Questions tagged [o-minimal]
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If a subset $X$ of a $C^k$ manifold $M$ is semialgebraic in the charts of $M$, is it Whitney stratifiable?
Let $M$ be a $C^k$ manifold for some $k\geq 1$ and $X$ be a subset of $M$.
Assume that there is an atlas of charts $(\phi_\alpha, U_\alpha)_\alpha$ of $M$ such that in the coordinates of each of these ...
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Equivalence of bounded gradient flow solutions and uniformly bounded gradient descent trajectories for definable functions
I am reading paper [1] by C. Josz regarding the global convergence of the gradient method. The main result is the following:
$\textbf{Theorem}$: For a definable differentiable function $f : \mathbb{R}^...
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Is there a largest o-minimal structure all of whose definable functions are analytic?
In the paper "Quasianalytic Denjoy-Carleman classes and o-minimality" by J.-P. Rolin, P. Speissegger and A. J. Wilkie, it is proven that there is no largest o-minimal structure on the real ...
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First order formula describing connected components
I ask this question here after no answer came up in the original MathSE question.
Let $\mathcal{L}$ be the language $\{+,-,\cdot,0,1,P\}$ where $P$ is some $n$-ary relation symbol. Is there a formula $...
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When do volumes depend real-analytically on the parameters defining the regions?
Suppose $f_1, \dots, f_n$ are real-valued real analytic functions defined on an open set $B=(0,1)^d$ of $\mathbb{R}^d$.
For $r \in \mathbb{R}$, let $S_r$ be the sub-level set in $B$ defined by the ...
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Definable constructions in o-minimal geometry
Recently I've been working with o-minimal expansions of $(\mathbb{R},\times,+)$, and I want to work "internally" to the language of o-minimal sets instead of working with "definable ...
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Indiscernible sequences in o-minimal structures
Is there an explicit characterisation of indiscernible sequences in o-minimal structures ? Say for expansions of the reals or $R_{an}$ ?
Is there a characterisation of o-minimality in terms of ...
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Is the intrinsic volume always positive for maximum dimension?
The intrinsic volume functions on $\mathbb{R}^d$ known from the Steiner formula and Hadwiger's Theorem can be extended to the domain of definable sets of an o-minimal structure $\text{Def}(\mathbb{R}^...
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Comparing an integral to zero, by slicing and stacking
Let $f \colon [0,1] \rightarrow {\mathbb R}$ be a nice function -- real-analytic, or maybe definable in some o-minimal structure, let's say. Let $0 < \alpha_1 < \cdots < \alpha_n < 1$ be ...
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Is the order arithmetic of the positive reals o-minimal?
Consider the structure of the positive real numbers $(0, \infty) $ with its unit $1$, its addition $+$, its multiplication $\times $, and its strict ordering $> $.
Is this structure
$$( (0, \infty)...
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O-minimality and forcing
It is well-known that the structure $(\mathbb{R}, +, \cdot, <, 0, 1)$ is an o-minimal structure and hence the set of integers $\mathbb{Z}$ is not definable in it.
In an ongoing project with Will ...
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Definable functions in $o$-minimal structures
Consider the field of real numbers $(\mathbb R,+,\cdot)$. An expansion of $(\mathbb R,+,\cdot)$ is a tuple $(\mathbb R,+,\cdot,S)$, where $S$ is a collection of subsets of $\mathbb R^n$ for $n \in \...
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Classification of $n$-dimensional Nash-submanifolds of $\mathbb{R}^n$
Let $M,N\subset \mathbb{R}^n$ be two open semi-algebraic subsets, and assume that $M$ and $N$ are $C^\infty$ diffeomorphic, i.e. isomorphic as smooth submanifolds of $\mathbb{R}^n$. Does this imply ...
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Unboundedness of number of solutions of intersection of bivariate polynomial with graph of function from an o-minimal structure
I am trying to understand a construction sketched in the paper by Gwozdziewicz, Kurdyka and Parusinski in the Proceedings of the AMS 1999 (paper here) and I'd like to request some help. The ...
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Sheafs in O-minimal Structures
Let $\mathcal{N} = (N, <, \ldots)$ be an o-minimal structure and let $X \subset N^m$ be a definable set. Following the procedure stablished by Edmundo, Jones and Peatfield in "Sheaf cohomology in o-...
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Undefinability of $\mathbb{Z}$ in the reals
It is a well-known fact that $\mathbb{Z}$ is not definable in the structure $\mathcal{R}=(\mathbb{R}, +, ., < , 0, 1)$. This follows from Tarski's quantifier elimination, and in fact, we can ...
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O-minimal spectrum is a spectral space
I'm trying to understand a proof on "Sheaves of Continuous Definable Functions" (Pillay, Anand. "Sheaves of continuous definable functions." The Journal of symbolic logic 53.04 (1988): 1165-1169.)
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Limits of definable maps
For sequences of semialgebraic maps there is the following result:
Let $(f_{n}: ]0,1[^d \to ]0,1[)_{n \in \mathbb{N}}$ be a sequence of continuous semialgebraic maps of bounded degree such that $(...
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Are definable sets in an o-minimal expansion of the real field locally analytic?
I have a strong suspicion that yes, but as I am not a specialist in o-minimal structures, I thought that I might have overlooked some corner case.
The precise statement is as follows: let $X \subset ...
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What is the precise relationship between o-minimal theory and Grothendieck's "Esquisse d'un programme"?
I have seen various references in the literature to such a connection but they tend to assume that the reader is familiar with the connection, and limit themselves to providing additional detail. So ...
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What classes of complex manifolds are known to be definable in an o-minimal expansion of the real field?
It is a widely known (perhaps slightly folkloric) fact that compact complex manifolds, understood as first-order structures with a predicate for each analytic subset, are definable in an expansion of ...
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If $f$ is $C^{\infty}$ and $f^2$ is analytic, then $f$ is analytic
Assume that $f:\mathbb{C}^n\rightarrow \mathbb{C}$ is a $C^{\infty}$ function such that $f^2$ is (complex) analytic. Then one can show that $f$ is analytic. (Note: Liviu Nicolaescu and Alexandre ...
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Representing quasianalytic functions in several variables
For functions in a quasianalytic Denjoy-Carleman class we have the property that their Taylor expansions at a point (the origin) determines the function. For classes that don't only contain analytic ...
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Importance of Denjoy-Carleman classes as a class.
Denjoy-Carleman classes of differentiable functions, say in Roumieu's form:
Given a log-convex sequence $M_n$ of positive number denote by $C_M=C_M(\mathbb{R}^n,0)$ the ring of germs of $C^\infty(\...
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Is there any o-minimal expansion of the real field with functions of growth higher than exponential?
Let $\bar{\mathbb{R}}$ be the structure of the real field, that is $(\mathbb{R},0,1,+,-,*,<)$ . We say that a function $f$ is of growth higher than exponential if for all $N\in \mathbb{N}$ there $f(...
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Intuition behind o-minimal structures.
This is very much the same post as I posted at math.stackexchange.
I am following the definitions of an o-minimal system in "Tame Topology and O-minimal Structures" by Lou van den Dries.
It is ...
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Why is definable compact equivalent to bounded and closed for sets with o-minimal structures?
We have $M$ an o-minimal structure. $X \in M^n$ with the induced topology.
I'm reading an article which shows that $X \in M^n$ is definable compact is
equivalent to $X$ being bounded and closed.
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