a numerical study of the 3 periodic wave solutions to kdv type equations abstract in this paper by using the direct method of calculating periodic wave solutions proposed by akira nakamura we present a numerical process to calculate the 3 periodic wave solutions to several kdv type equations the korteweg de vries equation the sawada koterra equation the boussinesq equation the ito equation the hietarinta equation and the 2 1 dimensional kadomtsev petviashvili equation some detailed numerical examples are given to show the existence of the three periodic wave solutions numerically

on schur flows for the finite schur dmkdv flows a non local poisson structure is introduced and shown to be linked via backlund darboux transformations to linear and quadratic poisson structures for the toda lattice two different lax representations for the schur flows are used one to construct backlund darboux transformations and the other to solve the cauchy problem via the trigonometric moment problem

scaling of the average weighted receiving time on a family of double weighted hierarchical networks abstract in this paper we introduce a family of double weighted hierarchical networks which is depended on the initial complete bipartite graph and two weight factors r w 0 r 1 0 w 1 then we consider the biased walk in the double weighted hierarchical networks what is more we define two weighted times the mean weighted first passing time mwfpt and the average weighted receiving time awrt according to the definition we calculate the exact expressions of the mwfpt and the awrt on the double weighted hierarchical networks the expressions we obtain in this paper illustrate that the awrt tends to a constant when the size of the networks increases while if n 2 4 n 1 n 2 1 w 2 where n 1 and n 2 denote the number of nodes in initial complete bipartite graph and n n 1 n 2 holds the awrt grows as a power law function of network order with the exponent denoted by log n n k

exact solution for mean first passage time on a pseudofractal scale free web the explicit determinations of the mean first passage time mfpt for trapping problem are limited to some simple structure e g regular lattices and regular geometrical fractals and determining mfpt for random walks on other media especially complex real networks is a theoretical challenge in this paper we investigate a simple random walk on the the pseudofractal scale free web psfw with a perfect trap located at a node with the highest degree which simultaneously exhibits the remarkable scale free and small world properties observed in real networks we obtain the exact solution for the mfpt that is calculated through the recurrence relations derived from the structure of psfw the rigorous solution exhibits that the mfpt approximately increases as a power law function of the number of nodes with the exponent less than 1 we confirm the closed form solution by direct numerical calculations we show that the structure of psfw can improve the efficiency of transport by diffusion compared with some other structure such as regular lattices sierpinski fractals and t graph the analytical method can be applied to other deterministic networks making the accurate computation of mfpt possible

mapping between hopping on hierarchical structures and diffusion on a family of fractals the authors present a simple analytical approach to studying diffusion on hierarchical and fractal structures they show that hopping on one dimensional hierarchical structures can be mapped onto diffusion on a family of loopless fractals

global mean first passage times of random walks on complex networks we present a general framework applicable to a broad class of random walks on complex networks which provides a rigorous lower bound for the mean first passage time of a random walker to a target site averaged over its starting position the so called global mean first passage time gmfpt this bound is simply expressed in terms of the equilibrium distribution at the target and implies a minimal scaling of the gmfpt with the network size we show that this minimal scaling which can be arbitrarily slow is realized under the simple condition that the random walk is transient at the target site and independently of the small world scale free or fractal properties of the network last we put forward that the gmfpt to a specific target is not a representative property of the network since the target averaged gmfpt satisfies much more restrictive bounds

schur flow for orthogonal polynomials on the unit circle and its integrable discretization a one parameter deformation of the measure of orthogonality for orthogonal polynomials on the unit circle is considered the corresponding dynamics of the schur parameters of the orthogonal polynomials is shown to be characterized by the complex semi discrete modified kdv equation namely the schur flow a discrete analogue of the miura transformation is found an integrable discretization of the schur flow enables us to compute a pade approximation of the caratheodory functions or equivalently to compute a perron caratheodory continued fraction in a polynomial time

