kardar parisi zhang universality class and the anchored toom interface we revisit the anchored toom interface and use kardar parisi zhang scaling theory to argue that the interface fluctuations are governed by the airy1 process with the role of space and time interchanged the predictions which contain no free parameter are numerically well confirmed for space time statistics in the stationary state in particular the spatial fluctuations of the interface computed numerically agree well with those given by the goe edge distribution of tracy and widom

gaps between integers with the same prime factors we give numerical and theoretical evidence in support of the conjecture of dressler that between any two positive integers having the same prime factors there is a prime in particular it is shown that the abc conjecture implies that the gap between two consecutive such numbers a a 1 2 and it is shown that this lower bound is best possible dressler s conjecture is verified for values of a and c up to 7 10 13

fluctuation properties of the tasep with periodic initial configuration we consider the joint distributions of particle positions for the continuous time totally asymmetric simple exclusion process tasep they are expressed as fredholm determinants with a kernel defining a signed determinantal point process we then consider certain periodic initial conditions and determine the kernel in the scaling limit this result has been announced first in a letter by one of us sasamoto in j phys a 38 l549 l556 2005 and here we provide a self contained derivation connections to last passage directed percolation and random matrices are also briefly discussed

on asymptotics for the airy process the airy process t a t introduced by prahofer and spohn is the limiting stationary process for a polynuclear growth model adler and van moerbeke found a pde in the variables s1 s2 and t for the probability pr a 0 s1 a t s2 using this they were able assuming the truth of a certain conjecture and appropriate uniformity to obtain the first few terms of an asymptotic expansion for this probability as t with fixed s1 and s2 we shall show that the expansion can be obtained by using the fredholm determinant representation for the probability the main ingredients are formulas obtained by the author and c a tracy in the derivation of the painleve ii representation for the distribution function f2 plus a few others obtained in the same way

polynuclear growth on a flat substrate and edge scaling of goe eigenvalues we consider the polynuclear growth png model in 1 1 dimension with flat initial condition and no extra constraints through the robinson schensted knuth rsk construction one obtains the multilayer png model which consists of a stack of non intersecting lines the top one being the png height the statistics of the lines is translation invariant and at a fixed position the lines define a point process we prove that for large times the edge of this point process suitably scaled has a limit this limit is a pfaffian point process and identical to the one obtained from the edge scaling of the gaussian orthogonal ensemble goe of random matrices our results give further insight to the universality structure within the kpz class of 1 1 dimensional growth models

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

the airy1 process is not the limit of the largest eigenvalue in goe matrix diffusion using a systematic approach to evaluate fredholm determinants numerically we provide convincing evidence that the airy1 process arising as a limit law in stochastic surface growth is not the limit law for the evolution of the largest eigenvalue in goe matrix diffusion

spatial correlations of the 1d kpz surface on a flat substrate we study the spatial correlations of the one dimensional kpz surface for the flat initial condition it is shown that the multi point joint distribution for the height is given by a fredholm determinant with its kernel in the scaling limit explicitly obtained this may also describe the dynamics of the largest eigenvalue in the goe dyson s brownian motion model our analysis is based on a reformulation of the determinantal green s function for the totally asep in terms of a vicious walk problem

local behavior and hitting probabilities of the text airy _1 process we obtain a formula for the n dimensional distributions of the text airy 1 process in terms of a fredholm determinant on l 2 mathbb r as opposed to the standard formula which involves extended kernels on l 2 1 dots n times mathbb r the formula is analogous to an earlier formula of prahofer and spohn j stat phys 108 5 6 1071 1106 2002 for the text airy 2 process using this formula we are able to prove that the text airy 1 process is holder continuous with exponent frac 1 2 and that it fluctuates locally like a brownian motion we also explain how the same methods can be used to obtain the analogous results for the text airy 2 process as a consequence of these two results we derive a formula for the continuum statistics of the text airy 1 process analogous to that obtained in corwin et al commun math phys 2011 to appear for the text airy 2 process

some examples related to the abc conjecture for algebraic number fields we present a numerical method for finding extreme examples of identities related to the uniform abc conjecture for algebraic number fields

a pfaffian representation for flat asep we obtain a fredholm pfaffian formula for an appropriate generating function of the height function of the asymmetric simple exclusion process starting from flat periodic initial data formal asymptotics lead to the goe tracy widom distribution 2016 wiley periodicals inc

directed last passage percolation and random matrices this paper is a survey of various techniques recent results and applications of directed last passage percolation model and its relation to random matrices the connection can be explained mainly via the universality of famous tracy widom distribution and airy process some unsolved problems are also surveyed

