first order intrinsic autoregressions and the de wijs process summary we discuss intrinsic autoregressions for a first order neighbourhood on a two dimensional rectangular lattice and give an exact formula for the variogram that extends known results to the asymmetric case we obtain a corresponding asymptotic expansion that is more accurate and more general than previous ones and use this to derive the de wijs variogram under appropriate averaging a result that can be interpreted as a two dimensional spatial analogue of brownian motion obtained as the limit of a random walk in one dimension this provides a bridge between geostatistics where the de wijs process was once the most popular formulation and markov random fields and also explains why statistical analysis using intrinsic autoregressions is usually robust to changes of scale we briefly describe corresponding calculations in the frequency domain including limiting results for higher order autoregressions the paper closes with some practical considerations including applications to irregularly spaced data

cartan decomposition subgroups of 2 for g sl 3 r and g so 2 n we give explicit practical conditions that determine whether or not a closed connected subgroup h of g has the property that there exists a compact subset c of g with chc g to do this we fix a cartan decomposition g ka k of g and then carry out an approximate calculation of khk n a for each closed connected subgroup h of g

on conditional and intrinsic autoregressions summary gaussian conditional autoregressions have been widely used in spatial statistics and bayesian image analysis where they are intended to describe interactions between random variables at fixed sites in euclidean space the main appeal of these distributions is in the markovian interpretation of their full conditionals intrinsic autoregressions are limiting forms that retain the markov property despite being improper they can have advantages over the standard autoregressions both conceptually and in practice for example they often avoid difficulties in parameter estimation without apparent loss or exhibit appealing invariances as in texture analysis however on small arrays and in nonlattice applications both forms of autoregression can lead to undesirable second order characteristics either in the variables themselves or in contrasts among them this paper discusses standard and intrinsic autoregressions and describes how the problems that arise can be alleviated using dempster s 1972 algorithm or an appropriate modification the approach represents a partial synthesis of standard geostatistical and gaussian markov random field formulations some nonspatial applications are also mentioned

parameter estimates for fractional autoregressive spatial processes a binomial type operator on a stationary gaussian process is introduced in order to model long memory in the spatial context consistent estimators of model parameters are demonstrated in particular it is shown that d n d o p log n 3 n where d d 1 d 2 denotes the long memory parameter

closed asymptotic couples the derivation of a hardy field induces on its value group a certain function if a hardy field extends the real field and is closed under powers then its value group is also a vector space over such ordered vector spaces with function are called h couples we define closed h couples and show that every h couple can be embedded into a closed one the key fact is that closed h couples have an elimination theory solvability of an arbitrary system of equations and inequalities built up from vector space operations the function parameters and the unknowns to be solved for is equivalent to an effective condition on the parameters of the system the h couple of a maximal hardy field is closed and this is also the case for the h couple of the field of logarithmic exponential series over we analyze in detail finitely generated extensions of a given h couple

vapnik chervonenkis density in some theories without the independence property ii we study the vapnik chervonenkis vc density of denable families in certain stable rst order theories in particular we obtain uniform bounds on vc density of denable families in nite u rank theories without the nite cover property and we characterize those abelian groups for which there exist uniform bounds on the vc density of denable families

the asymptotic couple of the field of logarithmic transseries abstract the derivation on the differential valued field t log of logarithmic transseries induces on its value group log a certain map the structure log is a divisible asymptotic couple we prove that the theory t log th log admits elimination of quantifiers in a natural first order language all models of t log have an important discrete subset 0 we give explicit descriptions of all definable functions on and prove that is stably embedded in

new fast algorithms for polynomial interpolation and evaluation on the chebyshev node set abstract for a polynomial p x of a degree n we study its interpolation and evaluation on a set of chebyshev nodes x cos 2 1 2n 2 0 1 n this is easily reduced to applying discrete fourier transforms dfts to the auxiliary polynomial q n p x where 2 x 1 exp 1 2n we show the back and forth transition between p x and q based on the respective back and forth transformations of the variable 1 z 1 z y x 1 x 1 y z 2 all these transformations like the dfts are performed by using o n log n arithmetic operations which thus suffice in order to support both interpolation and evaluation of p x on the chebychev set as well as on some related sets of nodes this improves by factor log n the known arithmetic time bound for chebyshev interpolation and gives an alternative supporting algorithm for the record estimate of o n log n for chebyshev evaluation obtained by gerasoulis in 1987 and based on a distinct algorithm

