 t u u 1 m 1u 2 m 2 v v 1 n 1v 2 n 2 1 nv 2 n f 2 t u u 1 m 1u 2 m 2 v v 1 n 1v 2 n 2 a 0u 2 i 0 b 0u 2 i 1 0 a 1u 2 i 1 b 1u 2 i 1 1 0 i 0 1 m 1 a 0v 2 j 0 b 0v 2 j 1 0 a 1v 2 j 1 b 1v 2 j 1 1 0 j 0 1 n 1 here f 1 f 2 c 0 1 xr m n r r 0 our hypotheses imposed on the nonlinearities f 1 and f 2 are formulated in terms of nonnegative matrices we use fixed point index theory to establish our main results based on a priori estimates of positive solutions

on the existence and nonexistence of positive solutions for nonlinear sturm liouville boundary value problems abstract in this paper the existence and nonexistence results of positive solutions are obtained for sturm liouville boundary value problem p x u q x u f x u x 0 1 a u 0 b p 0 u 0 0 c u 1 d p 1 u 1 0 where p c 1 0 1 q c 0 1 p x 0 q x 0 for x 0 1 f c 0 1 r a b c d 0 are constants and satisfy a b c d 0 the discussion is based on the positivity estimation for the green s function of associated linear boundary value problem and the fixed point index theory in cones

positive solutions of two point boundary value problems for second order differential equations with the nonlinearity dependent on the derivative abstract the existence of positive solutions is established for two point boundary value problems for second order differential equations with the nonlinearity dependent on the derivative l x h x f x x 0 x 1 r 1 1 0 1 0 0 r 2 2 1 2 1 0 where l x p x x q x x conditions are given in terms of the relative behaviors of the quotient f u v u v for u v near 0 and

statistical mechanics of the vertex cover problem we review recent progress in the study of the vertex cover problem vc the vc belongs to the class of np complete graph theoretical problems which plays a central role in theoretical computer science on ensembles of random graphs vc exhibits a coverable uncoverable phase transition very close to this transition depending on the solution algorithm easy hard transitions in the typical running time of the algorithms occur we explain a statistical mechanics approach which works by mapping the vc to a hard core lattice gas and then applying techniques such as the replica trick or the cavity approach using these methods the phase diagram of the vc could be obtained exactly for connectivities c e the solution of the vc exhibits full replica symmetry breaking the statistical mechanics approach can also be used to study analytically the typical running time of simple complete and incomplete algorithms for the vc finally we describe recent results for the vc when studied on other ensembles of finite and infinite dimensional graphs

performance of modularity maximization in practical contexts although widely used in practice the behavior and accuracy of the popular module identification technique called modularity maximization is not well understood in practical contexts here we present a broad characterization of its performance in such situations first we revisit and clarify the resolution limit phenomenon for modularity maximization second we show that the modularity function q exhibits extreme degeneracies it typically admits an exponential number of distinct high scoring solutions and typically lacks a clear global maximum third we derive the limiting behavior of the maximum modularity q max for one model of infinitely modular networks showing that it depends strongly both on the size of the network and on the number of modules it contains finally using three real world metabolic networks as examples we show that the degenerate solutions can fundamentally disagree on many but not all partition properties such as the composition of the largest modules and the distribution of module sizes these results imply that the output of any modularity maximization procedure should be interpreted cautiously in scientific contexts they also explain why many heuristics are often successful at finding high scoring partitions in practice and why different heuristics can disagree on the modular structure of the same network we conclude by discussing avenues for mitigating some of these behaviors such as combining information from many degenerate solutions or using generative models

multiplicity results for second order two point boundary value problems with superlinear or sublinear nonlinearities we consider boundary value problems for nonlinear second order differential equations of the form u a t f u 0 t epsilon 0 1 u 0 u 1 0 where a epsilon c 0 1 0 infinity and f r r is continuous and satisfies f s s 0 for s not equal 0 we establish existence and multiplicity results for nodal solutions to the problems if either f 0 0 f infinity infinity or f 0 infinity f 0 0 where f s s approaches f 0 and f infinity as s approaches 0 and infinity respectively we use bifurcation techniques to prove our main results c 2004 elsevier inc all rights reserved

positive solutions of sub superlinear sturm liouville problems in this paper we introduce the notion of a strict lower upper solution to nonlinear sturm liouville boundary value problems based on the maximum principles we establish a result of leray schauder degree on the ordered intervals induced by the pairs of strict lower and upper solutions applying the result and the fixed point index theory in cones we obtain the global existence results of positive solutions for sub superlinear sturm liouville problems

