Subsets of the Plane with Small Linear Sections and Invariant Extensions of the Two-Dimensional Lebesgue Measure

1999 ◽  
Vol 6 (5) ◽  
pp. 441-446
Author(s):  
A. Kharazishvili
Keyword(s):  
Lebesgue Measure ◽  
Euclidean Plane ◽  
Point Of View ◽  
Two Dimensional ◽  
Linear Sections ◽  
Dimensional Lebesgue Measure

Abstract We consider some subsets of the Euclidean plane 𝐑2, having small linear sections (in all directions), and investigate those sets from the point of view of measurability with respect to certain invariant extensions of the classical Lebesgue measure on 𝐑2.

STARLIKENESS AND CONVEXITY OF CAUCHY TRANSFORMS ON REGULAR POLYGONS

2020 ◽  
pp. 1-12
Author(s):  
PENG-FEI ZHANG ◽  
XIN-HAN DONG
Keyword(s):  
Lebesgue Measure ◽  
Cauchy Transform ◽  
Two Dimensional ◽  
Regular Polygons ◽  
Cauchy Transforms ◽  
Dimensional Lebesgue Measure

Abstract For $n\geq 3$ , let $Q_n\subset \mathbb {C}$ be an arbitrary regular n-sided polygon. We prove that the Cauchy transform $F_{Q_n}$ of the normalised two-dimensional Lebesgue measure on $Q_n$ is univalent and starlike but not convex in $\widehat {\mathbb {C}}\setminus Q_n$ .

The Measure of Plane Sets

1943 ◽  
Vol 39 (1) ◽  
pp. 51-53
Author(s):  
P. A. P. Moran
Keyword(s):  
Finite Number ◽  
Coordinate System ◽  
Lebesgue Measure ◽  
Upper Bound ◽  
Two Dimensional ◽  
Image Position ◽  
Bounded Sets ◽  
Dimensional Lebesgue Measure

We consider bounded sets in a plane. If X is such a set, we denote by Pθ(X) the projection of X on the line y = x tan θ, where x and y belong to some fixed coordinate system. By f(θ, X) we denote the measure of Pθ(X), taking this, in general, as an outer Lebesgue measure. The least upper bound of f (θ, X) for all θ we denote by M. We write sm X for the outer two-dimensional Lebesgue measure of X. Then G. Szekeres(1) has proved that if X consists of a finite number of continua,Béla v. Sz. Nagy(2) has obtained a stronger inequality, and it is the purpose of this paper to show that these results hold for more general classes of sets.

On Nonmeasurable Functions of Two Variables and Iterated Integrals

2009 ◽  
Vol 16 (4) ◽  
pp. 705-710
Author(s):  
Alexander Kharazishvili
Keyword(s):  
Lebesgue Measure ◽  
Two Dimensional ◽  
Iterated Integrals ◽  
Functions Of Two Variables ◽  
Dimensional Lebesgue Measure

Abstract Following the paper of Pkhakadze [Trudy Tbiliss. Mat. Inst. Razmadze 20: 167–209, 1954], we consider some properties of real-valued functions of two variables, which are not assumed to be measurable with respect to the two-dimensional Lebesgue measure on the plane 𝐑2, but for which the corresponding iterated integrals exist and are equal to each other. Close connections of these properties with certain set-theoretical axioms are emphasized.

$\beta$-transformation, natural extension and invariant measure

2000 ◽  
Vol 20 (5) ◽  
pp. 1271-1285 ◽  
Author(s):  
GAVIN BROWN ◽  
QINGHE YIN
Keyword(s):  
Invariant Measure ◽  
Lebesgue Measure ◽  
Natural Extension ◽  
Connected Region ◽  
Two Dimensional ◽  
Constant Multiple ◽  
Unit Square ◽  
Simply Connected Region ◽  
Simply Connected ◽  
Dimensional Lebesgue Measure

For $\beta>1$, consider the $\beta$-transformation $T_\beta$. When $\beta$ is an integer, the natural extension of $T_\beta$ can be represented explicitly as a map on the unit square with an invariant measure: the corresponding two-dimensional Lebesgue measure. We show that, under certain conditions on $\beta$, the natural extension is defined on a simply connected region and an invariant measure is a constant multiple of the Lebesgue measure.We characterize those $\beta$ in terms of the $\beta$-expansion of one, and study the structure and size of the set of all such $\beta$.