relational queries over interpreted structures we rework parts of the classical relational theory when the underlying domain is a structure with some interpreted operations that can be used in queries we identify parts of the classical theory that go through as before when interpreted structure is present parts that go through only for classes of nicely behaved structures and parts that only arise in the interpreted case the first category include a number of results on language equivalence and expressive power characterizations for the active domain semantics for a variety of logics under this semantics quantifiers range over elements of a relational database the main kind of results we prove here are generic collapse results for generic queries adding operations beyond order does not give us extra power the second category includes results on the natural semantics under which quantifiers range over the entire interpreted structure we prove for a variety of structures natural active collapse results showing that using unrestricted quantification does not give us any extra power moreover for a variety of structures including the real field we give a set of algorithms for eliminating unbounded quantifications in favor of bounded ones furthermore we extend these collapse results to a new class of higher order logics that mix unbounded and bounded quantification we give a set of normal forms for these logics under special conditions on the interpreted structures as a by product we obtain an elementary proof of the fact that parity test is not definable in the relational calculus with polynomial inequality constraints we also give examples of structures with nice model theoretic properties over which the natural active collapse fails

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

real polynomial root finding by means of matrix and polynomial iterations frequently one seeks approximation to all r real roots of a polynomial of degree n with real coefficients which also has nonreal roots we split a polynomial into two factors one of which has degree r and has r real roots we approximate them at a low cost and then decrease the arithmetic time of the known algorithms for this popular problem by roughly a factor of n k if k iterations prepare splitting k is a small integer unless some nonreal roots lie close to the real axis but even if there nonreal roots near the real axis we substantially accelerate the known algorithms we also propose a dual algorithm operating with the associated structured matrices at the price of minor increase of the arithmetic time it facilitates numerical implementation our analysis and tests demonstrate the efficiency of our approach

definable sets in ordered structures i this paper introduces and begins the study of a well behaved class of linearly ordered structures the minimal structures the definition of this class and the corresponding class of theories the strongly dlminlmal theories is made in analogy with the notions from stability theory of minimal structures and strongly minimal theories theorems 2 1 and 2 3 respectively provide characterizations of 41 minimal ordered groups and rings several other simple results are collected in 3 the primary tool in the analysis of minimal structures is a strong analogue of ii forking symmetry given by theorem 4 2 this result states that any parametri cally definable unary function in an yminimal structure is piecewise either constant or an ore er preserving or reversing bijection of intervals the results that follow include the existence and uniqueness of prime models over sets theorem 5 1 and a characterization of all bto categorical aminimal structures theorem 6 1 1 introduction the class of linearly ordered structures has long been an im portant subject of concern to model theorists impressive results have been obtained in the study of models of several particular theories that extend the theory of linear order among those theories that have been approached successfully are peano arithmetic the theory of ordered abelian groups that of real closed fields and that of linear order itself yet very little has been done in the way of developing a general model theory for ordered structures in this paper we develop the model theory for a class of linearly ordered structures that we isolate by demanding that a structure in this class satisfy a condition whose effect is that the linear ordering and the algebraic part of the structure behave quite well with respect to one another let l be a finitary first order language and w an l structure a set of n tuples a c x is said to be parametrically definable if there is some l formula

the differential rank of a differential valued field we develop a notion of principal differential rank for differential valued fields in analog of the exponential rank developed in kuhlmann the fields institute monograph series 12 2000 and of the difference rank developed in kuhlmann groups modules and model theory surveys and recent developments in memory of rdiger gbel pp 399 414 2017 we give several characterizations of this rank we then give a method to define a derivation on a field of generalized power series and use this method to show that any totally ordered set can be realized as the principal differential rank of a h field

rigorous uniform approximation of d finite functions using chebyshev expansions a wide range of numerical methods exists for computing polynomial approximations of solutions of ordinary differential equations based on chebyshev series expansions or chebyshev interpolation polynomials we consider the application of such methods in the context of rigorous computing where we need guarantees on the accuracy of the result and from the complexity point of view it is well known that the order n truncation of the chebyshev expansion of a function over a given interval is a near best uniform polynomial approximation of the function on that interval in the case of solutions of linear differential equations with polynomial coefficients the coefficients of the expansions obey linear recurrence relations with polynomial coefficients unfortunately these recurrences do not lend themselves to a direct recursive computation of the coefficients owing among other things to a lack of initial conditions we show how they can nevertheless be used as part of a validated process to compute good uniform approximations of d finite functions together with rigorous error bounds and we study the complexity of the resulting algorithms our approach is based on a new view of a classical numerical method going back to clenshaw combined with a functional enclosure method