typical approximation performance for maximum coverage problem this study investigated the typical performance of approximation algorithms known as belief propagation the greedy algorithm and linear programming relaxation for maximum coverage problems in sparse biregular random graphs after we used the cavity method for a corresponding hard core lattice gas model results showed that two distinct thresholds of replica symmetry and its breaking exist in the typical performance threshold of belief propagation in the low density region the superiority of three algorithms in terms of a typical performance threshold is obtained by some theoretical analyses although the greedy algorithm and linear programming relaxation have the same approximation ratio in worst case performance their typical performance thresholds are mutually different indicating the importance of typical performance results of numerical simulations validate the theoretical analyses and imply further mutual relations of approximation algorithms

analysis of four parameter fractional derivative model of real solid materials the introduction of fractional derivatives into the constitutive equation of the differential operator type of linear solid materials has led to the development of the so called fractional derivative models one of these models characterized by four parameters has been found usable for describing the variation of dynamics elastic and damping properties in a wide frequency range provided that there is only one loss peak in this paper this four parameter model is theoretically analyzed the effect of the parameters on the frequency curves is demonstrated and it is shown that there is a strict relation between the dispersion of the dynamic modulus the loss peak and the slope of the frequency curves the half value bandwidth of the loss modulus frequency curve is investigated and conditions are developed to establish the applicability of the model for a class of materials moreover it is shown that the model can be used to predict the frequency dependences of dynamic properties for a wide range even if measurements are made in only a narrow frequency range around the loss peak

a threshold of ln n for approximating set cover given a collection f of subsets of s 1 n setcover is the problem of selecting as few as possiblesubsets from f such that their union covers s and maxk cover is the problem of selecting k subsets from f such that their union has maximum cardinality both these problems arenp hard we prove that 1 o 1 ln n is a threshold below which setcover cannot be approximated efficiently unless np has slightlysuperpolynomial time algorithms this closes the gap up to low orderterms between the ratio of approximation achievable by the greedyalogorithm which is 1 o 1 lnn and provious results of lund and yanakakis that showed hardness ofapproximation within a ratio of log 2 n 2s0 72 ln n for max k cover we show an approximationthreshold of 1 1 e up tolow order terms under assumption that p np

statistical mechanics of the minimum vertex cover problem in stochastic block models the minimum vertex cover min vc problem is a well known np hard problem earlier studies illustrate that the problem defined over the erdos renyi random graph with a mean degree c exhibits computational difficulty in searching the min vc set above a critical point c e 2 718 here we address how this difficulty is influenced by the mesoscopic structures of graphs for this we evaluate the critical condition of difficulty for the stochastic block model we perform a detailed examination of the specific cases of two equal size communities characterized by in and out degrees which are denoted by c in and c out respectively our analysis based on the cavity method indicates that the solution search once becomes difficult when c in c out exceeds e from below but becomes easy again when c out is sufficiently larger than c in in the region c out e experiments based on various search algorithms support the theoretical prediction

spin glass model for the c dismantling problem c dismantling cd problem aims at finding the minimum vertex set d of a graph g v e after removing which the remaining graph will break into connected components with the size not larger than c in this paper we introduce a spin glass model with c 1 integer value states into the cd problem and then study the properties of this spin glass model by the belief propagation bp equations under the replica symmetry ansatz we give the lower bound rho c of the relative size of d with finite c on regular random graphs and erdos renyi random graphs we find rho c will decrease gradually with growing c and it converges to rho infty as c to infty the cd problem is called dismantling problem when c is a small finite fraction of v therefore rho infty is also the lower bound of the dismantling problem when v to infty to reduce the computation complexity of the bp equations taking the knowledge of the probability of a random selected vertex belonging to a remaining connected component with the size a the original bp equations can be simplified to one with only three states when c to infty the simplified bp equations are very similar to the bp equations of the feedback vertex set spin glass model h j zhou eur phys j b 86 455 2013 at last we develop two practical belief propagation guide decimation algorithms based on the original bp equations cd bpd and the simplified bp equations scd bpd to solve the cd problem on a certain graph our bpd algorithms and two other state of art heuristic algorithms are applied on various random graphs and some real world networks computation results show that the cd bpd is the best in all tested algorithms in the case of small c but considering the performance and computation consumption we recommend using scd bpd for the network with small clustering coefficient when c is large

