a preparation theorem for weierstrass systems it is shown that lion and rolin s preparation theorem for globally subanalytic functions holds for the collection of definable functions in any expansion of the real ordered field by a weierstrass system

implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps convergence theorems for approximation of common fixed points of strictly pseudocontractive mappings of browder petryshyn type are proved in banach spaces using an implicit iteration scheme recently introduced by xu and ori numer funct anal optim 22 2001 767 773

local inequalities for plurisubharmonic functions the main objective of this paper is to prove a new inequality for plurisubharmonic functions estimating their supremum over a ball by their supremum over a measurable subset of the ball we apply this result to study local properties of polynomial algebraic and analytic functions the paper has much in common with an earlier paper br of the author

approximate generalizations and computational experiments in this paper i demonstrate how one can generalize finitely many examples to statements about infinite classes of economic models if there exist upper bounds on the number of connected components of one dimensional linear subsets of the set of parameters for which a conjecture is true one can conclude that it is correct for all parameter values in the class considered except for a small residual set once one has verified the conjecture for a predetermined finite set of points i show how to apply this insight to computational experiments and spell out assumptions on the economic fundamentals that ensure that the necessary bounds on the number of connected components exist

convergence of the modified mann s iteration method for asymptotically strict pseudo contractions abstract let c be a closed convex subset of a real hilbert space h and assume that t is an asymptotically strict pseudo contraction on c with a fixed point for some 0 1 given an initial guess x 0 c and given also a real sequence n in 0 1 the modified mann s algorithm generates a sequence x n via the formula x n 1 n x n 1 n t n x n n 0 it is proved that if the control sequence n is chosen so that n 1 for some 0 1 then x n converges weakly to a fixed point of t we also modify this iteration method by applying projections onto suitably constructed closed convex sets to get an algorithm which generates a strongly convergent sequence

pointwise remez and nikolskii type inequalities for exponential sums let e n biggl f f t a 0 sum n j 1 a j e lambda j t quad a j lambda j in bbb r biggr so e n is the collection of all n 1 term exponential sums with constant first term we prove the following two theorems theorem 1 remez type inequality for e n at 0 let s in left 0 frac 12 right there are absolute constants c 1 0 and c 2 0 such that exp c 1ns leq sup f f 0 leq exp c 2ns where the supremum is taken for all f in e n satisfying m left left x in 1 1 f x leq 1 right right geq 2 s theorem 2 nikolskii type inequality for e n there are absolute constants c 1 0 and c 2 0 such that begin eqnarray c 1 1 1 q left frac 1 qn min y a b y right 1 q leq sup 0 neq f in e n frac f y f l q a b leq left frac c 2 1 qn min y a b y right 1 q end eqnarray for every a y b and q 0 it is quite remarkable that in the above remez and nikolskii type inequalities e n behaves like mathcal p n where mathcal p n denotes the collection of all algebraic polynomials of degree at most n with real coefficients

demiclosedness principle and convergence theorems for strictly pseudocontractive mappings of browder petryshyn type abstract let e be a real q uniformly smooth banach space which is also uniformly convex for example lp or lp spaces 1

small amplitude limit cycles and the distribution of zeros of families of analytic functions we estimate the expected number of limit cycles of a planar polynomial vector field situated in a neighbourhood of the origin provided that the field in a larger neighbourhood is close enough to a linear center our main tool is a distributional inequality for the number of zeros of some families of univariate holomorphic functions depending analytically on a parameter we obtain this inequality by methods of pluripotential theory this inequality also implies versions of a strong law of large numbers and the central limit theorem for a probabilistic scheme associated with the distribution of zeros

valuation theory of exponential hardy fields i we describe the valuation theoretic properties of the hardy fields associated to models of t exp where t is the theory of a polynomially bounded o minimal expansion of the reals and exp is the real exponential function we deduce that t exp has levels with parameters and is exponentially bounded we establish a maximality property of h mathbb r rm an powers the hardy field of the expansion by the restricted analytic functions and power functions

crossover behavior in driven cascades we propose a model which explains how power law crossover behavior can arise in a system which is capable of experiencing cascading failure in our model the susceptibility of the system to cascades is described by a single number the propagation power which measures the ease with which cascades propagate physically such a number could represent the density of unstable material in a system its internal connectivity or the mean susceptibility of its component parts to failure we assume that the propagation power follows an upward drifting brownian motion between cascades and drops discontinuously each time a cascade occurs cascades are described by a continuous state branching process with distributional properties determined by the value of the propagation power when they occur in common with many cascading models pure power law behavior is exhibited at a critical level of propagation power and the mean cascade size diverges this divergence constrains large systems to the subcritical region we show that as a result crossover behavior appears in the cascade distribution when an average is performed over the distribution of propagation power we are able to analytically determine the exponents before and after the crossover

