spatial models for species area curves the relationship between species number and area is an old problem in biology we propose here an interacting particle system the multitype voter model with mutation as a mathematical model to study this problem we analyze the species area curves of this model as the mutation rate a tends to zero we obtain two basic types of behavior depending on the size of the spatial region under consideration if the region is a square with area r r 1 then for small the number of species is of order l r log 2 whereas if r 1 the number of species is bounded

survival and coexistence for a multitype contact process we study the ergodic theory of a multitype contact process with equal death rates and unequal birth rates on the d dimensional integer lattice and regular trees we prove that for birth rates in a certain interval there is coexistence on the tree which by a result of neuhauser is not possible on the lattice we also prove a complete convergence result when the larger birth rate falls outside of this interval

on optimal spatial subsample size for variance estimation we consider the problem of determining the optimal block or subsample size for a spatial subsampling method for spatial processes observed on regular grids we derive expansions for the mean square error of the subsampling variance estimator which yields an expression for the theoretically optimal block size the optimal block size is shown to depend in an intricate way on the geometry of the spatial sampling region as well as characteristics of the underlying random field final expressions for the optimal block size make use of some nontrivial estimates of lattice point counts in shifts of convex sets optimal block sizes are computed for sampling regions of a number of commonly encountered shapes numerical studies are performed to compare subsampling methods as well as procedures for estimating the theoretically best block size

weyl asymptotics of bisingular operators and dirichlet divisor problem we consider a class of pseudodifferential operators with crossed vector valued symbols defined on the product of two closed manifolds we study the asymptotic expansion of the counting function of positive selfadjoint operators in this class using a general theorem of aramaki we can determine the first term of the asymptotic expansion of the counting function and in a special case we are able to find the second term we give also some examples emphasizing connections with problems of analytic number theory in particular with dirichlet divisor function

score statistics of global sequence alignment from the energy distribution of a modified directed polymer and directed percolation problem sequence alignment is one of the most important bioinformatics tools for modern molecular biology the statistical characterization of gapped alignment scores has been a long standing problem in sequence alignment research using a variant of the directed path in random media model we investigate the score statistics of global sequence alignment taking into account in particular the compositional bias of the sequences compared such statistics are used to distinguish accidental similarity due to compositional similarity from biologically significant similarity to accommodate the compositional bias we introduce an extra parameter p indicating the probability for positive matching scores to occur when p is small a high scoring alignment obviously cannot come from compositional similarity when p is large the highest scoring point within a global alignment tends to be close to the end of both sequences in which case we say the system percolates by applying finite size scaling theory on percolating probability functions of various sizes sequence lengths the critical p at infinite size is obtained for alignment of length t the fact that the score fluctuation grows as chi t 1 3 is confirmed upon investigating the scaling form of the alignment score using the kolmogorov smirnov statistics test we show that the random variable if properly scaled follows the tracy widom distributions gaussian orthogonal ensemble for p slightly larger than pc and gaussian unitary ensemble for larger p although these results deepen our understanding of the distribution of alignment scores the use of these results in practical applications remains somewhat heuristic and needs to be further developed nevertheless the possibility of characterizing score statistics for modest system size sequence lengths via proper reparametrization of alignment scores is illustrated

a new proof of weyl s formula on the asymptotic distribution of eigenvalues one of the more perplexing problems of finite particle quantum mechanics is to find the appropriate quantum mechanical description of a given classical mechanical system unfortunately from the mathematical point of view this problem is not terribly well posed suppose the classical system in question is represented by a symplectic manifold x and a classical observable p x r suppose also for simplicity that this system is a confined system in the sense that the sets p 1 are all compact let h be a hilbert space and let p be a self adjoint operator on h representing a quantum observable without specifying anything further about p for instance its relation to other observables there is practically no way of ascertaining whether p is an appropriate quantum counterpart of p at first glance the only requirements this appears to impose on p are that the spectrum of p be discrete and that the number of eigenvalues n a less than 1 be finite for all 1 this is the analogue of the condtion that p 1 be compact by examining the statistical properties of a large aggregate of classical systems each a copy of x p one can in fact say a little bit more about the relation p to p by comparing classical and quantum partition functions one can show that for iz large

