a berry esseen bound for finite population student s statistic a general and precise berry esseen bound is proved for the studentized mean based on n random observations drawn without replacement from a finite population the bound yields the optimal rate o n 1 2 under minimal conditions if the erdos renyi condition holds this bound implies the asymptotic normality of student s statistic and the self normalized sum

a unifying local semilocal convergence analysis and applications for two point newton like methods in banach space abstract we provide a local as well as a semilocal convergence analysis for two point newton like methods in a banach space setting under very general lipschitz type conditions our equation contains a frechet differentiable operator f and another operator g whose differentiability is not assumed using more precise majorizing sequences than before we provide sufficient convergence conditions for newton like methods to a locally unique solution of equation f x g x 0 in the semilocal case we show under weaker conditions that our error estimates on the distances involved are finer and the information on the location of the solution at least as precise as in earlier results in the local case a larger radius of convergence is obtained several numerical examples are provided to show that our results compare favorably with earlier ones as a special case we show that the famous newton kantorovich hypothesis is weakened under the same hypotheses as the ones contained in the newton kantorovich theorem

convergence of newton s method and inverse function theorem in banach space under the hypothesis that the derivative satisfies some kind of weak lipschitz condition a proper condition which makes newton s method converge and an exact estimate for the radius of the ball of the inverse function theorem are given in a banach space also the relevant results on premises of kantorovich and smale types are improved in this paper

an edgeworth expansion for u statistics based on samples from finite populations suppose that u is a u statistic of degree 2 based on a simple random sample of size n selected without replacement from a finite population of n elements a bound for the difference between the distribution function of a standardized version of u and its single term edgeworth expansion is given we apply these results to obtain an edgeworth equation for the variance estimator in a finite population some simulation results are reported in this case

degeneration through coalescence of the q painleve vi equation starting from the q discrete form of the painleve vi equation we obtain its degenerate forms by applying the procedure of coalescence of singularities the whole cascade of degenerate forms is thus obtained leading to new forms for the discrete painleve iv and v equations the lax pairs of these discrete painleve equations are explicitly constructed thus confirming their integrability

ultradiscrete painleve ii equation and a special function solution ultradiscretization with parity variables which keeps the information of original variables sign is applied to the q painleve ii equation of type a6 q pii a special solution of the resulting ultradiscrete system which corresponds to the special function solution of q pii is constructed ultradiscrete analogues of the q airy equation and its special solutions are also discussed in the process

painleve equations nonlinear special functions the six painleve equations pi pvi were first discovered about a hundred years ago by painleve and his colleagues in an investigation of nonlinear second order ordinary differential equations recently there has been considerable interest in the painleve equations primarily due to the fact that they arise as reductions of the soliton equations which are solvable by inverse scattering consequently the painleve equations can be regarded as completely integrable equations and possess solutions which can be expressed in terms of solutions of linear integral equations despite being nonlinear equations although first discovered from strictly mathematical considerations the painleve equations have arisen in a variety of important physical applications including statistical mechanics plasma physics nonlinear waves quantum gravity quantum field theory general relativity nonlinear optics and fibre optics the painleve equations may be thought of a nonlinear analogues of the classical special functions they possess hierarchies of rational solutions and one parameter families of solutions expressible in terms of the classical special functions for special values of the parameters further the painleve equations admit symmetries under affine weyl groups which are related to the associated backlund transformations in this paper i discuss some of the remarkable properties which the painleve equations possess including connection formulae backlund transformations associated discrete equations and hierarchies of exact solutions in particular the second painleve equation pii is used to illustrate these properties and some of the applications of pii are also discussed

