Fourier Transforms of Surface Area Measure on Convex Surfaces in R 3

1989 ◽  
Vol 111 (4) ◽  
pp. 633 ◽  
Author(s):  
Jong-Guk Bak ◽  
David McMichael ◽  
James Vance ◽  
Stephen Wainger
Keyword(s):  
Surface Area ◽  
Fourier Transforms ◽  
Area Measure ◽  
Surface Area Measure ◽  
Convex Surfaces

scholarly journals Characterizations of central symmetry for convex bodies in Minkowski spaces

2009 ◽  
Vol 46 (4) ◽  
pp. 493-514
Author(s):  
Gennadiy Averkov ◽  
Endre Makai ◽  
Horst Martini
Keyword(s):  
Convex Body ◽  
Surface Area ◽  
Analogous Result ◽  
Area Measure ◽  
Minkowski Metric ◽  
Surface Area Measure ◽  
Shadow Boundary ◽  
Dimensional Minkowski Space ◽  
Centrally Symmetric ◽  
Definition Of

K. Zindler [47] and P. C. Hammer and T. J. Smith [19] showed the following: Let K be a convex body in the Euclidean plane such that any two boundary points p and q of K , that divide the circumference of K into two arcs of equal length, are antipodal. Then K is centrally symmetric. [19] announced the analogous result for any Minkowski plane \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{M}^2$$ \end{document}, with arc length measured in the respective Minkowski metric. This was recently proved by Y. D. Chai — Y. I. Kim [7] and G. Averkov [4]. On the other hand, for Euclidean d -space ℝ d , R. Schneider [38] proved that if K ⊂ ℝ d is a convex body, such that each shadow boundary of K with respect to parallel illumination halves the Euclidean surface area of K (for the definition of “halving” see in the paper), then K is centrally symmetric. (This implies the result from [19] for ℝ 2 .) We give a common generalization of the results of Schneider [38] and Averkov [4]. Namely, let \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{M}^d$$ \end{document} be a d -dimensional Minkowski space, and K ⊂ \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{M}^d$$ \end{document} be a convex body. If some Minkowskian surface area (e.g., Busemann’s or Holmes-Thompson’s) of K is halved by each shadow boundary of K with respect to parallel illumination, then K is centrally symmetric. Actually, we use little from the definition of Minkowskian surface area(s). We may measure “surface area” via any even Borel function ϕ: Sd −1 → ℝ, for a convex body K with Euclidean surface area measure dSK ( u ), with ϕ( u ) being dSK ( u )-almost everywhere non-0, by the formula B ↦ ∫ B ϕ( u ) dSK ( u ) (supposing that ϕ is integrable with respect to dSK ( u )), for B ⊂ Sd −1 a Borel set, rather than the Euclidean surface area measure B ↦ ∫ BdSK ( u ). The conclusion remains the same, even if we suppose surface area halving only for parallel illumination from almost all directions. Moreover, replacing the surface are a measure dSK ( u ) by the k -th area measure of K ( k with 1 ≦ k ≦ d − 2 an integer), the analogous result holds. We follow rather closely the proof for ℝ d , which is due to Schneider [38].

Section means, integral transforms, and Boolean models

1996 ◽  
Vol 28 (2) ◽  
pp. 332-333
Author(s):  
Paul Goodey ◽  
Markus Kiderlen ◽  
Wolfgang Well
Keyword(s):  
Surface Area ◽  
Support Function ◽  
Convex Geometry ◽  
Convex Bodies ◽  
Integral Transforms ◽  
Boolean Models ◽  
Area Measure ◽  
Surface Area Measure ◽  
Left Hand ◽  
The Mean

For a stationary particle process X with convex particles in ℝdd ≧ 2, a mean body M(X) can be defined by where h(M,·) denotes the support function of the convex body M, γ the intensity of X, and P0 is the distribution of the typical particle of X (a probability measure on the set of convex bodies with Steiner point at the origin). Replacing the support function h(M,·) by the surface area measure S(M,·) (see Schneider (1993), for the basic notions from convex geometry), we get the Blaschke body B(X) of X, After normalization, the left-hand side represents the mean normal distribution of X. The main problem discussed here is whether B(X) (respectively S(B(X), ·)) is uniquely determined by the mean bodies M(X ∩ E) in random planar sections X ∩ E of X. From more general results in Weil (1995), it follows that the expectation ES(M(X ∩ E), ·) (taken w.r.t. the uniform distribution of two-dimensional subspaces E in ℝd) equals the surface area measure of a section mean B2(B(X)) of B(X). Thus, the formulated stereological question can be reduced to the injectivity of the transform B2 : K ↦ B2(K).