analytic p adic cell decomposition and integrals we prove a conjecture of denef on parameterized p adic analytic integrals using an analytic cell decomposition theorem which we also prove in this paper this cell decomposition theorem describes piecewise the valuation of analytic functions and more generally of subanalytic functions the pieces being geometrically simple sets called cells we also classify silbanalytic sets up to subanalytic bijection

an explicit link between gaussian fields and gaussian markov random fields the stochastic partial differential equation approach continuously indexed gaussian fields gfs are the most important ingredient in spatial statistical modelling and geostatistics the specification through the covariance function gives an intuitive interpretation of the field properties on the computational side gfs are hampered with the big n problem since the cost of factorizing dense matrices is cubic in the dimension although computational power today is at an all time high this fact seems still to be a computational bottleneck in many applications along with gfs there is the class of gaussian markov random fields gmrfs which are discretely indexed the markov property makes the precision matrix involved sparse which enables the use of numerical algorithms for sparse matrices that for fields in r 2 only use the square root of the time required by general algorithms the specification of a gmrf is through its full conditional distributions but its marginal properties are not transparent in such a parameterization we show that using an approximate stochastic weak solution to linear stochastic partial differential equations we can for some gfs in the matern class provide an explicit link for any triangulation of r d between gfs and gmrfs formulated as a basis function representation the consequence is that we can take the best from the two worlds and do the modelling by using gfs but do the computations by using gmrfs perhaps more importantly our approach generalizes to other covariance functions generated by spdes including oscillating and non stationary gfs as well as gfs on manifolds we illustrate our approach by analysing global temperature data with a non stationary model defined on a sphere less

independence in computable algebra abstract we give a sufficient condition for an algebraic structure to have a computable presentation with a computable basis and a computable presentation with no computable basis we apply the condition to differentially closed real closed and difference closed fields with the relevant notions of independence to cover these classes of structures we introduce a new technique of safe extensions that was not necessary for the previously known results of this kind we will then apply our techniques to derive new corollaries on the number of computable presentations of these structures the condition also implies classical and new results on vector spaces algebraically closed fields torsion free abelian groups and archimedean ordered abelian groups

hardy type derivations on generalised series fields abstract we consider the valued field k r of generalised series with real coefficients and monomials in a totally ordered multiplicative group we investigate how to endow k with a series derivation that is a derivation that satisfies some natural properties such as commuting with infinite sums strong linearity and an infinite version of leibniz rule we characterise when such a derivation is of hardy type that is when it behaves like differentiation of germs of real valued functions in a hardy field we provide a necessary and sufficient condition for a series derivation of hardy type to be surjective

definable sets in ordered structures on introduit la notion de theorie o minimale des structures ordonnees une telle theorie etant telle que les sous ensembles definissables de ses modeles soient particulierement simples

evidence for conformal invariance of crop yields the aim of this paper is to study the nature of spatial correlation of yields of agricultural crops the focus is primarily on natural or non anthropogenic spatial variation patterns that cannot be explained by topography by variety or treatment effects or by agricultural practices conformal invariance implies stationarity and isotropy and also determines the rate of decay of spatial correlations the resulting gaussian model is studied empirically to see whether it describes satisfactorily the pattern of spatial correlations observed in field trials of various crops by embedding the law in a larger

o minimal hauptvermutung for polyhedra i milnor discovered two compact polyhedra which are homeomorphic but not pl homeomorphic a counterexample to the hauptvermutung he constructed the homeomorphism by a finite procedure repeated infinitely often informally we call a procedure constructive if it consists of an explicit procedure that is repeated only finitely many times in this sense milnor did not give a constructive procedure to define the homeomorphism between the two polyhedra in the case where the homeomorphism is semialgebraic the author and yokoi proved that the polyhedra in r n are pl homeomorphic in that article the required pl homeomorphism was not constructively defined from the given homeomorphism in the present paper we obtain the pl homeomorphism by a constructive procedure starting from the homeomorphism we prove in fact that for any ordered field r equipped with any o minimal structure two definably homeomorphic compact polyhedra in r n are pl homeomorphic the o minimal hauptvermutung theorem 1 1 together with the fact that any compact definable set is definably homeomorphic to a compact polyhedron we can say that o minimal topology is tame

