scholarly journals Asymptotic Properties of Discrete Minimal s,logt-Energy Constants and Configurations

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 932
Author(s):  
Nichakan Loesatapornpipit ◽  
Nattapong Bosuwan
Keyword(s):  
Euclidean Space ◽  
Asymptotic Properties ◽  
High Symmetry ◽  
Minimal Energy ◽  
Compact Metric Space ◽  
Point Configurations ◽  
Metric Distance ◽  
Energy Constants ◽  
Very High ◽  
Differentiability Properties

We investigated the energy of N points on an infinite compact metric space (A,d) of a diameter less than 1 that interact through the potential (1/ds)(log1/d)t, where s,t≥0 and d is the metric distance. With Elogts(A,N) denoting the minimal energy for such N-point configurations, we studied certain continuity and differentiability properties of Elogts(A,N) in the variable s. Then, we showed that in the limits, as s→∞ and as s→s0>0,N-point configurations that minimize the s,logt-energy tends to an N-point best-packing configuration and an N-point configuration that minimizes the s0,logt-energy, respectively. Furthermore, we considered when A are circles in the Euclidean space R2. In particular, we proved the minimality of N distinct equally spaced points on circles in R2 for some certain s and t. The study on circles shows a possibility for the utilization of N points generated through such new potential to uniformly discretize on objects with very high symmetry.

Mechanistische Untersuchungen zur Ligandenfluktuation in den Sandwich-Komplexen [M([12]Krone-4)2]++ / Mechanistic Studies on the Ligand Fluctuations in the Sandwich Complexes [M([12]crown-4)2]++

1984 ◽  
Vol 39 (12) ◽  
pp. 1755-1758 ◽  
Author(s):  
Franz L. Dickert ◽  
Walter Gumbrecht ◽  
Manfred Waidhas
Keyword(s):  
Crown Ether ◽  
Metal Ions ◽  
Reaction Order ◽  
Outer Sphere ◽  
High Symmetry ◽  
Mechanistic Studies ◽  
Sandwich Complexes ◽  
Associative Mechanism ◽  
M 12 ◽  
Very High

The metal ions Co++ , Ni++ , Mg++ , and Zn++ form the sandwich complexes [M([12]crown-4)2]++ with [12]crown-4 in CD3NO2. From the reaction order of the crown ether exchange a strong outer sphere association between complex and ligand can be inferred. Ligand fluctuation processes occur via an intramolecular associative mechanism from which a very high symmetry of the complexes results.

General shape distributions in a plane

1991 ◽  
Vol 23 (2) ◽  
pp. 259-276 ◽  
Author(s):  
I. L. Dryden ◽  
K. V. Mardia
Keyword(s):  
Closed Form ◽  
Normal Approximation ◽  
Asymptotic Properties ◽  
Two Dimensions ◽  
General Shape ◽  
Shape Distribution ◽  
Inference Problems ◽  
Point Configurations ◽  
Exact Shape

In this paper we investigate the exact shape distribution for general Gaussian labelled point configurations in two dimensions. The shape density is written in a closed form, in terms of Kendall's or Bookstein's shape variables. The distribution simplifies considerably in certain cases, including the complex normal, isotropic, circular Markov and equal means cases. Various asymptotic properties of the distribution are investigated, including a large variation distribution and the normal approximation for small variations. The triangle case is considered in particular detail, and we compare the density with simulated densities for some examples. Finally, we consider inference problems, with an application in biology.

Pentacoordination and ionic character: crystal structure of Ph4SbBr

1989 ◽  
Vol 67 (1) ◽  
pp. 63-70 ◽  
Author(s):  
Osvald Knop ◽  
Beverly R. Vincent ◽  
T. Stanley Cameron
Keyword(s):  
Crystal Structure ◽  
Axial Position ◽  
Ionic Character ◽  
High Symmetry ◽  
Trans Effect ◽  
Bond Lengths ◽  
Axial Bond ◽  
The Mean ◽  
High Degree ◽  
Very High

