on the thermodynamic formalism for the gauss map we study the generalized transfer operator open image in new window of the gauss maptx 1 x mod 1 on the unit interval this operator which for 1 is the familiar perron frobenius operator oft can be defined for re 1 2 as a nuclear operator either on the banach spacea d of holomorphic functions over a certain discd or on the hilbert space open image in new window of functions belonging to some hardy class of functions over the half planeh 1 2 the spectra of open image in new window on the two spaces are identical on the space open image in new window is isomorphic to an integral operator open image in new window with kernel the bessel function mathfrak f 2 beta 1 2 sqrt st and hence to some generalized hankel transform this shows that open image in new window has real spectrum for real 1 2 on the spacea d the operator open image in new window can be analytically continued to the entire plane with simple poles at beta beta k 1 k 2 k 0 1 2 and residue the rank 1 operator mathcal n k f tfrac 1 2 1 k f k 0 from this similar analyticity properties for the fredholm determinant open image in new window of open image in new window and hence also for ruelle s zeta function follow another application is to the function zeta m beta sum limits n 1 infty n beta where n denotes the irrational n n n2 4 1 2 2 m extends to a meromorphic function in the plane with the only poles at 1 both with residue 1

relevance of soft modes for order parameter fluctuations in the two dimensional xy model we analyse the spin wave approximation for the two dimensional xy model directly in reciprocal space in this limit the model is diagonal and the normal modes are statistically independent despite this simplicity non trivial critical properties are observed and exploited we confirm that the observed asymmetry for the probability density function for order parameter fluctuations comes from the divergence of the mode amplitudes across the brillouin zone we show that the asymmetry is a many body effect despite the importance played by the zone centre the precise form of the function is dependent on the details of the microscopic generating function for the modes we discuss the relevance of this to more complex or non equilibrium systems where similar distributions are observed

constructing locally connected non computable julia sets a locally connected quadratic siegel julia set has a simple explicit topological model such a set is computable if there exists an algorithm to draw it on a computer screen with an arbitrary resolution we constructively produce parameter values for siegel quadratics for which the julia sets are non computable yet locally connected

line graphs link partitions and overlapping communities in this paper we use a partition of the links of a network in order to uncover its community structure this approach allows for communities to overlap at nodes so that nodes may be in more than one community we do this by making a node partition of the line graph of the original network in this way we show that any algorithm that produces a partition of nodes can be used to produce a partition of links we discuss the role of the degree heterogeneity and propose a weighted version of the line graph in order to account for this

critical functions for complex analytic maps critical functions measure the width of the domain of stability around a given fixed point or an invariant circle for complex analytic and area preserving maps the author discusses their dependence on the rotation number of the invariant curves and proposes some new methods to determine them based on the existence of critical points and on some properties of quasiconformal maps by means of the majorant series method some rigorous estimates are given for complex area preserving maps like the semistandard map and the modulated singular map in particular the author makes use of the brjuno function to interpolate critical maps and proves that the convergence of the brjuno function is a necessary and sufficient condition for the existence of an analytic invariant curve of a given rotation number the author also discusses the optimality of the rigorous bounds obtained

finding community structure in networks using the eigenvectors of matrices we consider the problem of detecting communities or modules in networks groups of vertices with a higher than average density of edges connecting them previous work indicates that a robust approach to this problem is the maximization of the benefit function known as modularity over possible divisions of a network here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix which plays a role in community detection similar to that played by the graph laplacian in graph partitioning calculations this result leads us to a number of possible algorithms for detecting community structure as well as several other results including a spectral measure of bipartite structure in networks and a centrality measure that identifies vertices that occupy central positions within the communities to which they belong the algorithms and measures proposed are illustrated with applications to a variety of real world complex networks