an estimate for the siciak extremal function subanalytic geometry approach we study the ojasiewicz siciak inequality in cn this concerns the estimating from below the siciak extremal function which is very much related to the green function in c as an application we present some results describing how fast holomorphic functions defined in a neighbourhood of a compact holomorphic polyhedron can be approximated uniformly on this polyhedron by complex polynomials

the fine structure of the kasparov groups i continuity of the kk pairing in this paper it is demonstrated that the kasparov pairing is continuous with respect to the natural topology on the kasparov groups so that a kk equivalence is an isomorphism of topological groups in addition we demonstrate that the groups have a natural pseudopolonais structure and we prove that various kk structural maps are continuous

how much information can one get from a wireless ad hoc sensor network over a correlated random field new large deviations results that characterize the asymptotic information rates for general d dimensional d d stationary gaussian fields are obtained by applying the general results to sensor nodes on a two dimensional 2 d lattice the asymptotic behavior of ad hoc sensor networks deployed over correlated random fields for statistical inference is investigated under a 2 d hidden gauss markov random field model with symmetric first order conditional autoregression and the assumption of no in network data fusion the behavior of the total obtainable information nats and energy efficiency nats j defined as the ratio of total gathered information to the required energy is obtained as the coverage area node density and energy vary when the sensor node density is fixed the energy efficiency decreases to zero with rate theta area 1 2 and the per node information under fixed per node energy also diminishes to zero with rate o nt 1 3 as the number nt of network nodes increases by increasing the coverage area as the sensor spacing dn increases the per node information converges to its limit d with rate d radic dn e alphadn for a given diffusion rate alpha when the coverage area is fixed and the node density increases the per node information is inversely proportional to the node density as the total energy et consumed in the network increases the total information obtainable from the network is given by o loget for the fixed node density and fixed coverage case and by theta et 2 3 for the fixed per node sensing energy and fixed density and increasing coverage case

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

a fast algorithm for the multiplication of generalized hilbert matrices with vectors on presente un algorithme rapide de complexite logarithmique pour la multiplication des matrices de hilbert generalisees par des vecteurs

elimination of quantifiers in algebraic structures one of the first techniques used in the study of the logical properties of algebraic objects was that of elimination of quantifiers it was for example used by tarski 161 to prove the decidability of the theory of any given algebraically closed field and the decidability of the theory of real closed ordered fields the theories of various other algebraic structures e g the padic numbers divisible ordered abelian groups may also be shown to eliminate quantifiers in appropriate languages the choice of the language is especially important in these dealings it must be chosen to lie close to natural algebraic phenomena any structure may be made to admit elimination of quantifiers by the choice of a sufficiently complex language the structure may be skolemized or morleyized 151 clearly the resulting languages are too far from the natural relations on the objects to be of immediate algebraic interest in this paper we will show that for many natural languages those structures which have been shown in the past to admit elimination of quantifiers are in fact the only structures of their type to admit elimination of quantifiers in that language in particular we will show that apart from the finite fields and the algebraically closed fields there are no other fields which admit elimination of quantifiers in the language of rings with identity i e the language with primitives 0 l we will show that the only theory of ordered fields that admits elimination of quantifiers in the natural language of ordered domains is the theory of real closed ordered fields we show that if k is a nontrivially valued field admitting elimination of quantifiers in the language of valued fields then k is algebraically closed now the theory of the p adic numbers does not admit elimination of quantifiers in the pure language of valued fields cf 121 however if we include certain auxilary predicates p x such that p x holds if and only if x has an nth

the six grothendieck operations on o minimal sheaves in this paper we develop the formalism of the grothendieck six operations on o minimal sheaves the grothendieck formalism allows us to obtain o minimal versions of i derived projection formula ii universal coefficient formula iii derived base change formula iv kunneth formula v local and global verdier duality

generically stable and smooth measures in nip theories we formulate the measure analogue of generically stable types in first order theories with nip without the independence property giving several characterizations answering some questions from 9 and giving another treatment of uniqueness results from 9 we introduce a notion of generic compact domination relating it to stationarity of keisler measures and also giving group versions we also prove the approximate definability of arbitrary borel probability measures on definable sets in the real and p adic fields