discrete polynuclear growth and determinantal processes we consider a discrete polynuclear growth png process and prove a functional limit theorem for its convergence to the airy process this generalizes previous results by prahofer and spohn the result enables us to express the f1 goe tracy widom distribution in terms of the airy process we also show some results and give a conjecture about the transversal fluctuations in a point to line last passage percolation problem furthermore we discuss a rather general class of measures given by products of determinants and show that these measures have determinantal correlation functions

p harmonic functions with boundary data having jump discontinuities and baernstein s problem for p harmonic functions on unweighted r 2 with 1 andlt p andlt infinity we show that if the boundary values f has a jump at an asymptotic corner point zo then the perron solution pf is asymptotically a b arg z z 0 o vertical bar z z 0 vertical bar we use this to obtain a positive answer to baernsteins problem on the equality of the p harmonic measure of a union g of open arcs on the boundary of the unit disc and the p harmonic measure of g over bar we also obtain various invariance results for functions with jumps and perturbations on small sets for p andgt 2 these results are new also for continuous functions finally we look at generalizations to r n and metric spaces

a note on the abc conjecture in this note we have generalized the abc conjecture for integers and extended mason s theorem to polynomials in m by dint of nevanlinna s value distribution theory 2002 wiley periodicals inc

c1 α local regularity of weak solutions of degenerate elliptic equations abstract it is demonstrated the local c 1 alpha nature of weak solutions of elliptic equations of the type 1 1 in the introduction under the degeneracy or singularity assumptions a sub 1 a sub 3

universal fluctuations in kardar parisi zhang growth on one dimensional flat substrates we present a numerical study of the evolution of height distributions hds obtained in interface growth models belonging to the kardar parisi zhang kpz universality class the growth is done on an initially flat substrate the hds obtained for all investigated models are very well fitted by the theoretically predicted gaussian orthogonal ensemble goe distribution the first cumulant has a shift that vanishes as t 1 3 while the cumulants of order 2 n 4 converge to goe as t 2 3 or faster behaviors previously observed in other kpz systems these results yield evidences for the universality of the goe distribution in kpz growth on flat substrates finally we further show that the surfaces are described by the airy 1 process

primes in arithmetic progressions strengthening work of rosser schoenfeld and mccurley we establish explicit chebyshev type estimates in the prime number theorem for arithmetic progressions for all moduli k 72 and other small moduli

pdes for the joint distributions of the dyson airy and sine processes in a celebrated paper dyson shows that the spectrum of an n x n random hermitian matrix diffusing according to an ornstein uhlenbeck process evolves as n noncolliding brownian motions held together by a drift term the universal edge and bulk scalings for hermitian random matrices applied to the dyson process lead to the airy and sine processes in particular the airy process is a continuous stationary process describing the motion of the outermost particle of the dyson brownian motion when the number of particles gets large with space and time appropriately rescaled in this paper we answer a question posed by kurt johansson to find a pde for the joint distribution of the airy process at two different times similarly we find a pde satisfied by the joint distribution of the sine process this hinges on finding a pde for the joint distribution of the dyson process which itself is based on the joint probability of the eigenvalues for coupled gaussian hermitian matrices the pde for the dyson process is then subjected to an asymptotic analysis consistent with the edge and bulk rescalings the pdes enable one to compute the asymptotic behavior of the joint distribution and the correlation for these processes at different times t 1 and t 2 when t 2 t 1 infinity as illustrated in this paper for the airy process this paper also contains a rigorous proof that the extended hermite kernel governing the joint probabilities for the dyson process converges to the extended airy and sine kernels after the appropriate rescalings

some remarks on the abc conjecture let r x be the product of all distinct primes dividing a nonzero integer x the abc conjecture says that if a b c are nonzero relatively prime integers such that a b c 0 then the biggest limit point of the numbers logmax lal ibl cil log r abc equals 1 we show that in a natural anologue of this conjecture for n 3 integers the largest limit point should be replaced by at least 2n 5 we present an algorithm leading to numerous examples of triples a b c for which the above quotients strongly deviate from the conjectural value 1

large time asymptotics of growth models on space like paths ii png and parallel tasep we consider the polynuclear growth png model in 1 1 dimension with flat initial condition and no extra constraints the joint distributions of surface height at finitely many points at a fixed time moment are given as marginals of a signed determinantal point process the long time scaling limit of the surface height is shown to coincide with the airy1 process this result holds more generally for the observation points located along any space like path in the space time plane we also obtain the corresponding results for the discrete time tasep totally asymmetric simple exclusion process with parallel update

on p harmonic mappings in the plane abstract to every p harmonic vector field in the plane with p 2 there corresponds a quasilinear system of first order pdes which couples the complex gradients of the coordinate functions of the field the ellipticity of such system is proved a relation between planar quasiregular mappings and p harmonic fields is discussed the p harmonic conjugate problem is stated