tangent cones and c 1 regularity of definable sets abstract let x r n be a connected locally closed set which is definable in an o minimal structure we prove that the following three statements are equivalent i x is a c 1 manifold ii the tangent cone and the paratangent cone of x coincide at every point in x iii for every x x the tangent cone of x at the point x is a k dimensional linear subspace of r n k does not depend on x varies continuously in x and the density x x 3 2

tropical totally positive matrices abstract we investigate the tropical analogues of totally positive and totally nonnegative matrices these arise when considering the images by the nonarchimedean valuation of the corresponding classes of matrices over a real nonarchimedean valued field like the field of real puiseux series we show that the nonarchimedean valuation sends the totally positive matrices precisely to the monge matrices this leads to explicit polyhedral representations of the tropical analogues of totally positive and totally nonnegative matrices we also show that tropical totally nonnegative matrices with a finite permanent can be factorized in terms of elementary matrices we finally determine the eigenvalues of tropical totally nonnegative matrices and relate them with the eigenvalues of totally nonnegative matrices over nonarchimedean fields

the field of reals with multisummable series and the exponential function we show that the field of real numbers with multisummable real power series is model complete o minimal and polynomially bounded further expansion by the exponential function yields again a model complete and o minimal structure which is exponentially bounded and in which the gamma function on the positive real line is definable 2000 mathematics subject classification primary 03c10 32b05 32b20 secondary 26e05

on the self concordance of the universal barrier function let k be a regular convex cone in hbox bbb r n and let f x be its universal barrier function let d kf x h ldots h be the k th order directional derivative at the point x in k 0 and direction h in hbox bbb r n we show that for every m ge3 there exists a constant c m 0 depending only on m such that d mf x h ldots h le c m d 2f x h h m 2 for m 3 this is the self concordance inequality of nesterov and nemirovskii our proof uses a powerful recent result of bourgain

quasi analytic solutions of analytic ordinary differential equations and o minimal structures it is well known that the non spiraling leaves of real analytic foliations of codimension 1 all belong to the same o minimal structure naturally the question arises if the same statement is true for non oscillating trajectories of real analytic vector fields we show under certain assumptions that such a trajectory generates an o minimal and model complete structure together with the analytic functions the proof uses the asymptotic theory of irregular singular ordinary differential equations in order to establish a quasi analyticity result from which the main theorem follows as applications we present an infinite family of o minimal structures such that any two of them do not admit a common extension and we construct a non oscillating trajectory of a real analytic vector field in dimension 5 that is not definable in any o minimal extension of the reals

smooth functions in o minimal structures abstract fix an o minimal expansion of the real exponential field that admits smooth cell decomposition we study the density of definable smooth functions in the definable continuously differentiable functions with respect to the definable version of the whitney topology this implies that abstract definable smooth manifolds are affine moreover abstract definable smooth manifolds are definably c diffeomorphic if and only if they are definably c 1 diffeomorphic

muntz systems and orthogonal muntz legendre polynomials the muntz legendre polynomials arise by orthogonalizing the muntz system xio x l with respect to lebesgue measure on 0 1 in this paper differential and integral recurrence formulae for the muntz legendre polynomials are obtained interlacing and lexicographical properties of their zeros are studied and the smallest and largest zeros are universally estimated via the zeros of laguerre polynomials the uniform convergence of the christoffel functions is proved equivalent to the nondenseness of the muntz space on 0 1 which implies that in this case the orthogonal muntz legendre polynomials tend to 0 uniformly on closed subintervals of 0 1 some inequalities for muntz polynomials are also investigated most notably a sharp l2 markov inequality is proved