on the number of sets definable by polynomials abstract we show that the known algorithms used to re write any first order quantifier free formula over an algebraically closed field into its normal disjunctive form are essentially optimal this result follows from an estimate of the number of sets definable by equalities and inequalities of fixed polynomials finally we apply our results to obtain similar estimates in the real case

on the decidability of termination of query evaluation in transitive closure logics for polynomial constraint databases the formalism of constraint databases in which possibly infinite data sets are described by boolean combinations of polynomial inequality and equality constraints has its main application area in spatial databases the standard query language for polynomial constraint databases is first order logic over the reals because of the limited expressive power of this logic with respect to queries that are important in spatial data base applications various extensions have been introduced we study extensions of first order logic with different types of transitive closure operators and we are in particular interested in deciding the termination of the evaluation of queries expressible in these transitive closure logics it turns out that termination is undecidable in general however we show that the termination of the transitive closure of a continuous function graph in the two dimensional plane viewed as a binary relation over the reals is decidable and even expressible in first order logic over the reals based on this result we identify a particular transitive closure logic for which termination of query evaluation is decidable and which is more expressive than first order logic over the reals furthermore we can define a guarded fragment in which exactly the terminating queries of this language are expressible

isomorphism theorem for bss recursively enumerable sets over real closed fields the main result of this paper lies in the framework of bss computability it shows roughly that any recursively enumerable set s in rn n where r is a real closed field is isomorphic to rdims by a bijection which is decidable over s moreover the map s is computable some related matters are also considered like characterization of the real closed fields with a r e set of infinitesimals and the dimension of r e sets

semialgebraic expansions of c we prove no nontrivial expansion of the field of complex numbers can be obtained from a reduct of the field of real numbers

on a system of two dimensional recurrence equations summary the solution of a system of two dimensional recurrence equations is discussed particularly in the context of the auto correlation structure of a stationary first order autonormal scheme on the doubly infinite rectangular lattice a simple approximation in the region of primary interest is described

on confidence intervals for spatial parameters estimated from nonreplicated data statistical analysis of replicated planar samples provides important information about variability of point estimates computed from spatial patterns such as botanical mosaics however replication can be done only at a cost there are instances where only a single sample is available for example the pattern of vegetation within a specific region in the present note we discuss methods for constructing conservative confidence intervals using a single planar sample a variant of the technique is also applicable to replicated data and may prove useful when the number of replications is small we assume that the spatial pattern is homogeneous and that there is negligible interaction between properties of the pattern at points sufficiently far apart statistical tests based on correlation in spatial maps are used to estimate an upper bound d to the minimum separation at which there is no interaction then the sample is partitioned into a lattice with edge width d and patterns within individual faces of the lattice are analyzed as though patterns within nonneighbouring faces were independent a bootstrap argument is used to construct confidence intervals based on independent pieces of pattern and then intervals are combined using a bonferroni inequality our techniques may be used to construct confidence intervals for parameters that can be estimated unbiasedly from data within general regions these include porosities e g pielou 1964 serra 1982 p 487 parameters based on curvature e g mullins 1976 1978 kellerer 1983 and cell counts in tissue sections our techniques have analogues for weakly dependent point processes see diggle 1983 for biological examples where confidence intervals for intensity may be constructed our assumption about negligible interaction at some distance d is valid in the case of patterns that are well approximated by independent and identically distributed bounded random sets centred at points of a homogeneous poisson process the latter is often termed a boolean model and has the important characteristic that sets are allowed to overlap one example concerning porosity is studied in section 4 and others may be found in serra 1982 part 4 on the other hand spatial patterns in which sets are rigid and do not overlap can exhibit interactions that are strong over an unexpectedly long range and have oscillating

bayesian analysis of agricultural field experiments the paper describes bayesian analysis for agricultural field experiments a topic that has received very little previous attention despite a vast frequentist literature adoption of the bayesian paradigm simplifies the interpretation of the results especially in ranking and selection also complex formulations can be analysed with comparative ease by using markov chain monte carlo methods a key ingredient in the approach is the need for spatial representations of the unobserved fertility patterns this is discussed in detail problems caused by outliers and by jumps in fertility are tackled via hierarchical f formulations that may find use in other contexts the paper includes three analyses of variety trials for yield and one example involving binary data none is entirely straightforward some numerical comparisons with frequentist analyses are made

