scholarly journals Random affine code tree fractals and Falconer–Sloan condition

2014 ◽  
Vol 36 (5) ◽  
pp. 1516-1533 ◽  
Author(s):  
ESA JÄRVENPÄÄ ◽  
MAARIT JÄRVENPÄÄ ◽  
BING LI ◽  
ÖRJAN STENFLO
Keyword(s):  
Natural Number ◽  
General Class ◽  
Satisfy Condition ◽  
The Family ◽  
General Systems ◽  
Real Anal ◽  
Probabilistic Version

We calculate the almost sure dimension for a general class of random affine code tree fractals in $\mathbb{R}^{d}$. The result is based on a probabilistic version of the Falconer–Sloan condition $C(s)$ introduced in Falconer and Sloan [Continuity of subadditive pressure for self-affine sets. Real Anal. Exchange 34 (2009), 413–427]. We verify that, in general, systems having a small number of maps do not satisfy condition $C(s)$. However, there exists a natural number $n$ such that for typical systems the family of all iterates up to level $n$ satisfies condition $C(s)$.

Convergence of the Ishikawa Iterates for Multi-Valued Mappings in Convex Metric Spaces

2008 ◽  
Vol 15 (1) ◽  
pp. 39-43
Author(s):  
Ljubomir B. Ćirić ◽  
Nebojša T. Nikolić
Keyword(s):  
Metric Space ◽  
Convex Subset ◽  
Metric Spaces ◽  
Satisfy Condition ◽  
Convex Metric Space ◽  
Convex Metric Spaces ◽  
The Family

Abstract Let (𝑋, 𝑑) be a convex metric space, 𝐶 be a closed and convex subset of 𝑋 and let 𝐵(𝐶) be the family of all nonempty bounded subsets of 𝐶. In this paper some results are obtained on the convergence of the Ishikawa iterates associated with a pair of multi-valued mappings 𝑆,𝑇 : 𝐶 → 𝐵(𝐶) which satisfy condition (2.1) below.

scholarly journals DIFFERENTIAL INEQUALITIES AND A MARTY-TYPE CRITERION FOR QUASI-NORMALITY

2018 ◽  
Vol 105 (1) ◽  
pp. 34-45
Author(s):  
JÜRGEN GRAHL ◽  
TOMER MANKET ◽  
SHAHAR NEVO
Keyword(s):  
Natural Number ◽  
Holomorphic Functions ◽  
Differential Inequalities ◽  
The Family ◽  
Type Criterion ◽  
Image Position

We show that the family of all holomorphic functions $f$ in a domain $D$ satisfying $$\begin{eqnarray}\frac{|f^{(k)}|}{1+|f|}(z)\leq C\quad \text{for all }z\in D\end{eqnarray}$$ (where $k$ is a natural number and $C>0$) is quasi-normal. Furthermore, we give a general counterexample to show that for $\unicode[STIX]{x1D6FC}>1$ and $k\geq 2$ the condition $$\begin{eqnarray}\frac{|f^{(k)}|}{1+|f|^{\unicode[STIX]{x1D6FC}}}(z)\leq C\quad \text{for all }z\in D\end{eqnarray}$$ does not imply quasi-normality.

Explicit Constructions of Rödl's Asymptotically Good Packings and Coverings

2000 ◽  
Vol 9 (3) ◽  
pp. 265-276 ◽  
Author(s):  
N. N. KUZJURIN
Keyword(s):  
Natural Number ◽  
Time And Space ◽  
Explicit Constructions ◽  
The Family ◽  
Element Set

For any fixed l < k we present families of asymptotically good packings and coverings of the l-subsets of an n-element set by k-subsets, and an algorithm that, given a natural number i, finds the ith k-subset of the family in time and space polynomial in log n.

On the generalization of density topologies on the real line

2014 ◽  
Vol 64 (5) ◽  
Author(s):  
Jacek Hejduk ◽  
Renata Wiertelak
Keyword(s):  
Real Line ◽  
Density Theorem ◽  
Density Topology ◽  
The Real ◽  
Lebesgue Density ◽  
The Family ◽  
Real Anal ◽  
Similarities And Differences ◽  
Relevant Operator

AbstractThe paper concerns the density points with respect to the sequences of intervals tending to zero in the family of Lebesgue measurable sets. It shows that for some sequences analogue of the Lebesgue density theorem holds. Simultaneously, this paper presents proof of theorem that for any sequence of intervals tending to zero a relevant operator ϕJ generates a topology. It is almost but not exactly the same result as in the category aspect presented in [WIERTELAK, R.: A generalization of density topology with respect to category, Real Anal. Exchange 32 (2006/2007), 273–286]. Therefore this paper is a continuation of the previous research concerning similarities and differences between measure and category.

scholarly journals Generalized Mutual Information

Stats ◽  
2020 ◽  
Vol 3 (2) ◽  
pp. 158-165
Author(s):  
Zhiyi Zhang
Keyword(s):  
Information Theory ◽  
Positive Integer ◽  
Mutual Information ◽  
General Class ◽  
Building Blocks ◽  
Potential Utility ◽  
First Order ◽  
The Family ◽  
Random Elements

Mutual information is one of the essential building blocks of information theory. It is however only finitely defined for distributions in a subclass of the general class of all distributions on a joint alphabet. The unboundedness of mutual information prevents its potential utility from being extended to the general class. This is in fact a void in the foundation of information theory that needs to be filled. This article proposes a family of generalized mutual information whose members are indexed by a positive integer n, with the nth member being the mutual information of nth order. The mutual information of the first order coincides with Shannon’s, which may or may not be finite. It is however established (a) that each mutual information of an order greater than 1 is finitely defined for all distributions of two random elements on a joint countable alphabet, and (b) that each and every member of the family enjoys all the utilities of a finite Shannon’s mutual information.