on the numerical evaluation of fredholm determinants some significant quantities in mathematics and physics are most naturally expressed as the fredholm determinant of an integral operator most notably many of the distribution functions in random matrix theory though their numerical values are of interest there is no systematic numerical treatment of fredholm determinants to be found in the literature instead the few numerical evaluations that are available rely on eigenfunction expansions of the operator if expressible in terms of special functions or on alternative numerically more straightforwardly accessible analytic expressions e g in terms of painleve transcendents that have masterfully been derived in some cases in this paper we close the gap in the literature by studying projection methods and above all a simple easily implementable general method for the numerical evaluation of fredholm determinants that is derived from the classical nystrom method for the solution of fredholm equations of the second kind using gauss legendre or clenshaw curtis as the underlying quadrature rule we prove that the approximation error essentially behaves like the quadrature error for the sections of the kernel in particular we get exponential convergence for analytic kernels which are typical in random matrix theory the application of the method to the distribution functions of the gaussian unitary ensemble gue in the bulk scaling limit and the edge scaling limit is discussed in detail after extending the method to systems of integral operators we evaluate the two point correlation functions of the more recently studied airy and airy 1 processes

a determinantal formula for the goe tracy widom distribution investigating the long time asymptotics of the totally asymmetric simple exclusion process sasamoto obtains rather indirectly a formula for the tracy widom distribution for the gaussian orthogonal ensemble we establish that his novel formula indeed agrees with more standard expressions

simultaneous rational approximation to binomial functions we apply pade approximation techniques to deduce lower bounds for simultaneous rational approximation to one or more algebraic numbers in particular we strengthen work of osgood fel dman and rickert proving for example that max 2 p1 q 3 p2 q q 1 79155 for q q0 where the latter is an effective constant some of the diophantine consequences of such bounds will be discussed specifically in the direction of solving simultaneous pell s equations and norm form equations 0 introduction in 1964 baker 1 2 utilized the method of pade approximation to hypergeometric functions to obtain explicit improvements upon liouville s theorem on rational approximation to algebraic numbers by way of example he showed that 3 2 pq 10 6q 2 955 0 1 for all positive integers p and q and used such bounds to solve related diophantine equations chudnovsky 6 subsequently refined baker s results primarily through a detailed analysis of the arithmetical properties of certain pade approximants analogous to 0 1 he proved that 3 2 pq q 2 42971 0 2 for all integers p and q with q greater than some effectively computable constant q0 by working out the implicit constants in 0 2 easton 8 deduced 3 2 pq 6 6 10 6q 2 795 for positive integers p and q as well as related bounds for other cubic irrationalities similar results exist for simultaneous approximation to several algebraic numbers in particular baker 3 derived bounds of the form max 1 u m u pu q q 0 3 received by the editors june 30 1994 and in revised form january 31 1995 1991 mathematics subject classification primary 11j68 11j82 secondary 11d57

on p harmonic functions in the plane and their stream functions on considere l equation p harmonique p div p 2 0 1 p dans le plan on developpe un certain nombre d analogies avec la theorie de base des fonctions complexes dans c

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

effective irrationality measure for certain algebraic numbers a result of chudnovsky concerning rational approximation to certain algebraic numbers is reworked to provide a quantitative result in which all constants are explicitly given more particularly pade approximants to the function 1 x 113 are employed to show for certain integers a and b that j a b 1 3 p q cq k when q 0 here c and k are given as functions of a and b only in 1964 baker 1 improving a technique used by siegel 8 was able to obtain effective irrationality measures for the function 1 x m n evaluated at certain rational points in particular he was able to show that for integers p q we have 1 121 3 p q 10 6q 2955 when q 0 the technique was further refined by chudnovsky 2 whose results when applied to 21 3 imply that for any 0 there exists a positive integer qo c such that for integers p q we have 2 121 3 p qi q 2 429 e when q q0 chudnovsky s result is effective in the sense that it is possible in principle to work through the proof and compute for any particular value of a qo c for which 2 holds however chudnovsky does not undertake such computations in this article we rework baker s proof using chudnovsky s refinement together with a chebyshev type result for primes in arithmetical progressions due to mccurley 6 and obtain the following quantitative result theorem let a b be integers with 0 b a define d by 0 if 3 4 a b 3 d i1 if 311 a b 13 2 otherwise further define a k c and q0 by 4 x 2328 3d al1 2 b1 2 2 received may 9 1985 1980 mathematics subject classification primary iof25 1986 american mathematical society 0025 5718 86 1 00 25 per page 613 this content downloaded from 207 46 13 124 on wed 22 jun 2016 05 36 01 utc all use subject to http about jstor org terms