REMARKS ON SELF-AFFINE FRACTALS WITH POLYTOPE CONVEX HULLS

Fractals ◽  
2010 ◽  
Vol 18 (04) ◽  
pp. 483-498 ◽  
Author(s):  
IBRAHIM KIRAT ◽  
ILKER KOCYIGIT
Keyword(s):  
Convex Hull ◽  
Lebesgue Measure ◽  
Spectral Radius ◽  
Sufficient Conditions ◽  
Convex Hulls ◽  
Compact Set ◽  
Joint Spectral Radius ◽  
Digit Set ◽  
Necessary And Sufficient ◽  
Digit Expansions

Suppose that the set [Formula: see text] of real n × n matrices has joint spectral radius less than 1. Then for any digit set D = {d1, …, dq} ⊂ ℝn, there exists a unique non-empty compact set [Formula: see text] satisfying [Formula: see text], which is typically a fractal set. We use the infinite digit expansions of the points of F to give simple necessary and sufficient conditions for the convex hull of F to be a polytope. Additionally, we present a technique to determine the vertices of such polytopes. These answer some of the related questions of Strichartz and Wang, and also enable us to approximate the Lebesgue measure of such self-affine sets. To show the use of our results, we also give several examples including the Levy dragon and the Heighway dragon.

scholarly journals Explicit volume formula for a hyperbolic tetrahedron in terms of edge lengths

2021 ◽  
Author(s):  
Nikolay Abrosimov ◽  
Bao Vuong
Keyword(s):  
Convex Hull ◽  
Hyperbolic Space ◽  
General Type ◽  
Integral Formula ◽  
Sufficient Conditions ◽  
Gram Matrix ◽  
Dihedral Angles ◽  
Hyperbolic Tetrahedron ◽  
Volume Formula ◽  
Necessary And Sufficient

We consider a compact hyperbolic tetrahedron of a general type. It is a convex hull of four points called vertices in the hyperbolic space [Formula: see text]. It can be determined by the set of six edge lengths up to isometry. For further considerations, we use the notion of edge matrix of the tetrahedron formed by hyperbolic cosines of its edge lengths. We establish necessary and sufficient conditions for the existence of a tetrahedron in [Formula: see text]. Then we find relations between their dihedral angles and edge lengths in the form of a cosine rule. Finally, we obtain exact integral formula expressing the volume of a hyperbolic tetrahedron in terms of the edge lengths. The latter volume formula can be regarded as a new version of classical Sforza’s formula for the volume of a tetrahedron but in terms of the edge matrix instead of the Gram matrix.

Best Polynomial Approximation with Linear Constraints

1992 ◽  
Vol 44 (6) ◽  
pp. 1289-1302
Author(s):  
K. Pan ◽  
E. B. Saff
Keyword(s):  
Asymptotic Behavior ◽  
Polynomial Approximation ◽  
Asymptotic Relation ◽  
Sufficient Conditions ◽  
Linear Constraints ◽  
Necessary And Sufficient Conditions ◽  
Compact Set ◽  
Approximation By Polynomials ◽  
The Matrix ◽  
Necessary And Sufficient

AbstractLet A be a (k + 1) × (k + 1) nonzero matrix. For polynomials p ∈ Pn, set and . Let E ⊂ C be a compact set that does not separate the plane and f be a function continuous on E and analytic in the interior of E. Set and . Our goal is to study approximation to f on E by polynomials from Bn(A). We obtain necessary and sufficient conditions on the matrix A for the convergence En(A,f) → 0 to take place. These results depend on whether zero lies inside, on the boundary or outside E and yield generalizations of theorems of Clunie, Hasson and Saff for approximation by polynomials that omit a power of z. Let be such that . We also study the asymptotic behavior of the zeros of and the asymptotic relation between En(f) and En(A,f).

Aubry set for asymptotically sub-additive potentials

2016 ◽  
Vol 16 (02) ◽  
pp. 1660009 ◽  
Author(s):  
Eduardo Garibaldi ◽  
João Tiago Assunção Gomes
Keyword(s):  
Dynamical Systems ◽  
Spectral Radius ◽  
Sufficient Conditions ◽  
Optimization Theory ◽  
Joint Spectral Radius ◽  
Maximum Value ◽  
Ergodic Optimization ◽  
Aubry Set ◽  
Central Result ◽  
Topological Dynamical Systems

Given a topological dynamical systems [Formula: see text], consider a sequence of continuous potentials [Formula: see text] that is asymptotically approached by sub-additive families. In a generalized version of ergodic optimization theory, one is interested in describing the set [Formula: see text] of [Formula: see text]-invariant probabilities that attain the following maximum value [Formula: see text] For this purpose, we extend the notion of Aubry set, denoted by [Formula: see text]. Our central result provides sufficient conditions for the Aubry set to be a maximizing set, i.e. [Formula: see text] belongs to [Formula: see text] if, and only if, its support lies on [Formula: see text]. Furthermore, we apply this result to the study of the joint spectral radius in order to show the existence of periodic matrix configurations approaching this value.