existence and iteration of positive solutions to a class of sturm liouville like p laplacian boundary value problems abstract this paper deals with the existence and iteration of positive solutions for the following p laplacian differential equation p u t q t f t u t u t 0 t 0 1 subject to the following sturm liouville like four point boundary condition u 0 u 0 u 1 u 0 where 0 1 by applying the monotone iterative technique and without the assumption of the existence of lower and upper solutions we not only obtain the existence of positive solutions for the problems but also establish iterative schemes for approximating the solutions

existence of positive solutions for second order boundary value problem with one parameter abstract in the case of not requiring f t u to be nonnegative by transforming the boundary value problem into the integral equation system and applying the fixed point index theory the author studies the following second order boundary value problem with one parameter u u f t u 0 t 1 u 0 u 1 0 and obtains the results on the existence of its positive solution

statistical mechanics of the hitting set problem in this paper we present a detailed study of the hitting set hs problem this problem is a generalization of the standard vertex cover to hypergraphs one seeks a configuration of particles with minimal density such that every hyperedge of the hypergraph contains at least one particle it can also be used in important practical tasks such as the group testing procedures where one wants to detect defective items in a large group by pool testing using a statistical mechanics approach based on the cavity method we study the phase diagram of the hs problem in the case of random regular hypergraphs depending on the values of the variables and tests degrees different situations can occur the hs problem can be either in a replica symmetric phase or in a one step replica symmetry breaking one in these two cases we give explicit results on the minimal density of particles and the structure of the phase space these problems are thus in some sense simpler than the original vertex cover problem where the need for a full replica symmetry breaking has prevented the derivation of exact results so far finally we show that decimation procedures based on the belief propagation and the survey propagation algorithms provide very efficient strategies to solve large individual instances of the hitting set problem

positive periodic solutions for nonlinear difference equations with periodic coefficients abstract we prove the existence of positive solutions of second order nonlinear difference equations on a finite discrete segment with periodic boundary conditions obtained results yield positive periodic solutions of the equation on the whole discrete axis provided that the coefficients are periodic

a global bifurcation phenomenon for second order singular boundary value problems we establish an existence result of unbounded continuum of solutions from the trivial branch for a second order singular boundary value problem as an application we prove several existence and multiplicity results of positive solutions for certain singular boundary value problems

abstract existence theorems of positive solutions for nonlinear boundary value problems let rn be a bounded domain h l2 l d l h h be a unbounded linear operator and f r r be a continuous function this paper deals with the existence of positive solutions of the operator equation lu f x u u d l which is the abstract form of differential equations boundary value problems the essential conditions on l and f guaranteeing the existence of positive solution are characterized the discussion is based on the krasnoselskii s fixed point theorem of cone mapping and the fixed point index theory in cones the abstract results are applied to some differential equations boundary value problems and some known results are extended and improved

existence and nonexistence results for positive solutions of an integral boundary value problem abstract this paper deals with the existence and nonexistence of positive solutions for the integral boundary value problem a u b u g t f t u cos 0 u 0 sin 0 u 0 0 1 u d cos 1 u 1 sin 1 u 1 0 1 u d we provide sufficient conditions for the existence of either none or one or more positive solutions of the above problem the main tool used in the proofs of existence results is a fixed point theorem in a cone due to krasnoselskii and zabreiko the results presented substantially unify and extend the existing results in the literature

number of guards needed by a museum a phase transition in vertex covering of random graphs in this letter we study the np complete vertex cover problem on finite connectivity random graphs when the allowed size of the cover set is decreased a discontinuous transition in solvability and typical case complexity occurs this transition is characterized by means of exact numerical simulations as well as by analytical replica calculations the replica symmetric phase diagram is in excellent agreement with numerical findings up to average connectivity e where replica symmetry becomes locally unstable

curvilinear component analysis a self organizing neural network for nonlinear mapping of data sets we present a new strategy called curvilinear component analysis cca for dimensionality reduction and representation of multidimensional data sets the principle of cca is a self organized neural network performing two tasks vector quantization vq of the submanifold in the data set input space and nonlinear projection p of these quantizing vectors toward an output space providing a revealing unfolding of the submanifold after learning the network has the ability to continuously map any new point from one space into another forward mapping of new points in the input space or backward mapping of an arbitrary position in the output space

multiplicity of solutions to two point boundary value problems for second order impulsive differential equations abstract in this paper we obtain some sufficient condition for the existence of at least three solutions to two point boundary value problems for second order impulsive differential equations by using leray schauder degree theory we assume the existence of two pairs of strict lower and upper solutions and the nonlinear term f satisfies a nagumo s growth condition with respect to the first order derivative

