Five Mutually Tangent Spheres and Various Associated Configurations

1982 ◽  
Vol 25 (2) ◽  
pp. 149-163
Author(s):  
J. F. Rigby
Keyword(s):  
Inversive Plane ◽  
Geometrical Progression

AbstractIf five spheres σ0, σ1, …, σ4 touch each other externally and have radii in geometrical progression, there is a dilative rotation mapping σ0, σ1, σ2, σ3, to σ1, σ2, σ3, σ4; the dilatation factor is shown to be negative. The ten points of contact of the spheres lie by fours on 15 circles, forming a (154106) configuration in inversive space. In the corresponding configuration in the inversive plane, the 15 circles meet again in 60 points, which lie by fours on 45 circles touching by threes at each of the 60 points, and forming a configuration isomorphic to that of 60 Pascal lines (associated with six points on a conic) meeting by fours at 45 points. The 45 circles arise from ten Money-Coutts configurations of nine anti-tangent cycles. Conjectures are made about other circles through the 60 points.

scholarly journals A Comprehensive Study on Gamma Rays and Fast Neutron Sensing Properties of GAGOC and CMO Scintillators for Shielding Radiation Applications

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Shams A. M. Issa ◽  
M. I. Sayyed ◽  
M. H. M. Zaid ◽  
K. A. Matori
Keyword(s):  
Energy Range ◽  
Fast Neutron ◽  
Gamma Ray ◽  
Mean Free Path ◽  
Nuclear Radiation ◽  
Space Technology ◽  
Effective Electron ◽  
Geometrical Progression ◽  
Broad Energy Range ◽  
Removal Cross Section

The WinXCom program has been used to calculate the mass attenuation coefficients (μm), effective atomic numbers (Zeff), effective electron densities (Nel), half-value layer (HVL), and mean free path (MFP) in the energy range 1 keV–100 GeV for Gd3Al2Ga3O12Ce (GAGOC) and CaMoO4 (CMO) scintillator materials. The geometrical progression (G-P) method has been used to compute the exposure buildup factors (EBF) and gamma ray energy absorption (EABF) in the photon energy range 0.015–15 MeV and up to a 40 penetration depth (mfp). In addition, the values of the removal cross section for a fast neutron ∑R have been calculated. The computed data observes that GAGOC showed excellent γ-rays and neutrons sensing a response in the broad energy range. This work could be useful for nuclear radiation sensors, detectors, nuclear medicine applications (medical imaging and mammography), nuclear engineering, and space technology.

Biological Significance and Fate of Atrazine under Aquifer Conditions

Weed Science ◽  
1983 ◽  
Vol 31 (5) ◽  
pp. 610-618 ◽  
Author(s):  
Glenn R. Wehtje ◽  
Roy F. Spalding ◽  
Orvin C. Burnside ◽  
Stephen R. Lowry ◽  
J. Robert C. Leavitt
Keyword(s):  
Avena Sativa ◽  
Seasonal Pattern ◽  
Biological Significance ◽  
Contaminated Groundwater ◽  
Monitoring Wells ◽  
Progression Model ◽  
Geometrical Progression ◽  
Partial Degradation ◽  
Contamination Levels

Concentrations of atrazine [2-chloro-4-(ethylamino)-6-(isopropylamino)-s-triazine] in contaminated groundwater recovered from 41 monitoring wells in Hall and Buffalo counties, Nebraska range from 0.01 to 8.29 μg/L. Over a 1-year period concentrations fluctuated sufficiently in a seasonal pattern to indicate that atrazine dissipation occurs. This reduction in concentration can be attributed to adsorption, dispersion, and degradation. A limited amount of atrazine degraded chemically to hydroxyatrazine [2-hydroxy-4-(ethylamino)-6-(isopropylamino)-s-triazine] under simulated aquifer conditions; microbial degradation could not be detected. A geometrical progression model for predicting long term residue accumulation indicates that current contamination levels probably reflect a steady-state situation between the amount that yearly enters into, and the partial degradation that occurs within, the aquifer. Oat (Avena sativaL. ‘Neal’) bioassays indicate that present levels of groundwater contamination remain well below the threshold necessary for phytotoxicity.

