generalized lame functions i the elliptic case we present and study a class of functions associated with the two particle quantum relativistic calogero moser system with elliptic interactions the functions may be viewed as joint eigenfunctions of two independent commuting analytic difference operators one of which is the defining quantum dynamics the second one is obtained by interchanging the step size and the imaginary period the functions depend on parameters that are dense in the natural parameter domain in essence they consist of products of weierstrass functions and plane waves the zeros of the functions satisfy a constraint system encoding both schrodinger equations at once

spectral analysis for a family of treelike networks abstract for a network knowledge of its laplacian eigenvalues is central to understand its structure and dynamics in this paper we study the laplacian spectra and their applications for a family of treelike networks firstly in order to obtain the laplacian spectra we calculate the constant term and monomial coefficient of characteristic polynomial of the laplacian matrix for a family of treelike networks by using the vieta theorem we then obtain the sum of reciprocals of all nonzero eigenvalues of laplacian matrix finally we determine some interesting quantities that are related to the sum of reciprocals of all nonzero eigenvalues of laplacian matrix such as kirchhoff index global mean first passage time gmfpt

nonparametric trellis equalization in the presence of non gaussian interference we consider the problem of trellis equalization of the intersymbol interference channel in the presence of thermal noise and cochannel interference cci conventional maximum likelihood sequence estimation mlse and maximum a posteriori probability map trellis equalizers treat the sum of noise and interference as additive white gaussian noise while cci is generally a colored non gaussian process we propose a novel nonparametric approach based on the estimation of the probability density function of the noise plus interference given the availability of a limited volume of data the density is estimated by kernel smoothing techniques the use of a whitening filter in the presence of temporally colored disturbance is also addressed simulation results are provided for the global system for mobile communications gsm showing a significant performance improvement with respect to the equalizer based on the gaussian assumption major advantages of the proposed strategy are its intrinsic robustness and general applicability to those cases where accurate modeling of the interference is difficult or a model is not available

nonparametric multivariate density estimation a comparative study the paper algorithmically and empirically studies two major types of nonparametric multivariate density estimation techniques where no assumption is made about the data being drawn from any of known parametric families of distribution the first type is the popular kernel method and several of its variants which uses locally tuned radial basis e g gaussian functions to interpolate the multidimensional density the second type is based on an exploratory projection pursuit technique which interprets the multidimensional density through the construction of several 1d densities along highly interesting projections of multidimensional data performance evaluations using training data from mixture gaussian and mixture cauchy densities are presented the results show that the curse of dimensionality and the sensitivity of control parameters have a much more adverse impact on the kernel density estimators than on the projection pursuit density estimators

on sampling rate analog prefiltering and sufficient statistics for digital receivers we consider the joint sequence estimation timing and phase recovery for linear modulation the paper differs from the classical ones in the sense that time discrete algorithms suitable for fully digital receivers are discussed sufficient conditions are given such that the signal samples represent sufficient statistics these conditions involve signal bandwidth sampling symbol rate and the analog prefilter characteristics it is shown that the sampling rate need not be an exact multiple of the symbol rate i e the samples can be taken from a free running oscillator all subsequent signal processing operations in the receiver then operate with the clock of this free running oscillator timing recovery is then performed by a time variant linear digital interpolator and a decimator carrier recovery and sequence estimation are performed at an average rate of one symbol per sample the digital matched filter for this case is derived for an arbitrary colored noise spectrum

elliptic ruijsenaars operators and functional equations we study mutually commutative difference operators introduced by ruijsenaars which are regarded as elliptic analogs of macdonald operators and consist of the jacobi theta functions and shift operators these operators are constructed in terms of r operators due to shibukawa and ueno or root algebras introduced by cherednik which give rise to a set of functional equations we investigate solutions of these functional equations for generalizations of elliptic ruijsenaars operators

essential self adjointness of the elliptic ruijsenaars models we study the elliptic ruijsenaars models associated with arbitrary root systems which are difference analogs of the calogero moser model we give a dense subspace in the space of square integrable functions invariant under the action of the weyl group on a torus as a domain of its hamiltonian and prove its essential self adjointness by using perturbation theory it is also clarified that these models consist of pure point spectrum

ruijsenaars commuting difference operators and invariant subspace spanned by theta functions we study a family of mutually commutative difference operators introduced by ruijsenaars the conjugations of these operators with an appropriate function give the hamiltonians of some relativistic quantum systems these operators can be regarded as elliptic analogs of the macdonald operators and their coefficients consist of the jacobi theta functions we show that these operators act on the space of meromorphic functions on the cartan subalgebra of affine lie algebras and that the space spanned by characters of a fixed positive level is invariant under the action of these operators

