Topological differential fields and dimension functions

2012 ◽  
Vol 77 (4) ◽  
pp. 1147-1164 ◽  
Author(s):  
Nicolas Guzy ◽  
Françoise Point
Keyword(s):  
Dimension Function ◽  
Dimension Functions ◽  
Differential Fields

AbstractWe construct a fibered dimension function in some topological differential fields.

scholarly journals Cell decomposition and dimension function in the theory of closed ordered differential fields

2009 ◽  
Vol 159 (1-2) ◽  
pp. 111-128 ◽  
Author(s):  
Thomas Brihaye ◽  
Christian Michaux ◽  
Cédric Rivière
Keyword(s):  
Cell Decomposition ◽  
Dimension Function ◽  
Differential Fields

scholarly journals APPLICATIONS OF DIVERGENCE POINTS TO LOCAL DIMENSION FUNCTIONS OF SUBSETS OF $\mathbb{R}^{d}$

2005 ◽  
Vol 48 (1) ◽  
pp. 213-218 ◽  
Author(s):  
L. Olsen
Keyword(s):  
Continuous Function ◽  
Hausdorff Dimension ◽  
Short Proof ◽  
Primary 28A80 ◽  
Subject Classification ◽  
Dimension Function ◽  
Local Dimension ◽  
Divergence Points ◽  
Dimension Functions

AbstractFor a subset $E\subseteq\mathbb{R}^{d}$ and $x\in\mathbb{R}^{d}$, the local Hausdorff dimension function of $E$ at $x$ is defined by$$ \mathrm{dim}_{\mathrm{loc}}(x,E)=\lim_{r\searrow0}\mathrm{dim}(E\cap B(x,r)), $$where ‘dim’ denotes the Hausdorff dimension. Using some of our earlier results on so-called multifractal divergence points we give a short proof of the following result: any continuous function $f:\mathbb{R}^{d}\to[0,d]$ is the local dimension function of some set $E\subseteq\mathbb{R}^{d}$. In fact, our result also provides information about the rate at which the dimension $\mathrm{dim}(E\cap B(x,r))$ converges to $f(x)$ as $r\searrow0$.AMS 2000 Mathematics subject classification: Primary 28A80

scholarly journals Some Results on Fuzzy ω-Covering Dimension Function in Fuzzy Topological Space

2021 ◽  
pp. 961-971
Author(s):  
Munir Abdul Khalik AL-Khafaji ◽  
Gazwan Haider Abdulhusein
Keyword(s):  
Topological Space ◽  
Dimension Function ◽  
Covering Dimension ◽  
New Class ◽  
Fuzzy Topological Space ◽  
Dimension Functions

     The purpose of this paper is to study a new class of fuzzy covering dimension functions, called fuzzy

scholarly journals STRONG DENSITY OF DEFINABLE TYPES AND CLOSED ORDERED DIFFERENTIAL FIELDS

2019 ◽  
Vol 84 (3) ◽  
pp. 1099-1117 ◽  
Author(s):  
QUENTIN BROUETTE ◽  
PABLO CUBIDES KOVACSICS ◽  
FRANÇOISE POINT
Keyword(s):  
Strong Form ◽  
Dimension Function ◽  
Definable Set ◽  
Definable Type ◽  
Differential Fields

AbstractThe following strong form of density of definable types is introduced for theories T admitting a fibered dimension function d: given a model M of T and a definable set X ⊆ Mn, there is a definable type p in X, definable over a code for X and of the same d-dimension as X. Both o-minimal theories and the theory of closed ordered differential fields (CODF) are shown to have this property. As an application, we derive a new proof of elimination of imaginaries for CODF.

Dimension Functions of Self-Affine Scaling Sets

2013 ◽  
Vol 56 (4) ◽  
pp. 745-758
Author(s):  
Xiaoye Fu ◽  
Jean-Pierre Gabardo
Keyword(s):  
Affine Scaling ◽  
Dimension Function ◽  
Expansive Dilation ◽  
Expansive Matrix ◽  
Dimension Functions

Abstract.In this paper, the dimension function of a self-affine generalized scaling set associated with an n×n integral expansive dilation A is studied. More specifically, we consider the dimension function of an A-dilation generalized scaling set K assuming that K is a self-affine tile satisfying BK = (K+d1)[ (K + d2), where B = At , A is an n×n integral expansive matrix with |det A| = 2, and d1, d2 ∊ ℝn. We show that the dimension function of K must be constant if either n = 1 or 2 or one of the digits is 0, and that it is bounded by 2|K| for any n.