10e9 solution to the elliptic painleve equation a function formalism for sakai s elliptic painleve equation is presented this establishes the equivalence between the two formulations by sakai and by ohta ramani grammaticos we also give a simple geometric description of the elliptic painleve equation as a non autonomous deformation of the addition formula on elliptic curves by using these formulations we construct a particular solution of the elliptic painleve equation expressed in terms of the elliptic hypergeometric function 10e9

self normalized processes exponential inequalities moment bounds and iterated logarithm laws self normalized processes arise naturally in statistical applications being unit free they are not affected by scale changes moreover self normalization often eliminates or weakens moment assumptions in this paper we present several exponential and moment inequalities particularly those related to laws of the iterated logarithm for self normalized random variables including martingales tail probability bounds are also derived for random variables bt 0 and at let yt exp at 2bt2 2 we develop inequalities for the moments of at bt or supt 0at bt log logbt 1 2 and variants thereof when eyt 1 or when yt is a supermartingale for all belonging to some interval our results are valid for a wide class of random processes including continuous martingales with at mt and b t sqrt langle m rangle t and sums of conditionally symmetric variables di with at i 1tdi and b t sqrt sum i 1 t d i 2 a sharp maximal inequality for conditionally symmetric random variables and for continuous local martingales with values in rm m 1 is also established another development in this paper is a bounded law of the iterated logarithm for general adapted sequences that are centered at certain truncated conditional expectations and self normalized by the square root of the sum of squares the key ingredient in this development is a new exponential supermartingale involving i 1tdi and i 1tdi2 a compact law of the iterated logarithm for self normalized martingales is also derived in this connection

improved local convergence of newton s method under weak majorant condition we provide a local convergence analysis for newton s method under a weak majorant condition in a banach space setting our results provide under the same information a larger radius of convergence and tighter error estimates on the distances involved than before 14 special cases and numerical examples are also provided in this study

gravitational instantons with faster than quadratic curvature decay iii this is our third paper in a series on the gravitational instantons in this paper we classify alg and alh gravitational instantons in alg case we extend tian yau hein s construction slightly and show that it is the only alg gravitational instanton in alh case we prove a torelli type theorem

on the lattice geometry and birational group of the six point multi ratio equation the inherent self consistency properties of the six point multi ratio equation allow it to be considered on a domain associated with a t shaped coxeter dynkin diagram this extends the kadomtsev pe

distribution of periodic orbits for the casati prosen map on rational lattices abstract we study numerically the periodic orbits of the casati prosen map a two parameter reversible map of the torus with zero entropy for rational parameter values this map preserves rational lattices and each lattice decomposes into periodic orbits we consider the distribution function of the periods over prime lattices and its dependence on the parameters of the map based on extensive numerical evidence we conjecture that asymptotically almost all orbits are symmetric and that for a set of rational parameters having full density the distribution function approaches the gamma distribution r x 1 e x 1 x these properties which have been proved to hold for random reversible maps were previously thought to require a stronger form of deterministic randomness such as that displayed by rational automorphisms over finite fields furthermore we show that the gamma distribution is the limit of a sequence of singular distributions which are observed on certain lines in parameter space our experiments reveal that the convergence rate to r is highly non uniform in parameter space being slowest in sharply defined regions reminiscent of resonant zones in hamiltonian perturbation theory

a semilocal convergence analysis for directional newton methods a semilocal convergence analysis for directional newton methods in n variables is provided in this study using weaker hypotheses than in the elegant related work by y levin and a ben israel and introducing the center lipschitz condition we provide under the same computational cost as in levin and ben israel a semilocal convergence analysis with the following advantages weaker convergence conditions larger convergence domain finer error estimates on the distances involved and an at least as precise information on the location of the zero of the function a numerical example where our results apply to solve an equation but not the ones in levin and ben israel is also provided in this study

sigma function solution of the initial value problem for somos 5 sequences the somos 5 sequences are a family of sequences defined by a fifth order bilinear recurrence relation with constant coefficients for particular choices of coefficients and initial data integer sequences arise by making the connection with a second order nonlinear mapping with a first integral we prove that the two subsequences of odd even index terms each satisfy a somos 4 fourth order recurrence this leads directly to the explicit solution of the initial value problem for the somos 5 sequences in terms of the weierstrass sigma function for an associated elliptic curve