on the convergence of implicit iteration process with error for a finite family of asymptotically nonexpansive mappings abstract the purpose of this paper is to study the weak and strong convergence of implicit iteration process with errors to a common fixed point for a finite family of asymptotically nonexpansive mappings and nonexpansive mappings in banach spaces the results presented in this paper extend and improve the corresponding results of h bauschke the approximation of fixed points of compositions of nonexpansive mappings in hilbert space j math anal appl 202 1996 150 159 b halpern fixed points of nonexpansive maps bull amer math soc 73 1967 957 961 p l lions approximation de points fixes de contractions c r acad sci paris ser a 284 1977 1357 1359 s reich strong convergence theorems for resolvents of accretive operators in banach spaces j math anal appl 75 1980 287 292 z h sun strong convergence of an implicit iteration process for a finite family of asymptotically quasi nonexpansive mappings j math anal appl 286 2003 351 358 r wittmann approximation of fixed points of nonexpansive mappings arch math 58 1992 486 491 h k xu m g ori an implicit iterative process for nonexpansive mappings numer funct anal optimiz 22 2001 767 773 y y zhou s s chang convergence of implicit iterative process for a finite family of asymptotically nonexpansive mappings in banach spaces numer funct anal optimiz 23 2002 911 921

ilyashenko algebras based on transserial asymptotic expansions abstract we construct a hardy field that contains ilyashenko s class of germs at of almost regular functions found in 12 as well as all log exp analytic germs this implies non oscillatory behaviour of almost regular germs with respect to all log exp analytic germs in addition each germ in this hardy field is uniquely characterized by an asymptotic expansion that is an le series as defined by van den dries et al 7 as these series generally have support of order type larger than the notion of asymptotic expansion itself needs to be generalized

transcendence measures and algebraic growth of entire functions in this paper we obtain estimates for certain transcendence measures of an entire function f using these estimates we prove bernstein doubling and markov inequalities for a polynomial p z w in 2 along the graph of f these inequalities provide in turn estimates for the number of zeros of the function p z f z in the disk of radius r in terms of the degree of p and of r

strong convergence of an implicit iteration process for a finite family of asymptotically quasi nonexpansive mappings in the present paper we prove that the modified implicit iteration sequence for a finite family of asymptotically quasi nonexpansive mappings converges strongly to a common fixed point of the family in a uniformly convex banach space requiring one member t in the family to be semi compact our results extend and improve some recent results

complements of subanalytic sets and existential formulas for analytic functions we show that the complement of a subanalytic set defined by real analytic functions from any subalgebra closed under differentiation is a subanalytic set defined by the functions from the same subalgebra this result has an equivalent formulation in logic consider an expression built from functions as above using equalities and inequalities as well as existential and universal quantifiers such an expression is equivalent to an existential expression involving functions from the same class provided that the variables approach neither infinity nor the boundary of the domain

time optimal boundary controls for the heat equation the fact that the time optimal controls for parabolic equations have the bang bang property has been recently proved for controls distributed inside the considered domain interior control the main result in this article asserts that the boundary controls for the heat equation have the same property at least in rectangular domains this result is proved by combining methods from traditionally distinct fields the lebeau robbiano strategy for null controllability and estimates of the controllability cost in small time for parabolic systems on one side and a remez type inequality for muntz spaces and a generalization of turan s inequality on the other side

bernstein inequalities and applications to analytic geometry and differential equations we introduce new methods of complex analysis inequalities of bernstein type to study projections of analytic sets these techniques are applied to problems of bifurcations of periodic orbits of differential equations such as the local hilbert s 16th problem

markov s inequality and polynomial mappings markov s inequality is a certain estimate for the norm of the derivative of a polynomial in terms of the degree and the norm of this polynomial it has many interesting applications in approximation theory constructive function theory and in analysis for instance to sobolev inequalities or whitney type extension problems one of the purposes of this paper is to give a solution to an old problem studied among others by baran and ple niak and concerning the invariance of markov s inequality under polynomial mappings polynomial images we also address the issue of preserving markov s inequality when taking polynomial preimages lastly we give a sufficient condition for a subset of a markov set to be a markov set

comparison theorems for o minimal singular co homology fct grant sfrh bpd 6015 2001 european research and training network hprn ct 2001 00271 raag

the real field with convergent generalized power series we construct a model complete and o minimal expansion of the field of real numbers in which each real function given on 0 1 by a series cnxn with 0 n and cn r n 1 is definable this expansion is polynomially bounded

graphs of zeros of analytic families let f f be a family of holomorphic functions in a domain d c depending holomorphically on u c we study the distribution of zeros of f in a subdomain r d whose boundary is a closed non singular analytic curve as an application we obtain several results about distributions of zeros of families of generalized exponential polynomials and displacement maps related to certain ode s

explicit averaging cyclic algorithm for common fixed points of asymptotically strictly pseudocontractive maps weak and strong convergence theorems for the approximation of common fixed points of a finite family of asymptotically strictly pseudocontractive mappings are proved in hilbert spaces using an explicit averaging cyclic algorithm