quasi exact solvability of inozemtsev models finite dimensional spaces which are invariant under the action of the hamiltonian of the bcn inozemtsev model are introduced and it is shown that the commuting operators of conserved quantities also preserve the finite dimensional spaces the relationship between the finite dimensional spaces of the bcn inozemtsev models and the theta type invariant spaces of the bcn ruijsenaars models is clarified the degeneration of the bcn inozemtsev models and the correspondence of their invariant spaces are considered

standard random walks and trapping on the koch network with scale free behavior and small world effect a vast variety of real life networks display the ubiquitous presence of scale free phenomenon and small world effect both of which play a significant role in the dynamical processes running on networks although various dynamical processes have been investigated in scale free small world networks analytical research about random walks on such networks is much less in this paper we will study analytically the scaling of the mean first passage time mfpt for random walks on scale free small world networks to this end we first map the classical koch fractal to a network called koch network according to this proposed mapping we present an iterative algorithm for generating the koch network based on which we derive closed form expressions for the relevant topological features such as degree distribution clustering coefficient average path length and degree correlations the obtained solutions show that the koch network exhibits scale free behavior and small world effect then we investigate the standard random walks and trapping issue on the koch network through the recurrence relations derived from the structure of the koch network we obtain the exact scaling for the mfpt we show that in the infinite network order limit the mfpt grows linearly with the number of all nodes in the network the obtained analytical results are corroborated by direct extensive numerical calculations in addition we also determine the scaling efficiency exponents characterizing random walks on the koch network

nonlinear techniques for the joint estimation of cochannel signals cochannel interference occurs when two or more signals overlap in frequency and are present concurrently unlike in spread spectrum multiple access systems where the different users necessarily share the same channel cochannel interference is a severe hindrance to frequency and time division multiple access communications and is typically minimized by interference rejection suppression techniques rather than using interference suppression we are interested in the joint estimation of the information bearing narrow band cochannel signals novel joint estimators are proposed that employ a single input demodulator with oversampling to compensate for timing uncertainties assuming finite impulse response channel characteristics maximum likelihood ml and maximum a posteriori map criteria are used to derive cochannel detectors of varying complexities and degrees of performance in particular a suboptimal two stage joint map symbol detector jmapsd is introduced that has a lower complexity than the single stage estimators while accruing only a marginal loss in error rate performance at high signal to interference ratios assuming only reliable estimates of the primary and secondary signal powers a blind adaptive jmapsd algorithm for a priori unknown channels is also derived the performance of these nonlinear joint estimation algorithms is studied through example computer simulations for two cochannel sources

quantizing for minimum average misclassification risk in pattern classification a decision rule is a labeled partition of the observation space where labels represent classes a way to establish a decision rule is to attach a label to each code vector of a vector quantizer vq when a labeled vq is adopted as a classifier we have to design it in such a way that high classification performance is obtained by a given number of code vectors in this paper we propose a learning algorithm which optimizes the position of labeled code vectors in the observation space under the minimum average misclassification risk criterion

scaling of average sending time on weighted koch networks random walks on weighted complex networks especially scale free networks have attracted considerable interest in the past but the efficiency of a hub sending information on scale free small world networks has been addressed less in this paper we study random walks on a class of weighted koch networks with scaling factor 0 r 1 we derive some basic properties for random walks on the weighted koch networks based on which we calculate analytically the average sending time ast defined as the average of mean first passage times mfpts from a hub node to all other nodes excluding the hub itself the obtained result displays that for 0 r 1 in large networks the ast grows as a power law function of the network order with the exponent represented by log43r 1r and for r 1 in large networks the ast grows with network order as n ln n which is larger than the linear scaling of the average receiving time defined as the average of mfpts for random walks to a given hub node averaged over all starti

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

eigentime identities for random walks on a family of treelike networks and polymer networks abstract in this paper we investigate the eigentime identities quantifying as the sum of reciprocals of all nonzero normalized laplacian eigenvalues on a family of treelike networks and the polymer networks firstly for a family of treelike networks it is shown that all their eigenvalues can be obtained by computing the roots of several small degree polynomials defined recursively we obtain the scalings of the eigentime identity on a family of treelike with network size n n is n n ln n n then for the polymer networks we apply the spectral decimation approach to determine analytically all the eigenvalues and their corresponding multiplicities using the relationship between the generation and the next generation of eigenvalues we obtain the scalings of the eigentime identity on the polymer networks with network size n n is n n ln n n by comparing the eigentime identities on these two kinds of networks their scalings with network size n n are all n n ln n n