Fundamental group of tiles associated to quadratic canonical number systems

2008 ◽  
Vol 58 (2) ◽  
Author(s):  
Benoit Loridant
Keyword(s):  
Fundamental Group ◽  
Lebesgue Measure ◽  
Affine Plane ◽  
Quadratic Polynomial ◽  
Two Dimensional ◽  
Closed Disk ◽  
Number Systems ◽  
Complete Set ◽  
Dimensional Lebesgue Measure

AbstractIf A is a 2 × 2 expanding matrix with integral coefficients, and ⊂ ℤ2 a complete set of coset representatives of ℤ2/Aℤ2 with |det(A)| elements, then the set ℐ defined by Aℐ = ℐ + is a self-affine plane tile of ℝ2, provided that its two-dimensional Lebesgue measure is positive.It was shown by Luo and Thuswaldner that the fundamental group of such a tile is either trivial or uncountable.To a quadratic polynomial x 2 + Ax + B, A, B ∈ ℤ such that B ≥ 2 and −1 ≤ A ≤ B, one can attach a tile ℐ. Akiyama and Thuswaldner proved the triviality of the fundamental group of this tile for 2A < B + 3, by showing that a tile of this class is homeomorphic to a closed disk. The case 2A ≥ B + 3 is treated here by using the criterion given by Luo and Thuswaldner.

scholarly journals Lossless Digital Filter Overflow Oscillations; Approximation of Invariant Fractals

1997 ◽  
Vol 07 (11) ◽  
pp. 2603-2610 ◽  
Author(s):  
Peter Ashwin ◽  
W. Chambers ◽  
G. Petkov
Keyword(s):  
Lebesgue Measure ◽  
Digital Filters ◽  
Measure Zero ◽  
Two Dimensional ◽  
Finite Precision ◽  
Present Evidence ◽  
Sizeable Fraction ◽  
Finite Precision Arithmetic ◽  
Dimensional Lebesgue Measure ◽  
Overflow Oscillations

We investigate second order lossless digital filters with two's complement overflow. We numerically approximate the fractal set D of points that iterate arbitrarily close to the discontinuity. For the case of eigenvalues of the associated linear map of the form eiθ with θ/π ∉ Q we present evidence that D has positive two dimensional Lebesgue measure. For θ/π ∈ Q we confirm that D has Lebesgue measure zero. As a by-product we get estimates of the exterior dimension of D. These results imply that if such filters are realized using finite-precision arithmetic then they will have a sizeable fraction of orbits that are periodic with high period overflows.

scholarly journals Cutoff AdS3 versus $$ T\overline{T} $$ CFT2 in the large central charge sector: correlators of energy-momentum tensor

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Yi Li ◽  
Yang Zhou
Keyword(s):  
Field Theory ◽  
Euclidean Plane ◽  
Energy Momentum Tensor ◽  
Central Charge ◽  
Momentum Tensor ◽  
Two Dimensional ◽  
Conformal Field ◽  
Operator Identity ◽  
Pure Gravity ◽  
Charge Sector

Abstract In this article we probe the proposed holographic duality between $$ T\overline{T} $$ T T ¯ deformed two dimensional conformal field theory and the gravity theory of AdS3 with a Dirichlet cutoff by computing correlators of energy-momentum tensor. We focus on the large central charge sector of the $$ T\overline{T} $$ T T ¯ CFT in a Euclidean plane and a sphere, and compute the correlators of energy-momentum tensor using an operator identity promoted from the classical trace relation. The result agrees with a computation of classical pure gravity in Euclidean AdS3 with the corresponding cutoff surface, given a holographic dictionary which identifies gravity parameters with $$ T\overline{T} $$ T T ¯ CFT parameters.

