a polynomial time approximation algorithm for the permanent of a matrix with nonnegative entries we present a polynomial time randomized algorithm for estimating the permanent of an arbitrary n n matrix with nonnegative entries this algorithm technically a fully polynomial randomized approximation scheme computes an approximation that is with high probability within arbitrarily small specified relative error of the true value of the permanent

no skew branes on non degenerate hyperquadrics we show that non degenerate hyperquadrics in r n 2 admit no skew n branes stated more traditionally a compact codimension one immersed submanifold of a non degenerate hyperquadric of euclidean space must have parallel tangent spaces at two distinct points similar results have been proven by others but except for ellipsoids in r 3 proc amer math soc 133 3687 3690 2005 always under c 2 smoothness and genericity assumptions we use neither assumption here

growth model for fractal scale free networks generated by a random walk abstract the diversity of networked systems with fractal structures suggests that knowing the underlying mechanism that generates the fractality is necessary for building a model of the development of complex networks in the present paper we propose a growth model of a network generated by a random walk and show that the evolving graph forms a fractal structure with various properties including the scale free property if the graph which provides a space where a random walk occurs by itself is formed by the random walk the proposed model is regulated by two parameters p v and p e which define the probability of either a roundabout path via a new vertex or a shortcut being formed by the random walk respectively the power law exponent describing the vertex degree distribution is determined by the ratio p e p v and is related to an internal factor f i via the relation 1 f i 1 where f i is a parameter that describes the local structure generated by the random walk a sufficiently small p v provides the small world property to the model network the small world property is usually considered to be incompatible with the fractal scaling property open m c l c d c where open m c is the average number of vertices which can be reached from a randomly chosen vertex in at most l c steps however we demonstrate that fractality can be reconciled with the small world property by introducing a size dependent fractal cluster dimension d c

exploring self similarity of complex cellular networks the edge covering method with simulated annealing and log periodic sampling abstract song et al self similarity of complex networks nature 433 2005 392 395 have recently used a version of the box counting method called the node covering method to quantify the self similar properties of 43 cellular networks the minimal number n v of boxes of size l needed to cover all the nodes of a cellular network was found to scale as the power law n v l 1 d v with a fractal dimension d v 3 53 0 26 we implement an alternative box counting method in terms of the minimum number n e of edge covering boxes which is well suited to cellular networks where the search over different covering sets is performed with the simulated annealing algorithm the method also takes into account a possible discrete scale symmetry to optimize the sampling rate and minimize possible biases in the estimation of the fractal dimension with this methodology we find that n e scales with respect to l as a power law n e l d e with d e 2 67 0 15 for the 43 cellular networks previously analyzed by song et al self similarity of complex networks nature 433 2005 392 395 bootstrap tests suggest that the analyzed cellular networks may have a significant log periodicity qualifying a discrete hierarchy with a scaling ratio close to 2

totally skew embeddings of manifolds we study a version of whitney s embedding problem in projective geometry what is the smallest dimension of an affine space that can contain an n dimensional submanifold without any pairs of parallel or intersecting tangent lines at distinct points this problem is related to the generalized vector field problem existence of non singular bilinear maps and the immersion problem for real projective spaces we use these connections and other methods to obtain several specific and general bounds for the desired dimension

a monte carlo algorithm for estimating the permanent let a be an n times n matrix with 0 1 valued entries and let operatorname per a be the permanent of a this paper describes a monte carlo algorithm that produces a good in the relative sense estimate of operatorname per a and has running time operatorname poly n 2 n 2 where operatorname poly n denotes a function that grows polynomially with n

the complexity of partial derivatives abstract let l denote the nonscalar complexity in k x 1 x n we prove l x 1 x n 3l using this we determine the complexity of single power sums single elementary symmetric functions the resultant and the discriminant as root functions up to order of magnitude also we linearly reduce matrix inversion to computing the determinant

the fractal dimension of percolation cluster hulls large simulations of 2d site percolation on a square lattice provide several estimates of the fractal dimension dh of the hull or outer boundary of percolation clusters close to criticality statistical estimates based on an ensemble of more than 106 clusters with varying sizes yield dh 1 74 or 0 02 geometrical measurements of the hull two point correlation for individual clusters with more than 105 sites give dh 1 76 or 0 01 the observed scaling behaviour is the same both slightly above and below pc