scholarly journals Valuations and surface area measures

2014 ◽  
Vol 2014 (687) ◽  
Author(s):  
Christoph Haberl ◽  
Lukas Parapatits
Keyword(s):  
Surface Area ◽  
Area Measure ◽  
Surface Area Measure

Abstract.We consider valuations defined on polytopes containing the origin which have measures on the sphere as values. We show that the classical surface area measure is essentially the only such valuation which is

scholarly journals One-sided Lebesgue Bernoulli maps of the sphere of degreen2and2n2

2000 ◽  
Vol 23 (6) ◽  
pp. 383-392 ◽  
Author(s):  
Julia A. Barnes ◽  
Lorelei Koss
Keyword(s):  
Invariant Measure ◽  
Surface Area ◽  
Julia Set ◽  
Maximal Entropy ◽  
Rational Maps ◽  
Area Measure ◽  
Surface Area Measure ◽  
Bernoulli Shifts ◽  
Finite Invariant Measure

We prove that there are families of rational maps of the sphere of degreen2(n=2,3,4,…)and2n2(n=1,2,3,…)which, with respect to a finite invariant measure equivalent to the surface area measure, are isomorphic to one-sided Bernoulli shifts of maximal entropy. The maps in question were constructed by Böettcher (1903--1904) and independently by Lattès (1919). They were the first examples of maps with Julia set equal to the whole sphere.

Local Boundary Behavior of Bounded Holomorphic Functions

1978 ◽  
Vol 30 (03) ◽  
pp. 583-592 ◽  
Author(s):  
Alexander Nagel ◽  
Walter Rudin
Keyword(s):  
Smooth Boundary ◽  
Boundary Behavior ◽  
Holomorphic Functions ◽  
Area Measure ◽  
Surface Area Measure ◽  
Smooth Curves ◽  
Local Boundary ◽  
Bounded Holomorphic Function ◽  
Bounded Holomorphic ◽  
Dimensional Lebesgue Measure

Let D ⊂⊂ Cn be a bounded domain with smooth boundary ∂D, and let F be a bounded holomorphic function on D. A generalization of the classical theorem of Fatou says that the set E of points on ∂D at which F fails to have nontangential limits satisfies the condition σ (E) = 0, where a denotes surface area measure. We show in the present paper that this result remains true when σ is replaced by 1-dimensional Lebesgue measure on certain smooth curves γ in ∂D. The condition that γ must satisfy is that its tangents avoid certain directions.

scholarly journals Hörmander's Carleson theorem for the ball

1985 ◽  
Vol 26 (1) ◽  
pp. 13-17 ◽  
Author(s):  
S. C. Power
Keyword(s):  
Unit Sphere ◽  
Carleson Measure ◽  
Borel Measure ◽  
Bergman Spaces ◽  
Pseudoconvex Domains ◽  
Positive Borel Measure ◽  
Area Measure ◽  
Surface Area Measure ◽  
Image Position ◽  
Special Case

Let denote the unit ball in ℂ2 and let Sdenote its boundary, the unit sphere. For z ∈ B and δ>0, the following non isotropic balls are defined, where A finite positive Borel measure μ, on B is called a Carleson measure if there exists a constant C for whichHere σ denotes normalized surface area measure on S. The following theorem was obtained by Hörmander [6] as a special case of more general variants for strictly pseudoconvex domains in ℂn. Recently Cima and Wogen [3] derived it from a Carleson measure theorem for Bergman spaces of the ball. A different direct approach to the Bergman context, and related settings, is given in Leucking [7].

scholarly journals Adsorption Kinetics and Isotherm Study of Basic Red 5 on Synthesized Silica Monolith Particles

