scholarly journals Multidimensional self-affine sets: non-empty interior and the set of uniqueness

2016 ◽  
pp. 1-10
Author(s):  
Kevin G. Hare ◽  
Nikita Sidorov
Keyword(s):  
Empty Interior ◽  
Set Of Uniqueness

scholarly journals A Full Description of ω-Limit Sets of Cournot Maps Having Non-Empty Interior and Some Economic Applications

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 452
Author(s):  
Antonio Linero-Bas ◽  
María Muñoz-Guillermo
Keyword(s):  
Economic Models ◽  
Full Description ◽  
Cournot Model ◽  
Empty Interior ◽  
Limit Sets ◽  
Economic Applications

Given a continuous Cournot map F(x,y)=(f2(y),f1(x)) defined from I2=[0,1]×[0,1] into itself, we give a full description of its ω-limit sets with non-empty interior. Additionally, we present some partial results for the empty interior case. The distribution of the ω-limits with non-empty interior gives information about the dynamics and the possible outputs of each firm in a Cournot model. We present some economic models to illustrate, with examples, the type of ω-limits that appear.

scholarly journals On the maximal Sobolev regularity of distributions supported by subsets of Euclidean space

2016 ◽  
Vol 15 (05) ◽  
pp. 731-770 ◽  
Author(s):  
D. P. Hewett ◽  
A. Moiola
Keyword(s):  
Maximal Regularity ◽  
Cantor Sets ◽  
Characteristic Functions ◽  
Bessel Potential ◽  
Empty Interior ◽  
Cartesian Products ◽  
Sobolev Regularity ◽  
Full Classification ◽  
Differential And Integral Equations

This paper concerns the following question: given a subset [Formula: see text] of [Formula: see text] with empty interior and an integrability parameter [Formula: see text], what is the maximal regularity [Formula: see text] for which there exists a non-zero distribution in the Bessel potential Sobolev space [Formula: see text] that is supported in [Formula: see text]? For sets of zero Lebesgue measure, we apply well-known results on set capacities from potential theory to characterize the maximal regularity in terms of the Hausdorff dimension of [Formula: see text], sharpening previous results. Furthermore, we provide a full classification of all possible maximal regularities, as functions of [Formula: see text], together with the sets of values of [Formula: see text] for which the maximal regularity is attained, and construct concrete examples for each case. Regarding sets with positive measure, for which the maximal regularity is non-negative, we present new lower bounds on the maximal Sobolev regularity supported by certain fat Cantor sets, which we obtain both by capacity-theoretic arguments, and by direct estimation of the Sobolev norms of characteristic functions. We collect several results characterizing the regularity that can be achieved on certain special classes of sets, such as [Formula: see text]-sets, boundaries of open sets, and Cartesian products, of relevance for applications in differential and integral equations.

Points of egress in problems of Hamiltonian dynamics

1991 ◽  
Vol 109 (2) ◽  
pp. 405-417 ◽  
Author(s):  
C. J. Amick ◽  
J. F. Toland
Keyword(s):  
Convex Set ◽  
Hamiltonian Dynamics ◽  
Empty Interior ◽  
Natural Assumption ◽  
Initial Value ◽  
Lipschitz Continuous ◽  
Convexity Assumptions ◽  
Image Position ◽  
Closed Convex Set ◽  
Well Posed

First we consider an elementary though delicate question about the trajectory in ℝn of a particle in a conservative field of force whose dynamics are governed by the equationHere the potential function V is supposed to have Lipschitz continuous first derivative at every point of ℝn. This is a natural assumption which ensures that the initial-value problem is well-posed. We suppose also that there is a closed convex set C with non-empty interior C° such that V ≥ 0 in C and V = 0 on the boundary ∂C of C. It is noteworthy that we make no assumptions about the degeneracy (or otherwise) of V on ∂C (i.e. whether ∇V = 0 on ∂C, or not); thus ∂C is a Lipschitz boundary because of its convexity but not necessarily any smoother in general. We remark also that there are no convexity assumptions about V and nothing is known about the behaviour of V outside C.