scholarly journals The regularity of the space of germs of Fréchet valued holomorphic functions and the mixed Hartog's theorem

2006 ◽  
Vol 99 (1) ◽  
pp. 119 ◽  
Author(s):  
Thai Thuan Quang
Keyword(s):  
Holomorphic Function ◽  
Sufficient Conditions ◽  
Holomorphic Functions ◽  
Schwartz Space ◽  
Necessary And Sufficient Conditions ◽  
Fréchet Space ◽  
Compact Set ◽  
Frechet Space ◽  
Set Of Uniqueness ◽  
Necessary And Sufficient

It is shown that $H(K, F)$ is regular for every reflexive Fréchet space $F$ with the property ($\mathrm{LB}_\infty)$ where $K$ is a compact set of uniqueness in a Fréchet-Schwartz space $E$ such that $E \in (\Omega)$. Using this result we give necessary and sufficient conditions for a Fréchet space $F$, under which every separately holomorphic function on $K \times F^*$ is holomorphic, where $K$ is as above.

Norm Convergence of Tn

1977 ◽  
Vol 29 (6) ◽  
pp. 1340-1344 ◽  
Author(s):  
Glenn R. Luecke
Keyword(s):  
Banach Space ◽  
Spectral Radius ◽  
Sufficient Conditions ◽  
Complex Banach Space ◽  
Necessary And Sufficient Conditions ◽  
Norm Convergence ◽  
Continuous Linear ◽  
Linear Transformations ◽  
Necessary And Sufficient

Throughout this paper X will denote a complex Banach space and all operators T will be assumed to be continuous linear transformations from X into X. If T is an operator then ┘(T), γ(T), and R(T) will denote the spectrum of T, the spectral radius of T, and range of T, respectively. This paper contains necessary and sufficient conditions for the (norm) convergence of {Tn} when T is an operator on X.

Left approximate identities in algebras of compact operators on Banach spaces

1986 ◽  
Vol 104 (1-2) ◽  
pp. 169-175 ◽  
Author(s):  
P. G. Dixon
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Finite Rank ◽  
Sufficient Conditions ◽  
Compact Operators ◽  
The Other ◽  
Compact Set ◽  
Approximate Identities ◽  
Necessary And Sufficient

SynopsisWe study the existence of left approximate units, left approximate identities and bounded left approximate identities in the algebras (X)of all compact operators on a Banach space X and ℱ(X)− of all operators uniformly approximable by finite rank operators. In the case of bounded left approximate identities, necessary and sufficient conditions on X are obtained. In the other cases, sufficient conditions are obtained, together with an example of non-existence using a space constructed by Szankowski. The possibility of the sufficient conditions being also necessary depends on the question of whether every compact set is contained in the closure of the image of the unit ball under an operator in (X)(or ℱ(X)−). Sufficient conditions on X are obtained for this to be true, but it is conjectured that the answer for general X is negative.

scholarly journals ON THE EXISTENCE OF NON-NORM-ATTAINING OPERATORS

2021 ◽  
pp. 1-13
Author(s):  
Sheldon Dantas ◽  
Mingu Jung ◽  
Gonzalo Martínez-Cervantes
Keyword(s):  
Sufficient Conditions ◽  
Compact Set ◽  
Closed Convex Hull ◽  
List Type ◽  
Weak Operator Topology ◽  
Compact Approximation ◽  
Weak Operator ◽  
Norm Attaining Operators ◽  
Norm Attaining ◽  
Necessary And Sufficient

Abstract In this article, we provide necessary and sufficient conditions for the existence of non-norm-attaining operators in $\mathcal {L}(E, F)$ . By using a theorem due to Pfitzner on James boundaries, we show that if there exists a relatively compact set K of $\mathcal {L}(E, F)$ (in the weak operator topology) such that $0$ is an element of its closure (in the weak operator topology) but it is not in its norm-closed convex hull, then we can guarantee the existence of an operator that does not attain its norm. This allows us to provide the following generalisation of results due to Holub and Mujica. If E is a reflexive space, F is an arbitrary Banach space and the pair $(E, F)$ has the (pointwise-)bounded compact approximation property, then the following are equivalent: (i) $\mathcal {K}(E, F) = \mathcal {L}(E, F)$ ; (ii) Every operator from E into F attains its norm; (iii) $(\mathcal {L}(E,F), \tau _c)^* = (\mathcal {L}(E, F), \left \Vert \cdot \right \Vert )^*$ , where $\tau _c$ denotes the topology of compact convergence. We conclude the article by presenting a characterisation of the Schur property in terms of norm-attaining operators.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

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