uniform asymptotic regularity for mann iterates abstract in numer funct anal optim 22 2001 641 656 we obtained an effective quantitative analysis of a theorem due to borwein reich and shafrir on the asymptotic behavior of general krasnoselski mann iterations for nonexpansive self mappings of convex sets c in arbitrary normed spaces we used this result to obtain a new strong uniform version of ishikawa s theorem for bounded c in this paper we give a qualitative improvement of our result in the unbounded case and prove the uniformity result for the bounded case under the weaker assumption that c contains a point x whose krasnoselski mann iteration x k is bounded we also consider more general iterations for which asymptotic regularity is known only for uniformly convex spaces groetsch we give uniform effective bounds for an extension of groetsch s theorem which generalize previous results by kirk martinez yanez and the author

on the bilinear structure associated to bezoutians abstract this paper is partly a survey of known results on quadratic forms that are hard to find in the literature our main focus is a twisted form of a construction due to bezout this skew bezoutian is a symplectic resp quadratic space associated to a pair of reciprocal or skew reciprocal coprime polynomials of same degree the isometry group of this space turns out to contain a certain associated hypergeometric group using the skew bezoutian we construct explicit isometries of bilinear spaces with given invariants such as the characteristic polynomial or jordan form and in the quadratic case the spinor norm

some computational aspects of metric fixed point theory abstract we show how bounds on asymptotic regularity for nonexpansive functions can effectively be converted into certain bounds on the convergence towards a fixed point

arithmeticity of certain symplectic hypergeometric groups we give a sufficient condition on a pair of primitive polynomials that the associated hypergeometric group monodromy group of the corresponding hypergeometric differential equation is an arithmetic subgroup of the integral symplectic group

general logical metatheorems for functional analysis in this paper we prove general logical metatheorems which state that for large classes of theorems and proofs in nonlinear functional analysis it is possible to extract from the proofs effective bounds which depend only on very sparse local bounds on certain parameters this means that the bounds are uniform for all parameters meeting these weak local boundedness conditions the results vastly generalize related theorems due to the second author where the global boundedness of the underlying metric space resp a convex subset of a normed space was assumed our results treat general classes of spaces such as metric hyperbolic cat 0 normed uniformly convex and inner product spaces and classes of functions such as nonexpansive holder lipschitz uniformly continuous bounded and weakly quasi nonexpansive ones we give several applications in the area of metric fixed point theory in particular we show that the uniformities observed in a number of recently found effective bounds by proof theoretic analysis can be seen as instances of our general logical results

on quantitative versions of theorems due to f e browder and r wittmann abstract this paper is another case study in the program of logically analyzing proofs to extract new typically effective information proof mining we extract explicit uniform rates of metastability in the sense of t tao from two ineffective proofs of a classical theorem of f e browder on the convergence of approximants to fixed points of nonexpansive mappings as well as from a proof of a theorem of r wittmann which can be viewed as a nonlinear extension of the mean ergodic theorem the first rate is extracted from browder s original proof that is based on an application of weak sequential compactness in addition to a projection argument wittmann s proof follows a similar line of reasoning and we adapt our analysis of browder s proof to get a quantitative version of wittmann s theorem as well in both cases one also obtains totally elementary proofs even for the strengthened quantitative forms of these theorems that neither use weak compactness nor the existence of projections anymore in this way the present article also discusses general features of extracting effective information from proofs based on weak compactness we then extract another rate of metastability of similar nature from an alternative proof of browder s theorem essentially due to halpern that already avoids any use of weak compactness the paper is concluded by general remarks concerning the logical analysis of proofs based on weak compactness as well as a quantitative form of the so called demiclosedness principle in a subsequent paper these results will be utilized in a quantitative analysis of baillon s nonlinear ergodic theorem

a new metastable convergence criterion and an application in the theory of uniformly convex banach spaces abstract we study a convergence criterion which generalises the notion of being monotonically decreasing and introduce a quantitative version of this criterion a so called metastable rate of asymptotic decreasingness we then present a concrete application in the fixed point theory of uniformly convex banach spaces in which we carry out a quantitative analysis of a convergence proof of kirk and sims more precisely we produce a rate of metastability in the sense of tao for the picard iterates of mappings which satisfy a variant of the convergence criterion and whose fixed point set has nonempty interior