width distribution for random walk interfaces roughening of a one dimensional interface is studied under the assumption that the interface configurations are continuous periodic random walks the distribution of the square of the width of interface mathit w 2 is found to scale as p mathit w 2 mathit w 2 mathrm mathrm ensuremath 1 ensuremath phi mathit w 2 mathit w 2 where mathit w 2 is the average of mathit w 2 we calculate the scaling function ensuremath phi x exactly and compare it both to exact enumerations for a discrete slope surface evolution model and to ensuremath phi s obtained in monte carlo simulations of equilibrium and driven interfaces of chemically reacting systems

line graphs as social networks abstract it was demonstrated recently that the line graphs are clustered and assortative these topological features are known to characterize some social networks m e j newman y park why social networks are different from other types of networks phys rev e 68 2003 036122 it was argued that this similarity reveals their cliquey character in the model proposed here a social network is the line graph of an initial network of families communities interest groups school classes and small companies these groups play the role of nodes and individuals are represented by links between these nodes the picture is supported by the data on the livejournal network of about 8 106 people

estimating order statistics of network degrees abstract we model the order statistics of network degrees of big data sets by a range of generalised beta distributions a three parameter beta distribution due to libby and novick 1982 is shown to give the best overall fit for at least four big data sets the fit of this distribution is significantly better than the fit suggested by olhede and wolfe 2012 across the whole range of order statistics for all four data sets

universal fluctuations and extreme value statistics we study the effect of long range algebraic correlations on extreme value statistics and demonstrate that correlations can produce a limit distribution which is indistinguishable from the ubiquitous bramwell holdsworth pinton distribution we also consider the square width fluctuations of the avalanche signal we find as recently predicted by antal et al for logarithmic correlated 1 f signals that these fluctuations follow the fisher tippett gumbel distribution from uncorrelated extreme value statistics

the brjuno function continuously estimates the size of quadratic siegel disks if a is an irrational number yoccoz defined the brjuno function by a ni 0 a0 a1 i i i an 1 log 1 an where a0 is the fractional part of a and an 1 is the fractional part of 1 an the numbers a such that a i are called the brjuno numbers the quadratic polynomial p a z iu e2i az z2 has an indifferent fixed point at the origin if p a is linearizable we let r a be the conformal radius of the siegel disk and we set r a 0 otherwise yoccoz y proved that a i if and only if r a 0 and that the restriction of a iu a log r a to the set of brjuno numbers is bounded from below by a universal constant in bc2 we proved that it is also bounded from above by a universal constant in fact marmi moussa and yoccoz mmy conjecture that this function extends to r as a hi older function of exponent 1 2 in this article we prove that there is a continuous extension to r

scale invariance of the png droplet and the airy process we establish that the static height fluctuations of a particular growth model the png droplet converges upon proper rescaling to a limit process which we call the airy process a y the airy process is stationary it has continuous sample paths its single time fixed y distribution is the tracy widom distribution of the largest eigenvalue of a gue random matrix and the airy process has a slow decay of correlations as y 2 roughly the airy process describes the last line of dyson s brownian motion model for random matrices our construction uses a multi layer version of the png model which can be analyzed through fermionic techniques specializing our result to a fixed value of y one reobtains the celebrated result of baik deift and johansson on the length of the longest increasing subsequence of a random permutation

renormalization flow for extreme value statistics of random variables raised to a varying power using a renormalization approach we study the asymptotic limit distribution of the maximum value in a set of independent and identically distributed random variables raised to a power qn that varies monotonically with the sample size n under these conditions a non standard class of max stable limit distributions which mirror the classical ones emerges furthermore a transition mechanism between the classical and the non standard limit distributions is brought to light if qn grows slower than a characteristic function q n the standard limit distributions are recovered while if qn behaves asymptotically as q n non standard limit distributions emerge