THE Ck-GORDIAN COMPLEX OF KNOTS

2006 ◽  
Vol 15 (01) ◽  
pp. 73-80 ◽  
Author(s):  
YOSHIYUKI OHYAMA
Keyword(s):  
Natural Number ◽  
Simplicial Complex ◽  
The Family

Hirasawa and Uchida defined the Gordian complex of knots which is a simplicial complex whose vertices consist of all knot types in S3by using "a crossing change". In this paper, we define the Ck-Gordian complex of knots which is an extension of the Gordian complex of knots. Let k be a natural number more than 2 and we show that for any knot K0and any given natural number n, there exists a family of knots {K0, K1,…, Kn} such that for any pair (Ki, Kj) of distinct elements of the family, the Ck-distance dCk(Ki, Kj) = 1.

Rho-GTPase Regulation by GEFs, GAPs and GDIs in Cell Migration

2010 ◽  
Author(s):  
Xavier Diego ◽  
Eugenio On˜ate ◽  
Wing Kam Liu
Keyword(s):  
Cell Migration ◽  
Experimental Evidence ◽  
Crucial Role ◽  
General Class ◽  
Rho Gtpase ◽  
Mathematical Basis ◽  
Temporal Coordination ◽  
The Family ◽  
Turing Instabilities

The family of small RhoGTPases plays a crucial role in the spatial and temporal coordination cell migration. GEFs, GAPs and GDIs are the enzymes that regulate their activity, although the mechanism is poorly understood. Regulation models proposed to date have focused on GEFs as the main modulators of RhoGTPase activity, leaving a passive role to GAPs and GDIs. In this work we show that this assumption leads to models with properties that may be inconsistent with observations, more precisely, appearance of Turing instabilities and reduced sensitivity to secondary stimuli. The mathematical basis of this behavior is established, and a general class of interaction schemes that bypass it by including GAP and GDI regulation, which is supported by experimental evidence, is proposed.

THE GORDIAN COMPLEX OF VIRTUAL KNOTS BY FORBIDDEN MOVES

2013 ◽  
Vol 22 (09) ◽  
pp. 1350051 ◽  
Author(s):  
SUMIKO HORIUCHI ◽  
YOSHIYUKI OHYAMA
Keyword(s):  
Natural Number ◽  
Simplicial Complex ◽  
Virtual Knot ◽  
Virtual Knots ◽  
The Family

Hirasawa and Uchida defined the Gordian complex of knots which is a simplicial complex whose vertices consist of all knot types in S3. In this paper, we define the Gordian complex of virtual knots by using forbidden moves. We show that for any virtual knot K0 and for any given natural number n, there exists a family of virtual knots {K0, K1, …, Kn} such that for any pair (Ki, Kj) of distinct elements of the family, the Gordian distance of virtual knots by forbidden moves dF(Ki, Kj) = 1.

On the paradox of grounded classes

1955 ◽  
Vol 20 (2) ◽  
pp. 140-140 ◽  
Author(s):  
Richard Montague
Keyword(s):  
Natural Number ◽  
Set Theory ◽  
Infinite Sequence ◽  
Axiom Of Choice ◽  
The Other ◽  
Other Hand ◽  
The Family ◽  
The One ◽  
The Axiom Of Choice ◽  
Regular Classes

Mr. Shen Yuting, in this Journal, vol. 18, no. 2 (June, 1953), stated a new paradox of intuitive set-theory. This paradox involves what Mr. Yuting calls the class of all grounded classes, that is, the family of all classes a for which there is no infinite sequence b such that … ϵ bn ϵ … ϵ b2ϵb1 ϵ a.Now it is possible to state this paradox without employing any complex set-theoretical notions (like those of a natural number or an infinite sequence). For let a class x be called regular if and only if (k)(x ϵ k ⊃ (∃y)(y ϵ k · ~(∃z)(z ϵ k · z ϵ y))). Let Reg be the class of all regular classes. I shall show that Reg is neither regular nor non-regular.Suppose, on the one hand, that Reg is regular. Then Reg ϵ Reg. Now Reg ϵ ẑ(z = Reg). Therefore, since Reg is regular, there is a y such that y ϵ ẑ(z = Reg) · ~(∃z)(z ϵ z(z = Reg) · z ϵ y). Hence ~(∃z)(z ϵ ẑ(z = Reg) · z ϵ Reg). But there is a z (namely Reg) such that z ϵ ẑ(z = Reg) · z ϵ Reg.On the other hand, suppose that Reg is not regular. Then, for some k, Reg ϵ k · [1] (y)(y ϵ k ⊃ (∃z)(z ϵ k · z ϵ y)). It follows that, for some z, z ϵ k · z ϵ Reg. But this implies that (ϵy)(y ϵ k · ~(ϵw)(w ϵ k · w ϵ y)), which contradicts [1].It can easily be shown, with the aid of the axiom of choice, that the regular classes are just Mr. Yuting's grounded classes.

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