multifractal analysis and topological properties of a new family of weighted koch networks abstract weighted complex networks especially scale free networks which characterize real life systems better than non weighted networks have attracted considerable interest in recent years studies on the multifractality of weighted complex networks are still to be undertaken in this paper inspired by the concepts of koch networks and koch island we propose a new family of weighted koch networks and investigate their multifractal behavior and topological properties we find some key topological properties of the new networks their vertex cumulative strength has a power law distribution there is a power law relationship between their topological degree and weight strength the networks have a high weighted clustering coefficient of 0 41004 which is independent of the scaling factor c in the limit of large generation t the second smallest eigenvalue 2 and the maximum eigenvalue n are approximated by quartic polynomials of the scaling factor c for the general laplacian operator while 2 is approximately a quartic polynomial of c and n 1 5 for the normalized laplacian operator then we find that weighted koch networks are both fractal and multifractal their fractal dimension is influenced by the scaling factor c we also apply these analyses to six real world networks and find that the multifractality in three of them are strong

on the minimal polynomial of gauss periods for prime powers for a positive integer m set m exp 2 i m and let z m denote the group of reduced residues modulo m fix a congruence group h of conductor m and of order f choose integers t 1 t e to represent the e m f cosets of h in z m the gauss periods j t j x m x h 1 j e corresponding to h are conjugate and distinct over q with minimal polynomial g x x e c 1 x e 1 c e 1 x c e to determine the coefficients of the period polynomial g x or equivalently its reciprocal polynomial g x x e g x 1 is a classical problem dating back to gauss previous work of the author and gupta and zagier primarily treated the case m p an odd prime with f 1 fixed in this setting it is known for certain integral power series a x and b x that for any positive integer n g x a x b x p 1 f mod x n holds in z x for all primes p 1 mod f except those in an effectively determinable finite set here we describe an analogous result for the case m p a prime power 1 the methods extend for odd prime powers p to give a similar result for certain twisted gauss periods of the form j i p x h t j x p t j x p 1 j e where p denotes the usual legendre symbol and i i p 1 2 4

algebro geometric solutions to the modified sawada kotera hierarchy based on solving the lenard recursion equation and the zero curvature equation we derive the modified sawada kotera sk hierarchy associated with a 3 3 matrix spectral problem resorting to the characteristic polynomial of lax matrix for the modified sk hierarchy we introduce a trigonal curve km 1 and present the corresponding baker akhiezer function and meromorphic function on it the abel map is introduced to straighten out the modified sk flows with the aid of the property of the baker akhiezer function and the meromorphic function and their asymptotic expansions we arrive at their explicit riemann theta function representations algebro geometric solutions of the entire modified sk hierarchy are obtained by using asymptotic expansion of the meromorphic function and its riemann theta function representation as an application we give two simple examples

mean first passage times for two biased walks on the weighted rose networks abstract compared with traditional random walk biased walks have been studied extensively over the past decade especially in the transport and communication networks in this paper we first introduce the weighted rose networks then for the weighted rose networks we focus on two biased walks maximal entropy walk and weight dependent walk and obtain the exact expressions of their stationary distributions and mean first passage times finally we find that the average receiving time for maximal entropy walk is a quadratic function of the weight parameter r while the average receiving time for weighted dependent walk is a linear function of the weight parameter r meanwhile for the maximal entropy walk the smaller the value of r is the more efficient the trapping process is for the weighted dependent walk the larger the value of r r r 0 2 6 is the more efficient for the weighted dependent walk the smaller the value of r r r 0 2 6 is the more efficient for the weight dependent walk

