In an article (which we will refer to as [BM]) of Boca and the fourth author of this paper, a new class of continued fraction expansions with odd partial quotients, …
In an article (which we will refer to as [BM]) of Boca and the fourth author of this paper, a new class of continued fraction expansions with odd partial quotients, parameterized by a parameter $\alpha\in [g,G]$, where $g=\tfrac{1}{2}(\sqrt{5}-1)$ and $G=g+1=1/g$ are the two golden mean numbers is introduced. In this article, by using operations called singularizations and insertions on the partial quotients of the odd continued fraction expansions under consideration, the natural extensions from [BM] are obtained, and it is shown that for each $\alpha,\alpha^*\in [g,G]$ the natural extensions from [BM] are metrically isomorphic. An immediate consequence of this is, that the entropy of all these natural extensions is equal for $\alpha\in [g,G]$, a fact already observed in [BM]. Furthermore, it is shown that this approach can be extended to values of $\alpha$ smaller than $g$, and that for values of $\alpha \in [\tfrac{1}{6}(\sqrt{13}-1), g]$ all natural extensions are still isomorphic. In the final section of this paper further attention is given to the entropy, as function of $\alpha\in [0,G]$. It is shown that in any neighborhood of $0$ we can find intervals on which the entropy is decreasing, intervals on which the entropy is increasing and intervals on which the entropy is constant. In order to prove this we use a phenomena called matching.
In an article (which we will refer to as [BM]) of Boca and the fourth author of this paper, a new class of continued fraction expansions with odd partial quotients, …
In an article (which we will refer to as [BM]) of Boca and the fourth author of this paper, a new class of continued fraction expansions with odd partial quotients, parameterized by a parameter $\alpha\in [g,G]$, where $g=\tfrac{1}{2}(\sqrt{5}-1)$ and $G=g+1=1/g$ are the two golden mean numbers is introduced. In this article, by using operations called singularizations and insertions on the partial quotients of the odd continued fraction expansions under consideration, the natural extensions from [BM] are obtained, and it is shown that for each $\alpha,\alpha^*\in [g,G]$ the natural extensions from [BM] are metrically isomorphic. An immediate consequence of this is, that the entropy of all these natural extensions is equal for $\alpha\in [g,G]$, a fact already observed in [BM]. Furthermore, it is shown that this approach can be extended to values of $\alpha$ smaller than $g$, and that for values of $\alpha \in [\tfrac{1}{6}(\sqrt{13}-1), g]$ all natural extensions are still isomorphic. In the final section of this paper further attention is given to the entropy, as function of $\alpha\in [0,G]$. It is shown that in any neighborhood of $0$ we can find intervals on which the entropy is decreasing, intervals on which the entropy is increasing and intervals on which the entropy is constant. In order to prove this we use a phenomena called matching.
Let $\beta \gt 1$ be a real number and $x \in [0,1)$ be an irrational number. We denote by $k_n(x)$ the exact number of partial quotients in the continued fraction …
Let $\beta \gt 1$ be a real number and $x \in [0,1)$ be an irrational number. We denote by $k_n(x)$ the exact number of partial quotients in the continued fraction expansion of $x$ given by the first $n$ digits in the $\beta$-expansion of $x$ ($n \in \mat
Let $\beta > 1$ be a real number and $x \in [0,1)$ be an irrational number. We denote by $k_n(x)$ the exact number of partial quotients in the continued fraction …
Let $\beta > 1$ be a real number and $x \in [0,1)$ be an irrational number. We denote by $k_n(x)$ the exact number of partial quotients in the continued fraction expansion of $x$ given by the first $n$ digits in the $\beta$-expansion of $x$ ($n \in \mathbb{N}$). It is known that $k_n(x)/n$ converges to $(6\log2\log\beta)/\pi^2$ almost everywhere in the sense of Lebesgue measure. In this paper, we improve this result by proving that the Lebesgue measure of the set of $x \in [0,1)$ for which $k_n(x)/n$ deviates away from $(6\log2\log\beta)/\pi^2$ decays to 0 exponentially as $n$ tends to $\infty$, which generalizes the result of Faivre \cite{lesFai97} from $\beta = 10$ to any $\beta >1$. Moreover, we also discuss which of the $\beta$-expansion and continued fraction expansion yields the better approximations of real numbers.