on sign changing solutions for nonlinear operator equations abstract in this paper the existence of sign changing solutions for nonlinear operator equations is discussed by using the topological degree and fixed point index theory the main theorems are some new three solution theorems which are different from the famous amann s and leggett williams three solution theorems as well as the results in f li g han generalization for amann s and leggett williams three solution theorems and applications j math anal appl 298 2004 638 654 these three solutions are all nonzero one of them is positive another is negative and the third one is a sign changing solution furthermore the theoretical results are successfully applied to both integral and differential equations

solutions of periodic boundary value problems for second order nonlinear differential equations via variational methods abstract by introducing a variational framework for a class of second order nonlinear differential equations with non separated periodic boundary value conditions some results on the existence of non trivial positive and negative solutions of the problems are obtained some results by atici guseinov graef kong etc obtained by topological degree methods are extended the resonant case of the problems where the nonlinearities are unbounded and satisfy ahmad lazer paul type conditions is also considered

covering problems and core percolations on hypergraphs we introduce two generalizations of core percolation in graphs to hypergraphs related to the minimum hyperedge cover problem and the minimum vertex cover problem on hypergraphs respectively we offer analytical solutions of these two core percolations for uncorrelated random hypergraphs whose vertex degree and hyperedge cardinality distributions are arbitrary but have nondiverging moments we find that for several real world hypergraphs their two cores tend to be much smaller than those of their null models suggesting that covering problems in those real world hypergraphs can actually be solved in polynomial time

positive solutions of two point boundary value problems for systems of nonlinear second order singular and impulsive differential equations abstract this paper is devoted to the study of multiple and single positive solutions of two point boundary value problems for nonlinear second order singular and impulsive differential systems by constructing a cone k 1 k 2 which is the cartesian product of two cones in the space c 0 1 and computing the fixed point index in the k 1 k 2 we establish the conditions for the existence of at least one or at least two positive solutions to two point boundary value problems for systems of nonlinear second order singular and impulsive differential equations

multiple positive solutions of some boundary value problems abstract we study the existence of multiple positive solutions of the equations u t u subject to linear boundary conditions we show that there are at least two positive solutions if t u is superlinear at one end point zero or infinity and sublinear at the other applications of these results are provided to yield multiple positive solutions of some elliptic boundary value problems on an annulus

two solutions to diluted p spin models and xorsat problems we derive analytical solutions for p spin models with finite connectivity at zero temperature these models are the statistical mechanics equivalent of p xorsat problems in theoretical computer science we give a full characterization of the phase diagram location of the phase transitions static and dynamic together with a description of the clustering phenomenon taking place in configurational space we use two alternative methods the cavity approach and a rigorous derivation

eigenvalue criteria for existence of multiple positive solutions to boundary value problems of second order delay differential equations abstract criteria are established for existence of multiple positive solutions to the boundary value problems expressed by second order delay differential equations of the form p t y t q t y t f t y t y t a t 0 1 y t y t t t 0 y t y t t t 1 1 a where 0 a 1 4 are suitably small p c 1 0 1 p 0 q c 0 1 and q 0

generalization of core percolation on complex networks we introduce a k leaf removal algorithm as a generalization of the so called leaf removal algorithm in this pruning algorithm vertices of degree smaller than k together with their first nearest neighbors and all incident edges are progressively removed from a random network as the result of this pruning the network is reduced to a subgraph which we call the generalized k core gk core performing this pruning for the sequence of natural numbers k we decompose the network into a hierarchy of progressively nested gk cores we present an analytical framework for description of gk core percolation for undirected uncorrelated networks with arbitrary degree distributions configuration model to confirm our results we also derive rate equations for the k leaf removal algorithm which enable us to obtain the structural characteristics of the gk cores in another way also we apply our algorithm to a number of real world networks and perform the gk core decomposition for them

on linear and nonlinear fourth order eigenvalue problems with indefinite weight abstract we determine the principal eigenvalues of the linear indefinite weight problem u x r g x u x x 0 1 u 0 u 1 u 0 u 1 0 moreover we investigate the existence of positive solutions for the corresponding nonlinear indefinite weight problem where g 0 1 r is a continuous function which attains both positive and negative values f c r r and r is a parameter

positive solutions of second order two delay differential systems with twin parameter abstract the paper is concerned with the existence of positive solutions for a second order two delay differential systems with twin parameter firstly we present some sufficient conditions with the twin parameter belonging to an open rectangular region eigenvalue region to ensure the existence of positive solutions for the differential systems with singularity secondly we deal with the superlinear semipositone problem of the differential systems and establish the existence of positive solutions for the systems the proofs are based on a fixed point theorem in cones