nonlinear complexity of random visibility graph and lempel ziv on multitype range intensity interacting financial dynamics abstract in an attempt to investigate the nonlinear complex evolution of financial dynamics a new financial price model the multitype range intensity contact mric financial model is developed based on the multitype range intensity interacting contact system in which the interaction and transmission of different types of investment attitudes in a stock market are simulated by viruses spreading two new random visibility graph vg based analyses and lempel ziv complexity lzc are applied to study the complex behaviors of return time series and the corresponding random sorted series the vg method is the complex network theory and the lzc is a non parametric measure of complexity reflecting the rate of new pattern generation of a series in this work the real stock market indices are considered to be comparatively studied with the simulation data of the proposed model further the numerical empirical study shows the similar complexity behaviors between the model and the real markets the research confirms that the financial model is reasonable to some extent

le spectre de certaines mosaiques poissoniennes du plan et l enveloppe convexe du pont brownien the spectral empirical function of a homogeneous isotropic poisson mosaic process is a functional of the perimeter of the convex hull of planar brownian bridge some geometrical identities follow

bounded imaginary powers of differential operators on manifolds with conical singularities we study the minimal and maximal closed extension of a differential operator a on a manifold b with conical singularities when a acts as an unbounded operator on weighted l p spaces over b 1 p under suitable ellipticity assumptions we can define a family of complex powers a z z we also obtain sufficient information on the resolvent of a to show the boundedness of the purely imaginary powers examples concern unique solvability and maximal regularity for the solution of the cauchy problem for the laplacian on conical manifolds as well as certain quasilinear diffusion equations

shape fluctuations and random matrices we study a certain random growth model in two dimensions closely related to the one dimensional totally asymmetric exclusion process the results show that the shape fluctuations appropriately scaled converges in distribution to the tracy widom largest eigenvalue distribution for the gaussian unitary ensemble gue

sign changes of the error term in weyl s law for heisenberg manifolds let r t be the error term in weyl s law for the 2l 1 dimensional heisenberg manifold hl gl in this paper several results on the sign changes and odd moments of r t are proved in particular it is proved that for some sufficiently large constant c r t changes sign in the interval t t c t for all large t moreover for a small constant c1 there exist infinitely many subintervals in t 2t of length c1 t log 5 t such that r t c1tl 1 4 holds on each of these subintervals

on the distribution of the length of the longest increasing subsequence of random permutations let sn be the group of permutations of 1 2 n if 7r e sn we say that 7 i1 7f ik is an increasing subsequence in 7r if il i2 ik and 7r ii 7r i2 7r ik let 1n r be the length of the longest increasing subsequence for example if n 5 and 7r is the permutation 5 1 3 2 4 in one line notation thus 7r 1 5 7r 2 1 then the longest increasing subsequences are 1 2 4 and 1 3 4 and n 3 equip sn with uniform distribution

unzipping of two random heteropolymers ground state energy and finite size effects we consider a pair of random heteropolymer chains with quenched primary sequences for this system we have analyzed the dependence of average ground state energy per monomer e on chain length n in the ensemble of chains with uniform distribution of primary sequences of monomers every monomer of the first second chain is randomly and independently chosen with the uniform probability distribution p 1 c from a set of c different types a b c d a b c d monomers of the first chain could form saturating reversible bonds with monomers of the second chain the bonds between similar monomer types such as a a b b c c etc have the attraction energy u while the bonds between different monomer types such as a b a d b d etc have the attraction energy v the main attention is paid to the computation of the normalized free energy e n for intermediate chain lengths n and different ratios a v u at sufficiently low temperatures when the entropic contribution of the loop formation is negligible compared to direct energetic interactions between chain monomers and when the partition function of the chains is dominated by the ground state the performed analysis allows one to derive the force f x which is necessary to apply for unzipping of two random heteropolymers of equal lengths whose ends are separated by the distance x averaged over all equally distributed primary structures at low temperatures for fixed values a and c

efficiency of characterizing ellipses and ellipsoids by discrete moments in this paper our studies are focused on ellipses and problems related to their representation and reconstruction from the data resulting from their digitization the main result of the paper is that a finite number of discrete moments corresponded to digital ellipses is in one to one correspondence with digital ellipses which enables coding of digital ellipses with an asymptotically optimal amount of memory in addition the problem of reconstruction based on the same parameters is considered since the digitization of real shapes causes an inherent loss of information about the original objects the precision of the original shape estimation from the corresponding digital data is limited we derive a sharp upper bound for the errors in reconstruction of the center position and half axes of the ellipse in function of the applied picture resolution i e the number of pixels per unit an extension of these results to the 3d case is also given