introduction to stochastic processes preface to second edition preface to first edition preliminaries introduction linear differential equations linear difference equations exercises finite markov chains definitions and examples large time behavior and invariant probability classification of states return times transient states examples exercises countable markov chains introduction recurrence and transience positive recurrence and null recurrence branching process exercises continuous time markov chains poisson process finite state space birth and death processes general case exercises optimal stopping optimal stopping of markov chains optimal stopping with cost optimal stopping with discounting exercises martingales conditional expectation definition and examples optional sampling theorem uniform integrability martingale convergence theorem maximal inequalities exercises renewal processes introduction renewal equation discrete renewal processes m g 1 and g m 1 queues exercises reversible markov chains reversible processes convergence to equilibrium markov chain algorithms a criterion for recurrence exercises brownian motion introduction markov property zero set of brownian motion brownian motion in several dimensions recurrence and transience fractal nature of brownian motion scaling rules brownian motion with drift exercises stochastic integration integration with respect to random walk integration with respect to brownian motion ito s formula extensions if ito s formula continuous martingales girsanov transformation feynman kac formula black scholes formula simulation exercises suggestions for further reading index

on local behavior of holomorphic functions along complex submanifolds of ℂ n in this paper we establish some general results on local behavior of holomorphic functions along complex submanifolds of n as a corollary we present multi dimensional generalizations of an important result of coman and poletsky on bernstein type inequalities on transcendental curves in 2

tauberian theorems for k summability with respect to an analytic germ abstract the goal of this article is to establish tauberian theorems for the k summability processes defined by germs of analytic functions in several complex variables the proofs are based on the tauberian theorems for k summability in one variable and in monomials and a method of monomialization of germs of analytic functions

canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant this article contains an elementary constructive proof of resolution of singularities in characteristic zero our proof applies in particular to schemes of finite type and to analytic spaces so we recover the great theorems of hironaka we introduce a discrete local invariant rm inv x a whose maximum locus determines a smooth centre of blowing up leading to desingularization to define rm inv x we need only to work with a category of local ringed spaces x x cal o x satisfying certain natural conditions if a in x then rm inv x a depends only on widehat cal o x a more generally rm inv x is defined inductively after any sequence of blowings up whose centres have only normal crossings with respect to the exceptional divisors and lie in the constant loci of rm inv x cdot the paper is self contained and includes detailed examples one of our goals is that the reader understand the desingularization theorem rather than simply know it is true

approximate solutions and micro regularity in the denjoy carleman classes abstract we begin with a sequence m of positive real numbers and we consider the denjoy carleman class c m we show how to construct m approximate solutions for complex vector fields with c m coefficients we then use our construction to study micro local properties of boundary values of approximate solutions in general m involutive structures of codimension one where the approximate solution is defined in a wedge whose edge where the boundary value exists is a maximally real submanifold we also obtain a c m version of the edge of the wedge theorem

muntz spaces and remez inequalities two relatively long standing conjectures concerning muntz polynomials are resolved the central tool is a bounded remez type inequality for non dense muntz spaces

approximation and existence of schauder bases in muntz spaces of l1 functions abstract an approximation of functions in muntz spaces m l 1 of l 1 functions is studied with the help of fourier series muntz spaces are considered with the muntz condition and the gap condition imposed it is proved that up to an isomorphism such spaces are contained in weil nagy s class moreover existence of schauder bases in muntz spaces m l 1 is investigated

remez type inequalities and their applications abstract the remez inequality gives a sharp uniform bound on 1 1 for real algebraic polynomials p of degree at most n if the lebesgue measure of the subset of 1 1 where p is at most 1 is known remez type inequalities give bounds for classes of functions on a line segment on a curve or on a region of the complex plane given that the modulus of the functions is bounded by 1 on some subset of prescribed measure this paper offers a survey of the extensive recent research on remez type inequalities for polynomials generalized nonnegative polynomials exponentials of logarithmic potentials and muntz polynomials remez type inequalities play a central role in proving other important inequalities for the above classes the paper illustrates the power of remez type inequalities by giving a number of applications

geometry of holomorphic mappings and holder continuity of the pluricomplex green function we provide a solution to a long standing open problem that lives in the interface of pluripotential theory and multivariate approximation theory the problem is to characterize the holomorphic maps which preserve holder continuity of the pluricomplex green function associated with a compact subset of mathbb c n we also prove under mild restrictions that nondegenerate holomorphic maps preserve markov s inequality for polynomials

