ramanujan formula for the generalized stirling approximation abstract the aim of this paper is to improve ramanujan s formula for approximation of the factorial function starting from burnside s formula in contradistinction with the classical formula that starts from stirling s formula

estimating and understanding exponential random graph models we introduce a method for the theoretical analysis of exponential random graph models the method is based on a large deviations approximation to the normalizing constant shown to be consistent using theory developed by chatterjee and varadhan european j combin 32 2011 1000 1017 the theory explains a host of difficulties encountered by applied workers many distinct models have essentially the same mle rendering the problems practically ill posed we give the first rigorous proofs of degeneracy observed in these models here almost all graphs have essentially no edges or are essentially complete we supplement recent work of bhamidi bresler and sly 2008 ieee 49th annual ieee symposium on foundations of computer science focs 2008 803 812 ieee showing that for many models the extra sufficient statistics are useless most realizations look like the results of a simple erd s renyi model we also find classes of models where the limiting graphs differ from erd s renyi graphs a limitation of our approach inherited from the limitation of graph limit theory is that it works only for dense graphs

growing scale free networks with tunable clustering we extend the standard scale free network model to include a triad formation step we analyze the geometric properties of networks generated by this algorithm both analytically and by numerical c

asymptotic expansions related to hyperfactorial function and glaisher kinkelin constant in this paper by the bernoulli numbers and the exponential complete bell polynomials we establish two general asymptotic expansions related to the hyperfactorial function and the glaisher kinkelin constant where the coefficients in the series of the expansions can be determined by recurrences moreover the explicit expressions of the coefficients are studied and some special asymptotic expansions are presented

on the capacity of vector gaussian channels with bounded inputs the capacity of a deterministic multiple input multiple output channel under the peak and average power constraints is investigated for the identity channel matrix the approach of shamai et al is generalized to the higher dimension settings to derive the necessary and sufficient conditions for the optimal input probability density function this approach prevents the usage of the identity theorem of the holomorphic functions of several complex variables which seems to fail in the multi dimensional scenarios it is proved that the support of the capacity achieving distribution is a finite set of hyper spheres with mutual independent phases and amplitude in the spherical domain subsequently it is shown that when the average power constraint is relaxed if the number of antennas is large enough the capacity has a closed form solution and constant amplitude signaling at the peak power achieves it moreover it will be observed that in a discrete time memoryless gaussian channel the average power constrained capacity which results from a gaussian input distribution can be closely obtained by an input where the support of its magnitude is a discrete finite set finally we investigate some upper and lower bounds for the capacity of the non identity channel matrix and evaluate their performance as a function of the condition number of the channel

network transitivity and matrix models this paper is a step towards a systematic theory of the transitivity clustering phenomenon in random networks a static framework is used with adjacency matrix playing the role of the dynamical variable hence our model is a matrix model where matrices are random but their elements take values 0 and 1 only confusion present in some papers where earlier attempts to incorporate transitivity in a similar framework have been made is hopefully dissipated inspired by more conventional matrix models analytic techniques to develop a static model with nontrivial clustering are introduced computer simulations complete the analytic discussion

constrained markovian dynamics of random graphs we introduce a statistical mechanics formalism for the study of constrained graph evolution as a markovian stochastic process in analogy with that available for spin systems deriving its basic properties and highlighting the role of the mobility the number of allowed moves for any given graph as an application of the general theory we analyze the properties of degree preserving markov chains based on elementary edge switchings we give an exact yet simple formula for the mobility in terms of the graph s adjacency matrix and its spectrum this formula allows us to define acceptance probabilities for edge switchings such that the markov chains become controlled glauber type detailed balance processes designed to evolve to any required invariant measure representing the asymptotic frequencies with which the allowed graphs are visited during the process as a corollary we also derive a condition in terms of simple degree statistics sufficient to guarantee that in the limit where the number of nodes diverges even for state independent acceptance probabilities of proposed moves the invariant measure of the process will be uniform we test our theory on synthetic graphs and on realistic larger graphs as studied in cellular biology showing explicitly that for instances where the simple edge swap dynamics fails to converge to the uniform measure a suitably modified markov chain instead generates the correct phase space sampling