the positive and negative camassa holm γ hierarchies zero curvature representations bi hamiltonian structures and algebro geometric solutions in this paper we extend the camassa holm ch equation to two different hierarchies of nonlinear evolution equations by resorting to two lenard recursion sequences one is of positive order ch hierarchy including a harry dym type equation while the other is of negative order ch hierarchy of integrodifferential equations with nonzero integration constants including the ch equation we provide the zero curvature representations for both hierarchies by solving a key matrix equation moreover the lenard sequences are used to construct a lax matrix that satisfies a stationary zero curvature equation which enables us to establish bi hamiltonian structures for both hierarchies by applying trace identity in addition the lenard sequence and lax pair are used to systematically derive the its matveev trace formula and the dubrovin type equations associated with the ch equation the hyperelliptic curve and abel jacobi coordinates are then introduced to linearize the associated flow from which the algebro geometric solutions to the ch equation are constructed by using standard jacobi inversion technique in this paper we extend the camassa holm ch equation to two different hierarchies of nonlinear evolution equations by resorting to two lenard recursion sequences one is of positive order ch hierarchy including a harry dym type equation while the other is of negative order ch hierarchy of integrodifferential equations with nonzero integration constants including the ch equation we provide the zero curvature representations for both hierarchies by solving a key matrix equation moreover the lenard sequences are used to construct a lax matrix that satisfies a stationary zero curvature equation which enables us to establish bi hamiltonian structures for both hierarchies by applying trace identity in addition the lenard sequence and lax pair are used to systematically derive the its matveev trace formula and the dubrovin type equations associated with the ch equation the hyperelliptic curve and abel jacobi coordinates are then introduced to linearize the associated flow from which the a

a unified construction for the algebro geometric quasiperiodic solutions of the lotka volterra and relativistic lotka volterra hierarchy in this paper a new type of integrable differential difference hierarchy namely the generalized relativistic lotka volterra grlv hierarchy is introduced this hierarchy is closely related to lotka volterra lattice and relativistic lotka volterra lattice which allows us to provide a unified and effective way to obtain some exact solutions for both the lotka volterra hierarchy and the relativistic lotka volterra hierarchy in particular we shall construct algebro geometric quasiperiodic solutions for the lv hierarchy and the rlv hierarchy in a unified manner on the basis of the finite gap integration theory

scaling of average weighted shortest path and average receiving time on weighted hierarchical networks abstract recent work on the networks has focused on the weighted hierarchical networks that are significantly different from the un weighted hierarchical networks in this paper we study a family of weighted hierarchical networks which are recursively defined from an initial uncompleted graph in which the weights of edges have been assigned to different values with certain scale firstly we study analytically the average weighted shortest path awsp on the weighted hierarchical networks using a recursive method we determine explicitly the awsp the obtained rigorous solution shows that the networks grow unbounded but with the logarithm of the network size while the weighted shortest paths stay bounded then depending on a biased random walk we research the mean first passage time mfpt between a hub node and any peripheral node finally we deduce the analytical expression of the average of mfpts for a random walker originating from any node to first visit to a hub node which is named as the average receiving time art the obtained result shows that art is bounded or grows sublinearly with the network order relating to the number of initial nodes and the weighted factor or grows quadratically with the iteration

random walks on deterministic scale free networks exact results we study the random walk problem on a class of deterministic scale free networks displaying a degree sequence for hubs scaling as a power law with an exponent gamma log 3 log 2 we find exact results concerning different first passage phenomena and in particular we calculate the probability of first return to the main hub these results allow to derive the exact analytic expression for the mean time to first reach the main hub whose leading behavior is given by tau approximately v1 1 gamma where v denotes the size of the structure and the mean is over a set of starting points distributed uniformly over all the other sites of the graph interestingly the process turns out to be particularly efficient we also discuss the thermodynamic limit of the structure and some local topological properties