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

new progress in real and complex polynomial root finding matrix methods are increasingly popular for polynomial root finding the idea is to approximate the roots as the eigenvalues of the companion or generalized companion matrix associated with an input polynomial the algorithms also solve secular equation qr algorithm is the most customary method for eigen solving but we explore the inverse rayleigh quotient iteration instead which turns out to be competitive with the most popular root finders because of its excellence in exploiting matrix structure to advance the iteration we preprocess the matrix and incorporate newton s linearization repeated squaring homotopy continuation techniques and some heuristics the resulting algorithms accelerate the known numerical root finders for univariate polynomial and secular equations and are particularly well suited for the acceleration by using parallel processing furthermore even on serial computers the acceleration is dramatic for numerical approximation of the real roots in the typical case where they are much less numerous than all complex roots

definable sets in algebraically closed valued fields elimination of imaginaries it is shown that if k is an algebraically closed valued field with valuation ring r then th k has elimination of imaginaries if sorts are added whose elements are certain cosets in k n of certain definable r submodules of k n for all n 1 the proof involves the development of a theory of independence for unary types which play the role of 1 types followed by an analysis of germs of definable functions from unary sets to the sorts contents

linear equations in variables which lie in a multiplicative group let k be a field of characteristic 0 and let n be a natural number let r be a subgroup of the multiplicative group k n of finite rank r given a 1 a n e k write a a 1 a n for the number of solutions x x 1 x n of the equation a 1 x 1 a n x n 1 such that no proper subsum of a 1 x 1 a n x n vanishes we derive an explicit upper bound for a a 1 a n r which depends only on the dimension n and on the rank r

rings of finite representation type and modules of finite morley rank this paper deals with rings whose modules have very pleasant decomposition properties the right pure semisimple rings the category of right modules over such a ring is in many ways similar to the category of modules over a semisimple artinian ring yet it is still an open question whether or not pure semisimplicity actually is a two sided property and hence 30 coincides with being of finite representation type most of the arguments in this paper are model theoretic algebraic in nature it seems that ideas from model theory and from the model theory of modules find natural application in the context considered here the paper is organized into three sections the first is introductory and with it i attempt to make the other sections accessible to a reasonably wide readership in the second section are gathered together various equivalents to right pure semisimplicity their equivalence is given a comparatively short unified essentially model theoretic algebraic proof the main theorem in the third section is that a ring is of finite representation type if and only if all its right modules have finite morley rank a number of related results are developed most of these involve some model theory in their statement but many are in essense algebraic since a good deal of information has been compressed into the introductory section this section should perhaps first be read through quickly and then referred back to as the need arises this section also is meant to serve as background to a sequel in which i consider pp formulas and types in terms of the corresponding matrices and in which the main aim is more purely algebraic the global conventions are r denotes a ring with identity module

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

groups measures and the nip we discuss measures invariant measures on definable groups and genericity often in an nip failure of the independence property environment we complete the proof of the third author s conjectures relating definably compact groups g in saturated o minimal structures to compact lie groups we also prove some other structural results about such g for example the existence of a left invariant finitely additive probability measure on definable subsets of g we finally introduce a new notion compact domination domination of a definable set by a compact space and raise some new conjectures in the o minimal case

how to avoid a compact set abstract a first order expansion of the r vector space structure on r does not define every compact subset of every r n if and only if topological and hausdorff dimension coincide on all closed definable sets equivalently if a r k is closed and the hausdorff dimension of a exceeds the topological dimension of a then every compact subset of every r n can be constructed from a using finitely many boolean operations cartesian products and linear operations the same statement fails when hausdorff dimension is replaced by packing dimension

totally categorical structures a first order theory is totally categorical if it has exactly one model in each infinite power we prove here that every such theory admits a finite language and is finitely axiomatizable in that language modulo axioms stating that the structure is infinite this was conjectured by vaught we also show that every n0 stable n0 categorical structure is a reduct of one that has finitely many models in small uncountable powers in the case of structures of disintegrated type we nearly find an explicit structure theorem and show that the remaining obstacle resides in certain nilpotent automorphism groups

randomizing a model abstract a randomization of a first order structure m is a new structure with certain closure properties whose universe is a setkof random elements of m randomizations assign probabilities to sentences of the language of m with new constants fromk our main theorem shows that all randomizations of m are models of the same first order theoryt which has a nice set of axioms and admits elimination of quantifiers moreover the class of substructures of models oftis characterized by a natural setvof universal axioms oft so thattis the model completion ofv