some logical metatheorems with applications in functional analysis in previous papers we have developed proof theoretic techniques for extracting effective uniform bounds from large classes of ineffective existence proofs in functional analysis here uniform means independence from parameters in compact spaces a recent case study in fixed point theory systematically yielded uniformity even w r t parameters in metrically bounded but noncompact subsets which had been known before only in special cases in the present paper we prove general logical metatheorems which cover these applications to fixed point theory as special cases but are not restricted to this area at all our theorems guarantee under general logical conditions such strong uniform versions of non uniform existence statements moreover they provide algorithms for actually extracting effective uniform bounds and transforming the original proof into one for the stronger uniformity result our metatheorems deal with general classes of spaces like metric spaces hyperbolic spaces cat 0 spaces normed linear spaces uniformly convex spaces as well as inner product spaces

advances in modeling model discrepancy comment on wu and browne 2015 wu and browne psychometrika 79 2015 have proposed an innovative approach to modeling discrepancy between a covariance structure model and the population that the model is intended to represent their contribution is related to ongoing developments in the field of uncertainty quantification uq on modeling and quantifying effects of model discrepancy we provide an overview of basic principles of uq and some relevant developments and we examine the wu browne work in that context we view the wu browne contribution as a seminal development providing a foundation for further work on the critical problem of model discrepancy in statistical modeling in psychological research

algebraicity of the appell lauricella and horn hypergeometric functions we extend schwarz list of irreducible algebraic gauss functions to the four classes of appell lauricella functions in several variables and the 14 complete horn functions in two variables this gives an example of a family of functions such that for any number of variables there are infinitely many algebraic functions namely the lauricella fc functions

connected linear groups as differential galois groups theorem 1 1 let c be an algebraically closed field of characteristic zero g a connected linear algebraic group defined o er c and k a differential field containing c as its field of constants and of finite nonzero transcendence degree o er c then g can be realized as the galois group of a picard vessiot extension of k w x previous work by kovacic 17 18 reduced this problem to the case of powers of a simple connected linear algebraic group our contribution is to show that one is able to realize any connected semisimple group as a galois group and when k s c x x9 s 1 to control the number and types of singularities when one constructs a system y 9 s ay a g m c x realizing an arbitrary connected linear algebraic group as its n

linear differential equations with finite differential galois group abstract for a finite irreducible subgroup h psl c n and an irreducible h invariant curve z p c n such that c z h c t a standard differential operator l s t c t d d t is constructed for n 2 this is essentially klein s work for n 2 an actual calculation of l s t is done by computing an evaluation of invariants c x 1 x n h c t and applying a scalar form of a theorem of e compoint in a procedure also in some cases where z is unknown evaluations are produced this new method is tested for n 2 and for three irreducible subgroups of sl 3 this supplements 18 the theory developed here relates to and continues classical work of h a schwarz g fano f klein and a hurwitz

bounds on kuhfittig s iteration schema in uniformly convex hyperbolic spaces abstract the purpose of this paper is to extract an explicit effective and uniform bound on the rate of asymptotic regularity of an iteration schema involving a finite family of nonexpansive mappings the results presented in this paper contribute to the general project of proof mining as developed by the second author as well as generalize and improve various classical and corresponding quantitative results in the current literature more precisely we give a rate of asymptotic regularity of an iteration schema due to kuhfittig for finitely many nonexpansive mappings in the context of uniformly convex hyperbolic spaces the rate only depends on an upper bound on the distance between the starting point and some common fixed point a lower bound 1 n n 1 n the error 0 and a modulus of uniform convexity

on rigid tuples in linear groups of odd dimension abstract we construct examples of linearly rigid tuples which lead to regular galois realizations over q for linear and unitary groups gl2m 1 q and u2m 1 q2 where q is odd and m q 1 resp m q 1 the notion of linear rigidity was introduced by strambach and volklein j reine angew math to appear as an analogue of katz physically rigid systems see n m katz rigid local systems annals of mathematics studies vol 139 princeton univ press princeton nj 1996

image of the burau representation at d th roots of unity we show that the image of the braid group under the monodromy action on the homology of a cyclic covering of degree d of the projective line is an arithmetic group provided the number of branch points is suciently large compared to the degree d this is deduced by proving the arithmeticity of the image of the braid group on n 1 letters under the burau representation evaluated at d th roots of unity when n 2d