real space renormalization from a geometric fractal perspective abstract an introduction to real space renormalization transformation rt is presented with geometric fractal aspects as focus critical exponents are equivalent to fractal dimensions details of the construction rule of deterministic fractals and the definition of their fractal dimensions illustrate differences in approach between field theory and condensed matter physics the appearance of fractal structures is typically less obvious within the context of partition functions equilibrium statistical mechanics and field theory and master equations dynamic processes an rt amounts to a reformulation of those construction rules into a form similar to the conventional ones for deterministic fractals but on an algebraic level without ever explicitly referring to the geometric structure this is illustrated explicitly in the context of the sierpinsky gasket its construction rule can be reformulated as a growth process and also as a partition function

random sequential renormalization of networks application to critical trees we introduce the concept of random sequential renormalization rsr for arbitrary networks rsr is a graph renormalization procedure that locally aggregates nodes to produce a coarse grained network it is analogous to the quasi parallel renormalization schemes introduced by c song et al c song et al nature london 433 392 2005 and studied by f radicchi et al f radicchi et al phys rev lett 101 148701 2008 but much simpler and easier to implement here we apply rsr to critical trees and derive analytical results consistent with numerical simulations critical trees exhibit three regimes in their evolution under rsr i for n 0 ensuremath nu ensuremath lesssim nl n 0 where n is the number of nodes at some step in the renormalization and n 0 is the initial size of the tree rsr is described by a mean field theory and fluctuations from one realization to another are small the exponent ensuremath nu 1 2 is derived using random walk and other arguments the degree distribution becomes broader under successive steps reaching a power law p k 1 k ensuremath gamma with ensuremath gamma 2 and a variance that diverges as n 0 1 2 at the end of this regime both of these latter results are obtained from a scaling theory ii for n 0 ensuremath nu mathrm star ensuremath lesssim n ensuremath lesssim n 0 1 2 with ensuremath nu mathrm star ensuremath approx 1 4 hubs develop and fluctuations between different realizations of the rsr are large trees are short and fat with an average radius that is mathcal o 1 crossover functions exhibiting finite size scaling in the critical region n n 0 1 2 ensuremath rightarrow ensuremath infty connect the behaviors in the first two regimes iii for n ensuremath lesssim n 0 ensuremath nu mathrm star star configurations appear with a central hub surrounded by many leaves the distribution of stars is broadly distributed over this range the scaling behaviors found under rsr are identified with a continuous transition in a process called agglomerative percolation ap with the coarse grained nodes in rsr corresponding to clusters in ap that grow by simultaneously attaching to all their neighboring clusters

dynamics of numerical discretization in a delayed diffusive nicholson s blowflies equation in this article we study the dynamics of a diffusive nicholson s blowflies equation with a finite delay and dirichlet boundary condition by the forward and backward euler difference schemes and crank nicolson scheme the existence of hopf bifurcation at the equilibrium is obtained through analyzing the distribution of the eigenvalues we find that the unstable equilibrium state without dispersion may become stable with dispersion under certain conditions our results show that crank nicolson and the backward euler difference schemes are superior to the forward euler difference scheme under the means of describing approximately the dynamics of the original system finally numerical examples are provided to illustrate the analytical results

fractality and scale free effect of a class of self similar networks abstract in this paper given an initial directed graph as a self similar pattern and fix two nodes in the pattern we can iterate the graph by replacing any directed edge with the initial graph of pattern and identifying the fixed nodes of pattern with the endpoints of directed edge using the iteration again and again we obtain a family of growing self similar networks modify these networks to be undirected ones we obtain growing self similar undirected networks we obtain the fractality of our self similar networks and find out the scale free effect in terms of the matrix related to two fixed nodes in the initial graph

stability and bifurcation analysis in van der pol s oscillator with delayed feedback abstract the classical van der pol equation with delayed feedback and a modified equation where a delayed term provides the damping are considered linear stability of the equations is investigated by analyzing the associated characteristic equations it is found that there exist the stability switches when delay varies and the hopf bifurcation occurs when the delay passes through a sequence of critical values the bifurcation diagram is drawn in e k plane and the stability and direction of the hopf bifurcation are determined by applying the normal form theory and the center manifold theorem

