scholarly journals Effective Cylindrical Cell Decompositions for Restricted Sub-Pfaffian Sets

2020 ◽  
Author(s):  
Gal Binyamini ◽  
Nicolai Vorobjov
Keyword(s):  
Betti Numbers ◽  
Cell Decomposition ◽  
Upper Bounds ◽  
Geometric Complexity ◽  
Cylindrical Cell ◽  
Cell Decompositions ◽  
Minimal Structure ◽  
Natural Measure ◽  
Definition Of

Abstract The o-minimal structure generated by the restricted Pfaffian functions, known as restricted sub-Pfaffian sets, admits a natural measure of complexity in terms of a format  ${{\mathcal{F}}}$, recording information like the number of variables and quantifiers involved in the definition of the set, and a degree  $D$, recording the degrees of the equations involved. Khovanskii and later Gabrielov and Vorobjov have established many effective estimates for the geometric complexity of sub-Pfaffian sets in terms of these parameters. It is often important in applications that these estimates are polynomial in $D$. Despite much research done in this area, it is still not known whether cell decomposition, the foundational operation of o-minimal geometry, preserves polynomial dependence on $D$. We slightly modify the usual notions of format and degree and prove that with these revised notions, this does in fact hold. As one consequence, we also obtain the first polynomial (in $D$) upper bounds for the sum of Betti numbers of sets defined using quantified formulas in the restricted sub-Pfaffian structure.

scholarly journals L2-BETTI NUMBERS OF DISCRETE MEASURED GROUPOIDS

2005 ◽  
Vol 15 (05n06) ◽  
pp. 1169-1188 ◽  
Author(s):  
ROMAN SAUER
Keyword(s):  
Probability Space ◽  
Betti Numbers ◽  
Free Action ◽  
Countable Group ◽  
Equivalence Relations ◽  
Dimension Theory ◽  
Discrete Groups ◽  
Type Ii ◽  
Orbit Equivalent ◽  
Definition Of

There are notions of L2-Betti numbers for discrete groups (Cheeger–Gromov, Lück), for type II1-factors (recent work of Connes-Shlyakhtenko) and for countable standard equivalence relations (Gaboriau). Whereas the first two are algebraically defined using Lück's dimension theory, Gaboriau's definition of the latter is inspired by the work of Cheeger and Gromov. In this work we give a definition of L2-Betti numbers of discrete measured groupoids that is based on Lück's dimension theory, thereby encompassing the cases of groups, equivalence relations and holonomy groupoids with an invariant measure for a complete transversal. We show that with our definition, like with Gaboriau's, the L2-Betti numbers [Formula: see text] of a countable group G coincide with the L2-Betti numbers [Formula: see text] of the orbit equivalence relation [Formula: see text] of a free action of G on a probability space. This yields a new proof of the fact the L2-Betti numbers of groups with orbit equivalent actions coincide.

scholarly journals Sharp upper bounds for the Betti numbers of a given Hilbert polynomial

2013 ◽  
Vol 7 (5) ◽  
pp. 1019-1064 ◽  
Author(s):  
Giulio Caviglia ◽  
Satoshi Murai
Keyword(s):  
Betti Numbers ◽  
Upper Bounds ◽  
Hilbert Polynomial

scholarly journals On a Counting Theorem for Weakly Admissible Lattices

2020 ◽  
Author(s):  
Reynold Fregoli
Keyword(s):  
Diophantine Approximation ◽  
General Setting ◽  
Upper Bounds ◽  
Lattice Points ◽  
Precise Estimate ◽  
Linear Subspaces ◽  
Minimal Structure ◽  
Partition Method

Abstract We give a precise estimate for the number of lattice points in certain bounded subsets of $\mathbb{R}^{n}$ that involve “hyperbolic spikes” and occur naturally in multiplicative Diophantine approximation. We use Wilkie’s o-minimal structure $\mathbb{R}_{\exp }$ and expansions thereof to formulate our counting result in a general setting. We give two different applications of our counting result. The 1st one establishes nearly sharp upper bounds for sums of reciprocals of fractional parts and thereby sheds light on a question raised by Lê and Vaaler, extending previous work of Widmer and of the author. The 2nd application establishes new examples of linear subspaces of Khintchine type thereby refining a theorem by Huang and Liu. For the proof of our counting result, we develop a sophisticated partition method that is crucial for further upcoming work on sums of reciprocals of fractional parts over distorted boxes.

Upper bounds for the betti numbers of graded ideals of a given length in the exterior algebra

1999 ◽  
Vol 27 (9) ◽  
pp. 4607-4631 ◽  
Author(s):  
Marilena Crupi ◽  
Rosanna Utano
Keyword(s):  
Betti Numbers ◽  
Upper Bounds ◽  
Exterior Algebra

scholarly journals Upper bounds for Betti numbers of multigraded modules

2007 ◽  
Vol 316 (1) ◽  
pp. 453-458
Author(s):  
Amanda Beecher
Keyword(s):  
Betti Numbers ◽  
Upper Bounds ◽  
Multigraded Modules

scholarly journals The maximum diversity assortment selection problem

2021 ◽  
Author(s):  
Felix Prause ◽  
Kai Hoppmann-Baum ◽  
Boris Defourny ◽  
Thorsten Koch
Keyword(s):  
Mathematical Model ◽  
Hamming Distance ◽  
Upper Bounds ◽  
Selection Problem ◽  
Computational Results ◽  
Two Dimensional ◽  
Minimum Threshold ◽  
Maximum Area ◽  
Rectangular Container ◽  
Definition Of