In this paper we show how to apply various techniques and theorems (including Pincherle's theorem, an extension of Euler's formula equating infinite series and continued fractions, an extension of the …
In this paper we show how to apply various techniques and theorems (including Pincherle's theorem, an extension of Euler's formula equating infinite series and continued fractions, an extension of the corresponding transformation that equates infinite products and continued fractions, extensions and contractions of continued fractions and the Bauer-Muir transformation) to derive infinite families of in-equivalent polynomial continued fractions in which each continued fraction has the same limit. This allows us, for example, to construct infinite families of polynomial continued fractions for famous constants like $\pi$ and $e$, $\zeta{(k)}$ (for each positive integer $k\geq 2$), various special functions evaluated at integral arguments and various algebraic numbers. We also pose several questions about the nature of the set of real numbers which have a polynomial continued fraction expansion.
We give a heuristic technique to find a model for the natural extension of a piecewise homographic, or more generally projective, map on a domain of R or Rd. In …
We give a heuristic technique to find a model for the natural extension of a piecewise homographic, or more generally projective, map on a domain of R or Rd. In case of success, this gives explicit formula for an invariant density.
We state and prove three general formulas allowing to transform formal finite sums into formal continued fractions and apply them to generalize certain expansions in continued fractions given by Hone …
We state and prove three general formulas allowing to transform formal finite sums into formal continued fractions and apply them to generalize certain expansions in continued fractions given by Hone and Varona.
A natural abstraction of rationality is that of linear independence, while for algebraicconsideration, a natural abstraction of transcendence is that of algebraic independence. Alsowell-known is that each real number is …
A natural abstraction of rationality is that of linear independence, while for algebraicconsideration, a natural abstraction of transcendence is that of algebraic independence. Alsowell-known is that each real number is representable as a simple continued fraction. Combiningindependence with representation, there arises a natural problem of characterizing elements viatheir representations. The work in this thesis centers around these two concepts, namely,continued fraction representation and their independence, in the field of Laurent series over afinite field, referred to here as function field. The major part of the thesis are devoted to the establishing of two general independencecriteria, one for linear and the other for algebraic independence and to the extensivecomputation of intriguing examples. The linear independence criterion states roughly that ifthe partial quotients grow at a moderately fast rate, their continued fractions are linearlyindependent. This linear independence criterion when applied to the real case encompasses thatobtained by Hancl in 2002. As to algebraic independence, a Liouville-type sufficient conditionthrough rational approximations is proved. When applied to continued fractions, it yields anumber of criteria which show that exponentially growing partial quotients imply algebraicindependence. Two interesting types of explicit continued fractions are worked out as examplesillustrating the strength of the criteria so obtained
8) This holds under the more general condition that gk{z) is at most of the zero type of order 1, by Wigert's theorem.See beginning of §2.130 (3) Unpublished.The only published …
8) This holds under the more general condition that gk{z) is at most of the zero type of order 1, by Wigert's theorem.See beginning of §2.130 (3) Unpublished.The only published proof is in Petersson [2].(4) A discussion of other types of expansions of zero can be found in [6], (5) The factor cp(k) appears because each gk(n) occurs 0(£) times as h runs mod k, {h, k) =1.
8) This holds under the more general condition that gk{z) is at most of the zero type of order 1, by Wigert's theorem.See beginning of §2.130 (3) Unpublished.The only published …
8) This holds under the more general condition that gk{z) is at most of the zero type of order 1, by Wigert's theorem.See beginning of §2.130 (3) Unpublished.The only published proof is in Petersson [2].(4) A discussion of other types of expansions of zero can be found in [6], (5) The factor cp(k) appears because each gk(n) occurs 0(£) times as h runs mod k, {h, k) =1.