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

integrability of difference calogero moser systems a general class of n particle difference calogero moser systems with elliptic potentials is introduced besides the step size and two periods the hamiltonian depends on nine coupling constants we prove the quantum integrability of the model for n 2 and present partial results for n 3 in degenerate cases rational hyperbolic or trigonometric limit the integrability follows for arbitrary particle number from previous work connected with the multivariable q polynomials of koornwinder and macdonald liouville integrability of the corresponding classical systems follows as a corollary limit transitions lead to various well known models such as the nonrelativistic calogero moser systems associated with classical root systems and the relativistic calogero moser system

difference calogero moser systems and finite toda chains limits of a recently introduced n particle difference calogero moser system with elliptic potentials are studied we obtain hyperbolic and rational difference calogero moser systems with an eight parameter external field and finite difference toda chains with four parameter potentials acting on the boundary particles hamiltonians for a number of known integrable n particle systems such as ruijsenaars relativistic calogero moser and toda models and their generalizations associated with classical root systems can be seen as special cases of the hamiltonians considered in this paper

quantum knizhnik zamolodchikov equations and affine root systems quantum difference knizhnik zamolodchikov equations s1 fr are generalized for ther matrices from ch1 with the arguments in arbitrary root systems and their formal counterparts in particular qkz equations with certain boundary conditions are introducted the self consistency of the equations from fr and the cross derivative integrability conditions for ther matrix kz equations from ch2 are obtained as corollaries a difference counterpart of the quantum many body problem connected with macdonald s operators is defined as an application

complete integrability of relativistic calogero moser systems and elliptic function identities poincare invariant generalizations of the galilei invariant calogero moser a particle systems are studied a quantization of the classical integrals s1 sn is presented such that the operators si9 sn mutually commute as a corollary it follows that si9 9sn poisson commute these results hinge on functional equations satisfied by the weierstrass and 0 functions a generalized cauchy identity involving the function leads to an n x n matrix l whose symmetric functions are proportional to sl5 9sn

distribution characteristics of weighted bipartite evolving networks abstract motivated by an evolving model of online bipartite networks we introduce a model of weighted bipartite evolving networks in this model there are two disjoint sets of nodes called user node set and object node set edges only exist between two disjoint sets edge weights represent the usage amount between a couple of user node and object node this model not only clinches the bipartite networks internal mechanism of network growth but also takes into account the object strength deterioration over time step user strength and object strength follow power law distributions respectively the weighted bipartite evolving networks have scare free property in certain situations numerical simulations results agree with the theoretical analyses

topics in quantum integrable systems recent developments in the theory of quantum integrable particle systems in one dimension with inverse square interactions are reviewed first the yangian symmetry is introduced and the energy spectra of the related spin models are discussed the character of the su n 1 wznw theory is shown to be closely related with the rogers szego polynomial second the infinite dimensional representation for solutions of the yang baxter equation and the reflection equation is given based on the representation the dunkl operators associated with the classical root systems are constructed the macdonald polynomial and its generalization are discussed in connection with the eigenstates for the trigonometric case finally some results on short range interacting systems are mentioned

sequence detection and adaptive channel estimation for isi channels under class a impulsive noise in this paper sequence detection and channel estimation for frequency selective intersymbol interference isi producing channels under class a impulsive noise are considered we introduce a novel suboptimum sequence detection ssd scheme and show that although ssd employs a simplified metric it achieves practically the same performance as maximum likelihood sequence detection mlsd for both ssd and mlsd a lower bound on the achievable performance is derived which is similar to the classical matched filter bound for frequency selective fading channels under gaussian noise for channel estimation we adopt a minimum entropy criterion and derive efficient least mean entropy and recursive least entropy algorithms for both adaptive algorithms we analyze the steady state channel estimation error variance theoretical considerations and simulation results show that in class a impulsive noise the proposed sequence detection and adaptive channel estimation schemes yield significant performance gains over their respective conventional counterparts designed for gaussian noise although the novel algorithms require knowledge of the class a noise model parameters their computational complexity is comparable to that of the corresponding conventional algorithms

elliptic quantum groups and ruijsenaars models we construct symmetric and exterior powers of the vector representation of the elliptic quantum groupse sln the corresponding transfer matrices give rise to various integrable difference equations which could be solved in principle by the nested bethe ansatz method in special cases we recover the ruijsenaars systems of commuting difference operators