the critical contact process dies out by producing a finite box criterion for survival in a slab we show that the critical contact process dies out and that in two and more dimensions the critical parameter value coincides with the limit of slab thresholds using the techniques developed in this paper one may obtain the complete convergence and shape theorems for contact processes in all dimensions and for all infection rates our results apply also to the discrete time analogue of the contact process viz oriented percolation

new pathways and connections in number theory and analysis motivated by two incorrect claims of ramanujan abstract the focus of this paper commences with an examination of three not obviously related pages in ramanujan s lost notebook pages 336 335 and 332 in decreasing order of attention on page 336 ramanujan proposes two identities but the formulas are wrong each is vitiated by divergent series we concentrate on only one of the two incorrect identities which may have been devised to attack the extended divisor problem we prove here a corrected version of ramanujan s claim which contains the convergent series appearing in it the convergent series in ramanujan s faulty claim is similar to one used by g f voronoi g h hardy and others in their study of the classical dirichlet divisor problem this now brings us to page 335 which comprises two formulas featuring doubly infinite series of bessel functions the first being conjoined with the classical circle problem initiated by gauss and the second being associated with the dirichlet divisor problem the first and fourth authors along with sun kim have written several papers providing proofs of these two difficult formulas in different interpretations in this monograph we return to these two formulas and examine them in more general settings the aforementioned convergent series in ramanujan s identity is also similar to one that appears in a curious identity found in chapter 15 in ramanujan s second notebook written in a more elegant equivalent formulation on page 332 in the lost notebook this formula may be regarded as a formula for 1 2 and in 1925 s wigert obtained a generalization giving a formula for 1 k for any even integer k 2 we extend the work of ramanujan and wigert in this paper the voronoi summation formula appears prominently in our study in particular we generalize work of j r wilton and derive an analogue involving the sum of divisors function s n the modified bessel functions k s x arise in several contexts as do lommel functions we establish here new series and integral identities involving modified bessel functions and modified lommel functions among other results we establish a modular transformation for an infinite series involving s n and modified lommel functions we also discuss certain obscure related work of n s koshliakov we define and discuss two new related classes of integral transforms which we call koshliakov transforms because he first found elegant special cases of each

local average in hyperbolic lattice point counting with an appendix by niko laaksonen the hyperbolic lattice point problem asks to estimate the size of the orbit gamma z inside a hyperbolic disk of radius cosh 1 x 2 for gamma a discrete subgroup of hbox psl 2 mathbb r selberg proved the estimate o x 2 3 for the error term for cofinite or cocompact groups this has not been improved for any group and any center in this paper local averaging over the center is investigated for hbox psl 2 mathbb z the result is that the error term can be improved to o x 7 12 varepsilon the proof uses surprisingly strong input e g results on the quantum ergodicity of maas cusp forms and estimates on spectral exponential sums we also prove omega results for this averaging consistent with the conjectural best error bound o x 1 2 varepsilon in the appendix the relevant exponential sum over the spectral parameters is investigated

weighted divisor sums and bessel function series on page 335 in his lost notebook ramanujan records without proof an identity involving a finite trigonometric sum and a doubly infinite series of ordinary bessel functions we provide the first published proof of this result the identity yields as corollaries representations of weighted divisor sums in particular the summatory function for r 2 n the number of representations of the positive integer n as a sum of two squares

exponential sums and lattice points iii the gauss circle problem and the dirichlet divisor problem are special cases of the problem of counting the points of the integer lattice in a planar domain bounded by a piecewise smooth curve in the bombieri iwaniec mozzochi exponential sums method we must count the number of pairs of arcs of the boundary curve which can be brought into coincidence by an automorphism of the integer lattice these coincidences are parametrised by integer points close to certain plane curves the resonance curves this paper sets up an iteration step from a strong hypothesis about integer points close to curves to a bound for the discrepancy the number of integer points minus the area as in the latest work on single exponential sums the bombieri iwaniec mozzochi method itself gives bounds for the number of integer points close to a curve in part of the required range and it can in principle be used iteratively we use a bound obtained by swinnerton dyer s approximation determinant method in the discrepancy estimate o r k log r lambda in terms of the maximum radius of curvature r we reduce k from 2 3 classical and 46 73 paper ii in this series to 131 208 the corresponding exponent in the dirichlet divisor problem becomes k 2 131 416