time delayed feedback control of a flashing ratchet closed loop or feedback control ratchets use information about the state of the system to operate with the aim of maximizing the performance of the system in this paper we investigate the effects of a time delay in the feedback for a protocol that performs an instantaneous maximization of the center of mass velocity for the one and the few particle cases the flux decreases with increasing delay as an effect of the decorrelation of the present state of the system with the information that the controller uses but the delayed closed loop protocol succeeds to perform better than its open loop counterpart provided the delays are smaller than the characteristic times of the brownian ratchet for the many particle case we also show that for small delays the center of mass velocity decreases for increasing delays however for large delays we find the surprising result that the presence of the delay can improve the performance of the nondelayed feedback ratchet and the flux can attain the maximum value obtained with the optimal periodic protocol this phenomenon is the result of the emergence of a dynamical regime where the presence of the delayed feedback stabilizes one quasiperiodic solution or several multistability which resemble the solutions obtained in the so called threshold protocol our analytical and numerical results point towards the feasibility of an experimental implementation of a feedback controlled ratchet that performs equal or better than its optimal open loop version

the modified box dimension and average weighted receiving time of the weighted hierarchical graph abstract in this paper we study the weighted hierarchical graph which is generated from bipartite graph with n n 1 n 2 vertices in which the weights of edges have been assigned to different values with certain scale firstly we introduce the definition of the modified box dimension then for the weighted hierarchical graph we deduce the modified box dimension dim m b g n n n log r n depending on the weighted factor r and the number n of copies secondly we mainly study their two average weighted receiving times awrts t i n and t ii n of the weighted hierarchical graph on random walk we discuss two cases in the case of n 1 n 2 r n 2 n 1 we deduce both awrts grow as a power law function of the network size v g n with the postive exponent represented by log n n n 1 n 2 or log n r 1 dim m b g n n n which means that the bigger the value of the modified box dimension is the slower the process of receiving information is in the case of n 1 n 2 r n 2 n 1 both awrts tend to constant if n n 1 n 2 the awrts grow with increasing order as log n v g n if n n 1 n 2 and both awrts grow as a power law function of the network size v g n with the exponent represented by log n n n 1 n 2 0 if n n 1 n 2

on the quasi periodic solutions to the discrete nonlinear schrodinger equation the authors have obtained the quasi periodic solutions to the discrete non linear schrodinger equation by a variant of a method due to date and tanaka 1976 it is shown explicitly that the non linear field variables at the different lattice points can be determined in a recursive fashion in terms of combinations of riemann s theta functions depending on lattice position and time

domination number and minimum dominating sets in pseudofractal scale free web and sierpinski graph abstract the minimum dominating set mds problem is a fundamental subject of theoretical computer science and has found vast applications in different areas including sensor networks protein interaction networks and structural controllability however the determination of the size of a mds and the number of all mdss in a general network is np hard and it thus makes sense to seek particular graphs for which the mds problem can be solved analytically in this paper we study the mds problem in the pseudofractal scale free web and the sierpinski graph which have the same number of vertices and edges for both networks we determine explicitly the domination number as well as the number of distinct mdss we show that the pseudofractal scale free web has a unique mds and its domination number is only half of that for the sierpinski graph which has many mdss we argue that the scale free topology is responsible for the difference of the size and number of mdss between the two studied graphs which in turn indicates that power law degree distribution plays an important role in the mds problem and its applications in scale free networks

tutte polynomial of pseudofractal scale free web the tutte polynomial of a graph is a 2 variable polynomial which is quite important in both combinatorics and statistical physics it contains various numerical invariants and polynomial invariants such as the number of spanning trees the number of spanning forests the number of acyclic orientations the reliability polynomial chromatic polynomial and flow polynomial in this paper we study and obtain a recursive formula for the tutte polynomial of pseudofractal scale free web psfw and thus logarithmic complexity algorithm to calculate the tutte polynomial of the psfw is obtained although it is np hard for general graph by solving the recurrence relations derived from the tutte polynomial the rigorous solution for the number of spanning trees of the psfw is obtained therefore an alternative approach to determine explicitly the number of spanning trees of the psfw is given furthermore we analyze the all terminal reliability of the psfw and compare the results with those of the sierpinski gasket which has the same number of nodes and edges as the psfw in contrast with the well known conclusion that inhomogeneous networks e g scale free networks are more robust than homogeneous networks i e networks in which each node has approximately the same number of links with respect to random deletion of nodes the sierpinski gasket which is a homogeneous network as our results show is more robust than the psfw which is an inhomogeneous network with respect to random edge failures

