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

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.

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 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.

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).

scholarly journals On a problem of Rankin about the Epstein zeta-function

1959 ◽  
Vol 4 (2) ◽  
pp. 73-80 ◽  
Author(s):  
J. W. S. Cassels
Keyword(s):  
Quadratic Form ◽  
Zeta Function ◽  
Special Form ◽  
Positive Definite ◽  
Positive Definite Quadratic Form ◽  
Epstein Zeta Function ◽  
Image Position

Letbe a positive definite quadratic form with determinant αβ−X2 = 1. A special form of this kind isWe consider the Epstein zeta-functionthe series converging for s > 1. For s ≥ 1·035 Rankin [1] proved the followingSTatement R.The sign of equality is needed only when h is equivalent to Q.

Quadratic diophantine equations

1960 ◽  
Vol 253 (1026) ◽  
pp. 227-254 ◽  
Keyword(s):  
Positive Integer ◽  
Quadratic Form ◽  
Diophantine Equation ◽  
Diophantine Equations ◽  
Positive Definite ◽  
Positive Definite Quadratic Form ◽  
Positive Integers

Tartakowsky (1929) proved that a positive definite quadratic form, with integral coefficients, in 5 or more variables represents all but at most finitely many of the positive integers not excluded by congruence considerations. Tartakowsky’s argument does not lead to any estimate for a positive integer which, though not so excluded, is not represented by the quadratic form. Here estimates for such an integer are obtained, in terms of the coefficients of the quadratic form. To simplify the argument and improve the estimates, the problem is slightly generalized (by considering a Diophantine equation with linear terms). A combination of analytical and arithmetical methods is needed.

On the minimum of a positive quadratic form in n ( ≤ 8) variables (verification of Blichfeldt's calculations)

1966 ◽  
Vol 62 (4) ◽  
pp. 719-719 ◽  
Author(s):  
G. L. Watson
Keyword(s):  
Quadratic Form ◽  
Positive Definite ◽  
Positive Definite Quadratic Form ◽  
Positive Quadratic Form ◽  
Image Position

Let f = f(x1, …, xn) be a positive definite quadratic form in n ( ≤ 8) variables; then we consider the old problem of estimating the minimum of f (its least value for integers xi not all 0) in terms of the determinant Δ(f). Normalizing by supposing the minimum to be 1, the known results may be stated aseach inequality being best possible. Further, for each n, all forms for which equality holds in (1) have integral coefficients and are equivalent to each other.

scholarly journals Minimal theta functions

1988 ◽  
Vol 30 (1) ◽  
pp. 75-85 ◽  
Author(s):  
Hugh L. Montgomery
Keyword(s):  
Functional Equation ◽  
Quadratic Form ◽  
Zeta Function ◽  
Theta Functions ◽  
Binary Quadratic Form ◽  
Positive Definite ◽  
Epstein Zeta Function ◽  
Image Position

Let be a positive definite binary quadratic form with real coefficients and discriminant b2 − 4ac = −1.Among such forms, let . The Epstein zeta function of f is denned to beRankin [7], Cassels [1], Ennola [5], and Diananda [4] between them proved that for every real s > 0,We prove a corresponding result for theta functions. For real α > 0, letThis function satisfies the functional equation(This may be proved by using the formula (4) below, and then twice applying the identity (8).)

scholarly journals Note On Extreme Forms

1955 ◽  
Vol 7 ◽  
pp. 150-154 ◽  
Author(s):  
E. S. Barnes
Keyword(s):  
Quadratic Form ◽  
Small Variation ◽  
Positive Definite ◽  
Positive Definite Quadratic Form

Letƒ(x1, … ,xn) = Σaijxixjbe a positive definite quadratic form of determinantD= |aij|, and letMbe the minimum offfor integralx1, … ,xnnot all zero. The formƒis said to beextremeif the ratioMn/Ddoes not increase when the coefficients aijoffsuffer any sufficiently small variation.

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