on lattice sums and wigner limits abstract wigner limits are given formally as the difference between a lattice sum associated to a positive definite quadratic form and a corresponding multiple integral to define these limits which arose in work of wigner on the energy of static electron lattices in a mathematically rigorous way one commonly truncates the lattice sum and the corresponding integral and takes the limit along expanding hypercubes or other regular geometric shapes we generalize the known mathematically rigorous two and three dimensional results regarding wigner limits as laid down in 3 to integer lattices of arbitrary dimension in doing so we also resolve a problem posed in 6 chapter 7 for the sake of clarity we begin by considering the simpler case of cubic lattice sums first before treating the case of arbitrary quadratic forms we also consider limits taken along expanding hyperballs with respect to general norms and connect with classical topics such as gauss s circle problem appendix a is included to recall certain properties of epstein zeta functions that are either used in the paper or serve to provide perspective

exact solution for the kardar parisi zhang equation with flat initial conditions we provide the first exact calculation of the height distribution at arbitrary time t of the continuum kardar parisi zhang kpz growth equation in one dimension with flat initial conditions we use the mapping onto a directed polymer with one end fixed one free and the bethe ansatz for the replicated attractive boson model we obtain the generating function of the moments of the directed polymer partition sum as a fredholm pfaffian our formula valid for all times exhibits convergence of the free energy i e kpz height distribution to the gaussian orthogonal ensemble tracy widom distribution at large time

the δn statistic for the β hermite ensemble abstract the fluctuation n of the n th unfolded eigenvalue was recently characterized for the classical gaussian ensembles of n n random matrices goe gue gse it is investigated here for the hermite ensemble as a function of the reciprocal of the temperature by monte carlo simulations the ensemble averaged fluctuation n 2 and the autocorrelation function vary logarithmically with n for any 0 1 n n the simple logarithmic behavior of the higher order moments of n reported in the literature for the goe 1 and the gue 2 holds for any 0 and is accounted for by gaussian distributions whose variances depend linearly on ln n

the central limit theorem for the right edge of supercritical oriented percolation on demontre un theoreme central limite pour la croissance des bords droits d une percolation orientee supercritique la technique de demonstration utilisee est de trouver des points avec des proprietes d un type de regeneration appeles points de ruptures cette technique peut etre appliquee a d autres processus

stochastic spatial models in the models we will consider space is represented by a grid of sites that can be in one of a finite number of states and that change at rates that depend on the states of a finite number of sites our main aim here is to explain an idea of durrett and levin 1994 the behavior of these models can be predicted from the properties of the mean field ode i e the equations for the densities of the various types that result from pretending that all sites are always independent we will illustrate this picture through a discussion of eight families of examples from statistical mechanics genetics population biology epidemiology and ecology some of our findings are only conjectures based on simulation but in a number of cases we are able to prove results for systems with fast stirring by exploiting connections between the spatial model and an associated reaction diffusion equation

exact asymptotic results for the bernoulli matching model of sequence alignment finding analytically the statistics of the longest common subsequence lcs of a pair of random sequences drawn from c alphabets is a challenging problem in computational evolutionary biology we present exact asymptotic results for the distribution of the lcs in a simpler yet nontrivial variant of the original model called the bernoulli matching bm model we show that in the bm model for all c the distribution of the asymptotic length of the lcs suitably scaled is identical to the tracy widom distribution of the largest eigenvalue of a random matrix whose entries are drawn from a gaussian unitary ensemble

ergodic theorems for the multitype contact process this paper studies a process involving competition of two types of particles 1 and 2 for the empty space 0 each site of the latticez d is therefore in one of three possible states 0 1 or 2 particles of each type die with rate 1 while an empty site becomes occupied by a particle of typei with rate i proportion of neighbors of typei the set of neighbors of a sitex is of the form y x y j for a positive integerj and a norm assuming there are only 0 s and 1 s present at the beginning the process reduces to the contact process with the critical rate of survival of 1 s being c the basic problem we address is the existence of equilibria in which both types of particles coexist without loss of generality one can restrict to the case 2 1 c and in this case we show 1 if 2 1 and the initial state is translation invariant and contains infinitely many 2 s then the 1 s go away and the process approaches the invariant measure of the contact process with only 2 s and 0 s present 2 if 2 1 andd 2 then clustering occurs starting from a translation invariant initial measure with no mass on all 0 s the process converges weakly to a convex combination of the two invariant measures obtained with only one type of particles present and 3 if 2 1 andd 3 then there is a one parameter family of invariant measures including both types