average receiving scaling of the weighted polygon koch networks with the weight dependent walk based on the weighted koch networks and the self similarity of fractals we present a family of weighted polygon koch networks with a weight factor r 0 r 1 we study the average receiving time art on weight dependent walk i e the walker moves to any of its neighbors with probability proportional to the weight of edge linking them whose key step is to calculate the sum of mean first passage times mfpts for all nodes absorpt at a hub node we use a recursive division method to divide the weighted polygon koch networks in order to calculate the art scaling more conveniently we show that the art scaling exhibits a sublinear or linear dependence on network order thus the weighted polygon koch networks are more efficient than expended koch networks in receiving information finally compared with other previous studies results i e koch networks weighted koch networks we find out that our models are more general

finite genus solutions for the ablowitz ladik hierarchy the question of constructing the finite genus quasiperiodic solutions for the ablowitz ladik hierarchy alh is considered by establishing relations between the alh and fay s identity for the functions it is shown that using a limiting procedure one can derive from the latter an infinite number of differential identities which can be arranged as an infinite set of differential difference equations coinciding with the equations of the alh and that the original fay s identity can be rewritten in a form similar to the functional equations representing the alh which have been derived in the previous works of the author this provides an algorithm for obtaining some class of quasiperiodic solutions for the alh which can be viewed as an alternative to the inverse scattering transform or the algebro geometrical approach

explicit solutions for a hierarchy of differential difference equations a new discrete isospectral problem and the corresponding hierarchy of nonlinear differential difference equations are proposed on the basis of the theory of algebraic curves the continuous flow and discrete flow related to the hierarchy of differential difference equations are straightened out using the abel jacobi coordinates the meromorphic function and the baker akhiezer function are introduced on the hyperelliptic curve quasi periodic solutions of the corresponding hierarchy of differential difference equations are constructed with the help of the asymptotic properties and the algebro geometric characters of the meromorphic function the baker akhiezer function and the hyperelliptic curve

rotation number associated with difference equations satisfied by polynomials orthogonal on the unit circle abstract the techniques of topological dynamics and differential dynamical systems are used to study polynomials orthogonal with respect to a measure supported on the unit circle it is assumed that the reflection coefficients associated with these polynomials form a stationary stochastic ergodic process in particular the techniques mentioned above are used to prove a gap labelling result

factorization of the tenth fermat number we describe the complete factorization of the tenth fermat number f10 by the elliptic curve method ecm f10 is a product of four prime factors with 8 10 40 and 252 decimal digits the 40 digit factor was found after about 140 mflop years of computation we also discuss the complete factorization of other fermat numbers by ecm and summarize the factorizations

periodic solutions of the camassa holm equation based on the bilinear form since the famous work of camassa and holm various analytic techniques have been used to find the exact solutions of camassa holm ch equation and many interesting results have been found by the different methods in this paper we first give an extended bilinear form of the ch equation which contains four more free parameters than those in the results proposed in parkers finding for soliton solutions thus we construct the smooth periodic solution of the ch equation in a parametric form based on the previous work of nakumura we also show that the periodic solution degenerates into the smooth solitary wave solution by limiting behavior of is the parameter appeared from the riemann theta function

explicit determination of mean first passage time for random walks on deterministic uniform recursive trees the determination of mean first passage time mfpt for random walks in networks is a theoretical challenge and is a topic of considerable recent interest within the physics community in this paper according to the known connections between mfpt effective resistance and the eigenvalues of graph laplacian we first study analytically the mfpt between all node pairs of a class of growing treelike networks which we term deterministic uniform recursive trees durts since one of its particular cases is a deterministic version of the famous uniform recursive tree the interesting quantity is determined exactly through the recursive relation of the laplacian spectra obtained from the special construction of durts the analytical result shows that the mfpt between all couples of nodes in durts varies as n text text ln text n for large networks with node number n second we study trapping on a particular network of durts focusing on a special case with the immobile trap positioned at a node having largest degree we determine exactly the average trapping time att that is defined as the average of fpt from all nodes to the trap in contrast to the scaling of the mfpt the leading behavior of att is a linear function of n interestingly we show that the behavior for att of the trapping problem is related to the trapping location which is in comparison with the phenomenon of trapping on fractal t graph although both networks exhibit tree structure finally we believe that the methods could open the way to exactly calculate the mfpt and att in a wide range of deterministic media