distinct clusterings and characteristic path lengths in dynamic small world networks with identical limit degree distribution many real world networks belong to a particular class of structures known as small world networks that display short distance between pair of nodes in this paper we introduce a simple family of growing small world networks where both addition and deletion of edges are possible by tuning the deletion probability q t the model undergoes a transition from large worlds to small worlds by making use of analytical or numerical means we determine the degree distribution clustering coefficient and average path length of our networks surprisingly we find that two similar evolving mechanisms which provide identical degree distribution under a reciprocal scaling as t goes to infinity can lead to quite different clustering behaviors and characteristic path lengths it is also worth noting that farey graphs constitute the extreme case q t 0 of our random construction

hopf bifurcation in numerical approximation of a class delay differential equations abstract in this paper we consider the numerical solution of a delay differential equation undergoing a hopf bifurcation we prove that if the delay differential equation has a hopf bifurcation point at then the numerical solution of the equation also has a hopf bifurcation point at h o h

typical ranks for 3 tensors nonsingular bilinear maps and determinantal ideals abstract let m n 3 m 1 n 1 2 p m n and u m n p the set r u n m of all real tensors with size u n m is one to one corresponding to the set of bilinear maps r m r n r u we show that r m n p has plural typical ranks p and p 1 if and only if there exists a nonsingular bilinear map r m r n r u we show that there is a dense open subset o of r u n m such that for any y o the ideal of maximal minors of a matrix defined by y in a certain way is a prime ideal and the real radical of that is the irrelevant maximal ideal if that is not a real prime ideal further we show that there is a dense open subset t of r n p m and continuous surjective open maps o r u p and t r u p where r u p is the set of u p matrices with entries in r such that if y t then rank t p if and only if the ideal of maximal minors of the matrix defined by y is a real prime ideal

orthogonal designs with maximal rates orthogonal designs have been used as space time block codes for wireless communications with multiple transmit antennas which can achieve full transmit diversity and have a very simple decoupled maximum likelihood decoding algorithm the rate of an orthogonal design is defined as the ratio of the number of transmitted information symbols in a block of channel uses to the length of the given block which reflects the bandwidth efficiency of the employed space time block code constructed from the orthogonal design this paper focuses on the analysis and synthesis of orthogonal designs with the maximum possible rates which may be real or complex and square or rectangular matrices we first provide several representations of orthogonal designs and their characterizations in terms of hurwitz radon matrix equations next we observe that the real orthogonal designs square or rectangular and the complex square orthogonal designs with maximal rates have been well understood from the existing results in the mathematics literature which can be dated back to 1890s however unfortunately it is not the case for the complex rectangular orthogonal designs with rates as high as possible we then construct a class of complex orthogonal designs for any number of transmit antennas the proposed complex orthogonal designs for the number of transmit antennas n 2m 1 and 2m have the same rate m 1 and 2m where m is any natural number finally we demonstrate that for n 2m 1 and 2m with any given natural number m the value m 1 and 2m is the maximum possible rate that the complex orthogonal designs square or rectangular with n transmit antennas can achieve

on highly regular embeddings a continuous map mathbb r d rightarrow mathbb r n is k regular if it maps any k pairwise distinct points to k linearly independent vectors our main result on k regular maps is the following lower bound for the existence of such maps between euclidean spaces in which k denotes the number of ones in the dyadic expansion of k

preservation of hopf bifurcation for neutral delay differential equations by θ methods this paper considers the preservation of hopf bifurcation of some neural delay differential equations by q method by analyzing the dynamics of the numerical discrete system derived by q method we show that q method could inherit the hopf bifurcation and the asymptotical stability for sufficiently small stepsize h 1 m where m is a positive integer in particular for q 1 2 the result holds for any stepsize h 1 m furthermore the stability of the closed invariant curve is established finally some numerical examples are illustrated to support the analytic results

fractality of multiple colored substitution networks abstract for multiple colored initial graphs we can construct evolving self similar networks by substitution rules in this paper we obtain the fractality of our substitution networks through the estimates on the cardinality of the minimal covering of boxes