scholarly journals General Theory of Subglacial Cavitation and Sliding of Temperate Glaciers

1968 ◽  
Vol 7 (49) ◽  
pp. 21-58 ◽  
Author(s):  
L. Lliboutry
Keyword(s):  
Frictional Force ◽  
Atmospheric Pressure ◽  
Hydraulic System ◽  
Water Pressure ◽  
Realistic Model ◽  
Point Of View ◽  
Small Scale ◽  
Sliding Velocity ◽  
Surface Slope ◽  
Geometrical Progression

AbstractEarlier theories of Weertman and the present author are reviewed and compared; both are insufficient to account for the facts observed at the tongue of the Allalingletscher.A calculation of the stresses and heat flow at the bed of a glacier with a sinusoidal profile is given which takes account of any degree of subglacial cavitation. The sliding due to plasticity and that due to pressure melting are related to this degree of cavitation and it is shown that these two terms are additive. There results an expression for the frictionfωin terms of the total sliding velocityuand the height of the bumpsa. For a given and large enough value ofu,fω(a) exhibits two maxima which are equal and independent ofu.The paper then considers a more realistic model of the bed consisting of a superposition of sine waves all having the same roughnessr, andadecreasing in a geometrical progression. The biggestamay be inferred from the overall profile of the bedrock; the resulting frictional force can be regarded either as part of the total frictional forcefin an overall view for whichf=ρghsinαholds, or else as a correction to such a value on the small scale (the best point of view for crevasse studies). To a first approximation Coulomb’s law of friction holds provided one takes account of the interstitial water pressure at the ice-rock interface.This interstitial pressurepis next related to the thickness of the glacierh. If the subglacial hydraulic system is at atmospheric pressure,pis proportional toh. Next, if the sliding velocity is not too large, the surface slope approaches 1.6r≈ 0.12 and kinematic waves (which move four times as fast as the ice) disappear rapidly. If the hydraulic system is not at atmospheric pressure the surface slope is smaller and flow instabilities can occur.

scholarly journals Symmetry in Regular Polyhedra Seen as 2D Möbius Transformations: Geodesic and Panel Domes Arising from 2D Diagrams

Symmetry ◽  
2018 ◽  
Vol 10 (9) ◽  
pp. 356 ◽  
Author(s):  
Jose Diaz-Severiano ◽  
Valentin Gomez-Jauregui ◽  
Cristina Manchado ◽  
Cesar Otero
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Space Structures ◽  
Inversive Plane ◽  
Delaunay Triangulations ◽  
Möbius Transformations ◽  
Regular Polyhedra ◽  
Complex Design ◽  
Elliptic Geometry ◽  
Mobius Transformations

This paper shows a methodology for reducing the complex design process of space structures to an adequate selection of points lying on a plane. This procedure can be directly implemented in a bi-dimensional plane when we substitute (i) Euclidean geometry by bi-dimensional projection of the elliptic geometry and (ii) rotations/symmetries on the sphere by Möbius transformations on the plane. These graphs can be obtained by sites, specific points obtained by homological transformations in the inversive plane, following the analogous procedure defined previously in the three-dimensional space. From the sites, it is possible to obtain different partitions of the plane, namely, power diagrams, Voronoi diagrams, or Delaunay triangulations. The first would generate geo-tangent structures on the sphere; the second, panel structures; and the third, lattice structures.