Schauder bases and diametrically complete sets with empty interior

2018 ◽  
Vol 463 (2) ◽  
pp. 452-460 ◽  
Author(s):  
Monika Budzyńska ◽  
Aleksandra Grzesik ◽  
Wiesława Kaczor ◽  
Tadeusz Kuczumow
Keyword(s):  
Empty Interior ◽  
Schauder Bases ◽  
Complete Sets

Cone-Monotone Functions: Differentiability and Continuity

2005 ◽  
Vol 57 (5) ◽  
pp. 961-982 ◽  
Author(s):  
Jonathan M. Borwein ◽  
Xianfu Wang
Keyword(s):  
Banach Spaces ◽  
Convex Cone ◽  
Continuous Functions ◽  
Monotone Functions ◽  
Empty Interior ◽  
Space Of Continuous Functions

AbstractWe provide a porosity-based approach to the differentiability and continuity of real-valued functions on separable Banach spaces, when the function is monotone with respect to an ordering induced by a convex cone K with non-empty interior. We also show that the set of nowhere K-monotone functions has a σ-porous complement in the space of continuous functions endowed with the uniform metric.

Transversals with Residue in Moderately Overlapping T(k)-Families of Translates

2009 ◽  
Vol 52 (3) ◽  
pp. 388-402
Author(s):  
Aladár Heppes
Keyword(s):  
Asymptotic Behavior ◽  
Compact Convex ◽  
Line Transversal ◽  
Type Result ◽  
Empty Interior ◽  
Disjoint Family ◽  
Straight Line ◽  
The Family ◽  
Centrally Symmetric ◽  
Line Meeting

AbstractLet K denote an oval, a centrally symmetric compact convex domain with non-empty interior. A family of translates of K is said to have property T(k) if for every subset of at most k translates there exists a common line transversal intersecting all of them. The integer k is the stabbing level of the family. Two translates Ki = K + ci and Kj = K + cj are said to be σ-disjoint if σK + ci and σK + cj are disjoint. A recent Helly-type result claims that for every σ > 0 there exists an integer k(σ) such that if a family of σ-disjoint unit diameter discs has property T(k)|k ≥ k(σ), then there exists a straight line meeting all members of the family. In the first part of the paper we give the extension of this theorem to translates of an oval K. The asymptotic behavior of k(σ) for σ → 0 is considered as well.Katchalski and Lewis proved the existence of a constant r such that for every pairwise disjoint family of translates of an oval K with property T(3) a straight line can be found meeting all but at most r members of the family. In the second part of the paper σ-disjoint families of translates of K are considered and the relation of σ and the residue r is investigated. The asymptotic behavior of r(σ) for σ → 0 is also discussed.

scholarly journals A Class of Star-Shaped Bodies

1959 ◽  
Vol 2 (3) ◽  
pp. 175-180 ◽  
Author(s):  
Z.A. Melzak
Keyword(s):  
Surface Area ◽  
Lipschitz Condition ◽  
Convex Bodies ◽  
Nonempty Interior ◽  
Empty Interior ◽  
Selection Principle ◽  
Positive Functions ◽  
Bounded Sets ◽  
Bounded Convex

The more important properties of the class κ of all bounded convex bodies in E3 with non-empty interior include: uniform approximability by polyhedra, existence of volume and surface area, and Blaschke's selection principle, [l ], [2 ]. In this note we define and consider a class ℋ of star-shaped bodies in E3, which enjoys many properties of κ, among them the above-mentioned ones, and is considerably larger. Roughly speaking, ℋ consists of closed bounded sets in E3 with nonempty interior, whose boundary is completely visible from every point of a set with non-empty interior. It turns out that ℋ is identifiable with the class of all real-valued positive functions on the sphere S3 which satisfy a Lipschitz condition.

scholarly journals Piecing Together the Empty Interior

2018 ◽  
Vol 43 (1) ◽  
pp. 9-18
Author(s):  
Pamela Salen
Keyword(s):  
Empty Interior
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