exactly solvable random graph ensemble with extensively many short cycles we introduce and analyse ensembles of 2 regular random graphs with a tuneable distribution of short cycles the phenomenology of these graphs depends critically on the scaling of the ensembles control parameters relative to the number of nodes a phase diagram is presented showing a second order phase transition from a connected to a disconnected phase we study both the canonical formulation where the size is large but fixed and the grand canonical formulation where the size is sampled from a discrete distribution and show their equivalence in the thermodynamical limit we also compute analytically the spectral density which consists of a discrete set of isolated eigenvalues representing short cycles and a continuous part representing cycles of diverging size

generation of uncorrelated random scale free networks uncorrelated random scale free networks are useful null models to check the accuracy and the analytical solutions of dynamical processes defined on complex networks we propose and analyze a model capable of generating random uncorrelated scale free networks with no multiple and self connections the model is based on the classical configuration model with an additional restriction on the maximum possible degree of the vertices we check numerically that the proposed model indeed generates scale free networks with no two and three vertex correlations as measured by the average degree of the nearest neighbors and the clustering coefficient of the vertices of degree k respectively complex networks constitute a general framework for the topological characterization of many natural and technological systems whose complexity prevents a more detailed microscopic description f1 3g within this framework these systems are represented in terms of networks or graphs f4g in which vertices stand for the units composing the system while edges among vertices represent the interactions or relations between pairs of units the focus is thus shift to the topological characterization of the representative network a task which is largely more feasible and yields nevertheless a noticeable amount of information on the structure and properties of the original system the empirical analysis of many real complex networks has unveiled the presence of several typical properties widely found in systems belonging to a large variety of realms one of the most relevant is given by the scale free nature of the degree distribution pskd f1 3 5g defined as the probability that a randomly chosen vertex has degree k si e it is connected to other k verticesd in mathematical terms the scale free property translates into a power law function of the form psk d k g

inequalities and asymptotic expansions associated with the ramanujan and nemes formulas for the gamma function we present some inequalities and asymptotic expansions associated with the ramanujan and nemes formulas for the gamma function

growing network with local rules preferential attachment clustering hierarchy and degree correlations the linear preferential attachment hypothesis has been shown to be quite successful in explaining the existence of networks with power law degree distributions it is then quite important to determine if this mechanism is the consequence of a general principle based on local rules in this work it is claimed that an effective linear preferential attachment is the natural outcome of growing network models based on local rules it is also shown that the local models offer an explanation for other properties like the clustering hierarchy and degree correlations recently observed in complex networks these conclusions are based on both analytical and numerical results for different local rules including some models already proposed in the literature

asymptotic expansions for the gamma function in terms of hyperbolic functions abstract the smith s approximation formula for the gamma function is given by x 1 2 2 x e x 2 x tanh 1 2 x x 2 1 o 1 x 5 as x in this paper by means of a little known power series we develop this formula to an asymptotic expansions x 1 2 2 x e x 2 x tanh 1 2 x x 2 exp n 3 2 n 2 n 1 2 2 n 1 1 2 1 2 n 2 n 2 n 1 2 n b 2 n x 2 n 1 as x and give an estimate of remainder in the above asymptotic series where b 2 n is the bernoulli number moreover as relevant results ramanujan type asymptotic expansions for the gamma function x 1 2 are obtained an asymptotic expansion for wallis ratio and an estimate of the remainder are established

integral manifolds of slow adaptation conditions are given under which an integral manifold slow manifold exists for a nonlinear system representing a broad class of adaptive algorithms the parameter update equation restricted to the manifold is an exact description of the slow adaptation process which can be approximately analyzed by averaging stability properties of the slow motion in the manifold are extended off the manifold by a set of conditions under which the manifold is shown to be attractive this two time scale analysis provides a geometrical visualization of instability phenomena observed in adaptive systems and generalizes earlier local stability results