elliptic painleve equations from next nearest neighbor translations on the e_8 1 lattice the well known elliptic discrete painleve equation of sakai is constructed by a standard translation on the lattice given by nearest neighbor vectors in this paper we give a new elliptic discrete painleve equation obtained by translations along next nearest neighbor vectors this equation is a generic 8 parameter version of a 2 parameter elliptic difference equation found by reduction from adler s partial difference equation the so called q4 equation we also provide a projective reduction of the well known equation of sakai

characterizing singular curves in parametrized families of biquadratics we consider families of biquadratic curves b 0 on defined with respect to arbitrarily many complex parameters due to the fact that these families include curve intersections across different parameter combinations they represent a generalization of the non intersecting foliations of one parameter invariant curves associated with the qrt mapping we use algebraic methods involving discriminants to provide a complete classification of the singular curves in these families in developing this classification we exploit the special symmetric nature of b namely that it is a quadratic in x and y whose reflection in the line y x is given by a simple change of parameters we also define a range of conditions in the biquadratic s parameters and demonstrate the manner in which they correspond to different geometric realizations of the singular curves

an improvement of the region of accessibility of chebyshev s method from newton s method a simple modification of chebyshev s method is presented so that the region of accessibility is extended to the one of newton s method

newton type methods of high order and domains of semilocal and global convergence we present the geometric construction of some classical iterative methods that have global convergence and infinite speed of convergence when they are applied to solve certain nonlinear equations f t 0 in particular for nonlinear equations with the degree of logarithmic convexity of f l f t f t f t f t 2 is constant a family of newton type iterative methods of high orders of convergence is constructed we see that this family of iterations includes the classical iterative methods the convergence of the family is studied in the real line and the complex plane and domains of semilocal and global convergence are located

a geometrical description of the discrete painleve vi and v equations we present a geometrical approach for the discrete painleve equations based on weyl groups the method relies on the bilinear formalism and assumes that the multidimensional function lives on the weight lattice of the appropriate affine weyl group the equations for the function a system of nonautonomous hirota miwa equations govern the evolution along the independent variable and the parameters of the equation the latter evolution induced by the schlesinger transformations in the present paper we analyse the case of the e 1 7 group using the geometrical description we derive the nonlinear discrete equations we find that in the case of the e 1 7 group these are the asymmetric q pvi and d pv that were recently proposed

singularity confinement and full deautonomisation a discrete integrability criterion abstract we present a new approach to singularity confinement which makes it an efficient and reliable discrete integrability detector our method is based on the full deautonomisation procedure which consists in analysing non autonomous extensions of a given discrete system obtained by adding terms that are initially absent but whose presence does not alter the singularity pattern a justification for this approach is given through an algebro geometric analysis we also introduce the notions of early and late confinement while the former is a confinement that may exist already for the autonomous system the latter corresponds to a singularity pattern longer than that of the autonomous case late confinement will be shown to play an important role in the singularity analysis of systems with non trivial gauge freedom for which the existence of an undetected gauge in conjunction with a sketchy analysis might lead to erroneous conclusions as to their integrability an algebro geometric analysis of the role of late confinement in this context is also offered this novel type of singularity confinement analysis will be shown to allow for the exact calculation of the algebraic entropy of a given mapping

tropical geometric interpretation of ultradiscrete singularity confinement using the interpretation of the ultradiscretization procedure as a non archimedean valuation we use results of tropical geometry to show how roots and poles manifest themselves in piece wise linear systems as points of non differentiability this will allow us to demonstrate a correspondence between singularity confinement for discrete integrable systems and ultradiscrete singularity confinement for ultradiscrete integrable systems

two variations on a3 a1 a1 1 type discrete painlev equations by considering the normalizers of reflection subgroups of types a 1 1 and a 1 3 in w d5 1 two subgroups w a3 a1 1 w a1 1 and w a1 a1 1 w a3 1 can be constructed from a a3a1a1 1 ty