mutually avoiding paths in random media and largest eigenvalues of random matrices recently it was shown that the probability distribution function pdf of the free energy of a single continuum directed polymer dp in a random potential equivalently to the height of a growing interface described by the kardar parisi zhang kpz equation converges at large scale to the tracy widom distribution the latter describes the fluctuations of the largest eigenvalue of a random matrix drawn from the gaussian unitary ensemble gue and the result holds for a dp with fixed end points i e for the kpz equation with droplet initial conditions a more general conjecture can be put forward relating the free energies of n 1 noncrossing continuum dp in a random potential to the sum of the nth largest eigenvalues of the gue here using replica methods we provide an important test of this conjecture by calculating exactly the right tails of both pdfs and showing that they coincide for arbitrary n

bethe ansatz in the bernoulli matching model of random sequence alignment for the bernoulli matching model of the sequence alignment problem we apply the bethe ansatz technique via an exact mapping to the five vertex model on a square lattice considering the terracelike representation of the sequence alignment problem we reproduce by the bethe ansatz the results for the averaged length of the longest common subsequence in the bernoulli approximation in addition we compute the average number of nucleation centers of the terraces

anisotropic ballistic deposition model with links to the ulam problem and the tracy widom distribution we compute exactly the asymptotic distribution of scaled height in a 1 1 dimensional anisotropic ballistic deposition model by mapping it to the ulam problem of finding the longest nondecreasing subsequence in a random sequence of integers using the known results for the ulam problem we show that the scaled height in our model has the tracy widom distribution appearing in the theory of random matrices near the edges of the spectrum our result supports the hypothesis that various growth models in 1 1 dimensions that belong to the kardar parisi zhang universality class perhaps all share the same universal tracy widom distribution for the suitably scaled height variables

weyl laws on open manifolds under suitable invertibility hypothesis the spectrum of the dirac operator on certain open spin riemannian manifolds is purely discrete and obeys a growth law depending qualitatively on the in finiteness of the volume

convergence of madelung like lattice sums we make a general study of the convergence properties of lattice sums involving potentials of the form occuring in mathematical chemistry and physics many specific examples are studied in detail the prototype is madelung s constant for nacl sum infty infty frac 1 n m p sqrt n 2 m 2 p 2 1 74756459 cdots presuming that one appropriately interprets the summation proccess

weakly parametric pseudodifferential operators and atiyah patodi singer boundary problems this paper introduces a class of pseudodifferential operators depending on a parameter in a particular way the main application is a complete expansion of the trace of the resolvent of a dirac type operator with nonlocal boundary conditions of the kind introduced by atiyah patodi and singer aps this extends the partial expansion in g2 to a complete one and extends the complete expansion in gs 1 to the case where the dirac operator does not have a product structure near the boundary a secondary application is to obtain a complete expansion of the resolvent of a bdo on a compact manifold essentially reproving a result of agranovich agr the resolvent expansion yields immediately an expansion of the trace of the heat kernel and determines the singularities of the zeta function moreover a pseudodifferential factor can be allowed a major motive for these expansions is to obtain index formulas for elliptic operators there are many such applications in the physics and geometry literature the index formula comes from one particular term in the expansion but each term is a spectral invariant and they have been used for other purposes as well as for the index in particular branson and gilkey have a number of papers e g bg and gi analyzing these invariants and drawing geometric consequences interest in the asymptotic behavior of the resolvent goes back to carleman c more recently agmon agm developed it extensively for analytic applications he introduced the fundamental idea of treating the resolvent parameter essentially as another cotangent variable this idea was developed in s1 to analyze the singularities of the zeta function of an elliptic odo on a compact manifold and in 3 to analyze the resolvent of a differential operator with differential boundary conditions the technique works smoothly for differential operators producing so called local invariants integrals over the underlying

mean field approximations to the longest common subsequence problem the longest common subsequence lcs problem is a fundamental problem of sequence comparison a natural approximation to this problem is a model in which every pair of letters of two sequences are matched independently of the other pairs with probability 1 s s representing the size of the alphabet this model is analogous to a mean field version of the lcs problem which can be solved with a cavity approach j boutet de monvel eur phys j b 7 293 1999 we refine here this approximation by incorporating in a systematic way correlations among the matches in the cavity calculation we obtain a series of closer and closer approximations to the lcs problem which we quantify in the large s limit both with a perturbative approach and by monte carlo simulations we find that as it happens in the expansion around mean field for other disordered systems the corrections to our approximations depend upon long ranged correlation effects that render the large s expansion nonperturbative