existence of positive solutions forboundary value problems of second order functional difference equations abstract we prove the existence of positive solutions for boundary value problems of second orderfunctional difference equations the result here is the generalization of a corresponding result for ordinary difference equations

dispersion curves of viscoelastic plane waves and rayleigh surface wave in high frequency range with fractional derivatives abstract the moduli of conventional elastic structural materials are extended to one of the viscoelastic materials through a modification whereby the dynamic moduli converge to the static moduli of elasticity as the fractional order approaches zero by plotting phase velocity curves and group velocity curves of plane waves and rayleigh surface wave for a viscoelastic material polyvinyl chloride foam the influence of the fractional order of viscoelasticity is examined the phase and group velocity curves in the high frequency range were derived for longitudinal transverse and rayleigh waves inherent to the viscoelastic material in addition the equation for the phase velocity was mathematically derived on the complex plane too and graphically illustrated a phenomenon was found that at the moment when the fractional order of the time derivative reaches an integer value 1 the curve on the complex plane becomes completely different exhibiting snap through behavior we examined the mechanism of the snap through mathematically numerical calculation examples were solved and good agreement was confirmed between the numerical calculation and the analytical expression mentioned above from the results of the numerical example regularities were derived for the absolute value of the complex phase and group velocities on the complex plane

phase transitions in semidefinite relaxations statistical inference problems arising within signal processing data mining and machine learning naturally give rise to hard combinatorial optimization problems these problems become intractable when the dimensionality of the data is large as is often the case for modern datasets a popular idea is to construct convex relaxations of these combinatorial problems which can be solved efficiently for large scale datasets semidefinite programming sdp relaxations are among the most powerful methods in this family and are surprisingly well suited for a broad range of problems where data take the form of matrices or graphs it has been observed several times that when the statistical noise is small enough sdp relaxations correctly detect the underlying combinatorial structures in this paper we develop asymptotic predictions for several detection thresholds as well as for the estimation error above these thresholds we study some classical sdp relaxations for statistical problems motivated by graph synchronization and community detection in networks we map these optimization problems to statistical mechanics models with vector spins and use nonrigorous techniques from statistical mechanics to characterize the corresponding phase transitions our results clarify the effectiveness of sdp relaxations in solving high dimensional statistical problems

on the existence of positive solutions for nonlinear differential equations with periodic boundary conditions we prove the existence of positive solutions of second order nonlinear differential equations on a finite interval with periodic boundary conditions and give upper and lower bounds for these positive solutions obtained results yield positive periodic solutions of the equation on the whole real axis provided that the coefficients are periodic

positive solutions of nonlinear singular two point boundary value problems for second order impulsive differential equations in this paper we study the positive solutions of nonlinear singular two point boundary value problems for second order impulsive differential equations the existence of one or two positive solutions are established by using the fixed point index theorem in cone

existence of solutions to a class of nonlinear second order two point boundary value problems in this paper the existence and multiplicity results of solutions are obtained for the second order two point boundary value problem u t f t u t for all t 0 1 subject to u 0 u 1 0 where f is continuous the monotone operator theory and critical point theory are employed to discuss this problem respectively in argument quadratic root operator and its properties play an important role

multiple solutions of impulsive sturm liouville boundary value problem via lower and upper solutions and variational methods abstract in this paper we prove the existence of multiple solutions for second order sturm liouville boundary value problem l u f x u x 0 1 r 1 u 0 r 2 u 0 where l u p x u q x u is a sturm liouville operator r 1 u u 0 u 0 r 2 u u 1 u 1 the technical approach is fully based on lower and upper solutions and variational methods the interesting point is that the existence of four solutions and seven solutions is given

verification of local kramers kronig relations for complex modulus by means of fractional derivative model abstract the local kramers kronig k k relations which link the damping properties of solid materials at one frequency to the rate of frequency variation of dynamic modulus are not exact the validity and accuracy of the local k k relations is theoretically investigated in this paper by means of material models especially the fractional zener model it is shown that the local k k relations qualitatively always properly predict the relation between the damping and the frequency dependence of dynamic modulus for any type of deformation and any linear mechanism of energy loss determining the frequency variations nevertheless the accuracy depends on the rate of frequency variation of dynamic properties mainly of the loss modulus and loss factor and the weaker the frequency dependence the better the accuracy the accuracy is better than 10 if the slope of frequency increase or decrease of loss functions plotted in a log log system is smaller than 0 35 the application of the local k k relations to some experimental data is presented

on the existence of positive solutions for semilinear elliptic equations in the annulus abstract we study the existence of positive radial solutions of u g x u 0 in annuli with dirichlet dirichlet neumann boundary conditions we prove that the problems have positive radial solutions on any annulus if is sublinear at 0 and