inverse problem for polynomials orthogonal on the unit circle polynomials orthogonal on the unit circle with random recurrence coefficients and finite band spectrum are investigated it is shown that the coefficients are in fact quasi periodic the measures associated with these quasi periodic coefficients are exhibited and necessary and sufficient conditions relating quasi periodicity and spectral measures of this type are given analogs for polynomials orthogonal on subsets of the real line are also presented

weighted fractal networks abstract in this paper we define a new class of weighted complex networks sharing several properties with fractal sets and whose topology can be completely analytically characterized in terms of the involved parameters and of the fractal dimension general networks with fractal or hierarchical structures can be set in the proposed framework that moreover could be used to provide some answers to the widespread emergence of fractal structures in nature

analysis of diffusion and trapping efficiency for random walks on non fractal scale free trees abstract in this paper the discrete random walks on the recursive non fractal scale free trees nfsft are studied and a kind of method to calculate the analytic solutions of the mean first passage time mfpt for any pair of nodes the mean trapping time mtt for any target node and mean diffusing time mdt for any starting node are proposed furthermore we compare the trapping efficiency and diffusion efficiency between any two nodes of nfsft by using the mtt and the mdt as the measures of trapping efficiency and diffusion efficiency respectively and find the best or worst trapping sites and the best or worst diffusion sites the results show that the two hubs of nfsft are not only the best trapping site but also the worst diffusion site and that the nodes which are the farthest nodes from the two hubs are not only the worst trapping sites but also the best diffusion sites furthermore we find that the ratio between the maximum and minimum of mtt grows logarithmically with network order but the ratio between the maximum and minimum of mdt is almost equal to 1 the results imply that the trap s position has great effect on the trapping efficiency but the position of starting node has little effect on diffusion efficiency finally the simulation for random walks on nfsft is done and it is consistent with the derived results

scaling of average weighted shortest path and average receiving time on the weighted cayley networks abstract in this paper we mainly study two important properties of weighted cayley networks depending on a weight factor r the weight factor makes it more difficult to calculate the topological characteristics such as average weighted shortest path awsp and average receiving time art of the networks we first calculate some basic network measurements such as average degree and average node strength then we derive the expression for awsp and show that it stays bounded with network order growing 0 r 1 in the limit of large network order at last we focus on random walks and trapping issue on the networks that is we calculate the art in more detail for 0 r 1 2 the art grows with increasing size n t as ln n t for r 1 2 the art grows with increasing size n t as ln 2 n t for 1 2 r 1 the art grows sublinearly with the network size n t for r 1 the art grows linearly with the network size n t

scaling of average receiving time and average weighted shortest path on weighted koch networks abstract in this paper we present weighted koch networks based on classic koch networks a new method is used to determine the average receiving time art whose key step is to write the sum of mean first passage times mfpts for all nodes to absorption at the trap located at a hub node as a recursive relation we show that the art exhibits a sublinear or linear dependence on network order thus the weighted koch networks are more efficient than classic koch networks in receiving information moreover average weighted shortest path awsp is calculated in the infinite network order limit the awsp depends on the scaling factor the weighted koch network grows unbounded but with the logarithm of the network size while the weighted shortest paths stay bounded