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

the viterbi algorithm and markov noise memory this work designs sequence detectors for channels with intersymbol interference isi and correlated and or signal dependent noise we describe three major contributions i first by modeling the noise as a finite order markov process we derive the optimal maximum likelihood sequence detector mlsd and the optimal maximum a posteriori map sequence detector extending to the correlated noise case the viterbi algorithm we show that when the signal dependent noise is conditionally gauss markov the branch metrics in the mlsd are computed from the conditional second order noise statistics we evaluate the branch metrics using a bank of finite impulse response fir filters ii second we characterize the error performance of the mlsd and map sequence detector the error analysis of these detectors is complicated by the correlation asymmetry of the channel noise we derive upper and lower bounds and computationally efficient approximations to these bounds based on the banded structure of the inverses of gauss markov covariance matrices an experimental study shows the tightness of these bounds iii finally we derive several classes of suboptimal sequence detectors and demonstrate how these and others available in the literature relate to the mlsd we compare their error rate performance and their relative computational complexity and show how the structure of the mlsd and the performance evaluation guide us in choosing a best compromise between several types of suboptimal sequence detectors

mlse for an unknown channel i optimality considerations the problem of performing joint maximum likelihood ml estimation of a digital sequence and unknown dispersive channel impulse response is considered starting from a continuous time ct model previous investigations of this problem have not considered the front end fe processing in detail rather a discrete time signal model has been assumed we show that a fractionally spaced whitened matched filter matched to the known data pulse provides a set of sufficient statistics when a tapped delay line channel model is assumed and that the problem is ill posed when the channel impulse response is generalized to a ct finite length model practical approximations are considered that circumvent this ill posed condition recursive computation of the joint ml metric is developed together the fe processing and metric recursion provide a receiver structure which may be interpreted as the theoretical foundation for the previously introduced technique of per survivor processing and they lead directly to generalizations several fe processors representative of those suggested in the literature are developed and related to the practically optimal fe

quantum integrability of the generalized elliptic ruijsenaars models the quantum integrability of the generalized elliptic ruijsenaars models is shown these models are mathematically related to the macdonald operator and the macdonald koornwinder operator which appeared in the q orthogonal polynomial theories we construct these integrable families by using the yang baxter equation and the reflection equation

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

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

generalized lame functions ii hyperbolic and trigonometric specializations in part i j math phys 40 1595 1999 we studied eigenfunctions of the quantum dynamics that defines the two particle relativistic calogero moser system with elliptic interaction in the present paper we consider the same system with hyperbolic and trigonometric interactions in these special regimes the eigenfunctions are shown to admit an elementary representation that is far more explicit than the zero representation of part i in particular the new representation can be exploited to prove that the hyperbolic eigenfunctions can be chosen to be symmetric under interchanging position and momentum variables self duality in the trigonometric case duality properties are derived too and several orthogonality and completeness results are obtained

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

eigentime identity of the weighted scale free triangulation networks for weight dependent walk abstract the eigenvalues of the normalized laplacian matrix of a network provide information on its structural properties and some relevant dynamical aspects in particular for weight dependent walk in order to get the eigentime identity for weight dependent walk we need to obtain the eigenvalues and their multiplicities of the laplacian matrix firstly the model of the weighted scale free triangulation networks is constructed then the eigenvalues and their multiplicities of transition weight matrix are presented after the recursive relationship of those eigenvalues at two successive generations are given consequently the laplacian spectrum is obtained finally the analytical expression of the eigentime identity indicating that the eigentime identity grows sublinearly with the network order is deduced

conserved operators of the generalized elliptic ruijsenaars models we construct the integrable families of the generalized elliptic ruijsenaars models by use of the yang baxter equation and the reflection equation the higher order conserved operators of these models are explicitly presented

on the performance of single threshold detectors for binary communications in the presence of gaussian mixture noise in this paper probability of error performance of single threshold detectors is studied for binary communications systems in the presence of gaussian mixture noise first sufficient conditions are proposed to specify when the sign detector is not an optimal detector among all the single threshold detectors then a monotonicity property of the error probability is derived for the optimal single threshold detector in addition a theoretical limit is obtained on the maximum ratio between the average probabilities of error for the sign detector and the optimal single threshold detector finally numerical examples are presented to investigate the theoretical results

commuting difference operators arising from the elliptic c2 1 face model we study a pair of commuting difference operators arising from the elliptic c2 1 face model the operators whose coefficients are expressed in terms of the jacobi s elliptic theta function act on the space of meromorphic functions on the weight space of the c2 type simple lie algebra we show that the space of functions spanned by the level one characters of the affine lie algebra sp 4 c is invariant under the action of the difference operators