Ph4SbBr (P21/n, a = 12.282(2) Å, b = 10.656(1) Å, c = 16.156(1) Å, β = 104.92(7)°, Z = 4) crystallizes with the same space group as Ph4SbCl, but the two compounds are not isostructural. The structure of Ph4SbBr consists of layers of Ph4SbBr molecules with the Br atoms almost exactly in the [Formula: see text] planes. The coordination figure of the Sb atom is a trigonal bipyramid (TBP) with Br in an axial position. The Sb—Br distance, 2.950(1) Å is the longest reported for Sb(V) to date. The Sb(V)—Hal bond lengths and the Ph4SbBr and Ph4SbCl structures are compared in detail. Analysis of a large sample of Ph4EX (E = P, As) structures and of all the available R4EX (E = Sb, Bi; R = Ph, Me) structures shows that the former are always ionic, R4E+X−, whereas in the latter the observed coordination figures represent R4EX trigonal bipyramids with various degrees of axial distortion. The inverse variation of the axial bond lengths (the trans effect) in the TBP Sb compounds can be quantified as Sb—Cax ~ 2.4(Sb—X)−1/9 to a very high degree of correlation. This and other geometric relationships show that there exists a limiting E—X bond length (~3.3 Å for E = Sb and ~3.5 Å for E = Bi when X is oxygen) at which the TBP coordination becomes unstable and changes over to a tetrahedral R4E. This change is accompanied by a rearrangement of the structure to an ionic R4E+X− solid (of high symmetry where permitted by the size and shape of the anion) with [Formula: see text] separations no smaller than ~4 Å when E = Sb or Bi and X = O. The distribution of the CEC angles in the Ph4EX (E = P, As) sample is analyzed; the mean (uncorrected) P—C and As—C bond lengths in Ph4E+ are found to be 1.789(11) and 1.902(12) Å, respectively. Keywords: crystal structure, organoantimony compounds, Ph4SbBr, pentacoordination, tetraphenylantimony bromide.

scholarly journals Continuity and differentiability properties of the Nemitskii operator in Hölder spaces

1988 ◽  
Vol 30 (1) ◽  
pp. 59-65 ◽  
Author(s):  
Rita Nugari
Keyword(s):  
Banach Space ◽  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Bounded Subset ◽  
Holder Space ◽  
Usual Norm ◽  
Image Position ◽  
Holder Spaces ◽  
Open Bounded Subset ◽  
Differentiability Properties

Let ℝn be the n-dimensional Euclidean space with the usual norm denoted by |·| In what follows 蒆 will denote an open bounded subset of ℝn, and its closure.For α ∊(0,1], is the space of all functions such that: is called the Holder space with exponent a and is a Banach space when endowed with the norm:where ‖u‖∞ is, as usual, defined by:

Fuzzy Interval Number K-Means Clustering for Region Division of Pork Market

2020 ◽  
Vol 12 (3) ◽  
pp. 43-61
Author(s):  
Xiangyan Meng ◽  
Muyan Liu ◽  
Ailing Qiao ◽  
Huiqiu Zhou ◽  
Jingyi Wu ◽  
...  
Keyword(s):  
Clustering Algorithm ◽  
Original Data ◽  
Interval Number ◽  
Evaluation Indexes ◽  
Region Division ◽  
Daily Data ◽  
Public Datasets ◽  
Metric Distance ◽  
Very High ◽  
Fuzzy Interval

This article proposes a new clustering algorithm named FINK-means. First, this article converts original data into a fuzzy interval number (FIN). Second, it proves the F that denotes the collection of FINs is a lattice. Finally, it introduces a novel metric distance on the lattice F. The contrast experiments about FINK-means, k-means, and FCM algorithm are carried out on two simulated datasets and four public datasets. The results show that the FINK-means algorithm has better clustering performance on three evaluation indexes including the purity, loss cost, and silhouette coefficient. FINK-means is applied to the task of region division of pork market in China based on the daily data of pork price for different provinces of China from August 9, 2017 to August 9, 2018. The results show that regions of pork market in China was divided into five categories, namely very low, low, medium, high, and very high. Every category has been discussed as well. At last, an additional experiment about region division in Canada was carried out to prove the efficiency of FINK-means further.

