scholarly journals On the strong separation conjecture

Author(s):  
François Lucas ◽  
Daniel Schaub ◽  
Mark Spivakovsky
Keyword(s):  
Fractals ◽  
2016 ◽  
Vol 24 (03) ◽  
pp. 1650036
Author(s):  
JUAN DENG ◽  
LIFENG XI

This paper studies the gap sequences of graph-directed sets satisfying the strong separation condition. An interesting application is to investigate the gap sequences of self-similar sets with overlaps.


2018 ◽  
Vol 167 (01) ◽  
pp. 193-207 ◽  
Author(s):  
ÁBEL FARKAS

AbstractWe show that for the attractor (K1, . . ., Kq) of a graph directed iterated function system, for each 1 ⩽ j ⩽ q and ϵ > 0 there exists a self-similar set K ⊆ Kj that satisfies the strong separation condition and dimHKj − ϵ < dimHK. We show that we can further assume convenient conditions on the orthogonal parts and similarity ratios of the defining similarities of K. Using this property as a ‘black box’ we obtain results on a range of topics including on dimensions of projections, intersections, distance sets and sums and products of sets.


2016 ◽  
Vol 160 (3) ◽  
pp. 537-563 ◽  
Author(s):  
MARIUSZ URBAŃSKI ◽  
ANNA ZDUNIK

AbstractWe deal with the question of continuity of numerical values of Hausdorff measures in parametrised families of linear (similarity) and conformal dynamical systems by developing the pioneering work of Lars Olsen and the work [SUZ]. We prove Hölder continuity of the function ascribing to a parameter the numerical value of the Hausdorff measure of either the corresponding limit set or the corresponding Julia set. We consider three cases. Firstly, we consider the case of parametrised families of conformal iterated function systems in $\mathbb{R}$k with k ⩾ 3. Secondly, we consider all linear iterated function systems consisting of similarities in $\mathbb{R}$k with k ⩾ 1. In either of these two cases, the strong separation condition is assumed. In the latter case the Hölder exponent obtained is equal to 1/2. Thirdly, we prove such Hölder continuity for analytic families of conformal expanding repellers in the complex plane $\mathbb{C}$. Furthermore, we prove the Hausdorff measure function to be piecewise real–analytic for families of naturally parametrised linear IFSs in $\mathbb{R}$ satisfying the strong separation condition. On the other hand, we also give an example of a family of linear IFSs in $\mathbb{R}$ for which this function is not even differentiable at some parameters.


2008 ◽  
Vol 38 (2) ◽  
pp. 226-238 ◽  
Author(s):  
Ronald Trosper ◽  
Harry Nelson ◽  
George Hoberg ◽  
Peggy Smith ◽  
William Nikolakis

This paper uses survey information to examine several common assertions about the institutional prerequisites for successful profitability when a First Nation enters an economic enterprise either independently or in joint effort with an outside firm. In the winter of 2004–2005, we interviewed managers on both the First Nations and private sides of joint ventures and other business alliances in Canada, to determine what affected their recent profitability experience. We gathered information on the ages, sizes, and activities of the firms. We also gathered information about the firms’ management structures and relationship with the First Nation, and the characteristics of the government of the First Nation. With a sample size of 40 firms that responded, we found that several institutional characteristics affected profit positively: strong separation of management from band governance, participation in management planning, and the use of staggered terms in band council elections. We found that the likelihood of profitability decreased if the band had been in third party management as well as if there was formal participation of elders or hereditary chiefs in decision making. We offer interpretations of these results.


2014 ◽  
Vol 511-512 ◽  
pp. 1185-1188
Author(s):  
Min Jin

Some undecidability on self-affine fractals have been supported. In this paper, we research on the decidability for self-similar fractal of Dubes type. In fact, we prove that the following problems are decidable to test if the Hausdorff dimension of a given Dubes self-similar set is equal to its similarity dimension, and to test if a given Dubes self-similar set satisfies the strong separation condition.


2007 ◽  
Vol 27 (5) ◽  
pp. 1419-1443 ◽  
Author(s):  
JULIEN BARRAL ◽  
MOUNIR MENSI

AbstractWe consider a class of Gibbs measures on self-affine Sierpiński carpets and perform the multifractal analysis of its elements. These deterministic measures are Gibbs measures associated with bundle random dynamical systems defined on probability spaces whose geometrical structure plays a central role. A special subclass of these measures is the class of multinomial measures on Sierpiński carpets. Our result improves the already known result concerning the multifractal nature of the elements of this subclass by considerably weakening and in some cases even eliminating a strong separation condition of geometrical nature.


2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Bhausaheb L. Pangarkar ◽  
Mukund G. Sane ◽  
Mahendra Guddad

In recent years, the increasing threat to groundwater quality due to human activities has become a matter of great concern. The groundwater quality problems present today are caused by contamination and by overexploitation, or by combination of both, which are faced by many Indian states. Today, reverse osmosis (RO) membranes are the leading technology for desalination of groundwater because of their strong separation capabilities and exhibiting a great potential for treatment of waters worldwide. However, the RO process had some problems due to the formation of polarization films because high pressure operation and by-products which may generate bacteria and fouling. Also, high energy consumption and brine disposal problem is faced in RO process due to the limited recovery of water. These problems may be overcome by other membrane thermal process such as a membrane distillation (MD). This paper addresses the outline of RO and MD process for desalination. RO has developed over the past 40 years and MD is an emerging technology for brackish water desalination and yet is not fully implemented in industry. The MD is the better alternative to RO for desalination theoretically found in the literature.


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