integrable mappings and soliton equations ii abstract some simple solutions of discrete soliton equations are shown to satisfy 2d mappings we show that these belong to a recently introduced 18 parameter family of integrable reversible mappings of the plane thus lending weight to a previous conjecture we also give an example of an integrable mapping occuring in an exactly solvable model in statistical mechanics finally we discuss the notion of generalized reversibility

a classification of two dimensional integrable mappings and rational elliptic surfaces we classify two dimensional integrable mappings by investigating the actions on the fiber space of rational elliptic surfaces while the qrt mappings can be restricted on each fiber there exist several classes of integrable mappings which exchange fibers we also show an equivalent condition when a generalized halphen surface becomes a halphen surface of index m

connection matrices for ultradiscrete linear problems we present theory outlining associated linear problems for ultradiscrete equations the appropriate domain for these problems is the max plus semiring our main result is that despite the restrictive nature of the max plus semiring it is still possible to define a theory of connection matrices analogous to that of birkhoff and his school for systems of linear difference equations we use such theory to provide evidence for the integrability of an ultradiscrete difference equation

globally nilpotent differential operators and the square ising model we recall various multiple integrals with one parameter related to the isotropic square ising model and corresponding respectively to the n particle contributions of the magnetic susceptibility to the lattice form factors to the two point correlation functions and to their extensions the univariate analytic functions defined by these integrals are holonomic and even g functions they satisfy fuchsian linear differential equations with polynomial coefficients and have some arithmetic properties we recall the explicit forms found in previous work of these fuchsian equations as well as their russian doll and direct sum structures these differential operators are selected fuchsian linear differential operators and their remarkable properties have a deep geometrical origin they are all globally nilpotent or sometimes even have zero p curvature we also display miscellaneous examples of globally nilpotent operators emerging from enumerative combinatorics problems for which no integral representation is yet known focusing on the factorized parts of all these operators we find out that the global nilpotence of the factors resp p curvature nullity corresponds to a set of selected structures of algebraic geometry elliptic curves modular curves curves of genus five six and even a remarkable weight 1 modular form emerging in the three particle contribution 3 of the magnetic susceptibility of the square ising model noticeably this associated weight 1 modular form is also seen in the factors of the differential operator for another n fold integral of the ising class 3 h for the staircase polygons counting and in ap ry s study of 3 g functions naturally occur as solutions of globally nilpotent operators in the case where we do not have g functions but hamburger functions one irregular singularity at 0 or that correspond to the confluence of singularities in the scaling limit the p curvature is also found to verify new structures associated with simple deformations of the nilpotent property

cramer type large deviations for samples from a finite population cramer type large deviations for means of samples from a finite population are established under weak conditions the results are comparable to results for the so called self normalized large deviation for independent random variables cramer type large deviations for the finite population student t statistic are also investigated

extending the newton kantorovich hypothesis for solving equations the famous newton kantorovich hypothesis kantorovich and akilov 1982 3 argyros 2007 2 argyros and hilout 2009 7 has been used for a long time as a sufficient condition for the convergence of newton s method to a solution of an equation in connection with the lipschitz continuity of the frechet derivative of the operator involved here using lipschitz and center lipschitz conditions and our new idea of recurrent functions we show that the newton kantorovich hypothesis can be weakened under the same information moreover the error bounds are tighter than the corresponding ones given by the dominating newton kantorovich theorem argyros 1998 1 2 7 ezquerro and hernandez 2002 11 3 proinov 2009 2010 16 17 numerical examples including a nonlinear integral equation of chandrasekhar type chandrasekhar 1960 9 as well as a two boundary value problem with a green s kernel argyros 2007 2 are also provided in this study