renormalization group theory for finite size scaling in extreme statistics we present a renormalization group rg approach to explain universal features of extreme statistics applied here to independent identically distributed variables the outlines of the theory have been described in a previous paper the main result being that finite size shape corrections to the limit distribution can be obtained from a linearization of the rg transformation near a fixed point leading to the computation of stable perturbations as eigenfunctions here we show details of the rg theory which exhibit remarkable similarities to the rg known in statistical physics besides the fixed points explaining universality and the least stable eigendirections accounting for convergence rates and shape corrections the similarities include marginally stable perturbations which turn out to be generic for the fisher tippett gumbel class distribution functions containing unstable perturbations are also considered we find that after a transitory divergence they return to the universal fixed line at the same or at a different point depending on the type of perturbation

incorporating variability into an approximation formula for bond percolation thresholds of planar periodic lattices approximation formulas to predict values for bond percolation thresholds of periodic graphs make use of certain features of lattice graphs such as dimension and average degree we show that a relationship exists between the average and second moment of the degree of a graph and the average degree of its line graph using this relationship together with the well known bond to site transformation between the bond percolation model on a graph and the site percolation model on its line graph we create a new approximation formula that improves the accuracy of bond percolation threshold predictions

universality classes for extreme value statistics the equilibrium low temperature physics of disordered systems is governed by the statistics of extremely low energy states it is thus relevant to discuss the possible universality classes for extreme value statistics we compare the usual probabilistic classification to the results of the replica approach we show in detail for several problems including the random energy model and the decaying burgers turbulence that one class of independent variables corresponds exactly to the so called one step replica symmetry breaking solution in the replica language we argue that this universality class holds if the correlations are sufficiently weak and propose a conjecture on the level of correlations which leads to different universality classes

analytic solutions for a class of differential equation with delays depending on state abstract in this paper we study existence of analytic solutions of a class of the differential equation with state dependent delays x z j 0 k t 1 c t j z x j z t g z in the complex field c by constructing a convergent power series solution y z of an auxiliary equation of the form y z y z j 0 k t 1 c t j y z y j z t g y z with g 0 c 0 invertible analytic solutions of the form y y 1 z for the original equation are obtained

synchronization landscapes in small world connected computer networks motivated by a synchronization problem in distributed computing we studied a simple growth model on regular and small world networks embedded in one and two dimensions we find that the synchronization landscape corresponding to the progress of the individual processors exhibits kardar parisi zhang like kinetic roughening on regular networks with short range communication links although the processors on average progress at a nonzero rate their spread the width of the synchronization landscape diverges with the number of nodes desynchronized state hindering efficient data management when random communication links are added on top of the one and two dimensional regular networks resulting in a small world network large fluctuations in the synchronization landscape are suppressed and the width approaches a finite value in the large system size limit synchronized state in the resulting synchronization scheme the processors make close to uniform progress with a nonzero rate without global intervention we obtain our results by simulating the simulations based on the exact algorithmic rules supported by coarse grained arguments

random line graphs and a linear law for assortativity for a fixed number n of nodes the number of links l in the line graph h n l can only appear in consecutive intervals called a band of l we prove that some consecutive integers can never represent the number of links l in h n l and they are called a bandgap of l we give the exact expressions of bands and bandgaps of l we propose a model which can randomly generate simple graphs which are line graphs of other simple graphs the essence of our model is to merge step by step a pair of nodes in cliques which we use to construct line graphs obeying necessary rules to ensure that the resulting graphs are line graphs two nodes to be merged are randomly chosen at each step if the cliques are all of the same size the assortativity of the line graphs in each step are close to 0 and the assortativity of the corresponding root graphs increases linearly from 1 to 0 with the steps of the nodal merging process if we dope the constructing elements of the line graphs the cliques of the same size with a relatively smaller number of cliques of different size the characteristics of the assortativity of the line graphs is completely altered we also generate line graphs with the cliques whose sizes follow a binomial distribution the corresponding root graphs with binomial degree distributions zero assortativity and semicircle eigenvalue distributions are equivalent to erd s renyi random graphs