Photochemisch initiierte Telomerisation von Äthylen mit Formamid / Photo-Initiated Telomerization of Ethylene with Formamide

1975 ◽  
Vol 30 (9-10) ◽  
pp. 740-747 ◽  
Author(s):  
H. P. Rath ◽  
A. Saus ◽  
B. Dederichs
Keyword(s):  
High Pressure ◽  
Reaction Time ◽  
Low Temperatures ◽  
Carbonic Acid ◽  
Uv Light ◽  
Electrical Power ◽  
Power Input ◽  
Elevated Pressure ◽  
Reaction Pressure ◽  
Geometrical Progression

Under the influence of UV light (high pressure HPK 125, Philips) and in presence of acetone ethylene and formamide react under elevated pressure (12.5-100 kp/cm2) to give odd-numbered n-alkane carbonic acid amides of chain lenghth C3-C19. Product yields improve with rising pressure. By increasing the concentration of aceton (0.1-8 mol-%), the reaction pressure the reaction time and at low temperatures the average mol.-weight is shifted to higher values. The chain distribution corresponds to a geometrical progression. Per 1 kwh electrical power input of the uv-equipment 0.5 kg of product is formed with chain length: 78% C3-C7, 12% C9, 10% C11-C19.

A Conformal Proof of a Jordan Curve Problem

1969 ◽  
Vol 12 (5) ◽  
pp. 673-674 ◽  
Author(s):  
G. Spoar ◽  
N.D. Lane
Keyword(s):  
Jordan Curve ◽  
Connected Region ◽  
Inversive Plane ◽  
Simply Connected Region ◽  
Simply Connected

The following theorem appears in [1].Let R be a closed simply connected region of the inversive plane bounded by a Jordan curve J, and let J be divided into three closed arcs A1, A2, A3. Then there exists a circle contained in R and having points in common with all three arcs.

scholarly journals On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. In a letter to Francis Baily, Esq. F. R. S. &c. By Benjamin Gompertz, Esq. F. R. S

1833 ◽  
Vol 2 ◽  
pp. 252-253 ◽  
Keyword(s):  
Causes Of Death ◽  
Value Of Life ◽  
Arithmetical Progression ◽  
Short Intervals ◽  
Law Of Nature ◽  
General Law ◽  
Geometrical Progression ◽  
History Of ◽  
The Law ◽  
Life Annuities

This paper, which professes to be a continuation of former researches on the same subject printed in the Transactions of the Royal Society, is divided into two chapters. In the first the author considers the nature of the law of those numbers in tables of mortality, which express the amount of persons living at the end of ages in regular arithmetical progression. He remarks that for short intervals the law approaches nearly to a decreasing geometrical progression, and that this must be the case whatever be the strict expression for the law of mortality, provided the intervals do not exceed certain limits. But he further remarks, that this property will be found to belong to very extensive portions of tables of mortality, and instances Deparcieux’s tables, where from the age of 25 to that of 45, the numbers living at the end of each year decrease very nearly in geometrical progression. Considering however the whole extent of such a table, it will be found that the ratio of this geometrical progression is not the same in all parts of the table. But before he enters on this consideration, the author draws some consequences from the hypothesis of a geometrical progression being the strict law of nature after a certain age. One of these is the equality of value of all life annuities commencing after that age. Another is, that the want of instances in history of persons living to very enormous ages (waving those of the patriarchs,) is no proof that such may not be the law of nature, as he shows by calculation, that out of 3,000,000 persons of 92, not more than one should on this supposition reach 192. This leads him to some general considerations on the causes of death, after which he resumes the consideration of the general law of the tables.

scholarly journals VI. On the exact form of waves near the surface of deep water

1863 ◽  
Vol 153 ◽  
pp. 127-138 ◽  
Keyword(s):  
Angular Velocity ◽  
Deep Water ◽  
Exact Form ◽  
Geometrical Progression

(1.) The investigations of the Astronomer Royal and of some other mathematicians on straight-crested parallel waves in a liquid, are based on the supposition that the displacements of the particles of the liquid are small compared with the length of a wave. Hence it has been very generally inferred that the results of those investigations are approximate only, when applied to waves in which the displacements, as compared with the length of a wave, are considerable. (2.) In the present paper I propose to prove that one of those results (viz., that in very deep water the particles move with a uniform angular velocity in vertical circles whose radii diminish in geometrical progression with increased depth, and consequently that surfaces of equal pressure, including the upper surface, are trochoidal) is exact for all displacements, how great soever.

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