a lower bound for the size of syntactically multilinear arithmetic circuits we construct an explicit polynomial f x 1 dots x n with coefficients in 0 1 such that the size of any syntactically multilinear arithmetic circuit computing f is at least omega n 4 3 log 2n the lower bound holds over any field

scale free effect of substitution networks abstract in this paper we construct the growing networks in terms of substitution rule roughly speaking we replace edges of different colors with different initial graphs then the evolving networks are constructed we obtained the free scale effect of our substitution networks

skew flat fibrations a fibration of mathbb r n by oriented copies of mathbb r p is called skew if no two fibers intersect nor contain parallel directions conditions on p and n for the existence of such a fibration were given by ovsienko and tabachnikov a classification of smooth fibrations of mathbb r 3 by skew oriented lines was given by salvai in analogue with the classification of oriented great circle fibrations of s 3 by gluck and warner we show that salvai s classification has a topological variation which generalizes to characterize all continuous fibrations of mathbb r n by skew oriented copies of mathbb r p we show that the space of fibrations of mathbb r 3 by skew oriented lines deformation retracts to the subspace of hopf fibration and therefore has the homotopy type of a pair of disjoint copies of s 2 we discuss skew fibrations in the complex and quaternionic setting and give a necessary condition for the existence of a fibration of mathbb c n respectively mathbb h n by skew oriented copies of mathbb c p respectively mathbb h p

entire spacelike hypersurfaces of constant gaus curvature in minkowski space we prove existence and stability of smooth entire strictly convex spacelike hypersurfaces of prescribed gaus curvature in minkowski space the proof is based on barrier constructions and local a priori estimates

topological properties and fractal analysis of a recurrence network constructed from fractional brownian motions many studies have shown that we can gain additional information on time series by investigating their accompanying complex networks in this work we investigate the fundamental topological and fractal properties of recurrence networks constructed from fractional brownian motions fbms first our results indicate that the constructed recurrence networks have exponential degree distributions the average degree exponent increases first and then decreases with the increase of hurst index h of the associated fbms the relationship between h and can be represented by a cubic polynomial function we next focus on the motif rank distribution of recurrence networks so that we can better understand networks at the local structure level we find the interesting superfamily phenomenon i e the recurrence networks with the same motif rank pattern being grouped into two superfamilies last we numerically analyze the fractal and multifractal properties of recurrence networks we find that the average fractal dimension db of recurrence networks decreases with the hurst index h of the associated fbms and their dependence approximately satisfies the linear formula db 2 h which means that the fractal dimension of the associated recurrence network is close to that of the graph of the fbm moreover our numerical results of multifractal analysis show that the multifractality exists in these recurrence networks and the multifractality of these networks becomes stronger at first and then weaker when the hurst index of the associated time series becomes larger from 0 4 to 0 95 in particular the recurrence network with the hurst index h 0 5 possesses the strongest multifractality in addition the dependence relationships of the average information dimension d 1 and the average correlation dimension d 2 on the hurst index h can also be fitted well with linear functions our results strongly suggest that the recurrence network inherits the basic characteristic and the fractal nature of the associated fbm series

average distances on substitution trees abstract in this paper we construct a deterministic class of evolving self similar trees in terms of initial directed tree using an integral on the self similar fractal we present the average geodesic distance on the fractal which is the limit of renormalized self similar trees we get the value of integral and thus obtain the asymptotic formula of average distances on self similar trees

stable geometric dimension of vector bundles over even dimensional real projective spaces in 6 the geometric dimension of all stable vector bundles over real projective spacep n was determined ifn is even and sufficiently large with respect to the order 2 e of the bundle in f

algebras with polynomial identities and computing the determinant in 1991 nisan proved an exponential lower bound on the size of an algebraic branching program abp that computes the determinant of a matrix in the noncommutative free algebra setting in which there are no nontrivial relationships between the matrix entries by contrast when the matrix entries commute there are polynomial size abps for the determinant this paper extends nisan s result to a much wider class of noncommutative algebras including all nontrivial matrix algebras over any field of characteristic 0 group algebras of all nonabelian finite groups over algebraically closed fields of characteristic 0 the quaternion algebra and the clifford algebras as a result we obtain more compelling evidence for the essential role played by commutativity in the efficient computation of the determinant the key to our approach is a characterization of noncommutative algebras by means of the polynomial identities that they satisfy extending nisan s lower bound framework we find that any reduction in complexity compared to the free algebra must arise from the ability of the identities to reduce the rank of certain naturally associated matrices using results from the theory of algebras with polynomial identities we are able to show that none of the identities of the above classes of algebras is able to achieve such a rank reduction