AbstractIn this article, we introduce the Maximum Diversity Assortment Selection Problem (MDASP), which is a generalization of the two-dimensional Knapsack Problem (2D-KP). Given a set of rectangles and a rectangular container, the goal of 2D-KP is to determine a subset of rectangles that can be placed in the container without overlapping, i.e., a feasible assortment, such that a maximum area is covered. MDASP is to determine a set of feasible assortments, each of them covering a certain minimum threshold of the container, such that the diversity among them is maximized. Thereby, diversity is defined as the minimum or average normalized Hamming distance of all assortment pairs. MDASP was the topic of the 11th AIMMS-MOPTA Competition in 2019. The methods described in this article and the resulting computational results won the contest. In the following, we give a definition of the problem, introduce a mathematical model and solution approaches, determine upper bounds on the diversity, and conclude with computational experiments conducted on test instances derived from the 2D-KP literature.

O-MINIMALISM

2014 ◽  
Vol 79 (2) ◽  
pp. 355-409 ◽  
Author(s):  
HANS SCHOUTENS
Keyword(s):  
Cell Decomposition ◽  
Dimension Theory ◽  
Pigeonhole Principle ◽  
Euler Characteristics ◽  
Grothendieck Ring ◽  
First Order ◽  
Minimal Structure ◽  
Related Notion ◽  
Depth Analysis

AbstractThis paper is devoted to o-minimalism, the study of the first-order properties of o-minimal structures. The main protagonists are the pseudo-o-minimal structures, that is to say, the models of the theory of all o-minimal L-structures, but we start with a more in-depth analysis of the well-known fragment DCTC (Definable Completeness/Type Completeness), and show how it already admits many of the properties of o-minimal structures: dimension theory, monotonicity, Hardy structures, and quasi-cell decomposition, provided one replaces finiteness by discreteness in all of these. Failure of cell decomposition leads to the related notion of a eukaryote structure, and we give a criterium for a pseudo-o-minimal structure to be eukaryote.To any pseudo-o-minimal structure, we can associate its Grothendieck ring, which in the non-o-minimal case is a nontrivial invariant. To study this invariant, we identify a third o-minimalistic property, the Discrete Pigeonhole Principle, which in turn allows us to define discretely valued Euler characteristics. As an application, we study certain analytic subsets, called Taylor sets.

Modeling overlapping structures

2013 ◽  
Author(s):  
Yves Marcoux ◽  
Michael Sperberg-McQueen ◽  
Claus Huitfeldt
Keyword(s):  
Directed Graphs ◽  
Upper Bounds ◽  
Directed Acyclic Graphs ◽  
Markup Language ◽  
Multiple Roots ◽  
Acyclic Graph ◽  
Xml Documents ◽  
Acyclic Graphs ◽  
Structured Document ◽  
Definition Of

The problem of overlapping structures has long been familiar to the structured document community. In a poem, for example, the verse and line structures overlap, and having them both available simultaneously is convenient, and sometimes necessary (for example for automatic analyses). However, only structures that embed nicely can be represented directly in XML. Proposals to address this problem include XML solutions (based essentially on a layer of semantics) and non-XML ones. Among the latter is TexMecs HS2003, a markup language that allows overlap (and many other features). XML documents, when viewed as graphs, correspond to trees. Marcoux M2008 characterized overlap-only TexMecs documents by showing that they correspond exactly to completion-acyclic node-ordered directed acyclic graphs. In this paper, we elaborate on that result in two ways. First, we cast it in the setting of a strictly larger class of graphs, child-arc-ordered directed graphs, that includes multi-graphs and non-acyclic graphs, and show that — somewhat surprisingly — it does not hold in general for graphs with multiple roots. Second, we formulate a stronger condition, full-completion-acyclicity, that guarantees correspondence with an overlap-only document, even for graphs that have multiple roots. The definition of fully-completion-acyclic graph does not in itself suggest an efficient algorithm for checking the condition, nor for computing a corresponding overlap-only document when the condition is satisfied. We present basic polynomial-time upper bounds on the complexity of accomplishing those tasks.

scholarly journals Upper Bounds on Betti Numbers of Tropical Prevarieties

2018 ◽  
Vol 4 (1) ◽  
pp. 127-136 ◽  
Author(s):  
Dima Grigoriev ◽  
Nicolai Vorobjov
Keyword(s):  
Betti Numbers ◽  
Upper Bounds

The Complexity of Everywhere Divergent Fourier Series

1991 ◽  
Vol 43 (2) ◽  
pp. 413-424
Author(s):  
T. I. Ramsamujh
Keyword(s):  
Fourier Series ◽  
Unit Circle ◽  
Polish Space ◽  
Continuous Functions ◽  
Rank Function ◽  
Alternative Definition ◽  
Closed Sets ◽  
Natural Measure ◽  
Definition Of ◽  
Divergent Fourier Series

AbstractA natural rank function is defined on the set DS of everywhere divergent sequences of continuous functions on the unit circle T. The rank function provides a natural measure of the complexity of the sequences in DS, and is obtained by associating a well-founded tree with each such sequence. The set DF of everywhere divergent Fourier series, and the set DT of everywhere divergent trigonometric series with coefficients that tend to zero, can be viewed as natural subsets of DS. It is shown that the rank function is a coanalytic norm which is unbounded in ω1 on DF. From this it follows that DF, DT and DS are not Borel subsets of the Polish space SC(T) of all sequences of continuous functions on T. Finally an alternative definition of the rank function is formulated by using nested sequences of closed sets.

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