average weighted receiving time on the non homogeneous double weighted fractal networks abstract in this paper based on actual road networks a model of the non homogeneous double weighted fractal networks is introduced depending on the number of copies s and two kinds of weight factors w i r i i 1 2 s the double weights represent the capacity flowing weights and the cost traveling weights respectively denote by w i j f the capacity flowing weight connecting the nodes i and j and denote by w i j c the cost traveling weight connecting the nodes i and j let w i j f be related to the weight factors w 1 w 2 w s and let w i j c be related to the weight factors r 1 r 2 r s assuming that the walker at each step starting from its current node moves to any of its neighbors with probability proportional to the capacity flowing weight of edge linking them the weighted time for two adjacency nodes is the cost traveling weight connecting the two nodes the average weighted receiving time awrt is defined on the non homogeneous double weighted fractal networks awrt depends on the relationships of the number of copies s and two kinds of weight factors w i r i i 1 2 s the obtained remarkable results display that in the large network the awrt grows as a power law function of the network size n g with the exponent represented by log s w 1 r 1 w 2 r 2 w s r s 1 when w 1 r 1 w 2 r 2 w s r s 1 which means that the smaller the value of w 1 r 1 w 2 r 2 w s r s is the more efficient the process of receiving information is especially when w 1 r 1 w 2 r 2 w s r s 1 awrt grows with increasing order n g as log n g or log n g 2 in the classic fractal networks the average receiving time art grows with linearly with the network size n g thus the non homogeneous double weighted fractal networks are more efficient than classic fractal networks in term of receiving information

fractal models for diffusion controlled aggregation a two dimensional fractal model is constructed for diffusion controlled deposition on a surface the fractal geometry of the deposit and the power law behaviour of the quantities characterising the non equilibrium cluster size distribution are shown to be consequences of the competition generally present in a nonlinear growth process a qualitative agreement with previous numerical results is found and the scaling laws for the critical exponents of the problem are shown to be satisfied exactly

analogs of the m function in the theory of orthogonal polynomials on the unit circle we show that the multitude of applications of the weyl titchmarsh m function leads to a multitude of different functions in the theory of orthogonal polynomials on the unit circle that serve as analogs of the m function

on computing factors of cyclotomic polynomials for odd square free n 1 the cyclotomic polynomial n x satisfies the identity of gauss 4 n x an 1 n 1 2nb2 n a similar identity of aurifeuille le lasseur and lucas is n 1 x c n nxd n or in the case that n is even and square free n 2 x c n nxd n here an x dn x are polynomials with integer coefficients we show how these coefficients can be computed by simple algorithms which require o n arithmetic operations and work over the integers we also give explicit formulae and generating functions for an x dn x and illustrate the application to integer factorization with some numerical examples 1991 mathematics subject classification primary 11 04 05a15 secondary 11t06 11t22 11t24 11y16 12 04 12e10 12y05

the architecture of complex weighted networks networked structures arise in a wide array of different contexts such as technological and transportation infrastructures social phenomena and biological systems these highly interconnected systems have recently been the focus of a great deal of attention that has uncovered and characterized their topological complexity along with a complex topological structure real networks display a large heterogeneity in the capacity and intensity of the connections these features however have mainly not been considered in past studies where links are usually represented as binary states i e either present or absent here we study the scientific collaboration network and the world wide air transportation network which are representative examples of social and large infrastructure systems respectively in both cases it is possible to assign to each edge of the graph a weight proportional to the intensity or capacity of the connections among the various elements of the network we define appropriate metrics combining weighted and topological observables that enable us to characterize the complex statistical properties and heterogeneity of the actual strength of edges and vertices this information allows us to investigate the correlations among weighted quantities and the underlying topological structure of the network these results provide a better description of the hierarchies and organizational principles at the basis of the architecture of weighted networks

algebro geometric solutions of the baxter szegő difference equation we derive theta function representations of algebro geometric solutions of a discrete system governed by a transfer matrix associated with an extension of the trigonometric moment problem studied by szeg and baxter we also derive a new hierarchy of coupled nonlinear difference equations satisfied by these algebro geometric solutions