h fields and their liouville extensions we introduce h fields as ordered differential fields of a certain kind hardy fields extending mathbb r as well as the field of logarithmic exponential series over mathbb r are h fields we study liouville extensions in the category of h fields as a step towards a model theory of h fields the main result is that an h field has at most two liouville closures

one basedness and reductions of elliptic curves over real closed fields building on the positive solution of pillay s conjecture we present a notion of intrinsic reduction for elliptic curves over a real closed eld k we compare such a notion with the traditional algebro geometric reduction and produce a classi cation of the group of k points of an elliptic curve e with three real roots according to the way e reduces algebro geometrically and the geometric complexity of the intrinsically reduced curve

fast and efficient parallel evaluation of the zeros of a polynomial having only real zeros abstract given a polynomial p x of degree n with integer coefficients between 2 m and 2 m it suffices to use o log 2 n log b parallel arithmetic steps and n 2 processors in order to compute all the zeros of p x with absolute errors at most 2 h provided that all the zeros are real and b m h the algorithm combines the recent techniques and results of 1 3 in order to improve the previous record processor bound of 1 at least on the factor of n preserving its upper bound on parallel time

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

relational expressive power of constraint query languages the expressive power of first order query languages with several classes of equality and inequality constraints is studied in this paper we settle the conjecture that recursive queries such as parity test and transitive closure cannot be expressed in the relational calculus augmented with polynomial inequality constraints over the reals furthermore noting that relational queries exhibit several forms of genericity we establish a number of collapse results of the following form the class of generic boolean queries expressible in the relational calculus augmented with a given class of constraints coincides with the class of queries expressible in the relational calculus with or without an order relation we prove such results for both the natural and active domain semantics as a consequence the relational calculus augmented with polynomial inequalities expresses the same classes of generic boolean queries under both the natural and active domain semantics in the course of proving these results for the active domin semantics we establish ramsey type theorems saying that any query involving certain kinds of constraints coincides with a constraint free query on databases whose elements come from a certain infinite subset of the domain to prove the collapse results for the natural semantics we make use of techniques from nonstandard analysis and from the model theory of ordered structures

externally definable sets and dependent pairs ii 2015 american mathematical society we continue investigating the structure of externally definable sets in nip theories and preservation of nip after expanding by new predicates most importantly types over finite sets are uniformly definable over a model a family of non forking instances of a formula with parameters ranging over a type definable set can be covered with finitely many invariant types we give some criteria for the boundedness of an expansion by a new predicate in a distal theory naming an arbitrary small indiscernible sequence preserves nip while naming a large one doesn t there are models of nip theories over which all 1 types are definable but not all n types

fast conversion algorithms for orthogonal polynomials we discuss efficient conversion algorithms for orthogonal polynomials we describe a known conversion algorithm from an arbitrary orthogonal basis to the monomial basis and deduce a new algorithm of the same complexity for the converse operation

on pillay s conjecture in the general case abstract let m be an arbitrary o minimal structure let g be a definably compact definably connected abelian definable group of dimension n here we compute i the new intrinsic o minimal fundamental group of g ii for each k 0 the k torsion subgroups of g iii the o minimal cohomology algebra over q of g as a corollary we obtain a new uniform proof of pillay s conjecture an o minimal analogue of hilbert s fifth problem relating definably compact groups to compact real lie groups extending the proof already known in o minimal expansions of ordered fields

structures having o minimal open core the open core of an expansion of a dense linear order is its reduct in the sense of definability generated by the collection of all of its open definable sets in this paper expansions of dense linear orders that have o minimal open core are investigated with emphasis on expansions of densely ordered groups the first main result establishes conditions under which an expansion of a densely ordered group has an o minimal open core specifically the following is proved let r be an expansion of a densely ordered group r that is definably complete and satisfies the uniform finiteness property then the open core of r is o minimal two examples of classes of structures that are not o minimal yet have o minimal open core are discussed dense pairs of o minimal expansions of ordered groups and expansions of o minimal structures by generic predicates in particular such structures have open core interdefinable with the original o minimal structure these examples are differentiated by the existence of definable unary functions whose graphs are dense in the plane a phenomenon that can occur in dense pairs but not in expansions by generic predicates the property of having no dense graphs is examined and related to uniform finiteness definable completeness and having o minimal open core