message passing for vertex covers constructing a minimal vertex cover of a graph can be seen as a prototype for a combinatorial optimization problem under hard constraints in this paper we develop and analyze message passing techniques namely warning and survey propagation which serve as efficient heuristic algorithms for solving these computational hard problems we show also how previously obtained results on the typical case behavior of vertex covers of random graphs can be recovered starting from the message passing equations and how they can be extended

generalized network structures the configuration model and the canonical ensemble of simplicial complexes simplicial complexes are generalized network structures able to encode interactions occurring between more than two nodes simplicial complexes describe a large variety of complex interacting systems ranging from brain networks to social and collaboration networks here we characterize the structure of simplicial complexes using their generalized degrees that capture fundamental properties of one two three or more linked nodes moreover we introduce the configuration model and the canonical ensemble of simplicial complexes enforcing respectively the sequence of generalized degrees of the nodes and the sequence of the expected generalized degrees of the nodes we evaluate the entropy of these ensembles finding the asymptotic expression for the number of simplicial complexes in the configuration model we provide the algorithms for the construction of simplicial complexes belonging to the configuration model and the canonical ensemble of simplicial complexes we give an expression for the structural cutoff of simplicial complexes that for simplicial complexes of dimension d 1 reduces to the structural cutoff of simple networks finally we provide a numerical analysis of the natural correlations emerging in the configuration model of simplicial complexes without structural cutoff

positive solutions to some equations with homogeneous operator abstract in this paper we discuss the positive solutions to the equation u u a a u b u u 0 where a is a positive linear completely continuous operator b is an homogeneous operator defined on a cone in a real banach space and u a b u by using the fixed point index theory when u 0 is sufficiently small the spectral radius r a 1 and 1 where sgn b we obtain a positive solution to the above equation under some appropriate conditions the new results generalize the previous research about the homogeneous operator equation as an application by using our main theorem we can obtain a symmetrical positive solution to the one dimensional kirchhoff equation

existence of positive solutions of a fourth order boundary value problem abstract we consider the fourth order boundary value problem u f t u u 0 t 1 u 0 u 1 u 0 u 1 0 where f t u p au b p u p near 0 0 and f t u p cu dp u p near we give conditions on the constants a b c d that guarantee the existence of positive solutions the proof of our main result is based upon global bifurcation techniques

on the number of solutions of two classes of sturm liouville boundary value problems abstract we employ global bifurcation theory of p h rabinowitz together with sturm comparison theorem to establish multiplicity results for some classes of sturm liouville boundary value problems

multiple solutions of some boundary value problems with parameters abstract in this paper we study the existence and multiplicity of nontrivial solutions for the following second order dirichlet nonlinear boundary value problem with odd order derivative u t a u t b u t f t u t for all t 0 1 with u 0 u 1 0 where a b r 1 f c 1 0 1 r 1 r 1 by using the morse theory we impose certain conditions on f which are able to guarantee that the problem has at least one nontrivial solution two nontrivial solutions and infinitely many solutions separately

positive solutions of second order boundary value problems with sign changing nonlinear terms abstract in this paper the existence results of positive solutions are obtained for second order boundary value problem u f t u t 0 1 u 0 u 1 0 where f c 0 1 r whose sign may change the proof is based on the fixed point index theory in cones

unbounded solutions of the singular boundary value problems for second order differential equations on the half line this paper presents some results on the existence of multiple unbounded solutions for the boundary value problems for second order differential equations on half line and on the uniqueness of unbounded solution

nontrivial solutions of singular sublinear sturm liouville problems abstract the singular superlinear sturm liouville problems l x h x f x 0 x 1 r 1 1 0 1 0 0 r 2 2 1 2 1 0 are considered under some conditions concerning the first eigenvalues corresponding to the relevant linear operators where l x p x x q x x and h x is allowed to be singular at both x 0 and x 1 in particular f is not necessary to be nonnegative the existence results of nontrivial solutions and positive solutions are given by means of the topological degree theory

positive solutions of a second order integral boundary value problem in this paper we study the existence of positive solutions of a second order integral boundary value problems for ordinary differential equations our results presented here unify generalize and substantially improve the existing results in the literature moreover it is worthwhile to point out that our method will dispense with constructing a new green function