integrable mappings of the plane preserving biquadratic invariant curves we provide a general framework to construct integrable mappings of the plane that preserve a one parameter family b x y k of biquadratic invariant curves where parametrization by k is very general these mappings are reversible by construction i e they are the composition of two involutions and can be shown to be measure preserving they generalize integrable maps previously given by mcmillan and quispel roberts and thompson by considering a transformation of the case of the symmetric biquadratic to a canonical form we provide a normal form for the symmetric integrable map acting on each invariant curve we give a lax pair for a large subclass of our symmetric integrable maps including at least a 10 parameter subfamily of the 12 parameter symmetric quispel roberts thompson maps

on the newton kantorovich hypothesis for solving equations the famous newton kantorovich hypothesis has been used for a long time as a sufficient condition for the convergence of newton s method to a solution of an equation in connection with the lipschitz continuity of the frechet derivative of the operator involved here using lipschitz and center lipschitz conditions we show that the newton kantorovich hypothesis can be weakened the error bounds obtained under our semilocal convergence result are more precise than the corresponding ones given by the dominating newton kantorovich theorem

a classification of four state spin edge potts models we classify four state spin models with interactions along the edges according to their behaviour under a specific group of symmetry transformations this analysis uses the measure of complexity of the action of the symmetries in the spirit of the study of discrete dynamical systems on the space of parameters of the models and aims at uncovering solvable ones we find that the action of these symmetries has low complexity polynomial growth zero entropy we obtain natural parametrizations of various models among which an unexpected elliptic parametrization of the four state chiral potts model which we use to localize possible integrability conditions associated with high genus curves

quantum discrete dubrovin equations the discrete equations of motion for the quantum mappings of kdv type are given in terms of the sklyanin variables which are also known as quantum separated variables both temporal discrete time evolutions and spatial along the lattice at a constant time level evolutions are considered in the classical limit the temporal equations reduce to the classical discrete dubrovin equations as given in a previous publication nijhoff f w 2000 chaos solitons fractals 11 19 28 the reconstruction of the original dynamical variables in terms of the sklyanin variables is also achieved

integrable mappings via rational elliptic surfaces we present a geometric description of the qrt map which is an integrable mapping introduced by quispel roberts and thompson in terms of the addition formula of a rational elliptic surface by this formulation we classify all the cases when the qrt map is periodic and show that its period is 2 3 4 5 or 6 a generalization of the qrt map which acts birationally on a pencil of k3 surfaces or calabi yau manifolds is also presented

isomonodromy transformations of linear systems of difference equations we introduce and study isomonodromy transformations of the matrix linear difference equation y z 1 a z y z with polynomial a z our main result is construction of an isomonodromy action of zm n 1 1 on the space of coefficients a z here m is the size of matrices and n is the degree of a z the birational action of certain rank n subgroups can be described by difference analogs of the classical schlesinger equations and we prove that for generic initial conditions these difference schlesinger equations have a unique solution we also show that both the classical schlesinger equations and the schlesinger transformations known in isomonodromy theory can be obtained as limits of our action in two different limit regimes similarly to the continuous case for m n 2 the difference schlesinger equations and their q analogs yield discrete painlev e equations examples include dpii dpiv dpv and q pvi

birational maps that send biquadratic curves to biquadratic curves recently many papers have begun to consider so called non quispel roberts thompson qrt birational maps of the plane compared to the qrt family of maps which preserve each biquadratic curve in a fibration of the plane non qrt maps send a biquadratic curve to another biquadratic curve belonging to the same fibration or to a biquadratic curve from a different fibration of the plane in this communication we give the general form of a birational map derived from a difference equation that sends a biquadratic curve to another the necessary and sufficient condition for such a map to exist is that the discriminants of the two biquadratic curves are the same and hence so are the j invariants the result allows existing examples in the literature to be better understood and allows some statements to be made concerning their generality

moderate deviations for finite population student s statistic in this paper moderate deviations for finite population are given under weaker conditions as by products some exponential inequalities for finite population are also established