A COMPUTATIONAL ERGODIC THEOREM FOR INFINITE ITERATED FUNCTION SYSTEMS

2008 ◽  
Vol 08 (03) ◽  
pp. 365-381 ◽  
Author(s):  
NGUYEN DINH CONG ◽  
DOAN THAI SON ◽  
STEFAN SIEGMUND
Keyword(s):  
Dynamical Systems ◽  
Euclidean Space ◽  
Ergodic Theorem ◽  
Error Bound ◽  
Metric Space ◽  
Iterated Function Systems ◽  
Random Dynamical Systems ◽  
Compact Metric Space ◽  
Iterated Function ◽  
Time Average

Iterated function systems are examples of random dynamical systems and became popular as generators of fractals like the Sierpinski Gasket and the Barnsley Fern. In this paper we prove an ergodic theorem for iterated function systems which consist of countably many functions and which are contractive on average on an arbitrary compact metric space and we provide a computational version of this ergodic theorem in Euclidean space which allows to numerically approximate the time average together with an explicit error bound. The results are applied to an explicit example.

General shape distributions in a plane

1991 ◽  
Vol 23 (02) ◽  
pp. 259-276 ◽  
Author(s):  
I. L. Dryden ◽  
K. V. Mardia
Keyword(s):  
Closed Form ◽  
Normal Approximation ◽  
Asymptotic Properties ◽  
Two Dimensions ◽  
General Shape ◽  
Shape Distribution ◽  
Inference Problems ◽  
Point Configurations ◽  
Exact Shape

In this paper we investigate the exact shape distribution for general Gaussian labelled point configurations in two dimensions. The shape density is written in a closed form, in terms of Kendall's or Bookstein's shape variables. The distribution simplifies considerably in certain cases, including the complex normal, isotropic, circular Markov and equal means cases. Various asymptotic properties of the distribution are investigated, including a large variation distribution and the normal approximation for small variations. The triangle case is considered in particular detail, and we compare the density with simulated densities for some examples. Finally, we consider inference problems, with an application in biology.

scholarly journals Simulation of neutron production in hadron-nucleus and nucleus-nucleus interactions in Geant4

2019 ◽  
Vol 204 ◽  
pp. 03004
Author(s):  
Aida Galoyan ◽  
Alberto Ribon ◽  
Vladimir Uzhinsky
Keyword(s):  
Experimental Data ◽  
Energy Range ◽  
Collision Energy ◽  
Asymptotic Properties ◽  
Slow Neutron ◽  
Neutron Production ◽  
Projectile Energy ◽  
Neutron Spectra ◽  
High Energies ◽  
Very High

Studying experimental data obtained at ITEP [1] on neutron production in interactions of protons with various nuclei in the energy range from 747 MeV up to 8.1 GeV, we have found that slow neutron spectra have scaling and asymptotic properties [2]. The spectra weakly depend on the collision energy at momenta of projectile protons larger than 5 – 6 GeV/c. These properties are taken into account in the Geant4 Fritiof (FTF) model. The improved FTF model describes as well as the Geant4 Bertini model the experimental data on neutron production by 1.2 GeV and 1.6 GeV protons on targets (Fe, Pb) [3] and by 3.0 GeV protons on various targets (Al, Fe, Pb) [4]. For neutron production in antiproton-nucleus interactions, it is demonstrated that the FTF results are in a satisfactory agreement with experimental data of the LEAR collaboration [5]. The FTF model gives promising results for neutron production in nucleus - nucleus interactions at projectile energy 1 – 2 GeV per nucleon [6]. The observed properties allow one to predict neutron yields in the nucleus-nucleus interactions at high and very high energies. Predictions for the NICA/MPD experiment at JINR are presented.

The absorption of very high frequency sound in dielectric solids

1957 ◽  
Vol 53 (3) ◽  
pp. 702-716 ◽  
Author(s):  
S. Simons
Keyword(s):  
Mean Free Path ◽  
Theoretical Treatment ◽  
High Symmetry ◽  
Interaction Theory ◽  
Very High Frequency ◽  
The Mean ◽  
Dielectric Solids ◽  
Solid Argon ◽  
Very High ◽  
Supply Information

ABSTRACTA theoretical treatment is given of the absorption of longitudinally polarized sound in dielectric crystals of high symmetry, due to interaction with the thermal phonous. The Q value for the absorption is calculated in terms of the deviation from Hooke's law, the result being applicable at low temperatures when the mean free path of the phonons is greater than the wavelength of the acoustic wave. The theory is applied to cubic crystals, and in particular to solid argon, where a minimum Q value of about 108–109 is calculated. It is suggested that measurement of the absorption (at present somewhat beyond experimental techniques) would supply information on the deviation from linear elasticity, as well as providing a satisfactory verification of the basic interaction theory. The effect of isotopes, and, in more complicated structures, of optic modes, is shown to be negligible.

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