boltzmann samplers for v balanced cycles this paper is devoted to the random generation of a new operator which we call v balanced colored cycle a cycle of which the components are assigned different colors with frequencies restricted by a vector v we propose an efficient approximate size sampler of linear expected time complexity following the boltzmann model introduced by duchon et al 2004 9 one of our motivations is to show that the absence of a decomposable specification which form the basis of the boltzmann model can be circumvented by mixing boltzmann samplers with other methods of random generation

on lame operators which are pull backs of hypergeometric ones we give a method that would allow one to calculate the number of lame operators 5n n e n with prescribed finite monodromy and do the calculation for the case n 1 we find a bound for the degree over q of the field of definition of the coefficients of a lame operator with prescribed finite monodromy and give examples of lame operators with finite monodromy finally we study lame operators with infinite monodromy and generic second order differential operators which are pull backs of hypergeometric ones under algebraic maps consider an ordinary differential operator 0 1 l a x an x where as x e k x and k is a number field over q what one can say about its solutions ly 0 and monodromy looking only at the coefficients as x in this paper we give some partial answers to this question for the special case of a lame operator one of the questions we will approach is the following for every n e n how many lame operators yn are there whose monodromy group is a dihedral group of order 2n n e n we will answer in the case n 1 2 14 2 15 the method we use for the proof works for general n for example the case n 2 can be found in co before summarizing our work we give a brief review of the known results since the work of schwarz on hypergeometric equations and the classification of the relative finite monodromy groups no basic improvements have been made recently the challenge of the grothendieck conjecture has stimulated some progress see ka ka1 ka2 ho be he dw1 we recall that this conjecture asserts that if in 0 1 almost all the reductions mod p prime ideal in the ring of integers of k have a complete set of solutions which is equivalent to l having zero p curvature for almost all p then equation 0 1 has only algebraic solutions on the other hand some improvements in the sense of klein and schwarz can be found in ba dw sing ba dw but we are still a long way from achieving the same level of understanding for general operators that we have attained in the hypergeometric case in received by the editors january 2 1991 1991 mathematics subject classification primary 34b30 secondary 14e20 the author was a visiting fellow at princeton university supported by c n r italy 1995 american mathematical society

nonexpansive iterations in hyperbolic spaces one of the most active research areas in nonlinear functional analysis is the asymptotics of nonexpansive mappings most of the results however have been obtained in normed linear spaces it is natural therefore to try to develop a theory of nonexpansive iterations in more general infinite dimensional manifolds this is the purpose of the present paper more specifically we propose the class of hyperbolic spaces as an appropriate background for the study of operator theory in general and of iterative processes for nonexpansive mappings in particular this class of metric spaces which is defined in section 2 includes all normed linear spaces and hadamard manifolds as well as the hilbert ball and the cartesian product of hilbert balls in section 3 we introduce co accretive operators and their resolvents and present some of their properties in the fourth section we discuss the concept of uniform convexity for hyperbolic spaces section 5 is devoted to two new geometric properties of infinite dimensional banach spaces theorem 5 6 provides a characterization of banach spaces having these properties in terms of nonlinear accretive operators in sections 6 7 and 8 we study explicit implict and continuous iterations repectively using the same approach in all three sections we illustrate this common approach with the following special case let c be a closed convex subset of a hyperbolic space x p let t c f c be a nonexpansive mapping and let x be a point in c in order to study the iteration t x n 0 1 2 we set z 1 l n x 0 l n t x k clco zj j i l and d inf p y ty y e c the first step is to show that p x k lim p x t x n d this leads to the convergence m of lz when x is uniformly convex and to the weak convergence of z when x is a banach space which is reflexive and strictly convex when t is an averaged mapping we are also able to establish the following triple equality for all k 2 1

calabi yau threefolds fibred by mirror quartic k3 surfaces abstract we study threefolds fibred by mirror quartic k3 surfaces we begin by showing that any family of such k3 surfaces is completely determined by a map from the base of the family to the moduli space of mirror quartic k3 surfaces this is then used to give a complete explicit description of all calabi yau threefolds fibred by mirror quartic k3 surfaces we conclude by studying the properties of such calabi yau threefolds including their hodge numbers and deformation theory