bryuno function and the standard map for the standard map the homotopically non trivial invariant curves of rotation number satisfying the bryuno condition are shown to be analytic in the perturbative parameter e provided e is small enough the radius of convergence of the lindstedt series sometimes called critical function of the standard map is studied and the relation with the bryuno function b is derived the quantity log 2 b is proved to be bounded uniformly in

parameter specification for the degree distribution of simulated barabasi albert graphs abstract the degree distribution of a simulated barabasi albert graph under linear preferential attachment is investigated specifically the parameters of the power law distribution are estimated and compared against the theoretical values derived using mean field theory least squares method and mle nonparametric method were utilized to estimate the distribution parameters on 1000 simulated barabasi albert graphs for edge parameter m 2 4 6 and size n 2 k k 5 6 14 15 goodness of fit metrics were computed on a second set of simulated graphs for the median of the estimated parameters and other hypothetical values for the distribution parameters the results suggest that the distribution of the parameters from simulated graphs are significantly different from the theoretical distribution and is also dependent on m further results confirm the finding that the parameter of the power law distribution increases as m increases

the order of magnitude for moments for certain cotangent sums abstract we settle a question on the rate of growth of the moments of cotangent sums considered by the authors in their previous papers 7 8

extreme statistics of complex random and quantum chaotic states complex random states have the statistical properties of the gaussian and circular unitary ensemble eigenstates of random matrix theory even though their components are correlated by the normalization constraint it is nevertheless possible to derive compact analytic formulas for their extreme values statistical properties for all dimensionalities the maximum intensity result slowly approaches the gumbel distribution even though the variables are bounded whereas the minimum intensity result rapidly approaches the weibull distribution since random matrix theory is conjectured to be applicable to chaotic quantum systems we calculate the extreme eigenfunction statistics for the standard map with parameters at which its classical map is fully chaotic the statistical behaviors are consistent with the finite n formulas

airy distribution function from the area under a brownian excursion to the maximal height of fluctuating interfaces the airy distribution function describes the probability distribution of the area under a brownian excursion over a unit interval surprisingly this function has appeared in a number of seemingly unrelated problems mostly in computer science and graph theory in this paper we show that this distribution function also appears in a rather well studied physical system namely the fluctuating interfaces we present an exact solution for the distribution p h m l of the maximal height h m measured with respect to the average spatial height in the steady state of a fluctuating interface in a one dimensional system of size l with both periodic and free boundary conditions for the periodic case we show that p h m l l 1 2f h m l 1 2 for all l 0 where the function f x is the airy distribution function this result is valid for both the edwards wilkinson ew and the kardar parisi zhang interfaces for the free boundary case the same scaling holds p h m l l 1 2f h m l 1 2 but the scaling function f x is different from that of the periodic case we compute this scaling function explicitly for the ew interface and call it the f airy distribution function numerical simulations are in excellent agreement with our analytical results our results provide a rather rare exactly solvable case for the distribution of extremum of a set of strongly correlated random variables some of these results were announced in a recent letter s n majumdar and a comtet phys rev lett 92 225501 2004

analytic solutions of a class of iterative functional differential equations analytic solutions are obtained for iterative functional differential equations that are natural extensions of x 1 x x we make use of neutral functional differential equations with proportional delays as well as neutral differential difference equations for achieving our purposes

statistics of extremal intensities for gaussian interfaces the extremal fourier intensities are studied for stationary edwards wilkinson type gaussian interfaces with power law dispersion we calculate the probability distribution of the maximal intensity and find that generically it does not coincide with the distribution of the integrated power spectrum i e roughness of the surface nor does it obey any of the known extreme statistics limit distributions the fisher tippett gumbel limit distribution is however recovered in three cases i in the nondispersive white noise limit ii for high dimensions and iii when only short wavelength modes are kept in the last two cases the limit distribution emerges in nonconventional scenarios