Water ◽  
2021 ◽  
Vol 13 (20) ◽  
pp. 2803
Author(s):  
Sultan Alam ◽  
Barkat Ullah ◽  
Muhammad Sufaid Khan ◽  
Najeeb ur Rahman ◽  
Luqman Khan ◽  
...  
Keyword(s):  
Surface Area ◽  
Removal Efficiency ◽  
Fourier Transforms ◽  
Kinetic Equations ◽  
Total Pore Volume ◽  
Sem Analysis ◽  
Adsorption Parameters ◽  
Isotherm Study ◽  
Silica Monolith ◽  
Adsorption Kinetics And Isotherm

The Silica monolith particles (SMP) were prepared from Tetra-Methyl-Ortho-Silicate (TMOS) and characterized by Fourier transforms infrared (FTIR), Scanning electron microscopy (SEM), energy dispersive X-ray spectroscopy (EDX), and surface area analyzer. FTIR analysis showed the Si−O stretching confirming SMP formation. SEM analysis provided information about the mean diameter of SMP (1−5 µm). EDX confirmed the presence of silicon and oxygen in the SMP. Moreover, the calculated surface area for SMP was found to be around 367 m2/g, whereas BJH pore size distributed particles were 87.15 along with the total pore volume and pore radius of 0.073 cm3/g and 16.627 Å, respectively. Besides, the removal efficiency was found to be about 96%. Various kinetic equations were used to calculate the adsorption parameters. Overall, the results show that the most appropriate model for the kinetics data was the pseudo-second order kinetics model while the mechanism of adsorption was best explained by the Langmuir isotherm. The highest removal of Basic Red 5 dye after 120 min at 298 K was 576 mg/g. Moreover, the thermodynamics parameters (Enthalpy, Gibb’s energy, and Entropy) were also estimated. The ΔH° (0.995 kJ/mol) value depicted the endothermic nature of the process. The non-spontaneous aspect of the process was evident from the ΔG° values which were 60.431, 328.93, and 339.5 kJ/mol at 293, 303, and 313 K, respectively. From the high removal efficiency value, it can be concluded that the prepared adsorbent can be a potential adsorbent in the reclamation of dyes from wastewater.

scholarly journals Facile Synthesis of Bio-Template Tubular MCo2O4 (M = Cr, Mn, Ni) Microstructure and Its Electrochemical Performance in Aqueous Electrolyte

Processes ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 343 ◽  
Author(s):  
Deepa Guragain ◽  
Camila Zequine ◽  
Ram K Gupta ◽  
Sanjay R Mishra
Keyword(s):  
Electrochemical Performance ◽  
Surface Area ◽  
Gas Adsorption ◽  
Fourier Transforms ◽  
Coulombic Efficiency ◽  
High Surface ◽  
High Surface Area ◽  
Carbon Matrix ◽  
Superior Performance ◽  
Volume Distribution

In this project, we present a comparative study of the electrochemical performance for tubular MCo2O4 (M = Cr, Mn, Ni) microstructures prepared using cotton fiber as a bio-template. Crystal structure, surface properties, morphology, and electrochemical properties of MCo2O4 are characterized using X-ray diffraction (XRD), gas adsorption, scanning electron microscopy (SEM), Fourier transforms infrared spectroscopy (FTIR), cyclic voltammetry (CV), and galvanostatic charge-discharge cycling (GCD). The electrochemical performance of the electrode made up of tubular MCo2O4 structures was evaluated in aqueous 3M KOH electrolytes. The as-obtained templated MCo2O4 microstructures inherit the tubular morphology. The large-surface-area of tubular microstructures leads to a noticeable pseudocapacitive property with the excellent electrochemical performance of NiCo2O4 with specific capacitance value exceeding 407.2 F/g at 2 mV/s scan rate. In addition, a Coulombic efficiency ~100%, and excellent cycling stability with 100% capacitance retention for MCo2O4 was noted even after 5000 cycles. These tubular MCo2O4 microstructure display peak power density is exceeding 7000 W/Kg. The superior performance of the tubular MCo2O4 microstructure electrode is attributed to their high surface area, adequate pore volume distribution, and active carbon matrix, which allows effective redox reaction and diffusion of hydrated ions.

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