Approximation of Definable Sets by Compact Families, and Upper Bounds on Homotopy and Homology
نویسنده
چکیده
We prove new upper bounds on homotopy and homology groups of o-minimal sets in terms of their approximations by compact o-minimal sets. In particular, we improve the known upper bounds on Betti numbers of semialgebraic sets defined by quantifier-free formulae, and obtain for the first time a singly exponential bound on Betti numbers of sub-Pfaffian sets.
منابع مشابه
Universal Approximation of Interval-valued Fuzzy Systems Based on Interval-valued Implications
It is firstly proved that the multi-input-single-output (MISO) fuzzy systems based on interval-valued $R$- and $S$-implications can approximate any continuous function defined on a compact set to arbitrary accuracy. A formula to compute the lower upper bounds on the number of interval-valued fuzzy sets needed to achieve a pre-specified approximation accuracy for an arbitrary multivariate con...
متن کاملTriangulations of Monotone Families I: Two-dimensional Families
Let K ⊂ Rn be a compact definable set in an o-minimal structure over R, e.g., a semi-algebraic or a subanalytic set. A definable family {Sδ| 0 < δ ∈ R} of compact subsets of K, is called a monotone family if Sδ ⊂ Sη for all sufficiently small δ > η > 0. The main result of the paper is that when dimK ≤ 2 there exists a definable triangulation of K such that for each (open) simplex Λ of the trian...
متن کاملOn the number of homotopy types of fibres of a definable map
In this paper we prove a single exponential upper bound on the number of possible homotopy types of the fibres of a Pfaffian map in terms of the format of its graph. In particular, we show that if a semi-algebraic set S ⊂ Rm+n, where R is a real closed field, is defined by a Boolean formula with s polynomials of degree less than d, and π : Rm+n → Rn is the projection on a subspace, then the num...
متن کاملVanishing Homology Guillaume
In this paper we introduce a new homology theory devoted to the study of families such as semi-algebraic or subanalytic families and in general to any family definable in an o-minimal structure (such as Denjoy-Carleman definable or ln − exp definable sets). The idea is to study the cycles which are vanishing when we approach a special fiber. This also enables us to derive local metric invariant...
متن کاملUpper and lower bounds of symmetric division deg index
Symmetric Division Deg index is one of the 148 discrete Adriatic indices that showed good predictive properties on the testing sets provided by International Academy of Mathematical Chemistry. Symmetric Division Deg index is defined by $$ SDD(G) = sumE left( frac{min{d_u,d_v}}{max{d_u,d_v}} + frac{max{d_u,d_v}}{min{d_u,d_v}} right), $$ where $d_i$ is the degree of vertex $i$ in graph $G$. In th...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2007