In [1], Boca and the fourth author of this paper introduced a new class of continued fraction expansions with odd partial quotients, parameterized by a parameter $ \alpha\in [g, G] …
In [1], Boca and the fourth author of this paper introduced a new class of continued fraction expansions with odd partial quotients, parameterized by a parameter $ \alpha\in [g, G] $, where $ g = \tfrac{1}{2}(\sqrt{5}-1) $ and $ G = g+1 = 1/g $ are the two golden mean numbers. Using operations called singularizations and insertions on the partial quotients of the odd continued fraction expansions under consideration, the natural extensions from [1] are obtained, and it is shown that for each $ \alpha, \alpha^*\in [g, G] $ the natural extensions from [1] are metrically isomorphic. An immediate consequence of this is, that the entropy of all these natural extensions is equal for $ \alpha\in [g, G] $, a fact already observed in [1]. Furthermore, it is shown that this approach can be extended to values of $ \alpha $ smaller than $ g $, and that for values of $ \alpha \in [\tfrac{1}{6}(\sqrt{13}-1), g] $ all natural extensions are still isomorphic. In the final section of this paper further attention is given to the entropy, as function of $ \alpha\in [0, G] $. It is shown that if there exists an ergodic, absolutely continuous $ T_{\alpha} $-invariant measure, in any neighborhood of $ 0 $ we can find intervals on which the entropy is decreasing, intervals on which the entropy is increasing and intervals on which the entropy is constant. Moreover, we identify the largest interval on which the entropy is constant. In order to prove this we use a phenomenon called matching.
We investigate several extension properties of Fréchet differentiable functions defined on closed sets for o-minimal expansions of real closed fields.
We investigate several extension properties of Fréchet differentiable functions defined on closed sets for o-minimal expansions of real closed fields.
We introduce the split torsor method to count rational points of bounded height on Fano varieties. As an application, we prove Maninâs conjecture for all nonsplit quartic del Pezzo surfaces …
We introduce the split torsor method to count rational points of bounded height on Fano varieties. As an application, we prove Maninâs conjecture for all nonsplit quartic del Pezzo surfaces of type $\mathbf {A}_{3}+\mathbf {A}_{1}$ over arbitrary number fields. The counting problem on the split torsor is solved in the framework of o-minimal structures.
The aim of this paper is to give a unifying description of various constructions of sites (subanalytic, semialgebraic, o-minimal) and consider the corresponding theory of sheaves. The method used applies …
The aim of this paper is to give a unifying description of various constructions of sites (subanalytic, semialgebraic, o-minimal) and consider the corresponding theory of sheaves. The method used applies to a more general context and gives new results in semialgebraic and o-minimal sheaf theory.
The classical Erdős–Littlewood–Offord theorem says that for nonzero vectors a1,…,an∈Rd, any x∈Rd, and uniformly random (ξ1,…,ξn)∈{−1,1}n, we have Pr(a1ξ1+⋯+anξn=x)=O(n−1/2). In this paper, we show that Pr(a1ξ1+⋯+anξn∈S)≤n−1/2+o(1) whenever S is definable …
The classical Erdős–Littlewood–Offord theorem says that for nonzero vectors a1,…,an∈Rd, any x∈Rd, and uniformly random (ξ1,…,ξn)∈{−1,1}n, we have Pr(a1ξ1+⋯+anξn=x)=O(n−1/2). In this paper, we show that Pr(a1ξ1+⋯+anξn∈S)≤n−1/2+o(1) whenever S is definable with respect to an o-minimal structure (e.g., this holds when S is any algebraic hypersurface), under the necessary condition that it does not contain a line segment. We also obtain an inverse theorem in this setting.
Let G be a group definable in an o-minimal structure M. We prove that the union of the Cartan subgroups of G is a dense subset of G. When M …
Let G be a group definable in an o-minimal structure M. We prove that the union of the Cartan subgroups of G is a dense subset of G. When M is an expansion of a real closed field, we give a characterization of Cartan subgroups of G via their Lie algebras which allow us to prove firstly that every Cartan subalgebra of the Lie algebra of G is the Lie algebra of a definable subgroup — a Cartan subgroup of G — and secondly that the set of regular points of G — a dense subset of G — is formed by points which belong to a unique Cartan subgroup of G.
Abstract This article presents Tarski's Address at the Princeton Bicentennial Conference on Problems of Mathematics, together with a separate summary. Two accounts of the discussion which followed are also included. …
Abstract This article presents Tarski's Address at the Princeton Bicentennial Conference on Problems of Mathematics, together with a separate summary. Two accounts of the discussion which followed are also included. The central topic of the Address and of the discussion is decision problems. The introductory note gives information about the Conference, about the background of the subjects discussed in the Address, and about subsequent developments to these subjects.
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 …
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 = KAK of G, and then carry out an approximate calculation of the intersection of KHK with A, for each closed, connected subgroup H of G.