on families of anticommuting matrices abstract let e 1 e k be complex n n matrices such that e i e j e j e i whenever i j we conjecture that rk e 1 2 rk e 2 2 rk e k 2 o n log n we show that i rk e 1 n rk e 2 n rk e k n o n log n ii if e 1 2 e k 2 0 then k o n iii if e 1 e k have full rank or at least n o n log n then k o log n i implies that the conjecture holds if e 1 2 e k 2 are diagonalisable or if e 1 e k are ii and iii show it holds when their rank is sufficiently large or sufficiently small

self similar scale free networks and disassortativity self similar networks with scale free degree distribution have recently attracted much attention since these apparently incompatible properties were reconciled in c song s havlin and h a makse nature 433 392 2005 by an appropriate box counting method that enters the measurement of the fractal dimension we study two genetic regulatory networks saccharomyces cerevisiae n m luscombe m m babu h yu m snyder s teichmann and m gerstein nature 431 308 2004 and escherichia coli http www ccg unam mx computationalgenomics regulondb datasets regulonnetdatasets html and http www gbf de systemsbiology and show their self similar and scale free features in extension to the datasets studied by c song s havlin and h a makse nature 433 392 2005 moreover by a number of numerical results we support the conjecture that self similar scale free networks are not assortative from our simulations so far these networks seem to be disassortative instead we also find that the qualitative feature of disassortativity is scale invariant under renormalization but it appears as an intrinsic feature of the renormalization prescription as even assortative networks become disassortative after a sufficient number of renormalization steps

the use of boundary locus plots in the identification of bifurcation points in numerical approximation of delay differential equations abstract we are interested in nonlinear delay differential equations which have a hopf bifurcation we assume zero is a steady state for the problem and so a hopf bifurcation point lies on the boundary of the region of asymptotic stability for the zero solution we investigate whether discrete versions of the nonlinear delay differential equation also exhibit hopf bifurcations we use the boundary locus method as a tool both for the delay differential equation and for numerical analogues we use the information obtained about the stability domain to assist in identifying hopf bifurcations we demonstrate the following for certain linear multistep methods the boundary of the region of stability for the zero solution of the original equation is approximated by the boundary of the region of stability for the zero solution of the numerical analogue equation to the order of the method the boundary locus method enables us to determine precise parameter values at which any hopf bifurcations arise in the discrete equations we prove that hopf bifurcation points for the true equation are approximated to the order of the method by corresponding points in the discrete scheme further calculations are necessary to determine the precise nature of bifurcation points identified in this way

numerical hopf bifurcation of linear multistep methods for a class of delay differential equations abstract in this paper we consider the discretization of parameter dependent delay differential equation of the form y t f y t y t 1 0 y r d it is shown that if the delay differential equation undergoes a hopf bifurcation at then the discrete scheme undergoes a hopf bifurcation at h o h p for sufficiently small step size h where p 1 is the order of the strictly stable linear multistep method the direction of numerical hopf bifurcation and stability of bifurcating invariant curve are the same as that of the corresponding delay differential equation

the laplacian spectrum and average trapping time for weighted dyson hierarchical network abstract in this paper we consider the weighted dyson hierarchical network that is a weighted fully connected network where the pattern of weights is ruled by a weight factor given that the laplacian operator is intrinsically implied in the analysis of dynamic processes occurring on complex networks laplacian spectrum allows addressing analytically a large class of problems exploiting the deterministic recursive structure we are able to derive explicitly all eigenvalues and their corresponding eigenvectors of laplacian matrix further we derive exact solutions of the trapping time tt and average trapping time att on the weighted dyson hierarchical network with weight dependent walk the obtained results show that tt and att are related to the weight factor of the weighted dyson hierarchical network