scaling of weighted spectral distribution in deterministic scale free networks abstract scale free networks are abundant in the real world in this paper we investigate the scaling properties of the weighted spectral distribution in several deterministic and stochastic models of evolving scale free networks first we construct a new deterministic scale free model whose node degrees have a unified format using graph structure features we derive a precise formula for the spectral metric in this model this formula verifies that the spectral metric grows sublinearly as network size i e the number of nodes grows additionally the mathematical reasoning of the precise formula theoretically provides detailed explanations for this scaling property finally we validate the scaling properties of the spectral metric using some stochastic models the experimental results show that this scaling property can be retained regardless of local world node deleting and assortativity adjustment

aurifeuillian factorizations and the period of the bell numbers modulo a prime we show that the minimum period modulo p of the bell exponential integers is p p 1 p 1 for all primes p 102 and several larger p our proof of this result requires the prime factorization of these periods for some primes p the factoring is aided by an algebraic formula called an aurifeuillian factorization we explain how the coefficients of the factors in these formulas may be computed

algebro geometric constructions of the discrete ablowitz ladik flows and applications resorting to the finite order expansion of the lax matrix the elliptic coordinates are introduced from which the discrete ablowitz ladik equations and the 2 1 dimensional toda lattice are decomposed into solvable ordinary differential equations the straightening out of the continuous flow and the discrete flow is exactly given through the abel jacobi coordinates as an application explicit quasiperiodic solutions for the 2 1 dimensional toda lattice are obtained

monomer dimer problem on some networks abstract zhang et al 2012 obtained the exact formula for the number of all possible monomer dimer arrangements and the asymptotic growth constant on a scale free small world network in this note we generalize this result and obtain the exact solution on the monomer dimer model on many networks particularly we prove that these networks have the same asymptotic growth constant of the number of monomer dimer arrangements

a difference equation arising from the trigonometric moment problem having random reflection coefficients an operator theoretic approach abstract operators associated with polynomials orthogonal on the unit circle and their doubly infinite analogs are considered we assume that the elements in the operators are obtained from a stationary stochastic process and we investigate the properties of the spectrum of these operators lyapunov exponents are introduced and we examine their relationship with the absolutely continuous components of the spectrum of the above operators

algebro geometric solutions for the gerdjikov ivanov hierarchy this paper is dedicated to provide theta function representations of algebro geometric solutions and related crucial quantities for the gerdjikov ivanov gi hierarchy our main tools include the polynomial recursive formalism the hyperelliptic curve with finite number of genus the baker akhiezer functions the meromorphic function the dubrovin type equations for auxiliary divisors and the associated trace formulas with the help of these tools the explicit representations of the baker akhiezer functions the meromorphic function and the algebro geometric solutions are obtained for the entire gi hierarchy

mean field diffusive dynamics on weighted networks diffusion is a key element of a large set of phenomena occurring on natural and social systems modeled in terms of complex weighted networks here we introduce a general formalism that allows to easily write down mean field equations for any diffusive dynamics on weighted networks we also propose the concept of annealed weighted networks in which such equations become exact we show the validity of our approach addressing the problem of the random walk process pointing out a strong departure of the behavior observed in quenched real scale free networks from the mean field predictions additionally we show how to employ our formalism for more complex dynamics our work sheds light on mean field theory on weighted networks and on its range of validity and warns about the reliability of mean field results for complex dynamics

generalisations and randomisation of the plane koch curve the koch curve evolves from a base equilateral triangle by the trisection of each side and the replication of the original triangle on the mid section the process being repeated ad infinitum by the addition of sets of successively smaller triangles the process is generalised to replace the trisectioning by 2k 1 sectioning it is shown that a square is the only other regular polygon on which the 2k 1 sectioning procedure can be implemented the koch curves thus generated are strictly self similar their fractal dimensions being similarity dimensions and enclose simply connected areas randomisation of the generating procedure is also discussed