phase transition for cutting plane approach to vertex cover problem we study the vertex cover problem which is a nondeterministic polynomial time hard optimization problem and a prototypical model exhibiting phase transitions on random graphs such as erd s renyi er random graphs these phase transitions coincide with changes of the solution space structure e g for the er ensemble at connectivity c e 2 7183 from replica symmetric to replica symmetry broken for the vertex cover problem the typical complexity of exact branch and bound algorithms which proceed by exploring the landscape of feasible configurations also changes close to this phase transition from easy to hard in this work we consider an algorithm which has a completely different strategy the problem is mapped onto a linear programming problem augmented by a cutting plane approach hence the algorithm operates in a space outside the space of feasible configurations until the final step where a solution is found here we show that this type of algorithm also exhibits an easy hard transition around c e which strongly indicates that the typical hardness of a problem is fundamental to the problem and not due to a specific representation of the problem

existence of positive solutions of lidstone boundary value problems abstract we show that there exists at least one positive solution for lidstone boundary value problem 1 n u 2 n t f t u t u t u 2 n 1 t 0 t 1 u 2 i 0 u 2 i 1 0 i 0 1 n 1 under some suitable conditions our proof is based upon global bifurcation techniques

existence of solutions of semilinear systems with gradient dependence via eigenvalue criteria abstract in this paper new criteria are established for the existence of positive radial solutions of a semilinear elliptic system depending on the gradient these criteria are determined by some relationships between the upper and lower bounds on suitable strips of r n of the nonlinearities of the system and the principal characteristic values of some associated linear hammerstein integral operators moreover using smoothing tools the totality of the involved cone is established

positive solutions and eigenvalue regions of two delay singular systems with a twin parameter abstract the paper is concerned with a two delay singular differential system with a twin parameter applying fixed point index theory we show the relationship between the asymptotic behaviors of nonlinearities at zero and infinity and the open regions eigenvalue regions of parameters which are correlated with delays such that the system has zero one and two positive solution s

an algorithm for set covering problem in this paper we present an algorithm for the set covering problem that combines problem reduction tests with dual ascent subgradient optimisation and linear programming computational results are presented for problems involving up to 400 rows and 4000 columns

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

generalization for amann s and leggett williams three solution theorems and applications abstract in this paper some new three solution theorems are obtained moreover the famous amann s and leggett williams three solution theorems in nonlinear functional analysis can be seen as their special cases namely these two famous theorems are united so they are both improved and that some new three solution theorems are applied to ordinary differential equations and some new existence results are obtained

multiple positive solutions of nonlinear two point boundary value problems we improve the results obtained by erbe hu and wang in a recent paper we show that there exist at least two positive solutions of two point boundary value problems under conditions weaker than those used by erbe hu and wang

dispersion curves for a viscoelastic timoshenko beam with fractional derivatives abstract the kramers kronig dispersion relation often used as a viscoelastic constitutive law for polymeric materials is based on purely mathematical properties of linearity convergence of improper integrals and causality thus it may also be valid as a viscoelastic constitutive law for general structural materials accordingly the motion equation of a timoshenko beam composed of conventional elastic structural materials is extended to one composed of viscoelastic materials from the derived governing equation a dispersive equation is derived for a viscoelastic timoshenko beam by plotting phase velocity curves and group velocity curves for a beam of solid circular cross section composed of a viscoelastic material polyvinyl chloride foam the influence of the fractional order of viscoelasticity is examined as a result it is found that in the high frequency range only the first mode of a timoshenko beam converged to the propagation velocity of the rayleigh wave which takes account of the fractional order of viscoelasticity in addition the phase velocity and the group velocity were found to increase as the fractional order approaches 0 and to decrease as the fractional order approaches 1 furthermore the rate of velocity change becomes greater as the fractional order approaches 0 and becomes smaller as the fractional order approaches 1

a variational method for identification of viscoelastic parameters from experimental data an identification method based on a variational algorithm is proposed for the determination of the viscoelastic properties of plates from a set of experimental data a pulse transmission configuration with parametric sources is used for the experimental measurements whereby two electrically mixed high frequency pulses are combined to generate a low frequency narrow beam signal to minimize the edge diffraction effect from a finite sized panel the variational method is based on an optimization technique adapted from an approach normally used in dynamics for the identification of motion parameters the fluid panel fluid system is initially represented by a transmission matrix model the model and the known experimental data are then used to build an optimization functional to be minimized by means of a collocation method based on the hamilton principle analogy the plate viscoelastic parameters are then calculated by application of well established numerical procedures to the resulting set of integral differential equations

a three point boundary value problem for nonlinear fourth order differential equations abstract in this paper the authors consider the nonlinear fourth order ordinary differential equation e u t g t f u 0 with the boundary conditions b u 0 u 1 u 0 u p u 1 0 some results on the existence and nonexistence of positive solutions to problem e b are obtained results on the existence of infinitely many positive solutions are also presented examples are included to demonstrate that the results are sharp