extreme value statistics of hierarchically correlated variables deviation from gumbel statistics and anomalous persistence we study analytically the distribution of the minimum of a set of hierarchically correlated random variables e1 e2 ellipsis e n where e i represents the energy of the ith path of a directed polymer on a cayley tree if the variables were uncorrelated the minimum energy would have an asymptotic gumbel distribution we show that due to the hierarchical correlations the forward tail of the distribution of the minimum energy becomes highly nonuniversal depends explicitly on the distribution of the bond energies epsilon and is generically different from the superexponential forward tail of the gumbel distribution the consequence of these results to the persistence of hierarchically correlated random variables is discussed and the persistence is also shown to be generically anomalous

width distribution of curvature driven interfaces a study of universality one dimensional interfaces with curvature driven growth kinetics are investigated we calculate the steady state distribution p mathit w 2 of the square of the width of the interface mathit w 2 and show that as in the case for random walk interfaces the result can be written in a scaling form mathit w 2 p mathit w 2 ensuremath phi mathit w 2 mathit w 2 where mathit w 2 is the average of mathit w 2 the scaling function ensuremath phi x is found to be distinct from that of random walk interfaces but as our monte carlo simulations indicate this function is universal for curvature driven growth it is argued that comparison of scaling functions can be a useful method for distinguishing between universality classes of growth processes

the minimal spectral radius of graphs with a given diameter the spectral radius of a graph i e the largest eigenvalue of its corresponding adjacency matrix plays an important role in modeling virus propagation in networks in fact the smaller the spectral radius the larger the robustness of a network against the spread of viruses among all connected graphs on n nodes the path pn has minimal spectral radius however its diameter d i e the maximum number of hops between any pair of nodes in the graph is the largest possible namely d n 1 in general communication networks are designed such that the diameter is small because the larger the number of nodes traversed on a connection the lower the quality of the service running over the network this leads us to state the following problem which connected graph on n nodes and a given diameter d has minimal spectral radius in this paper we solve this problem explicitly for graphs with diameter d fenced 1 2 fenced frac n 2 n 3 n 2 n 1 moreover we solve the problem for almost all graphs on at most 20 nodes by a computer search 2007 elsevier inc all rights reserved

degree distribution and assortativity in line graphs of complex networks topological characteristics of links of complex networks influence the dynamical processes executed on networks triggered by links such as cascading failures triggered by links in power grids and epidemic spread due to link infection the line graph transforms links in the original graph into nodes in this paper we investigate how graph metrics in the original graph are mapped into those for its line graph in particular we study the degree distribution and the assortativity of a graph and its line graph specifically we show both analytically and numerically the degree distribution of the line graph of an erdos renyi graph follows the same distribution as its original graph we derive a formula for the assortativity of line graphs and indicate that the assortativity of a line graph is not linearly related to its original graph additionally line graphs of various graphs e g erdos renyi graphs scale free graphs show positive assortativity in contrast we find certain types of trees and non trees whose line graphs have negative assortativity 2015 elsevier b v all rights reserved

extreme value statistics and traveling fronts various applications abstract an intriguing connection between extreme value statistics and traveling fronts has been found recently in a number of diverse problems in this short review we outline a few such problems and consider their various applications

distribution of the time at which n vicious walkers reach their maximal height we study the extreme statistics of n nonintersecting brownian motions vicious walkers over a unit time interval in one dimension using path integral techniques we compute exactly the joint distribution of the maximum m and of the time m at which this maximum is reached we focus in particular on nonintersecting brownian bridges watermelons without wall and nonintersecting brownian excursions watermelons with a wall we discuss in detail the relationships between such vicious walkers models in watermelon configurations and stochastic growth models in curved geometry on the one hand and the directed polymer in a disordered medium dprm with one free end point on the other hand we also check our results using numerical simulations of dyson s brownian motion and confront them with numerical simulations of the polynuclear growth model png and of a model of dprm on a discrete lattice some of the results presented here were announced in a recent letter j rambeau and g schehr europhys lett 91 60006 2010