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

pseudofractal scale free web we find that scale free random networks are excellently modeled by simple deterministic graphs our graph has a discrete degree distribution degree is the number of connections of a vertex which is characterized by a power law with exponent gamma 1 ln 3 ln 2 properties of this compact structure are surprisingly close to those of growing random scale free networks with gamma in the most interesting region between 2 and 3 we succeed to find exactly and numerically with high precision all main characteristics of the graph in particular we obtain the exact shortest path length distribution for a large network ln n 1 the distribution tends to a gaussian of width approximately sqrt ln n centered at l approximately ln n we show that the eigenvalue spectrum of the adjacency matrix of the graph has a power law tail with exponent 2 gamma

maximum likelihood sequence estimators a geometric view communication issues are described in terms of macro operations between the data and the observation spaces the problem of recovering the data is related to the inversion of an operator the channel mapping for this reason results available in linear algebra and functional analysis are applicable traditional concepts in communications are identified with these operations an approach to the maximum likelihood sequence estimator based on a sufficient statistic derived from these concepts is proposed intersymbol interference is removed by linear equalization and a viterbi like dynamic programming algorithm takes into account the correlated noise in the metric evaluation the performance of suboptimal receivers obtained by means of metric simplification is analyzed

ruijsenaars commuting difference operators as commuting transfer matrices for belavin s elliptic quantum r matrix we construct an l operator as a set of difference operators acting on functions on the type a weight space according to the fundamental relation rll llr taking the trace of the l operator gives a set of commuting difference operators we show that for the above mentioned l operator this approach gives macdonald type operators with elliptic theta function coefficient actually equivalent to ruijsenaars operators the relationship between the difference l operator and krichever s lax matrix is given and an explicit formula for elliptic commuting differential operators is derived we also study the invariant subspace for the system which is spanned by symmetric theta functions on the weight space

double affine hecke algebras and macdonald s conjectures in this paper we prove the main results about the structure of double affine hecke algebras announced in c1 c2 the technique is based on the realization of these algebras in terms of demazure lusztig operators bgg d l2 ls c3 and rather standard facts from the theory of affine weyl groups in particular it completes the proof partially published in c2 of the macdonald scalar product conjecture see ml 12 6 including the famous macdonald constant term conjecture the q t case we mainly follow the opdam paper 0 where the macdonald mehta conjectures in the degenerate differential case were deduced from certain properties of the heckman opdam operators ho and the existence of the shift operators heckman s interpretation of these operators via the so called dunkl operators see he and also c5 was important to our approach we note that the ho operators are closely related to the so called quantum many body problem calogero sutherland moser olshanetsky perelomov the conformal field theory knizhnik zamolodchikov equations the harmonic analysis on symmetric spaces harish chandra helgason etc and last but not the least the classic theory of the hypergeometric functions establishing the connection between the difference counterparts of heckman opdam operators introduced in c4 and the macdonald theory ml m2 including the construction of the difference shift operators is the main thrust of this paper once the connection is established it is not very difficult to calculate the scalar squares of the macdonald polynomials and to prove the constant term conjecture from his fundamental paper m3 to simplify the exposition we discuss the reduced root systems only and impose the relation q tk for k e z to avoid infinite products in the definition of macdonald s pairing the purpose of this work is to present a concrete application of the new technique arbitrary q t can be handled in a

a unified model for sierpinski networks with scale free scaling and small world effect abstract in this paper we propose an evolving sierpinski gasket based on which we establish a model of evolutionary sierpinski networks esns that unifies deterministic sierpinski network z z zhang s g zhou t zou l c chen j h guan eur phys j b 60 2007 259 and random sierpinski network z z zhang s g zhou z su t zou j h guan eur phys j b 65 2008 141 to the same framework we suggest an iterative algorithm generating the esns on the basis of the algorithm some relevant properties of presented networks are calculated or predicted analytically analytical solution shows that the networks under consideration follow a power law degree distribution with the distribution exponent continuously tuned in a wide range the obtained accurate expression of clustering coefficient together with the prediction of average path length reveals that the esns possess small world effect all our theoretical results are successfully contrasted by numerical simulations moreover the evolutionary prisoner s dilemma game is also studied on some limitations of the esns i e deterministic sierpinski network and random sierpinski network

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