mapping class group dynamics and the holonomy of branched affine structures we classify up to few exceptions the orbit closures of the mathrm mod sigma action on the affine character variety chi mathrm aff mathbb c we obtain from this classification that the only obstruction for a non abelian representation rho pi 1 sigma longrightarrow mathrm aff mathbb c to be the holonomy of a branched affine structure on sigma is to be euclidean and not to have positive volume where sigma is a closed oriented surface of genus g ge 2

transcendence in positive characteristic and special values of hypergeometric functions we prove a simple transcendence criterion suitable for function field arithmetic we apply it to show the transcendence of special values at non zero rational arguments or more generally at algebraic arguments which generate extension of the rational function field with less than q places at infinity of the entire hypergeometric functions in the function field over fq context and to obtain a new proof of the transcendence of special values at non natural p adic integers of the carlitz goss gamma function we also characterize in the balanced case the algebraicity of hypergeometric functions giving an analog of the result of f r villegas based on beukers heckman results in the classical hypergeometric case 0 introduction number theorists study the number fields and the function fields over finite fields together as global fields in view of the strong analogies that exist between them transcendental number theory originated with liouville s result of 1844 see 17 that an algebraic irrational cannot be well approximated by rationals k mahler 18 showed in 1949 that liouville s theorem holds for function fields over any field and also that the liouville s inequality is best possible if the ground field k is of positive characteristic p by considering t p i k 1 t which is algebraic of degree p as well as of the approximation exponent p in particular naive analog of roth s theorem fails see 30 cha 9 for a survey of general situation which is not even conjecturally fully understood yet so we need stronger approximations in function fields than in number fields counterparts to conclude transcendence by diophantine approximation on the other hand for many naturally occurring quantities in the function field arithmetic related to drinfeld modules and anderson s t motives such as periods special values of exponential logarithm gamma and zeta functions etc different techniques that exist algebraic group techniques more developed motivic machinery automata method strong diophantine approximation criterion of special kind etc in function fields have been used to prove very powerful results see 30 cha 10 11 for a survey of older results and 4 21 22 7 for some recent results many quantities of interest that occur naturally in the function field arithmetic have expansions that are fq linear or closely related to such so that they are date may 12 2010 1991 mathematics subject classification primary 11j81 11j93 secondary 11t55

ising n fold integrals as diagonals of rational functions and integrality of series expansions we show that the n fold integrals n of the magnetic susceptibility of the ising model as well as various other n fold integrals of the ising class or n fold integrals from enumerative combinatorics like lattice green functions correspond to a distinguished class of functions generalizing algebraic functions they are actually diagonals of rational functions as a consequence the power series expansions of the analytic at x 0 solutions of these linear differential equations derived from geometry are globally bounded which means that after just one rescaling of the expansion variable they can be cast into series expansions with integer coefficients we also give several results showing that the unique analytical solution of calabi yau odes and more generally picard fuchs linear odes with solutions of maximal weights are always diagonals of rational functions besides in a more enumerative combinatorics context generating functions whose coefficients are expressed in terms of nested sums of products of binomial terms can also be shown to be diagonals of rational functions we finally address the question of the relations between the notion of integrality series with integer coefficients or more generally globally bounded series and the modularity of odes

arithmeticity of some hypergeometric monodromy groups in sp 4 abstract the article 14 gives a list of 51 symplectic hypergeometric monodromy groups corresponding to primitive pairs of degree four polynomials which are products of cyclotomic polynomials and for which the absolute value of the leading coefficient of the difference polynomial is greater than 2 it follows from 12 and 14 that 12 of the 51 monodromy groups are arithmetic cf table 1 and the thinness of 13 of the remaining 39 monodromy groups follows from 3 cf table 2 in this article we show that 15 of the remaining 26 monodromy groups are arithmetic cf table 3

monodromy for the hypergeometric function nfn 1 la fonction hypergeometrique le groupe hypergeometrique la forme hermitienne invariante le cas imprimitif theorie de galois differentielle fonctions hypergeometriques algebriques