existence of analytic solutions of an iterative functional equation abstract this paper is concern analytic solutions of an iterative functional equation of the form f p z q f z h f z z c byconstructing a convergent power series solution of an auxiliary equation g 2 z p z h g z p z z c analytic solutions of the original equation are obtained we discuss not only these appeared in the auxiliary equation at the hyperbolic case 0

extremal point densities of interface fluctuations we introduce and investigate the stochastic dynamics of the density of local extrema minima and maxima of nonequilibrium surface fluctuations we give a number of analytic results for interface fluctuations described by linear langevin equations and for on lattice solid on solid surface growth models we show that in spite of the nonuniversal character of the quantities studied their behavior against the variation of the microscopic length scales can present generic features characteristic of the macroscopic observables of the system the quantities investigated here provide us with tools that give an unorthodox approach to the dynamics of surface morphologies a statistical analysis from the short wavelength end of the fourier decomposition spectrum in addition to surface growth applications our results can be used to solve the asymptotic scalability problem of massively parallel algorithms for discrete event simulations which are extensively used in monte carlo simulations on parallel architectures

order statistics of 1 fα signals order statistics of periodic gaussian noise with 1 f power spectrum is investigated using simulations and phenomenological arguments we find three scaling regimes for the average gap d k x k x k 1 between the kth and k 1 st largest values of the signal the result d k k 1 known for independent identically distributed variables remains valid for 0 5 the spectra of average ordered values e k x 1 x k k is also examined the exponent is derived from the gap scaling as well as by relating e k to the density of near extreme states known results for the density of near extreme states combined with scaling suggest that 2 1 2 4 3 2 and 2 are exact values we also show that parallels can be drawn between e k and the quantum mechanical spectra of a particle in power law potentials

the brjuno functions and their regularity properties we show that various possible versions of the brjuno function based on different kinds of continued fraction developments are all equivalent and we study their regularity l p bmo and holder properties through a systematic analysis of the functional equation which they fulfill

scaling and commonality in anomalous fluctuation statistics in models for turbulence and ferromagnetism recently portelli et al 2003 phys rev lett 90 104501 have semi numerically obtained a functional form of the probability distribution of fluctuations in the total energy flow in a model for fluid turbulence this follows earlier work suggesting that fluctuations in the total magnetization in the 2d xy model for a ferromagnet also follow this distribution here starting from the scaling ansatz that is the basis of the turbulence model we analytically derive the functional form of this distribution and find its single control parameter that depends upon the scaling exponents and system size of the model our analysis allows us to identify this explicitly with that of the xy model and suggest a possible generalization

local invertible analytic solutions for an iterative differential equation related to a discrete derivatives sequence abstract in this paper we are concerned with the existence of analytic solutions of a class of iterative differential equation f z 1 k f 1 z a 1 f 2 z a 2 f n z a n in the complex field c where k c 0 a i r f i z denotes ith iterate of f z i 1 2 n the above equation is closely related to a discrete derivatives sequence f m see y f s petermann jean luc remy ilan vardi discrete derivative of sequences adv in appl math 27 2001 562 584 we first give the existence of analytic solutions of the form of power functions for such an equation then by constructing a convergent power series solution y z of an auxiliary equation of the form x z k x z x z a 1 x 2 z a 2 x n z a n invertible analytic solutions of the form f z x x 1 z for the original equation are obtained we discuss not only the constant at resonance i e at a root of the unity but also those near resonance near a root of the unity under the brjuno condition

