The resolution of monoidal Cremona transformations of three-dimensional space

1938 ◽  
Vol 34 (1) ◽  
pp. 22-26
Author(s):  
D. W. Babbage
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Suitable Choice ◽  
Cremona Transformation ◽  
Cremona Transformations ◽  
Image Position ◽  
Three Dimensional Space

A Cremona transformation Tn, n′ between two three-dimensional spaces is said to be monoidal if the surfaces of order n in one space which form the homaloidal system corresponding to the planes of the second space have a fixed (n − 1)-ple point O. If the surfaces of order n′ forming the homaloidal system in the second space have a fixed (n′ − 1)-ple point O′, the transformation is said to be bimonoidal. A particularly simple bimonoidal transformation is that which transforms lines through O into lines through O′, and planes through O into planes through O′. Such a transformation we shall call an M-transformation. Its equations can, by suitable choice of coordinates, be expressed in the formwhere φn−1(x, y, z, w) = 0, φn(x, y, z, w) = 0 are monoids with vertex (0, 0, 0, 1).

A Geometrical treatment of the Correspondence between Lines in Threefold Space and Points of a Quadric in Fivefold space

1925 ◽  
Vol 22 (5) ◽  
pp. 694-699 ◽  
Author(s):  
H. W. Turnbull
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Quadratic Relation ◽  
Four Dimensions ◽  
Homogeneous Coordinates ◽  
Plücker Coordinates ◽  
Straight Line ◽  
Set Up ◽  
Image Position ◽  
Three Dimensional Space

§ 1. The six Plücker coordinates of a straight line in three dimensional space satisfy an identical quadratic relationwhich immediately shows that a one-one correspondence may be set up between lines in three dimensional space, λ, and points on a quadric manifold of four dimensions in five dimensional space, S5. For these six numbers pij may be considered to be six homogeneous coordinates of such a point.

The Covering Of Space By Spheres

1956 ◽  
Vol 8 ◽  
pp. 293-304 ◽  
Author(s):  
E. S. Barnes
Keyword(s):  
Quadratic Form ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Positive Definite ◽  
Positive Definite Quadratic Form ◽  
Inhomogeneous Minimum ◽  
Image Position ◽  
Three Dimensional Space

Bambah (1) has recently determined the most economical covering of three dimensional space by equal spheres whose centres form a lattice, the density of this covering being1.1.As is well known, this problem may be interpreted in terms of the inhomogeneous minimum of a positive definite quadratic form.

3-D Graphing of XRF Matrix Correction Equations

1994 ◽  
Vol 38 ◽  
pp. 649-656
Author(s):  
Anthony J. Klimasara
Keyword(s):  
Dimensional Space ◽  
Correction Term ◽  
Three Dimensional ◽  
Calibration Plot ◽  
Two Dimensional ◽  
Matrix Correction ◽  
Image Position ◽  
Correction Equations ◽  
Calibration Plane ◽  
Three Dimensional Space

Abstract The Lachance-Traill, and Lucas-Tooth-Price matrix correction equations/functions for XRF determined concentrations can be graphically interpreted with the help of three dimensional graphics. Statistically derived Lachance-Traill and Lucas-Tooth-Price matrix correction equations can be represented as follows: 1 where: Ci -elemental concentration of element “i” Ij -X-Ray intensity representing element “i” Ai0 -regression intercept for element “i” Ai -regression coefficient Zj -correction term defined below 2 Ai0, Aj , and Zi together represent the results of a multi-dimensional contribution. li, Ci, and Zi can be represented in three dimensional Cartesian space by X, Y and Z. These three variables are connected by a matrix correction equation that can be graphed as the function Y = F(X, Z), which represents a plane in three dimensional space. It can be seen that each chemical element will deliver a different set of coefficients in the equation of a plane that is called here a calibration plane. The commonly known and used two dimensional calibration plot is a “shadow” of the three dimensional real calibration points. These real (not shadow) points reside on a regression calibration plane in this three dimensional space. Lachance-Traill and Lucas-Tooth-Price matrix correction equations introduce the additional dimension(s) to the two dimensional flat image of uncorrected data. Illustrative examples generated by three dimensional graphics will be presented.

XVIII.—The Equation of Conduction of Heat

1926 ◽  
Vol 45 (3) ◽  
pp. 230-244 ◽  
Author(s):  
Marion C. Gray
Keyword(s):  
Differential Equation ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Infinite Cylinder ◽  
Image Position ◽  
Three Dimensional Space

The differential equation of the conduction of heat in ordinary three-dimensional space is generally written in the formwhere v denotes the temperature of the medium at time t. For a medium in which the temperature varies only in one direction, e.g. an infinite cylinder with the temperature varying along the axis, the equation is

XVII.—Some Hyperspace Harmonic Analysis Problems introducing Extensions of Mathieu's Equation

1927 ◽  
Vol 46 ◽  
pp. 206-209 ◽  
Author(s):  
Pierre Humbert
Keyword(s):  
Harmonic Analysis ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Orthogonal System ◽  
Mathieu’S Equation ◽  
Image Position ◽  
Hyperbolic Cylinders ◽  
Three Dimensional Space

It is well known that two problems of harmonic analysis in ordinary three-dimensional space can be solved by Mathieu's functions, namely, (a) harmonic analysis for an orthogonal system of elliptic (or hyperbolic) cylinders,(b) harmonic analysis for a system of confocal paraboloïds,

Extension of the Spatial PDI Model to Three Dimensions

1997 ◽  
Vol 84 (1) ◽  
pp. 176-178
Author(s):  
Frank O'Brien
Keyword(s):  
Population Density ◽  
Dimensional Space ◽  
Finite Lattice ◽  
Three Dimensional ◽  
Upper Bounds ◽  
Three Dimensions ◽  
Lower And Upper Bounds ◽  
Density Index ◽  
Three Dimensional Space

The author's population density index ( PDI) model is extended to three-dimensional distributions. A derived formula is presented that allows for the calculation of the lower and upper bounds of density in three-dimensional space for any finite lattice.

A Peptoid with Extended Shape in Water

2019 ◽  
Author(s):  
Jumpei Morimoto ◽  
Yasuhiro Fukuda ◽  
Takumu Watanabe ◽  
Daisuke Kuroda ◽  
Kouhei Tsumoto ◽  
...  
Keyword(s):  
Spectroscopic Analysis ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Biomolecular Recognition ◽  
Biological Application ◽  
New Class ◽  
Proteolytic Resistance ◽  
Pharmacokinetic Properties ◽  
Optically Pure ◽  
Three Dimensional Space

<div> <div> <div> <p>“Peptoids” was proposed, over decades ago, as a term describing analogs of peptides that exhibit better physicochemical and pharmacokinetic properties than peptides. Oligo-(N-substituted glycines) (oligo-NSG) was previously proposed as a peptoid due to its high proteolytic resistance and membrane permeability. However, oligo-NSG is conformationally flexible and is difficult to achieve a defined shape in water. This conformational flexibility is severely limiting biological application of oligo-NSG. Here, we propose oligo-(N-substituted alanines) (oligo-NSA) as a new peptoid that forms a defined shape in water. A synthetic method established in this study enabled the first isolation and conformational study of optically pure oligo-NSA. Computational simulations, crystallographic studies and spectroscopic analysis demonstrated the well-defined extended shape of oligo-NSA realized by backbone steric effects. The new class of peptoid achieves the constrained conformation without any assistance of N-substituents and serves as an ideal scaffold for displaying functional groups in well-defined three-dimensional space, which leads to effective biomolecular recognition. </p> </div> </div> </div>

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