a modification of the classical kantorovich conditions for newton s method in the classical kantorovich theorem on newton s method it is assumed that the second frechet derivative of the involved operator satisfies the condition f x k in an appropiate domain in this paper we study a modification of this condition assuming that f x x where is a continuous and non decreasing real function

an algebraic geometric approach to integrable maps of the plane we show that the dynamics of a birational map on an elliptic curve over a field is typically conjugate to addition by a point under the associated group law when the field is taken to be the function field of rational complex functions of one variable this amounts to an algebraic geometric version of the arnold liouville integrability theorem for planar integrable maps by products of this approach are that birational maps preserving foliations are necessarily the composition of two involutions and that relationships between birational maps preserving the same foliation can be described in terms of the respective points they add on the corresponding weierstrass curves when the result is applied to finite fields it helps explain some universal features of the periodic orbit distribution function for the reductions of integrable maps

new airy type solutions of the ultradiscrete painleve ii equation with parity variables the q difference painleve ii equation admits special solutions written in terms of determinants whose entries are the general solution of the q airy equation an ultradiscrete limit of the special solutions is studied using the procedure of ultradiscretization with parity variables then we obtain new airy type solutions of the ultradiscrete painleve ii equation with parity variables and the solutions have a richer structure than the known solutions

solution of the equivalence problem for the third painleve equation we find necessary conditions for a second order ordinary differential equation to be equivalent to the painleve iii equation under a general point transformation their sufficiency is established by reduction to known results for the equations of the form y f x y we consider separately the generic case and the case of reducibility to an autonomous equation the results are illustrated by the primary resonance equation

holonomy of the ising model form factors we study the ising model two point diagonal correlation function c n n by presenting an exponential and form factor expansion in an integral representation which differs from the known expansion of wu mccoy tracy and barouch we extend this expansion weighting by powers of a variable the j particle contributions f j n n in the form factor expansion the corresponding extension of the two point diagonal correlation function c n n is shown for arbitrary to be a solution of the sigma form of the painleve vi equation introduced by jimbo and miwa in their isomonodromic approach to the ising model fuchsian linear differential equations for the form factors f j n n are obtained for j 9 and shown to have both a russian doll nesting and a decomposition of the corresponding linear differential operators as a direct sum of operators equivalent to symmetric powers of the second order linear differential operator associated with the elliptic integral e from this we show that each f j n n is unexpectedly simple being expressed polynomially in terms of the elliptic integrals e and k in contrast we exhibit some mathematical objects built from these form factors f j n n which break the direct sum of symmetric powers decomposition with its associated polynomial expressions first we show that the scaling limit of these differential operators and form factors breaks the direct sum structure but not the russian doll structure secondly we show that the previous extension of two point diagonal correlation functions c n n is for singled out values cos m n m n integers also solutions of fuchsian linear differential equations these solutions of painleve vi are not polynomial in e and k but are actually algebraic functions being associated with modular curves

matrix painleve systems we study a class of the isomonodromic deformation equations which we call matrix painleve systems they can be written in hamiltonian form the hamiltonian is written as the trace of the hamiltonian of the sixth painleve equation with canonical variables replaced by matrices we also derive the matrix version of the fifth fourth third and second painleve equation from the sixth matrix painleve system by the degeneration

lie symmetries of birational maps preserving genus 0 fibrations we prove that any planar birational integrable map which preserves a fibration given by genus 0 curves has a lie symmetry and some associated invariant measures the obtained results allow to study in a systematic way the global dynamics of these maps using this approach the dynamics of several maps is described in particular we are able to give for particular examples the explicit expression of the rotation number function and the set of periods of the considered maps