In this paper we prove tight bounds on the combinatorial and topological complexity of sets defined in terms of n definable sets belonging to some fixed definable family of sets …
In this paper we prove tight bounds on the combinatorial and topological complexity of sets defined in terms of n definable sets belonging to some fixed definable family of sets in an o-minimal structure. This generalizes the combinatorial parts of similar bounds known in the case of semi-algebraic and semi-Pfaffian sets, and as a result vastly increases the applicability of results on combinatorial and topological complexity of arrangements studied in discrete and computational geometry. As a sample application, we extend a Ramsey-type theorem due to Alon et al. [Crossing patterns of semi-algebraic sets, J. Combin. Theory Ser. A 111 (2005), 310–326. MR 2156215 (2006k:14108)], originally proved for semi-algebraic sets of fixed description complexity to this more general setting.
We consider the Riemann mapping theorem in the case of a bounded simply connected and semianalytic domain. We show that the germ at 0 of the Riemann map (that is, …
We consider the Riemann mapping theorem in the case of a bounded simply connected and semianalytic domain. We show that the germ at 0 of the Riemann map (that is, biholomorphic map) from the upper half plane to such a domain can be realized in a certain quasianalytic class if the angle of the boundary at the point to which 0 is mapped is greater than 0. This quasianalytic class was introduced and used by Ilyashenko in his work on Hilbert's 16th problem. With this result, we can prove that the Riemann map from a bounded simply connected semianalytic domain onto the unit ball is definable in an o-minimal structure, provided that at singular boundary points the angles of the boundary are irrational multiples of π.
Let M be a polynomially bounded, o-minimal structure with archimedean prime model, for example if M is a real closed field. Let C be a convex and unbounded subset of …
Let M be a polynomially bounded, o-minimal structure with archimedean prime model, for example if M is a real closed field. Let C be a convex and unbounded subset of M. We determine the first order theory of the structure M expanded by the set C. We do this also over any given set of parameters from M, which yields a description of all subsets of M^n, definable in the expanded structure.
We explain how the field of logarithmic-exponential series constructed in \cite{DMM1} and \cite {DMM2} embeds as an exponential field in any field of exponential-logarithmic series constructed in \cite{KK1}, \cite {K} …
We explain how the field of logarithmic-exponential series constructed in \cite{DMM1} and \cite {DMM2} embeds as an exponential field in any field of exponential-logarithmic series constructed in \cite{KK1}, \cite {K} and \cite {KS}. On the other hand, we explain why no field of exponential-logarithmic series embeds in the field of logarithmic-exponential series. This clarifies why the two constructions are intrinsically different, in the sense that they produce non-isomorphic models of Th$(\R_{an, exp})$; the elementary theory of the ordered field of real numbers, with the exponential function and restricted analytic functions.
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 …
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.
In this article we study definable functions in tame expansions of algebraically closed valued fields. For a given definable function we have two types of results: of type (I), which …
In this article we study definable functions in tame expansions of algebraically closed valued fields. For a given definable function we have two types of results: of type (I), which hold at a neighborhood of infinity, and of type (II), which hold locally for all but finitely many points in the domain of the function. In the first part of the article, we show type (I) and (II) results concerning factorizations of definable functions over the value group. As an application, we show that tame expansions of algebraically closed valued fields having value group $\mathbb{Q}$ (like $\mathbb{C}_p$ and $\overline{\mathbb{F}_p}^{alg}(\!(t^\mathbb{Q})\!)$) are polynomially bounded. In the second part, under an additional assumption on the asymptotic behavior of unary definable functions of the value group, we extend these factorizations over the residue multiplicative structure $\mathrm{RV}$. In characteristic 0, we obtain as a corollary that the domain of a definable function $f\colon X\subseteq K\to K$ can be partitioned into sets $F\cup E\cup J$, where $F$ is finite, $f|E$ is locally constant and $f|J$ satisfies locally the Jacobian property.