the rational points of a definable set let x r n be a set that is definable in an o minimal structure over r this article shows that in a suitable sense there are very few rational points of x which do not lie on some connected semialgebraic subset of x of positive dimension

strong convergence of the cq method for fixed point iteration processes strong convergence theorems are obtained for the cq method for an ishikawa iteration process a contractive type iteration process for nonexpansive mappings and the proximal point algorithm for maximal monotone operators in hilbert spaces

generalizations of muntz s theorem via a remez type inequality for muntz spaces the principal result of this paper is a remez type inequality for m untz polynomials p x sum n i 0 a i x lambda i or equivalently for dirichlet sums p t sum n i 0 a i e lambda i t where 0 lambda 0 0 here cdot a denotes the supremum norm on a this remez type inequality allows us to resolve two reasonably long standing conjectures the first due to d j newman and dating from 1978 asserts that if sum infty i 1 1 lambda i infty then the set of products p 1 p 2 p 1 p 2 in text span x lambda 0 x lambda 1 ldots is not dense in c 0 1 the second is a complete extension of m untz s classical theorem on the denseness of m untz spaces in c 0 1 to denseness in c a where a subset 0 infty is an arbitrary compact set with positive lebesgue measure that is for an arbitrary compact set a subset 0 infty with positive lebesgue measure text span x lambda 0 x lambda 1 ldots is dense in c a if and only if sum infty i 1 1 lambda i infty several other interesting consequences are also presented

self organized critical forest fire model a forest fire model is introduced which contains a lightning probability f this leads to a self organized critical state in the limit f ensuremath rightarrow 0 provided that the time scales of tree growth and burning down of forest clusters are separated we derive scaling laws and calculate all critical exponents the values of the critical exponents are confirmed by computer simulations for a two dimensional system we show that the forest density in the critical state assumes its minimum possible value i e that energy dissipation is maximum

remez type inequality on sets with cusps abstract a remez type inequality is proved for a large family of sets with cusps in r n including compact fat and semialgebraic or subanalytic sets

demiclosedness principle and convergence theorems for k strictly asymptotically pseudocontractive maps let e be a real q uniformly smooth banach space which is also uniformly convex for example lp or lp spaces 1 p and k a nonempty closed convex not necessarily bounded subset of e let t k k be a k strictly asymptotically pseudocontractive map with a nonempty fixed point set it is proved that i t is demiclosed at 0 furthermore weak and strong convergence of an averaging iteration method to a fixed point of t are proved

definable zero sum stochastic games definable zero sum stochastic games involve a finite number of states and action sets and reward and transition functions that are definable in an o minimal structure prominent examples of such games are finite semi algebraic or globally subanalytic stochastic games we prove that the shapley operator of any definable stochastic game with separable transition and reward functions is definable in the same structure definability in the same structure does not hold systematically we provide a counterexample of a stochastic game with semi algebraic data yielding a non semi algebraic but globally subanalytic shapley operator our definability results on shapley operators are used to prove that any separable definable game has a uniform value in the case of polynomially bounded structures we also provide convergence rates using an approximation procedure we actually establish that general zero sum games with separable definable transition functions have a uniform value these results highlight the key role played by the tame structure of transition functions as particular cases of our main results we obtain that stochastic games with polynomial transitions definable games with finite actions on one side and definable games with perfect information or switching controls have a uniform value applications to nonlinear maps arising in risk sensitive control and perron frobenius theory are also given

the elementary theory of restricted analytic fields with exponentiation numbers with exponentiation is model complete when we combine this with hovanskii s finiteness theorem 9 it follows that the real exponential field is o minimal in o minimal expansions of the real field the definable subsets of r share many of the nice structural properties of semialgebraic sets for example definable subsets have only finitely many connected components definable sets can be stratified and triangulated and continuous definable maps are piecewise trivial see 5 in this paper we will prove a quantifier elimination result for the real field augmented by exponentiation and all restricted analytic functions and use this result to obtain o minimality we were led to this while studying work of ressayre 13 and several of his ideas emerge here in simplified form however our treatment is formally independent of the results of 16 17 9 and 13

iterative methods for strict pseudo contractions in hilbert spaces abstract let t i i 1 n be n strict pseudo contractions defined on a closed convex subset c of a real hilbert space h consider the problem of finding a common fixed point of these mappings and consider the parallel and cyclic algorithms for solving this problem we will prove the weak convergence of these algorithms moreover by applying additional projections we further prove that these algorithms can be modified to have strong convergence