super integrable four dimensional autonomous mappings a systematic investigation of the complete integrability of a fourth order autonomous difference equation of the type w n 4 w n f w n 1 w n 2 w n 3 is presented we identify seven distinct families of four dimensional mappings which are super integrable and have three independent integrals via a duality relation as introduced in a recent paper by quispel capel and roberts 2005 j phys a math gen 38 3965 80 it is observed that these seven families can be related to the four dimensional symplectic mappings with two integrals including all the four dimensional periodic reductions of the integrable double discrete modified korteweg devries and sine gordon equations treated in an earlier paper by two of us capel and sahadevan 2001 physica a 289 86 106

a new class of integrable discrete systems we derive a new class of integrable autonomous mappings which possess invariants biquartic in the dependent variables we link these mappings to discrete painleve equations we show how mappings with biquartic invariants can be naturally built up from the requirement that their solutions be parametrized in terms of elliptic functions

the theory of smale s point estimation and its applications abstract the main result of this paper is that we exact smale s point estimation theory i e without assuming k p z 1 p k z k k 2 being bounded by the point estimation convergence theorem of the ne method is set up by making use of the majorizing method the proof of the theorem is simple and precise while the required point estimation conditions are weaker than all those of known point estimation convergence theorems another result of this paper is an application of the above new theory to the durand kerner method we compare the point estimation conditions for the durand kerner method with other known point estimation conditions numerical results show that our results have evident advantages

self normalized cramer type large deviations for independent random variables let x 1 x 2 be independent random variables with zero means and finite variances it is well known that a finite exponential moment assumption is necessary for a cramer type large deviation result for the standardized partial sums in this paper we show that a cramer type large deviation theorem holds for self normalized sums only under a finite 2 th moment 0 1 in particular we show p s n v n x 1 x 1 o 1 1 x 2 d 2 n for 0 x d n where d n n i 1 ex 2 i 1 2 n i 1 e x i 2 1 2 and v n n i 1 x 2 i 1 2 applications to the studentized bootstrap and to the self normalized law of the iterated logarithm are discussed

an area preserving action of the modular group on cubic surfaces and the painleve vi equation we construct an area preserving action of the modular group on a general 4 parameter family of affine cubic surfaces we present a geometrical background behind this construction that is a natural symplectic structure on a moduli space of rank two linear monodromy representations over the 2 dimensional sphere with four punctures and a natural symplectic action upon it of the braid group on three strings studying this action as a discrete dynamical system will be important in discussing the monodromy of the painleve vi equation

on quispel roberts thompson extensions and integrable correspondences we present several novel examples of an extension of the quispel roberts thompson qrt mapping in which the coefficients are periodic functions of the independent variable and we prove that for period 2 this extension includes all so called asymmetric qrt mappings we establish the integrability of these new mappings by explicitly constructing rational invariants for them which can be parametrised in terms of elliptic functions moreover we show that the correspondences induced by these conserved quantities are all integrable i e exhibit polynomial growth of the number of images they produce and we present a conjecture pertaining to the degree of the growth the case of hirota kimura yahagi mappings and their associated correspondences is examined as well and we give examples of integrable correspondences associated with such mappings

discrete painleve equations with maximal periodicities we present the discrete painleve equations the coefficients of which have the maximal periodicity allowed by the geometric description in terms of affine weyl groups for each equation of the degeneration pattern obtained by sakai and which starts at the group we give the corresponding discrete painleve equation some of these systems were already known but here we proceed in a systematic way filling in all the gaps in particular we derive the equations with maximal periodicity associated to the affine weyl group which is the richest system in the sakai classification and which has yet to be studied in detail

expanding the applicability of newton s method using smale s α theory we present a tighter convergence analysis than earlier studies such as in cianciaruso 2007 guo 2007 shen and li 2010 smale 1986 1987 wang and zhao 1995 wang 1999 wang and han 1990 of newton s method using smale s a theory by introducing the notion of the center c 0 condition in particular in the semilocal convergence case we show that if the center c 0 condition is smaller than the c condition then the new majorizing sequence is tighter than the old majorizing sequence the new convergence criteria are weaker than the older convergence criteria furthermore in the local convergence case we obtain a larger radius of convergence and tighter error estimates on the distances involved these improvements are obtained under the same computational cost numerical examples and applications are also provided in this study to show that the older results cannot apply but the new results apply to solve equations