minimum vertex cover problems on random hypergraphs replica symmetric solution and a leaf removal algorithm the minimum vertex cover problems on random ensuremath alpha uniform hypergraphs are studied using two different approaches a replica method in statistical mechanics of random systems and a leaf removal algorithm it is found that there exists a phase transition at the critical average degree e ensuremath alpha ensuremath 1 below which a replica symmetric ansatz in the replica method holds and the algorithm estimates exactly the same solution of the problem as that by the replica method in contrast above the critical degree the replica symmetric solution becomes unstable and the leaf removal algorithm fails to estimate the optimal solution because of the emergence of a large size core these results strongly suggest a close relation between the replica symmetry and the performance of an approximation algorithm critical properties of the core percolation are also examined numerically by a finite size scaling

positive solutions of m point boundary value problems the second order m point boundary value problem x h x f x 0 0 x 1 0 0 1 i 1m 2ai i is considered under some conditions concerning the first eigenvalue of the relevant linear operator where i 0 1 with 0 1 2 m 2 1 ai 0 with i 1m 2ai 1 h x is allowed to be singular at x 0 and x 1 the existence of positive solutions and multiple positive solutions is obtained by means of fixed point index theory similarly conclusions hold for some other m point boundary value conditions

multiplicity results for second order m point boundary value problem by using fixed point theorems some new results for multiplicity of positive solutions for a class of second order m point boundary value problem are obtained the associated green s function of the problem is also given

stability analysis on the finite temperature replica symmetric and first step replica symmetry broken cavity solutions of the random vertex cover problem the vertex cover problem is a prototypical hard combinatorial optimization problem it was studied in recent years by physicists using the cavity method of statistical mechanics in this paper the stability of the finite temperature replica symmetric rs and the first step replica symmetry broken 1rsb cavity solutions of the vertex cover problem on random regular graphs of finite vertex degree k are analyzed by population dynamics simulations we found that 1 the lowest temperature for the rs solution to be stable t rs k is not a monotonic function of k 2 at relatively large connectivity k and temperature t slightly below the dynamic transition temperature t d k the 1rsb solutions with small but non negative complexity values are stable and 3 the dynamical transition temperature t d and kauzmann temperature t k is equal to each other similar results are obtained on random poissonian graphs

positive solutions of a 2nth order boundary value problem involving all derivatives via the order reduction method this paper is mainly concerned with the existence multiplicity and uniqueness of positive solutions for the 2nth order boundary value problem 1 nu 2 n f t u u 1 i 2 u i 1 n 1u 2 n 1 u 2 i 0 u 2 i 1 1 0 i 0 1 n 1 where n 2 and f c 0 1 xr 2 n r r 0 we first use the method of order reduction to transform the above problem into an equivalent initial value problem for a first order integro differential equation and then use the fixed point index theory to prove the existence multiplicity and uniqueness of positive solutions for the resulting problem based on a priori estimates achieved by developing spectral properties of associated parameterized linear integral operators finally as a byproduct our main results are applied for establishing the existence multiplicity and uniqueness of symmetric positive solutions for the lidstone problem involving all derivatives

existence of three positive solutions for some second order boundary value problems in this paper a new fixed point theorem of functional type in a cone is established with using the new fixed point theorem and imposing growth conditions on the nonlinearity the existence of three positive solutions for the boundary value problemx t f t x t x t 0 0 0 is continuous finally an example is given to illustrate the importance of results obtained

global bifurcation and multiple results for sturm liouville problems we consider the nonlinear sturm liouville boundary value problem lu t email protected t f u t 0

finite element analysis of vibrating linear systems with fractional derivative viscoelastic models abstract fractional derivative rheological models are known to be very effective in describing the viscoelastic behaviour of materials especially of polymers and when applied to dynamic problems the resulting equations of motion after a fractional state space expansion can still be studied in terms of modal analysis but the growth in matrix dimensions carried by this expansion is in general so fast to make the calculations exceedingly cumbersome this paper presents a method for reducing the computational effort due to finite element fe analysis of vibrating linear systems with a fractional derivative viscoelastic model namely the fractional kelvin voigt the proposed method may be applied also to problems involving other fractional derivative linear models and it takes under control the computational effort by reducing the main eigenproblem of large dimension to the solution of two standard related eigenproblems of lower size numerical examples are provided in order to validate both the accuracy and the efficiency of the proposed methodology