periodic homotopy groups of exceptional lie groups torsion free cases the v 1 periodic homotopy groups v 1 751 x p are computed explicitly for all pairs x p where x is an exceptional lie group whose integral homology has no p torsion this yields new lower bounds for p exponents of actual homotopy groups of these spaces delicate calculations with the unstable novikov spectral sequence are required in the proof

geometric fractal growth model for scale free networks we introduce a deterministic model for scale free networks whose degree distribution follows a power law with the exponent ensuremath gamma at each time step each vertex generates its offspring whose number is proportional to the degree of that vertex with proportionality constant m ensuremath 1 mg1 we consider the two cases first each offspring is connected to its parent vertex only forming a tree structure second it is connected to both its parent and grandparent vertices forming a loop structure we find that both models exhibit power law behaviors in their degree distributions with the exponent ensuremath gamma 1 mathrm ln 2m ensuremath 1 mathrm ln m thus by tuning m the degree exponent can be adjusted in the range 2l ensuremath gamma l3 we also solve analytically a mean shortest path distance d between two vertices for the tree structure showing the small world behavior that is d ensuremath sim mathrm ln n mathrm ln overline k where n is system size and overline k is the mean degree finally we consider the case that the number of offspring is the same for all vertices and find that the degree distribution exhibits an exponential decay behavior

generalized adjacency and laplacian spectra of the weighted corona graphs abstract recently corona product graphs have attracted a great deal of attention from scientific communities such as coding theory frequency assignment mathematical chemistry meanwhile weight factor gradually became a research hotpoint because of closing to reality hence we introduce the weighted corona graphs in this paper the weighted corona product graphs are the in depth and promotion of the corona product graphs linking the spectra between the initial graph and the product graph then we deduce the complete information of generalized adjacency and laplacian eigenvalues of the weighted product graph with different structures and we generalize the results obtained previously into product corona graph with m 1 iterations

introducing totally nonparallel immersions abstract an immersion of a smooth n dimensional manifold m r q is called totally nonparallel if for every distinct x y m the tangent spaces at f x and f y contain no parallel lines equivalently they span a 2n dimensional space given a manifold m we seek the minimum dimension tn m such that there exists a totally nonparallel immersion m r tn m in analogy with the totally skew embeddings studied by ghomi and tabachnikov we find that totally nonparallel immersions are related to the generalized vector field problem the immersion and embedding problems for real projective spaces and nonsingular symmetric bilinear maps our study of totally nonparallel immersions follows a recent trend of studying conditions which manifest on the configuration space f k m of k tuples of distinct points of m for example k regular embeddings k skew embeddings k neighborly embeddings and several others typically a map satisfying one of these configuration space conditions induces some s k equivariant map on the configuration space f k m or on a bundle thereof and obstructions can be computed in the form of stiefel whitney classes however the existence problem for such conditions is relatively unstudied our main result is a whitney type theorem every smooth n manifold m admits a totally nonparallel immersion into r 4 n 1 one dimension less than given by genericity we begin by studying the local problem which requires a thorough understanding of the space of nonsingular symmetric bilinear maps after which the main theorem is established using the removal of singularities h principle technique due to gromov and eliashberg when combined with a recent non immersion theorem of davis we obtain the exact value tn r p n 4 n 1 when n is a power of 2 this is the first optimal dimension result for any closed manifold m besides s 1 for any of the recently studied configuration space conditions

on the nonimmersion of products of real projective spaces in this paper we utilize bp a generalized cohomology theory associated with the brown peterson spectrum to prove a nonimmersion theorem for products of real projective spaces 1 statement of results let v 2a 2b 1 a and let a n denote the number of l s in the binary expansion of n let pn denote the n dimensional real projective space then our main theorem is theorem 1 1 if d and k are positive integers with a m 1 a mk d k and 2v m d 2k j 1 for each i then p2mi 2d p2m2 4d x xp 2kd cannot be immersed in r4 mi mk 2d k i to our knowledge the only published works on nonimmersions of products of real projective spaces are those of suzuki 18 and kobayashi 12 their methods seem to be effective for pm x pn with m 2r1 and n 2s 1 when s r 7 their nonimmersions are in dimensions m n 2s 1 see 12 corollary 4 we can apply theorem 1 1 to the case ml 2 2 2r 3 and 2 s3 m2 2s 2 when r 3 and s 3 are greater than 14 and we have k 2 and d 2 this manifold is a submanifold of theirs and it fails to immerse indimension m 2s i 2r5 which is an improvement of 2r 5 dimensions a more detailed analysis 17 of this special case lowers the above 14 to 4 from known immersions of single real projective spaces e g adem 2 lam 1 3 cohen 7 and randall 1 5 we see that nonimmersions in theorem 1 1 received by the editors july 18 1988 1980 mathematics subject classification 1985 revision primary 57r42 55n20 kei words and phrases real projective space bp theory euler class immersion the first author was partially supported by kosef the second author was partially supported by the n s f d 1990 american mathematical society 0002 9947 90 1 00 25 per page