Let ${\mathcal S}(\R)$ be an o-minimal structure over $\R$, $T \subset \R^{k_1+k_2+\ell}$ a closed definable set, and $$ \displaylines{π_1: \R^{k_1+k_2+\ell}\to \R^{k_1 + k_2}, π_2: \R^{k_1+k_2+\ell}\to \R^{\ell}, \ π_3: \R^{k_1 + …
Let ${\mathcal S}(\R)$ be an o-minimal structure over $\R$, $T \subset \R^{k_1+k_2+\ell}$ a closed definable set, and $$ \displaylines{π_1: \R^{k_1+k_2+\ell}\to \R^{k_1 + k_2}, π_2: \R^{k_1+k_2+\ell}\to \R^{\ell}, \ π_3: \R^{k_1 + k_2} \to \R^{k_2}} $$ the projection maps. For any collection ${\mathcal A} = \{A_1,...,A_n\}$ of subsets of $\R^{k_1+k_2}$, and $\z \in \R^{k_2}$, let $\A_\z$ denote the collection of subsets of $\R^{k_1}$, $\{A_{1,\z},..., A_{n,\z}\}$, where $A_{i,\z} = A_i \cap π_3^{-1}(\z), 1 \leq i \leq n$. We prove that there exists a constant $C = C(T) > 0,$ such that for any family ${\mathcal A} = \{A_1,...,A_n\}$ of definable sets, where each $A_i = π_1(T \cap π_2^{-1}(\y_i))$, for some $\y_i \in \R^{\ell}$, the number of distinct stable homotopy types of $\A_\z, \z \in \R^{k_2}$, is bounded by $ \displaystyle{C \cdot n^{(k_1+1)k_2},} $ while the number of distinct homotopy types is bounded by $ \displaystyle{C \cdot n^{(k_1+3)k_2}.} $ This generalizes to the general o-minimal setting, bounds of the same type proved in \cite{BV} for semi-algebraic and semi-Pfaffian families. One main technical tool used in the proof of the above results, is a topological comparison theorem which might be of independent interest in the study of arrangements.
It is shown that the extension of $\R$ by a generic smooth function restricted to the unit cube is o-minimal. The generalization to countably many generic smooth functions is indicated. …
It is shown that the extension of $\R$ by a generic smooth function restricted to the unit cube is o-minimal. The generalization to countably many generic smooth functions is indicated. Possible applications are sketched.
In this paper, we study a class of composite optimization problems, whose objective function is the summation of a bunch of nonsmooth nonconvex loss functions and a cardinality regularizer. Firstly …
In this paper, we study a class of composite optimization problems, whose objective function is the summation of a bunch of nonsmooth nonconvex loss functions and a cardinality regularizer. Firstly we investigate the optimality condition of these problems and then suggest a stochastic proximal subgradient method (SPSG) to solve them. Then we establish the almost surely subsequence convergence of SPSG under mild assumptions. We emphasize that these assumptions are satisfied by a wide range of problems arising from training neural networks. Furthermore, we conduct preliminary numerical experiments to demonstrate the effectiveness and efficiency of SPSG in solving this class of problems.
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Let 〈R, >,+,⋅〉 be a real closed field, and let M be an o-minimal expansion of R. We prove here several results regarding rings and groups which are definable in …
Let 〈R, >,+,⋅〉 be a real closed field, and let M be an o-minimal expansion of R. We prove here several results regarding rings and groups which are definable in M. We show that every M–definable ring without zero divisors is definably isomorphic to R, R(√(−l)) or the ring of quaternions over R. One corollary is that no model of Texp is interpretable in a model of Tan.
Abstract In this paper we study the welldefinedness of the central path associated to a nonlinear convex semidefinite programming problem with smooth objective and constraint functions. Under standard assumptions, we …
Abstract In this paper we study the welldefinedness of the central path associated to a nonlinear convex semidefinite programming problem with smooth objective and constraint functions. Under standard assumptions, we prove that the existence of the central path is equivalent to the nonemptiness and boundedness of the optimal set. Other equivalent conditions are given, such as the existence of a strictly dual feasible point or the existence of a single central point. The monotonic behavior of the primal and dual logarithmic barriers and of the primal and dual objective functions along the trajectory is also discussed. The existence and optimality of cluster points is established and finally, under the additional assumption of analyticity of the data functions, the convergence of the primal-dual trajectory is proved.
We describe an algorithm which reduces (the problem of finding regular solutions of) a system of [Formula: see text]-subanalytic equations to (the same problem for) a finite number of systems …
We describe an algorithm which reduces (the problem of finding regular solutions of) a system of [Formula: see text]-subanalytic equations to (the same problem for) a finite number of systems of subanalytic equations, by using the following operations: restriction to an open [Formula: see text]-subanalytic subset, smooth [Formula: see text]-subanalytic change of variables and restriction to a linear subspace. An application of this together with a theorem of Lion is that the structure [Formula: see text] is o-minimal.