self organized branching processes avalanche models with dissipation we explore in the mean field approximation robustness with respect to dissipation of self organized criticality in sandpile models to this end we generalize a recently introduced self organized branching process and show that the model self organizes not into a critical state but rather into a subcritical state when dissipation is present the dynamical fixed point does not coincide with the critical point thus the level of dissipation acts as a relevant parameter in the renormalization group sense we study the model numerically and compute analytically the critical exponents for the avalanche size and lifetime distributions and the scaling exponents for the corresponding cutoffs s1063 651x 96 08009 9

asymptotic properties of sobolev orthogonal polynomials abstract in this report we will survey some of the main ideas and tools which appeared recently in the study of the analytic properties of polynomials orthogonal with respect to inner products involving derivatives although some results on weak asymptotics are mentioned the strong outer asymptotics constitutes the core of the paper both the discrete and the continuous cases are considered and several open problems and conjectures are posed

centennial history of hilbert s 16th problem the second part of hilbert s 16th problem deals with polynomial differential equations in the plane it remains unsolved even for quadratic polynomials there were several attempts to solve it that failed yet the problem inspired significant progress in the geometric theory of planar differential equations as well as bifurcation theory normal forms foliations and some topics in algebraic geometry the dramatic history of the problem as well as related developments are presented below 1 the problem and its counterparts what may be said about the number and location of limit cycles of a planar polynomial vector field of degree n the limit cycle is an isolated closed orbit of a vector field this second part of hilbert s 16th problem appears to be one of the most persistent in the famous hilbert list h second only to the riemann function conjecture traditionally hilbert s question is split into three each one requiring a stronger answer problem 1 is it true that a planar polynomial vector field has but a finite number of limit cycles problem 2 is it true that the number of limit cycles of a planar polynomial vector field is bounded by a constant depending on the degree of the polynomials only the bound on the number of limit cycles in problem 2 is denoted by h n and known as the hilbert number linear vector fields have no limit cycles hence h 1 0 it is still unknown whether or not h 2 exists problem 3 give an upper bound for h n a solution to any of these problems implies a solution for the previous ones only the first problem is solved now the positive answer was established in e92 i91 there are analytic counterparts of problems 1 and 2 received by the editors december 2001 2000 mathematics subject classification primary 34cxx 34mxx 37f75

counting special points logic diophantine geometry and transcendence theory we expose a theorem of pila and wilkie on counting rational points in sets definable in o minimal structures and some applications of this theorem to problems in diophantine geometry due to masser peterzil pila starchenko and

strong convergence theorem for pseudo contractive mappings in hilbert spaces abstract the purpose of this paper is to construct an ishikawa type of hybrid algorithm for pseudo contractive mappings in hilbert spaces our results extend the recent ones announced by yao et al y h yao y c liou g marino a hybrid algorithm for pseudo contractive mappings nonlinear anal 71 2009 4997 5002 and many others

level spacing for non monotone anderson models we prove localization and probabilistic bounds on the minimum level spacing for a random block anderson model without monotonicity using a sequence of narrowing energy windows and associated schur complements we obtain detailed probabilistic information about the microscopic structure of energy levels of the hamiltonian as well as the support and decay of eigenfunctions

on local behavior of analytic functions we prove local inequalities for analytic functions defined on a convex body in r n which generalize well known classical inequalities for polynomials

quasianalytic denjoy carleman classes and o minimality we show that the expansion of the real field generated by the functions of a quasianalytic denjoy carleman class is model complete and o minimal provided that the class satisfies certain closure conditions some of these structures do not admit analytic cell decomposition and they show that there is no largest o minimal expansion of the real field

o minimality and the andre oort conjecture for mathbb c n we give an unconditional proof of the andre oort conjecture for arbitrary products of modular curves we establish two generalizations the first includes the manin mumford conjecture for arbitrary products of elliptic curves defined over q as well as lang s conjecture for torsion points in powers of the multiplicative group the second includes the manin mumford conjecture for abelian varieties defined over q our approach uses the theory of o minimal structures a part of model theory and follows a strategy proposed by zannier and implemented in three recent papers a new proof of the manin mumford conjecture by pila zannier a proof of a special but new case of pink s relative manin mumford conjecture by masser zannier and new proofs of certain known results of andre oort manin mumford type by pila 2010 mathematics subject classification 11g18 03c64