a detailed investigation into near degenerate exponential random graphs the exponential family of random graphs has been a topic of continued research interest despite the relative simplicity these models capture a variety of interesting features displayed by large scale networks and allow us to better understand how phases transition between one another as tuning parameters vary as the parameters cross certain lines the model asymptotically transitions from a very sparse graph to a very dense graph completely skipping all intermediate structures we delve deeper into this near degenerate tendency and give an explicit characterization of the asymptotic graph structure as a function of the parameters

on the asymptotic expansions of the gamma function related to the nemes gosper and burnside formulas in this paper we present new asymptotic expansions of the gamma function related to the nemes gosper and burnside formulas

monotonicity properties inequalities and asymptotic expansions associated with the gamma function we define x by the equality x 1 x e x 8 x 3 4 x 2 x 1 30 1 6 e x we call x the remainder of ramanujan s formula in this paper we present some properties for x including monotonicity properties inequalities and asymptotic expansions furthermore we present some full asymptotic expansions for the gamma function related to the nemes ramanujan and burnside formulas

on a general class of models for interaction this paper develops a class of probability models for configurations of interacting points in a domain the distributions depend on a function which may be viewed as giving the potential energy of the configurations examples include models for interaction in a spatial region and on lattices and graphs new models in the general class arise naturally an example being a spatial model for points of different categories some general methods including series expansions and a simulation method known from statistical mechanics can be useful for many of these models several applications of this kind are considered and some connections between the models and with the statistical mechanics are explored

universal hypotrochoidic law for random matrices with cyclic correlations the celebrated elliptic law describes the distribution of eigenvalues of random matrices with correlations between off diagonal pairs of elements having applications to a wide range of physical and biological systems here we investigate the generalization of this law to random matrices exhibiting higher order cyclic correlations between k tuples of matrix entries we show that the eigenvalue spectrum in this ensemble is bounded by a hypotrochoid curve with k fold rotational symmetry this hypotrochoid law applies to full matrices as well as sparse ones and thereby holds with remarkable universality we further extend our analysis to matrices and graphs with competing cycle motifs which are described more generally by polytrochoid spectral boundaries

some properties of the gamma and psi functions with applications in this paper some monotoneity and concavity properties of the gamma beta and psi functions are obtained from which several asymptotically sharp inequalities follow applying these properties the authors improve some well known results for the volume n of the unit ball b n r n the surface area n 1 of the unit sphere s n 1 and some related constants

a model for social networks social networks are organized into communities with dense internal connections giving rise to high values of the clustering coefficient in addition these networks have been observed to be assortative i e highly connected vertices tend to connect to other highly connected vertices and have broad degree distributions we present a model for an undirected growing network which reproduces these characteristics with the aim of producing efficiently very large networks to be used as platforms for studying sociodynamic phenomena the communities arise from a mixture of random attachment and implicit preferential attachment the structural properties of the model are studied analytically and numerically using the k clique method for quantifying the communities r 2006 elsevier b v all rights reserved

duplication divergence model of protein interaction network we investigate a very simple model describing the evolution of protein protein interaction networks via duplication and divergence the model exhibits a remarkably rich behavior depending on a single parameter the probability to retain a duplicated link during divergence when this parameter is large the network growth is not self averaging and an average node degree increases algebraically the lack of self averaging results in a great diversity of networks grown out of the same initial condition when less than a half of links are on average preserved after divergence the growth is self averaging the average degree increases very slowly or tends to a constant and a degree distribution has a power law tail the predicted degree distributions are in a very good agreement with the distributions observed in real protein networks