numerical hopf bifurcation for a class of delay differential equations abstract in this paper we consider discretization of parameter dependent delay differential equations of the form x t f x t x t r we show that if the delay differential equation undergoes a hopf bifurcation then the discrete scheme undergoes a hopf bifurcation of the same type the results of this paper extend the results of our previous analysis relating to the discretization of the delay logistic equation to a wider class of problems

scale free trees the skeletons of complex networks we investigate the properties of the spanning trees of various real world and model networks the spanning tree representing the communication kernel of the original network is determined by maximizing the total weight of the edges whose weights are given by the edge betweenness centralities we find that a scale free tree and shortcuts organize a complex network especially in ubiquitous scale free networks it is found that the scale free spanning tree shows very robust betweenness centrality distributions and the remaining shortcuts characterize the properties of the original network such as the clustering coefficient and the classification of scale free networks by the betweenness centrality distribution

fractality in complex networks critical and supercritical skeletons fractal scaling char22 a power law behavior of the number of boxes needed to tile a given network with respect to the lateral size of the box char22 is studied we introduce a box covering algorithm that is a modified version of the original algorithm introduced by song et al nature london 433 392 2005 this algorithm enables easy implementation fractal networks are viewed as comprising a skeleton and shortcuts the skeleton embedded underneath the original network is a special type of spanning tree based on the edge betweenness centrality it provides a scaffold for the fractality of the network when the skeleton is regarded as a branching tree it exhibits a plateau in the mean branching number as a function of the distance from a root for nonfractal networks on the other hand the mean branching number decays to zero without forming a plateau based on these observations we construct a fractal network model by combining a random branching tree and local shortcuts the scaffold branching tree can be either critical or supercritical depending on the small worldness of a given network for the network constructed from the critical supercritical branching tree the average number of vertices within a given box grows with the lateral size of the box according to a power law an exponential form in the cluster growing method the critical and supercritical skeletons are observed in protein interaction networks and the world wide web respectively the distribution of box masses i e the number of vertices within each box follows a power law p m m ensuremath sim m ensuremath ensuremath eta the exponent ensuremath eta depends on the box lateral size ensuremath ell b for small values of ensuremath ell b ensuremath eta is equal to the degree exponent ensuremath gamma of a given scale free network whereas ensuremath eta approaches the exponent ensuremath tau ensuremath gamma ensuremath gamma ensuremath 1 as ensuremath ell b increases which is the exponent of the cluster size distribution of the random branching tree finally we study the perimeter h ensuremath alpha of a given box ensuremath alpha i e the number of edges connected to different boxes from a given box ensuremath alpha as a function of the box mass m b ensuremath alpha it is obtained that the average perimeter over the boxes with box mass m b is likely to scale as h m b ensuremath sim m b irrespective of the box size ensuremath ell b

numerical dynamics of a nonstandard finite difference method for a class of delay differential equations abstract this paper proposes a nonstandard finite difference nsfd method for a class of delay differential equations we prove that for any step size h 1 m m z this numerical method could intrinsically preserve the qualitative behavior of the dynamical system including the local stability of equilibrium the existence and the direction of hopf bifurcation and the stability of bifurcating periodic solution finally to illustrate the analytic results the nsfd method is used for a model of the survival of red blood cells in animals