scholarly journals Dimension theory and fuzzy topological spaces

2006 ◽  
Vol 2006 ◽  
pp. 1-8
Author(s):  
S. S. Benchalli ◽  
B. M. Ittanagi ◽  
P. G. Patil
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Dimension Theory ◽  
Dimension Function ◽  
Inductive Dimension ◽  
Fuzzy Topological Space ◽  
Small Inductive Dimension ◽  
Dimension Functions ◽  
Fuzzy Topological Spaces

J. M. Aarts introduced and studied a new dimension function,Hind, in 1975 and obtained several results on this function. In this paper, a new local inductive dimension function called local huge inductive dimension function denoted bylocHindis introduced and studied. Furthermore, an effort is made to introduce and study dimension functions for fuzzy topological spaces. It has been possible to introduce and study the small inductive dimension functionindfXand large inductive dimension functionindfXfor a fuzzy topological spaceX.

Measure and dimension functions: measurability and densities

1997 ◽  
Vol 121 (1) ◽  
pp. 81-100 ◽  
Author(s):  
PERTTI MATTILA ◽  
R. DANIEL MAULDIN
Keyword(s):  
Hausdorff Measure ◽  
Measure Theory ◽  
Packing Dimension ◽  
Dimension Function ◽  
Packing Measure ◽  
Compact Sets ◽  
Analytic Sets ◽  
Dimension Functions ◽  
Definition Of ◽  
Borel Measurable

During the past several years, new types of geometric measure and dimension have been introduced; the packing measure and dimension, see [Su], [Tr] and [TT1]. These notions are playing an increasingly prevalent role in various aspects of dynamics and measure theory. Packing measure is a sort of dual of Hausdorff measure in that it is defined in terms of packings rather than coverings. However, in contrast to Hausdorff measure, the usual definition of packing measure requires two limiting procedures, first the construction of a premeasure and then a second standard limiting process to obtain the measure. This makes packing measure somewhat delicate to deal with. The question arises as to whether there is some simpler method for defining packing measure and dimension. In this paper, we find a basic limitation on this possibility. We do this by determining the descriptive set-theoretic complexity of the packing functions. Whereas the Hausdorff dimension function on the space of compact sets is Borel measurable, the packing dimension function is not. On the other hand, we show that the packing dimension functions are measurable with respect to the σ-algebra generated by the analytic sets. Thus, the usual sorts of measurability properties used in connection with Hausdorff measure, for example measures of sections and projections, remain true for packing measure.

scholarly journals Dade groups for finite groups and dimension functions

2021 ◽  
Vol 576 ◽  
pp. 146-196
Author(s):  
Matthew Gelvin ◽  
Ergün Yalçın
Keyword(s):  
Finite Groups ◽  
Dimension Functions

scholarly journals GBUO: “The Good, the Bad, and the Ugly” Optimizer

2021 ◽  
Vol 11 (5) ◽  
pp. 2042
Author(s):  
Hadi Givi ◽  
Mohammad Dehghani ◽  
Zeinab Montazeri ◽  
Ruben Morales-Menendez ◽  
Ricardo A. Ramirez-Mendoza ◽  
...  
Keyword(s):  
High Dimension ◽  
Essential Role ◽  
Optimization Problems ◽  
Optimization Algorithms ◽  
Stochastic Search ◽  
Objective Functions ◽  
Science And Engineering ◽  
Fixed Dimension ◽  
Dimension Functions ◽  
Simulation Results

Optimization problems in various fields of science and engineering should be solved using appropriate methods. Stochastic search-based optimization algorithms are a widely used approach for solving optimization problems. In this paper, a new optimization algorithm called “the good, the bad, and the ugly” optimizer (GBUO) is introduced, based on the effect of three members of the population on the population updates. In the proposed GBUO, the algorithm population moves towards the good member and avoids the bad member. In the proposed algorithm, a new member called ugly member is also introduced, which plays an essential role in updating the population. In a challenging move, the ugly member leads the population to situations contrary to society’s movement. GBUO is mathematically modeled, and its equations are presented. GBUO is implemented on a set of twenty-three standard objective functions to evaluate the proposed optimizer’s performance for solving optimization problems. The mentioned standard objective functions can be classified into three groups: unimodal, multimodal with high-dimension, and multimodal with fixed dimension functions. There was a further analysis carried-out for eight well-known optimization algorithms. The simulation results show that the proposed algorithm has a good performance in solving different optimization problems models and is superior to the mentioned optimization algorithms.

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