ishikawa and mann iterative processes with errors for nonlinear strongly accretive operator equations abstract the purpose of this paper is to revise the definitions of ishikawa and mann iterative processes with errors by using a new inequality to study the unique solution of the nonlinear strongly accretive operator equationtx fand the convergence problem of ishikawa and mann iterative sequences for strongly pseudo contractive mappings without the lipschitz condition the results presented in this paper improve and extend the corresponding results in 4 5 7 10 12 15 16 in the more general setting in particular the open problem mentioned by chidume in 5 has been given an affirmative answer

quantifying adventitious error in a covariance structure as a random effect we present an approach to quantifying errors in covariance structures in which adventitious error identified as the process underlying the discrepancy between the population and the structured model is explicitly modeled as a random effect with a distribution and the dispersion parameter of this distribution to be estimated gives a measure of misspecification analytical properties of the resultant procedure are investigated and the measure of misspecification is found to be related to the root mean square error of approximation an algorithm is developed for numerical implementation of the procedure the consistency and asymptotic sampling distributions of the estimators are established under a new asymptotic paradigm and an assumption weaker than the standard pitman drift assumption simulations validate the asymptotic sampling distributions and demonstrate the importance of accounting for the variations in the parameter estimates due to adventitious error two examples are also given as illustrations

some explicit solutions of the additive deligne simpson problem and their applications abstract in this paper we construct three infinite series and two extra triples e8 and e 8 of complex matrices b c and a b c of special spectral types associated to simpson s classification in amer math soc proc 1 1992 157 and magyar et al classification in adv math 141 1999 97 this enables us to construct fuchsian systems of differential equations which generalize the hypergeometric equation of gauss riemann in a sense they are the closest relatives of the famous equation because their triples of spectral flags have finitely many orbits for the diagonal action of the general linear group in the space of solutions in all the cases except for e8 we also explicitly construct scalar products such that a b and c are self adjoint with respect to them in the context of fuchsian systems these scalar products become monodromy invariant complex symmetric bilinear forms in the spaces of solutions when the eigenvalues of a b and c are real the matrices and the scalar products become real as well we find inequalities on the eigenvalues of a b and c which make the scalar products positive definite as proved by klyachko spectra of three hermitian or real symmetric matrices b c and a b c form a polyhedral convex cone in the space of triple spectra he also gave a recursive algorithm to generate inequalities describing the cone the inequalities we obtain describe non recursively some faces of the klyachko cone

isomonodromy aspects of the tt equations of cecotti and vafa ii riemann hilbert problem in guest et al arxiv 1209 2045 part i we computed the stokes data for the smooth solutions of the tt toda equations whose existence we had previously established by p d e methods here we formulate the existence problem as a riemann hilbert problem based on this stokes data and solve it under certain conditions theorem 5 4 in the process we compute the connection matrix for all smooth solutions thus completing the computation of the monodromy data theorem 5 5 we also give connection formulae relating the asymptotics at zero and infinity of all smooth solutions theorem 4 1 clarifying the region of validity of the formulae established earlier by tracy and widom finally we resolve some conjectures of cecotti and vafa concerning the positivity of s st where s is the stokes matrix and the unimodularity of the eigenvalues of the monodromy matrix theorem 5 6 in particular we show that unitarity implies regularity for the tt toda equations

bayesian calibration of computer models we consider prediction and uncertainty analysis for systems which are approximated using complex mathematical models such models implemented as computer codes are often generic in the sense that by a suitable choice of some of the model s input parameters the code can be used to predict the behaviour of the system in a variety of specific applications however in any specific application the values of necessary parameters may be unknown in this case physical observations of the system in the specific context are used to learn about the unknown parameters the process of fitting the model to the observed data by adjusting the parameters is known as calibration calibration is typically effected by ad hoc fitting and after calibration the model is used with the fitted input values to predict the future behaviour of the system we present a bayesian calibration technique which improves on this traditional approach in two respects first the predictions allow for all sources of uncertainty including the remaining uncertainty over the fitted parameters second they attempt to correct for any inadequacy of the model which is revealed by a discrepancy between the observed data and the model predictions from even the best fitting parameter values the method is illustrated by using data from a nuclear radiation release at tomsk and from a more complex simulated nuclear accident exercise

a differential equation with monodromy group 2 j2 abstract we construct a sixth order differential equation having the central extension of c 2 by the hall janko group j 2 as monodromy group moreover it arises from an iterated application of tensor products and convolution operations from a first order differential equation