statistical ensemble of scale free random graphs a thorough discussion of the statistical ensemble of scale free connected random tree graphs is presented methods borrowed from field theory are used to define the ensemble and to study analytically its properties the ensemble is characterized by two global parameters the fractal and the spectral dimensions which are explicitly calculated it is discussed in detail how the geometry of the graphs varies when the weights of the nodes are modified the stability of the scale free regime is also considered when it breaks down either a scale is spontaneously generated or else a singular node appears and the graphs become crumpled a new computer algorithm to generate these random graphs is proposed possible generalizations are also discussed in particular more general ensembles are defined along the same lines and the computer algorithm is extended to arbitrary degenerate scale free random graphs

tree networks with causal structure a geometry of networks endowed with a causal structure is discussed using the conventional framework of the equilibrium statistical mechanics the popular growing network models appear as particular causal models we focus on a class of tree graphs an analytically solvable case general formulas are derived describing the degree distribution the ancestor descendant correlation and the probability that a randomly chosen node lives at a given geodesic distance from the root it is shown that the hausdorff dimension d h of the causal networks is generically infinite in contrast to the maximally random trees where it is generically finite

effect of disorder strength on optimal paths in complex networks we study the transition between the strong and weak disorder regimes in the scaling properties of the average optimal path opt in a disordered erd s renyi er random network and scale free sf network each link i is associated with a weight ti expsarid where ri is a random number taken from a uniform distribution between 0 and 1 and the parameter a controls the strength of the disorder we find that for any finitea there is a crossover network size n sad at which the transition occurs for n n sad the scaling behavior of opt is in the strong disorder regime with opt n 1 3 for er networks and for sf networks with lo 4 and opt n sl 3d sl 1d for sf networks with 3 l 4 for n n sad the scaling behavior is in the weak disorder regime with opt ln n for er networks and sf networks with l 3 in order to study the transition we propose a measure which indicates how close or far the disordered network is from the limit of strong disorder we propose a scaling ansatz for this measure and demonstrate its validity we proceed to derive the scaling relation between n sad and a we find thatn sa d a 3 for er networks and for sf networks with lo 4 and n sa d a sl 1d sl 3d for sf networks with 3 l 4

a review of fractality and self similarity in complex networks we review recent findings of self similarity in complex networks using the box covering technique it was shown that many networks present a fractal behavior which is seemingly in contrast to their small world property moreover even non fractal networks have been shown to present a self similar picture under renormalization of the length scale these results have an important effect in our understanding of the evolution and behavior of such systems a large number of network properties can now be described through a set of simple scaling exponents in analogy with traditional fractal theory

shadows and convexity of surfaces we study the geometry and topology of immersed surfaces in euclidean 3 space whose gauss map satisfies a certain two piece property and solve the shadow problem formulated by h wente

the statistical geometry of scale free random trees the properties of random trees galton watson trees with scale free power like probability distribution of coordinations are investigated in the thermodynamic limit the scaling form of volume probability is found and the connectivity dimensions are determined and compared with other exponents which describe the growth the local spectral dimension is also determined through the study of the massless limit of the gaussian model on such trees

normal forms of non resonance and weak resonance double hopf bifurcation in the retarded functional differential equations and applications the norm forms of the double hopf bifurcation for a general dde is established the van der pol duffing oscillator with delayed position and velocity feedback is investigated the dynamical classification near the double hopf bifurcation point is studied the stable periodic doubling solutions and unstable three dimensional torus are found in this paper we firstly present the general framework of calculation of normal forms of non resonance and weak resonance double hopf bifurcation for the general retarded functional differential equations by using the normal form theory of delay differential equations due to faria and magalh a es then the dynamical behavior of van der pol duffing oscillator with delayed position and velocity feedback is considered specifically the dynamical classification near the double hopf bifurcation point is investigated by analyzing the obtained normal form finally the numerical simulations support the theoretical results and present some interesting phenomena

topological obstructions to totally skew embeddings following ghomi and tabachnikov s 2008 work we study the invariant n m n defined as the smallest dimension n such that there exists a totally skew embedding of a smooth manifold m n in r n this problem is naturally related to the question of estimating the geometric dimension of the stable normal bundle of the configuration space f 2 m n of ordered pairs of distinct points in m n we demonstrate that in a number of interesting cases the lower bounds on n m n obtained by this method are quite accurate and very close to the best known general upper bound n m n 1 n m n 4n 2 alpha n 1