The study of NIP theories has received much attention from model theorists in the last decade, fuelled by applications to o-minimal structures and valued fields. This book, the first to …
The study of NIP theories has received much attention from model theorists in the last decade, fuelled by applications to o-minimal structures and valued fields. This book, the first to be written on NIP theories, is an introduction to the subject that will appeal to anyone interested in model theory: graduate students and researchers in the field, as well as those in nearby areas such as combinatorics and algebraic geometry. Without dwelling on any one particular topic, it covers all of the basic notions and gives the reader the tools needed to pursue research in this area. An effort has been made in each chapter to give a concise and elegant path to the main results and to stress the most useful ideas. Particular emphasis is put on honest definitions, handling of indiscernible sequences and measures. The relevant material from other fields of mathematics is made accessible to the logician.
Abstract The complex field, equipped with the multivalued functions of raising to each complex power, is quasiminimal, proving a conjecture of Zilber and providing evidence towards his stronger conjecture that …
Abstract The complex field, equipped with the multivalued functions of raising to each complex power, is quasiminimal, proving a conjecture of Zilber and providing evidence towards his stronger conjecture that the complex exponential field is quasiminimal.
Let $\mathbb {V}$ be a polarized variation of Hodge structure over a smooth complex quasi-projective variety $S$ . In this paper, we give a complete description of the typical Hodge …
Let $\mathbb {V}$ be a polarized variation of Hodge structure over a smooth complex quasi-projective variety $S$ . In this paper, we give a complete description of the typical Hodge locus for such variations. We prove that it is either empty or equidistributed with respect to a natural differential form, the pull–push form . In particular, it is always analytically dense when the pull–push form does not vanish. When the weight is two, the Hodge numbers are $(q,p,q)$ and the dimension of $S$ is least $rq$ , we prove that the typical locus where the Picard rank is at least $r$ is equidistributed in $S$ with respect to the volume form $c_q^r$ , where $c_q$ is the $q$ th Chern form of the Hodge bundle. We obtain also several equidistribution results of the typical locus in Shimura varieties: a criterion for the density of the typical Hodge loci of a variety in $\mathcal {A}_g$ , equidistribution of certain families of CM points and equidistribution of Hecke translates of curves and surfaces in $\mathcal {A}_g$ . These results are proved in the much broader context of dynamics on homogeneous spaces of Lie groups which are of independent interest. The pull–push form appears in this greater generality, we provide several tools to determine it, and we compute it in many examples.
§1. Introduction . By and large, definitions of a differentiable structure on a set involve two ingredients, topology and algebra. However, in some cases, partial information on one or both …
§1. Introduction . By and large, definitions of a differentiable structure on a set involve two ingredients, topology and algebra. However, in some cases, partial information on one or both of these is sufficient. A very simple example is that of the field ℝ (or any real closed field) where algebra alone determines the ordering and hence the topology of the field: In the case of the field ℂ, the algebraic structure is insufficient to determine the Euclidean topology; another topology, Zariski, is associated with the ield but this will be too coarse to give a diferentiable structure. A celebrated example of how partial algebraic and topological data ( G a locally euclidean group) determines a differentiable structure ( G is a Lie group) is Hilbert's 5th problem and its solution by Montgomery-Zippin and Gleason. The main result which we discuss here (see [13] for the full version) is of a similar flavor: we recover an algebraic and later differentiable structure from a topological data. We begin with a linearly ordered set ⟨ M , <⟩, equipped with the order topology, and its cartesian products with the product topologies. We then consider the collection of definable subsets of M n , n = 1, 2, …, in some first order expansion ℳ of ⟨ M , <⟩.