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

eigenvalue tunneling and decay of quenched random network we consider the canonical ensemble of n vertex erd s renyi er random topological graphs with quenched vertex degree and with fugacity for each closed triple of bonds we claim complete defragmentation of large n graphs into the collection of p 1 almost full subgraphs cliques above critical fugacity c where p is the er bond formation probability evolution of the spectral density of the adjacency matrix with increasing leads to the formation of a multizonal support for c eigenvalue tunneling from the central zone to the side one means formation of a new clique in the defragmentation process the adjacency matrix of the network ground state has a block diagonal form where the number of vertices in blocks fluctuates around the mean value np the spectral density of the whole network in this regime has triangular shape we interpret the phenomena from the viewpoint of the conventional random matrix model and speculate about possible physical applications

time domain signal analysis using adaptive notch filter noise reduction and signal decomposition are among important and practical issues in time domain signal analysis this paper presents an adaptive notch filter anf to achieve both these objectives for noise reduction purpose the proposed adaptive filter successfully extracts a single sinusoid of a possibly time varying nature from a noise corrupted signal the paper proceeds with introducing a chain of filters which is capable of estimating the fundamental frequency of a signal composed of harmonically related sinusoids and of decomposing it into its constituent components the order of differential equations governing this algorithm is 2n 1 where n is the number of constituent sinusoids that should be extracted stability analysis of the proposed algorithm is carried out based on the application of the local averaging theory under the assumption of slow adaptation when compared with the conventional fourier analysis the proposed method provides instantaneous values of the constituting components moreover it is adaptive with respect to the fundamental frequency of the signal simulation results verify the validity of the presented algorithm and confirm its desirable transient and steady state performances as well as its desirable noise characteristics

adaptive algorithms for the rejection of sinusoidal disturbances with unknown frequency abstract two algorithms are presented for the rejection of sinusoidal disturbances with unknown frequency the first is an indirect algorithm where the frequency of the disturbance is estimated and the estimate is used in another adaptive algorithm that adjusts the magnitude and phase of the input needed to cancel the effect of the disturbance a direct algorithm that uses the concept of a phase locked loop is also presented in which frequency estimation and disturbance cancellation are performed simultaneously approximate analyses are presented for both schemes and the results are found useful for the selection of the design parameters simulations are given which demonstrate the validity of the analytical results and the ability of the algorithms to reject sinusoidal disturbances with unknown frequency the indirect algorithm is found to have a larger capture region for the parameter estimates whereas the direct algorithm has superior convergence properties locally about the optimum parameter estimates

network growth by copying we introduce a growing network model in which a new node attaches to a randomly selected node as well as to all ancestors of the target node this mechanism produces a sparse ultrasmall network where the average node degree grows logarithmically with network size while the network diameter equals 2 we determine basic geometrical network properties such as the size dependence of the number of links and the in and out degree distributions we also compare our predictions with real networks where the node degree also grows slowly with time the internet and the citation network of all physical review papers

on the asymptotic representation of the euler gamma function by ramanujan abstract the problem of approximation to the euler gamma function on the basis of some ramanujan s formulas is considered the function h x g x 6 8x3 4x2 x where g x e x x 1 x is studied it is proved that on the interval 1 the function h x is increasing monotonically from h 1 0 0111976 to h 1 30 0 0333

periodic disturbance cancellation with uncertain frequency this paper presents a new algorithm to cancel periodic disturbances with uncertain frequency the disturbances are cancelled using an internal model structure with adaptive frequency in parallel with a stabilizing controller the time varying internal model controller s states in steady state can be mapped to two time invariant variables the magnitude or energy of the internal model and frequency of the disturbance an additional integral controller then can be used to reduce the difference between the internal model controller imc and disturbance frequency to zero the stability of the feedback control system with this algorithm and convergence of the algorithm to the correct frequency with exact disturbance cancellation are justified by singular perturbation and averaging theories the algorithm is locally exponentially stable rather than asymptotically stable simulations demonstrate the performance of the algorithm the ability of this algorithm to identify the frequency of periodic disturbances and to reject periodic disturbances with uncertain frequency

improved asymptotic formulas for the gamma function the aim of this paper is to improve the ramanujan formula for approximation the gamma function a fast asymptotic series is constructed