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.

II.—On Fitting Polynomials to Data with Weighted and Correlated Errors

1935 ◽  
Vol 54 ◽  
pp. 12-16 ◽  
Author(s):  
A. C. Aitken
Keyword(s):  
Quadratic Form ◽  
Least Squares ◽  
Positive Definite ◽  
Correlated Errors ◽  
Positive Definite Quadratic Form ◽  
Matrix Notation ◽  
Image Position

This paper concludes the study of fitting polynomials by Least Squares, treated in two previous papers. The problem being concerned with the minimum of a positive definite quadratic form, it makes for conciseness to use matrix notation. We shall therefore adopt the following conventions :—The n values of the variable x, of the data u0, u1, …, un−1, of certain polynomials qr(x) entering into the solution, and so on, will be regarded compositely as vectors. They will be imagined as having their components or elements disposed in column array, but when written in full will be written horizontally, to save space, enclosed by curled brackets. Row vectors, when written out in full, will be enclosed by square brackets. In the shorter notation we shall write, for example, u, x for column vectors, u′, x′ for the row vectors obtained by transposition. The vectors occurring in the problem will be the following:—

scholarly journals Some extreme forms defined in terms of Abelian groups

1959 ◽  
Vol 1 (1) ◽  
pp. 47-63 ◽  
Author(s):  
E. S. Barnes ◽  
G. E. Wall
Keyword(s):  
Quadratic Form ◽  
Abelian Groups ◽  
Local Maximum ◽  
Positive Definite ◽  
Absolute Maximum ◽  
Positive Definite Quadratic Form ◽  
Image Position

Let be a positive definite quadratic form of determinant D, and let M be the minimum of f(x) for integral x ≠ 0. Then we set and the maximum being over all positive forms f in n variables. f is said to be extreme if y γn(f) is a local maximum for varying f, absolutely extreme if y γ(f) is an absolute maximum, i.e. if y γ(f) = γn.

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.

scholarly journals A lemma about the Epstein zeta-function

1964 ◽  
Vol 6 (4) ◽  
pp. 198-201 ◽  
Author(s):  
Veikko Ennola
Keyword(s):  
Quadratic Form ◽  
Zeta Function ◽  
Positive Definite ◽  
Simple Pole ◽  
Positive Definite Quadratic Form ◽  
Epstein Zeta Function ◽  
Image Position

Let h (m, n) = αm2 + 2δmn + βn2 be a positive definite quadratic form with determinant αβ–δ2 = 1. It may be put in the shapewith y > 0. We write (for s > 1)The function Zn(s) may be analytically continued over the whole s-plane. Its only singularity is a simple pole with residue π at s = 1.

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).

scholarly journals On the values of the Epstein zeta function

1973 ◽  
Vol 14 (1) ◽  
pp. 1-12 ◽  
Author(s):  
John Roderick Smart
Keyword(s):  
Quadratic Form ◽  
Zeta Function ◽  
Bernoulli Number ◽  
Positive Definite ◽  
Positive Definite Quadratic Form ◽  
The Riemann Zeta Function ◽  
Epstein Zeta Function ◽  
Upper Half Plane ◽  
Riemann Zeta ◽  
Image Position

Let ζ(s) = σn-s(Res >1) denote the Riemann zeta function; then, as is well known,, whereBmdenotes themth Bernoulli number, In this paper we investigate the possibility of similar evaluations of the Epstein zeta function ζq(s) at the rational integerss = k> 2. Letbe a positive definite quadratic form andwhere the summation is over all pairs of integers except (0, 0). In attempting to evaluate ζq(k) we are guided by Kronecker's first limit formula [11]where γ is Euler's constant,is the Dedekind eta-function, and τ is the complex number in the upper half plane, ℋ, associated with Q by the formulaOn the basis of (1.3) we would expect a formula involving functions of τ. This formula is stated in Theorem 1, (2.13).

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,

An Extreme Duodenary Form

1953 ◽  
Vol 5 ◽  
pp. 384-392 ◽  
Author(s):  
H. S. M. Coxeter ◽  
J. A. Todd
Keyword(s):  
Quadratic Form ◽  
Diophantine Equation ◽  
Positive Definite ◽  
Positive Definite Quadratic Form ◽  
Number Of Solutions ◽  
Minimum Value ◽  
Image Position

Let f(x1, … , xn) be a positive definite quadratic form of determinant Δ; let M be its minimum value for integers x1, … , xn not all zero; and let 2s be the number of times this minimum is attained, i.e., the number of solutions of the Diophantine equation

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