Abstract This paper focuses on the minimization of a possibly nonsmooth objective function over the Stiefel manifold. The existing approaches either lack efficiency or can only tackle prox-friendly objective functions. …
Abstract This paper focuses on the minimization of a possibly nonsmooth objective function over the Stiefel manifold. The existing approaches either lack efficiency or can only tackle prox-friendly objective functions. We propose a constraint dissolving function named NCDF and show that it has the same first-order stationary points and local minimizers as the original problem in a neighborhood of the Stiefel manifold. Furthermore, we show that the Clarke subdifferential of NCDF is easy to achieve from the Clarke subdifferential of the objective function. Therefore, various existing approaches for unconstrained nonsmooth optimization can be directly applied to nonsmooth optimization problems over the Stiefel manifold. We propose a framework for developing subgradient-based methods and establishing their convergence properties based on prior works. Furthermore, based on our proposed framework, we can develop efficient approaches for optimization over the Stiefel manifold. Preliminary numerical experiments further highlight that the proposed constraint dissolving approach yields efficient and direct implementations of various unconstrained approaches to nonsmooth optimization problems over the Stiefel manifold.
Abstract We prove that no restriction of the sine function to any (open and nonempty) interval is definable in 〈 R , +, ·, ×, <, exp, constants〉, and that …
Abstract We prove that no restriction of the sine function to any (open and nonempty) interval is definable in 〈 R , +, ·, ×, <, exp, constants〉, and that no restriction of the exponential function to an (open and nonempty) interval is definable in 〈 R , +, ·, <, sin 0 , constants〉, where sin 0 ( x ) = sin( x ) for x ∈ [— π, π ], and sin 0 ( x ) = 0 for all x ∉ [— π , π ].
existence of sheaf cohomology theory in arbitrary o-minimal structures, following the work of Edmundo, Jones and Peatfield.
existence of sheaf cohomology theory in arbitrary o-minimal structures, following the work of Edmundo, Jones and Peatfield.
Skolem (1956) studied the germs at infinity of the smallest class of real valued functions on the positive real line containing the constant $1$, the identity function $x$, and such …
Skolem (1956) studied the germs at infinity of the smallest class of real valued functions on the positive real line containing the constant $1$, the identity function $x$, and such that whenever $f$ and $g$ are in the set, $f+g,fg$ and $f^g$ are in the set. This set of germs is well ordered and Skolem conjectured that its order type is epsilon-zero. Van den Dries and Levitz (1984) computed the order type of the fragment below $2^{2^x}$. Here we prove that the set of asymptotic classes within any archimedean class of Skolem functions has order type $\omega$. As a consequence we obtain, for each positive integer $n$, an upper bound for the fragment below $2^{n^x}$. We deduce an epsilon-zero upper bound for the fragment below $2^{x^x}$, improving the previous epsilon-omega bound by Levitz (1978). A novel feature of our approach is the use of Conway's surreal number for asymptotic calculations.
I see model theory as becoming increasingly detached from set theory, and the Tarskian notion of set-theoretic model being no longer central to model theory. In much of modern mathematics, …
I see model theory as becoming increasingly detached from set theory, and the Tarskian notion of set-theoretic model being no longer central to model theory. In much of modern mathematics, the set-theoretic component is of minor interest, and basic notions are geometric or category-theoretic. In algebraic geometry, schemes or algebraic spaces are the basic notions, with the older “sets of points in affine or projective space” no more than restrictive special cases. The basic notions may be given sheaf-theoretically, or functorially. To understand in depth the historically important affine cases, one does best to work with more general schemes. The resulting relativization and “transfer of structure” is incomparably more flexible and powerful than anything yet known in “set-theoretic model theory”. It seems to me now uncontroversial to see the fine structure of definitions as becoming the central concern of model theory, to the extent that one can easily imagine the subject being called “Definability Theory” in the near future. Tarski's set-theoretic foundational formulations are still favoured by the majority of model-theorists, and evolution towards a more suggestive language has been perplexingly slow. None of the main texts uses in any nontrivial way the language of category theory, far less sheaf theory or topos theory. Given that the most notable interactions of model theory with geometry are in areas of geometry where the language of sheaves is almost indispensable (to the geometers), this is a curious situation, and I find it hard to imagine that it will not change soon, and rapidly.