affine weyl groups discrete dynamical systems and painleve equations a new class of representations of affine weyl groups on rational functions are constructed in order to formulate discrete dynamical systems associated with affine root systems as an application some examples of difference and differential systems of painleve type are discussed

linearizable qrt mappings we derive the members of the qrt family of mappings which are integrable through linearization we show how these mappings can be integrated and link them to known families of linearizable mappings we study their growth properties and comment on the relation of the latter to the singularity structure of the mappings the case of integrable linearizable mappings involving free functions of the independent variable is also discussed

on recurrence equations associated with invariant varieties of periodic points a recurrence equation is a discrete integrable equation whose solutions are all periodic and the period is fixed we show that infinitely many recurrence equations can be derived from the information about invariant varieties of periodic points of higher dimensional integrable maps

2 parameter tau τ function for the first painleve equation topological recursion and direct monodromy problem via exact wkb analysis we show that a 2 parameter family of tau functions for the first painleve equation can be constructed by the discrete fourier transform of the topological recursion partition function for a family of elliptic curves we also perform an exact wkb theoretic computation of the stokes multipliers of associated isomonodromy system assuming certain conjectures

time reversal symmetry in dynamical systems a survey abstract in this paper we survey the topic of time reversal symmetry in dynamical systems we begin with a brief discussion of the position of time reversal symmetry in physics after defining time reversal symmetry as it applies to dynamical systems we then introduce a major theme of our survey namely the relation of time reversible dynamical sytems to equivariant and hamiltonian dynamical systems we follow with a survey of the state of the art on the theory of reversible dynamical systems including results on symmetric periodic orbits local bifurcation theory homoclinic orbits and renormalization and scaling some areas of physics and mathematics in which reversible dynamical systems arise are discussed in an appendix we provide an extensive bibliography on the topic of time reversal symmetry in dynamical systems

monodromy preserving deformation of linear ordinary differential equations with rational coefficients iii abstract a unified treatment of monodromy and spectrum preserving deformations is presented in particular a general procedure is described to reduce the latter into the former consistently the concept of the function previously introduced for the former 2 is extended to the isospectral context it is shown that the general monodromy and spectrum preserving deformation equations can be written as hirota s bilinear differential equations by using the functions as dependent variables

bilinear equations and q discrete painleve equations satisfied by variables and coefficients in cluster algebras we construct cluster algebras the variables and coefficients of which satisfy the discrete mkdv equation the discrete toda equation and other integrable bilinear equations several of which lead to q discrete painleve equations these cluster algebras are obtained from quivers with an infinite number of vertices or with the mutation period property we will also show that a suitable transformation of quivers corresponds to a reduction of the difference equation

orthogonal decomposition of finite population statistics and its applications to distributional asymptotics we study orthogonal decomposition of symmetric statistics based on samples drawn without replacement from finite populations several applications to finite population statistics are given we establish one term edge worth expansions for general asymptotically normal symmetric statistics prove an efron stein inequality and the consistency of the jackknife estimator of variance our expansions provide second order a s approximations to wu s jackknife histogram

rational surfaces associated with affine root systems and geometry of the painleve equations we present a geometric approach to the theory of painleve equations based on rational surfaces our starting point is a compact smooth rational surface x which has a unique anti canonical divisor d of canonical type we classify all such surfaces x to each x there corresponds a root subsystem of e 1 8 inside the picard lattice of x we realize the action of the corresponding affine weyl group as the cremona action on a family of these surfaces we show that the translation part of the affine weyl group gives rise to discrete painleve equations and that the above action constitutes their group of symmetries by backlund transformations the six painleve differential equations appear as degenerate cases of this construction in the latter context x is okamoto s space of initial conditions and d is the pole divisor of the symplectic form defining the hamiltonian structure