magnetic fluctuations in the classical xy model the origin of an exponential tail in a complex system we study the probability density function for the fluctuations of the magnetic order parameter in the low temperature phase of the mathrm xy model of finite size in two dimensions this system is critical over the whole of the low temperature phase it is shown analytically and without recourse to the scaling hypothesis that in this case the distribution is non gaussian and of universal form independent of both system size and critical exponent ensuremath eta an exact expression for the generating function of the distribution is obtained which is transformed and compared with numerical data from high resolution molecular dynamics and monte carlo simulations the asymptotes of the distribution are calculated and found to be of exponential and double exponential form the calculated distribution is fitted to three standard functions a generalization of gumbel s first asymptote distribution from the theory of extremal statistics a generalized log normal distribution and a ensuremath chi 2 distribution the calculation is extended to general dimension and an exponential tail is found in all dimensions less than 4 despite the fact that critical fluctuations are limited to d 2 these results are discussed in the light of similar behavior observed in models of interface growth and for dissipative systems driven into a nonequilibrium steady state

random convex hulls and extreme value statistics in this paper we study the statistical properties of convex hulls of n random points in a plane chosen according to a given distribution the points may be chosen independently or they may be correlated after a non exhaustive survey of the somewhat sporadic literature and diverse methods used in the random convex hull problem we present a unifying approach based on the notion of support function of a closed curve and the associated cauchy s formulae that allows us to compute exactly the mean perimeter and the mean area enclosed by the convex polygon both in case of independent as well as correlated points our method demonstrates a beautiful link between the random convex hull problem and the subject of extreme value statistics as an example of correlated points we study here in detail the case when the points represent the vertices of n independent random walks in the continuum time limit this reduces to n independent planar brownian trajectories for which we compute exactly for all n the mean perimeter and the mean area of their global convex hull our results have relevant applications in ecology in estimating the home range of a herd of animals some of these results were announced recently in a short communication phys rev lett 103 140602 2009

a proof of the marmi moussa yoccoz conjecture for rotation numbers of high type marmi moussa and yoccoz conjectured that some error function upsilon related to the approximation of the size of siegel disks by some arithmetic function of the rotation number theta is a holder continuous function of theta with exponent 1 2 using the renormalization invariant class of inou and shishikura we prove this conjecture for the restriction of upsilon to a class of high type numbers

which graphs are determined by their spectrum for almost all graphs the answer to the question in the title is still unknown here we survey the cases for which the answer is known not only the adjacency matrix but also other types of matrices such as the laplacian matrix are considered

extreme statistics for time series distribution of the maximum relative to the initial value the extreme statistics of time signals is studied when the maximum is measured from the initial value in the case of independent identically distributed iid variables we classify the limiting distribution of the maximum according to the properties of the parent distribution from which the variables are drawn then we turn to correlated periodic gaussian signals with a 1 f ensuremath alpha power spectrum and study the distribution of the maximum relative height with respect to the initial height mathrm mrh mathrm i the exact mathrm mrh mathrm i distribution is derived for ensuremath alpha 0 iid variables ensuremath alpha 2 random walk ensuremath alpha 4 random acceleration and ensuremath alpha ensuremath infty single sinusoidal mode for other intermediate values of ensuremath alpha the distribution is determined from simulations we find that the mathrm mrh mathrm i distribution is markedly different from the previously studied distribution of the maximum height relative to the average height for all ensuremath alpha the two main distinguishing features of the mathrm mrh mathrm i distribution are the much larger weight for small relative heights and the divergence at zero height for ensuremath alpha g3 we also demonstrate that the boundary conditions affect the shape of the distribution by presenting exact results for some nonperiodic boundary conditions finally we show that for signals arising from time translationally invariant distributions the density of near extreme states is the same as the mathrm mrh mathrm i distribution this is used in developing a scaling theory for the threshold singularities of the two distributions

maximal and minimal height distributions of fluctuating interfaces maximal and minimal height distributions mahd mihd of two dimensional interfaces grown with the nonlinear equations of kardar parisi zhang kpz second order and of villain lai das sarma vlds fourth order are shown to be different two universal curves may be mahd or mihd of each class depending on the sign of the relevant nonlinear term which is confirmed by results of several lattice models in the kpz and vlds classes the difference between mahd and midh is connected with the asymmetry of the local height distribution a simple exactly solvable deposition erosion model is introduced to illustrate this feature the average extremal heights scale with the same exponent of the average roughness in contrast to other correlated systems generalized gumbel distributions do not fit those mahd and mihd nor those of edwards wilkinson growth