Introduction and notationLet L be the usual first order language of ordered rings together with a new unary function symbol e.We are interested in the L-structure R (R, 0, 1, …
Introduction and notationLet L be the usual first order language of ordered rings together with a new unary function symbol e.We are interested in the L-structure R (R, 0, 1, +,., -, <, e) consisting of the ordered field of real numbers with e(x) interpreted as the exponential function e (and we shall henceforth write e x for e(x) in any L-structure).We denote by T the L-theory of R e.This theory and its subtheories have been investigated by many authors and we refer the reader to Macintyre [4] for a comprehensive survey.We are con- cerned here with the problem of determining whether T is model complete, that is whether k, K T and k _ _ _ K imply k K, or equivalently k 1 K (i.e., existential formulas with parameters in k are preserved down from K to k).We shall prove the following: THEOREM 1. Suppose k, K Te, k c_C_ K and k is cofinal in K (i.e., if a K then b < a < c for some b, c k). Then k l K.(Unfortunately there seems to be no general model theoretic argument that allows us to deduce that k K here.)We shall actually prove a result slightly stronger than Theorem 1 which allows us to isolate a plausible conjecture that would imply the model completeness of T e.To state this result we require some notation.Let us fix a model K of T and a substructure k of K. We also assume that k is a field.For n N we denote by k[] the set of all terms of L(k) (defined as L together with a constant symbol for each element of k) in the variables ' x 1,..., x factored by the equivalence relation f-g iff Ter-Vf=g.Since it is known (see [4]) that f---g iff k Vf g it will be harmless to
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
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
This article points out some remarkable facts implicit in the results of Lojasiewicz [LI] and Gabrielov [Ga].An important consequence of Tarski's work [T] on the elementary theory of the reals …
This article points out some remarkable facts implicit in the results of Lojasiewicz [LI] and Gabrielov [Ga].An important consequence of Tarski's work [T] on the elementary theory of the reals is a characterization of the sets which are elementarily definable from addition and multiplication on R. Allowing arbitrary reals as constants, this characterization consists of the identification of the definable sets with the semialgebraic sets.(A semialgebraic subset of R m is by definition a finite union of sets of the form {x e R w : p(x) = 0, q x (x) > 0,...,q k (x) > 0} where p,q 1 ,...,q k are real polynomials.)The fact that the system of semialgebraic sets is closed under definability is also known as the Tarski-Seidenberg theorem, and this property, together with the topological finiteness phenomena that go with it-triangulability of semialgebraic sets [L2, Gi], generic triviality of semialgebraic maps [Ha]-make the theory of semialgebraic sets a useful analytic-topological tool.Below we extend the system of semialgebraic sets in such a way that the Tarski-Seidenberg property, i.e., closure under definability, and the topological finiteness phenomena are preserved.The polynomial growth property of semialgebraic functions is also preserved.This extended system contains the arctangent function on R, the sine function on any bounded interval, the exponential function e x on any bounded interval, but not the exponential function on all of R. (And it couldn't possibly contain the sine function on all of R without sacrificing the finiteness phenomena, and a lot more.)As a corollary we obtain that neither the exponential function on R, nor the set of integers, is definable from addition, multiplication, and the restrictions of the sine and exponential functions to bounded intervals.Questions of this type have puzzled logicians for a long time.(There still remain, of course, countless unsolved problems of this sort.)In a more positive spirit Tarski [T, p. 45] asked to extend his results so as to include, besides the algebraic operations on R, certain transcendental elementary functions like e x ; the theorem below is a partial answer.(More recently, Hovanskii [Ho, p. 562] and the author [VdDl, VdD2] asked similar questions, and in [VdD3] we
As a contribution to definability theory in the spirit of Tarski's classical work on ( R , <, 0, 1, +, ·) we extend here part of his results to …
As a contribution to definability theory in the spirit of Tarski's classical work on ( R , <, 0, 1, +, ·) we extend here part of his results to the structure Here exp ∣ [0, 1] and sin ∣ [0, π ] are the restrictions of the exponential and sine function to the closed intervals indicated; formally we identify these restricted functions with their graphs and regard these as binary relations on R . The superscript “RE” stands for “restricted elementary” since, given any elementary function, one can in general only define certain restrictions of it in R RE . Let ( R RE , constants) be the expansion of R RE obtained by adding a name for each real number to the language. We can now formulate our main result as follows. Theorem. ( R RE , constants) is strongly model-complete . This means that every formula ϕ ( X 1 , …, X m ) in the natural language of ( R RE , constants) is equivalent to an existential formula with the extra property that for each x ∈ R m such that ϕ ( x ) is true in R RE there is exactly one y ∈ R n such that ψ ( x, y ) is true in R RE . (Here ψ is quantifier free.)