Consistency conditions for a finite set of projections of a function

1979 ◽  
Vol 85 (1) ◽  
pp. 61-68 ◽  
Author(s):  
K. J. Falconer
Keyword(s):  
Lebesgue Measure ◽  
Convex Subset ◽  
Compact Convex ◽  
Unit Vector ◽  
Compact Convex Subset ◽  
Finite Collection ◽  
Finite Set ◽  
Almost All ◽  
Dimensional Lebesgue Measure ◽  
Unit Vectors

Let H(μ, θ) be the hyperplane in Rn (n ≥ 2) that is perpendicular to the unit vector 6 and perpendicular distance μ from the origin; that is, H(μ, θ) = (x ∈ Rn: x. θ = μ). (Note that H(μ, θ) and H(−μ, −θ) are the same hyperplanes.) Let X be a proper compact convex subset of Rm. If f(x) ∈ L1(X) we will denote by F(μ, θ) the projection of f perpendicular to θ; that is, the integral of f(x) over H(μ, θ) with respect to (n − 1)-dimensional Lebesgue measure. By Fubini's Theorem, if f(x) ∈ L1(X), F(μ, θ) exists for almost all μ for every θ. Our aim in this paper is, given a finite collection of unit vectors θ1, …, θN, to characterize the F(μ, θi) that are the projections of some function f(x) with support in X for 1 ≤ i ≤ N.

A STUDY OF TWO‐DIMENSIONAL HEAD WAVES IN FLUID AND SOLID SYSTEMS

Geophysics ◽  
1963 ◽  
Vol 28 (4) ◽  
pp. 563-581 ◽  
Author(s):  
John W. Dunkin
Keyword(s):  
Half Space ◽  
Vertical Displacement ◽  
Point Of View ◽  
Head Wave ◽  
Apparent Velocity ◽  
Two Dimensional ◽  
Multiple Reflections ◽  
Transient Wave ◽  
Line Load ◽  
Head Waves

The problem of transient wave propagation in a three‐layered, fluid or solid half‐plane is investigated with the point of view of determining the effect of refracting bed thickness on the character of the two‐dimensional head wave. The “ray‐theory” technique is used to obtain exact expressions for the vertical displacement at the surface caused by an impulsive line load. The impulsive solutions are convolved with a time function having the shape of one cycle of a sinusoid. The multiple reflections in the refracting bed are found to affect the head wave significantly. For thin refracting beds in the fluid half‐space the character of the head wave can be completely altered by the strong multiple reflections. In the solid half‐space the weaker multiple reflections affect both the rate of decay of the amplitude of the head wave with distance and the apparent velocity of the head wave by changing its shape. A comparison is made of the results for the solid half‐space with previously published results of model experiments.

Poly[[μ2-1,3-bis(pyridin-4-yl)propane](μ3-1,4-phenylenediacetato)cadmium]

2012 ◽  
Vol 68 (2) ◽  
pp. m34-m36 ◽  
Author(s):  
Dong Liu
Keyword(s):  
Title Complex ◽  
Point Of View ◽  
Solvothermal Reaction ◽  
Dimensional Chain ◽  
Two Dimensional ◽  
One Dimensional ◽  
Connected Network ◽  
Dimensional Network

Solvothermal reaction between Cd(NO3)2, 1,4-phenylenediacetate (1,4-PDA) and 1,3-bis(pyridin-4-yl)propane (bpp) afforded the title complex, [Cd(C10H8O4)(C13H14N2)]n. Adjacent carboxylate-bridged CdIIions are related by an inversion centre. The 1,4-PDA ligands adopt acisconformation and connect the CdIIions to form a one-dimensional chain extending along thecaxis. These chains are in turn linked into a two-dimensional network through bpp bridges. The bpp ligands adopt ananti–gaucheconformation. From a topological point of view, each bpp ligand and each pair of 1,4-PDA ligands can be considered as linkers, while the dinuclear CdIIunit can be regarded as a 6-connecting node. Thus, the structure can be simplified to a two-dimensional 6-connected network.

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