power fluctuations in a closed turbulent shear flow we study experimentally the power consumption p of a confined turbulent flow at constant reynolds number re we analyze in details its temporal dynamics and statistical properties in a setup that covers two decades in reynolds numbers we show that nontrivial power fluctuations occur over a wide range of amplitudes and that they involve coherent fluid motions over the entire system size as a result the power fluctuations do not result from averaging of independent subsystems and its probability density function pi p is strongly non gaussian the shape of pi p is reynolds number independant and we show that the relative intensity of fluctuations decreases very slowly as re increases these results are discussed in terms of an analogy with critical phenomena

universal asymptotic statistics of maximal relative height in one dimensional solid on solid models we study the probability density function p h m l of the maximum relative height h m in a wide class of one dimensional solid on solid models of finite size l for all these lattice models in the large l limit a central limit argument shows that for periodic boundary conditions p h m l takes a universal scaling form p h m l ensuremath sim sqrt 12 w l ensuremath 1 f mathbf h m sqrt 12 w l mathbf with w l the width of the fluctuating interface and f x the airy distribution function for one instance of these models corresponding to the extremely anisotropic ising model in two dimensions this result is obtained by an exact computation using the transfer matrix technique valid for any lg0 these arguments and exact analytical calculations are supported by numerical simulations which show in addition that the subleading scaling function is also universal up to a nonuniversal amplitude and simply given by the derivative of the airy distribution function f ensuremath x

explicit estimates of sums related to the nyman beurling criterion for the riemann hypothesis abstract we give an estimate for sums appearing in the nyman beurling criterion for the riemann hypothesis these sums contain the mobius function and are related to the imaginary part of the estermann zeta function the estimate is remarkably sharp in comparison to other sums containing the mobius function the bound is smaller than the trivial bound essentially the number of terms by a fixed power of that number the exponent is made explicit the methods intensively use tools from the theory of continued fractions and from the theory of fourier series

global fluctuations and gumbel statistics we explain how the statistics of global observables in correlated systems can be related to extreme value problems and to gumbel statistics this relationship then naturally leads to the emergence of the generalized gumbel distribution ga x with a real index a in the study of global fluctuations to illustrate these findings we introduce an exactly solvable nonequilibrium model describing an energy flux on a lattice with local dissipation in which the fluctuations of the global energy are precisely described by the generalized gumbel distribution

density of near extreme events we provide a quantitative analysis of the phenomenon of crowding of near extreme events by computing exactly the density of states dos near the maximum of a set of independent and identically distributed random variables we show that the mean dos converges to three different limiting forms depending on whether the tail of the distribution of the random variables decays slower than pure exponential faster than pure exponential or as a pure exponential function we argue that some of these results would remain valid even for certain correlated cases and verify it for power law correlated stationary gaussian sequences satisfactory agreement is found between the near maximum crowding in the summer temperature reconstruction data of western siberia and the theoretical prediction

maximal height statistics for 1 f alpha signals numerical and analytical results are presented for the maximal relative height distribution of stationary periodic gaussian signals one dimensional interfaces displaying a 1 f alpha power spectrum for 0 1 regime of strong correlations and a highly accurate picture gallery of distribution functions can be constructed numerically analytical results can be obtained in the limit alpha infinity and for large alpha by perturbation expansion furthermore using path integral techniques we derive a trace formula for the distribution function valid for alpha 2n even integer from the latter we extract the small argument asymptote of the distribution function whose analytic continuation to arbitrary alpha 1 is found to be in agreement with simulations comparison of the extreme and roughness statistics of the interfaces reveals similarities in both the small and large argument asymptotes of the distribution functions