statistical self similarity of one dimensional growth processes abstract for one dimensional growth processes we consider the distribution of the height above a given point of the substrate and study its scale invariance in the limit of large times we argue that for self similar growth from a single seed the universal distribution is the tracy widom distribution from the theory of random matrices and that for the growth from a flat substrate it is some other only numerically determined distribution in particular for the polynuclear growth model in the droplet geometry the height maps onto the longest increasing subsequence of a random permutation from which the height distribution is identified as the tracy widom distribution

the contact process on a finite set iii the critical case we show that in n is the time that the contact process on 1 n first hits the empty set then for c the critical value for the contact process on z n n and n n 4 0 in probability as n as a consequence of the result we get bounds on some critical exponents we also study the analogous problem for bond percolation in 1 n z and investigate the limit distribution of n e n

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

spectral form factors of rectangle billiards the berry tabor conjecture asserts that local statistical measures of the eigenvalues j of a generic integrable quantum system coincide with those of a poisson process we prove that in the case of a rectangle billiard with random ratio of sides the sum behaves for random and n large like a random walk in the complex plane with a non gaussian limit distribution the expectation value of the distribution is zero its variance which is essentially the average pair correlation function is one in accordance with the berry tabor conjecture but all higher moments 4 diverge the proof of the existence of the limit distribution uses the mixing property of a dynamical system defined on a product of hyperbolic surfaces the berry tabor conjecture and the existence of the limit distribution for a fixed generic rectangle are related to an equidistribution conjecture for long horocycles on this product space

the number of n point digital discs a digital disc is the set of all integer points inside some given disc let dn be the number of different digital discs consisting of n points different up to translation the upper bound d n o n2 was shown recently no corresponding lower bound is known in this paper we refine the upper bound to dn o n which seems to be the true order of magnitude and we show that the average dn d1 d2 dn n has upper and lower bounds which are of polynomial growth in n

long common subsequences and the proximity of two random strings let x 1 x 2 cdots x n and x 1 x 2 cdots x n be two strings from an alphabet mathcal a and let l n denote their longest common subsequence the probabilistic behavior of l n is studied under various probability models for the x s and x s

on the index of elliptic operators on a wedge abstract we propose an analytical approach to the index theory of pseudo differential operators on a manifold with edges it results in an intermediate algebraic index formula the latter permits much more freedom in homotopies and in particular can be transformed to the topological formula

on the recovery of a function on a circular domain we consider the problem of estimating a function f x y on the unit disk f x y x sup 2 y sup 2 spl les 1 given a discrete and noisy data recorded on a regular square grid an estimate of f x y based on a class of orthogonal and complete functions over the unit disk is proposed this class of functions has a distinctive property of being invariant to rotation of axes about the origin of coordinates yielding therefore a rotationally invariant estimate for radial functions the orthogonal set has a particularly simple form being related to the classical legendre polynomials we give the statistical accuracy analysis of the proposed estimate of f x y in the sense of the l sub 2 metric it is found that there is an inherent limitation in the precision of the estimate due to the geometric nature of a circular domain this is explained by relating the accuracy issue to the celebrated problem in the analytic number theory called the lattice points of a circle in fact the obtained bounds for the mean integrated squared error are determined by the best known result so far on the problem of lattice points within the circular domain

universal distributions for growth processes in 1 1 dimensions and random matrices we develop a scaling theory for kardar parisi zhang growth in one dimension by a detailed study of the polynuclear growth model in particular we identify three universal distributions for shape fluctuations and their dependence on the macroscopic shape these distribution functions are computed using the partition function of gaussian random matrices in a cosine potential

the average behaviour of the error term in a new kind of the divisor problem abstract we define a new kind of divisor function d 1 n by the nth coefficient of the dirichlet series s 2 and denote by 1 x the error term in the asymptotic formula for n x d 1 n then we compute the first moment of 1 x that is 1 x 1 x d x and consider discrete mean values n x 1 k n k 1 2 and deduce asymptotic formulas which are analogous to the results obtained by voronoi hardy and furuya

limiting distributions for a polynuclear growth model with external sources the purpose of this paper is to investigate the limiting distribution functions for a polynuclear growth model with two external sources which was considered by prahofer and spohn depending on the strength of the sources the limiting distribution functions are either the tracy widom functions of random matrix theory or a new explicit function which has the special property that its mean is zero moreover we obtain transition functions between pairs of the above distribution functions in suitably scaled limits there are also similar results for a discrete totally asymmetric exclusion process