scholarly journals Positive values of inhomogeneous quinary quadratic forms of type (4,1)

1981 ◽  
Vol 31 (2) ◽  
pp. 175-188 ◽  
Author(s):  
R. J. Hans-Gill ◽  
Madhu Raka
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Real Numbers ◽  
Image Position

AbstractHere it is proved that if Q(x, y, z, t, u) is a real indefinite quinary quadratic form of type (4,1) and determinant D, then given any real numbers x0, y0, z0, t0, u0 there exist integers x, y, z, t, u such thatAll critical forms are also obtained.

The inhomogeneous minimum of quadratic forms of signature ± 1

1981 ◽  
Vol 89 (2) ◽  
pp. 225-235 ◽  
Author(s):  
Madhu Raka
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Real Numbers ◽  
Inhomogeneous Minimum ◽  
Indefinite Quadratic Form ◽  
Indefinite Quadratic ◽  
Image Position

Let Qr be a real indefinite quadratic form in r variables of determinant D ≠ 0 and of type (r1, r2), 0 < r1 < r, r = r1 + r2, S = r1 − r2 being the signature of Qr. It is known (e.g. Blaney (3)) that, given any real numbers c1, c2,…, cr, there exists a constant C depending only on r and s such that the inequalityhas a solution in integers x1, x2, …, xr.

Inhomogeneous minimum of indefinite quadratic forms in six variables: A conjecture of Watson

1983 ◽  
Vol 94 (1) ◽  
pp. 1-8 ◽  
Author(s):  
Madhu Raka
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Real Numbers ◽  
Famous Conjecture ◽  
Inhomogeneous Minimum ◽  
Indefinite Quadratic Form ◽  
History Of ◽  
Indefinite Quadratic ◽  
Image Position

The famous conjecture of Watson(11) on the minima of indefinite quadratic forms in n variables has been proved for n ≤ 5, n ≥ 21 and for signatures 0 and ± 1. For the details and history of the conjecture the reader is referred to the author's paper(8). In the succeeding paper (9), we prove Watson's conjecture for signature ± 2 and ± 3 and for all n. Thus only one case for n = 6 (i.e. forms of type (1, 5) or (5, 1)) remains to he proved which we do here; thereby completing the case n = 6. This result is also used in (9) for proving the conjecture for all quadratic forms of signature ± 4. More precisely, here we prove:Theorem 1. Let Q6(x1, …, x6) be a real indefinite quadratic form in six variables of determinant D ( < 0) and of type (5, 1) or (1, 5). Then given any real numbers ci, 1 ≤ i ≤ 6, there exist integers x1,…, x6such that

Positive values of inhomogeneous quaternary quadratic forms, I

1968 ◽  
Vol 8 (1) ◽  
pp. 87-101 ◽  
Author(s):  
Vishwa Chander Dumir
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Real Numbers ◽  
Indefinite Quadratic Form ◽  
Indefinite Quadratic ◽  
Image Position

Let Q(x1, …, xn) be an indefinite quadratic form in n-variables with real coefficients, determinant D ≠ 0 and signature (r, s), r+s = n. Then it is known (e.g. see Blaney [2]) that there exist constants Γr, s depending only on r and s such for any real numbers c1, …, cn we can find integers x1, …, xn satisfying

Inhomogeneous minimum of indefinite quaternary quadratic forms

1967 ◽  
Vol 63 (2) ◽  
pp. 277-290 ◽  
Author(s):  
Vishwa Chander Dumir
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Real Numbers ◽  
Inhomogeneous Minimum ◽  
Indefinite Quadratic Form ◽  
Indefinite Quadratic ◽  
Image Position

Let Q (x1, …, xn) be a real indefinite quadratic form in n-variables x1,…, xn with signature (r, s),r + s = n and determinant D ≠ 0. Then it is known (see Blaney (2)) that there exists constant Cr, s depending only on r, s such that given any real numbers c1, …,cn we can find integers x1, …, xn satisfying

scholarly journals An Inhomogeneous Minimum For Non-Convex Star-Regions With Hexagonal Symmetry

1955 ◽  
Vol 7 ◽  
pp. 337-346 ◽  
Author(s):  
R. P. Bambah ◽  
K. Rogers
Keyword(s):  
Quadratic Form ◽  
Binary Quadratic Form ◽  
Elementary Geometry ◽  
Hexagonal Symmetry ◽  
Real Numbers ◽  
Inhomogeneous Minimum ◽  
Image Position ◽  
Indefinite Binary Quadratic Form

1. Introduction. Several authors have proved theorems of the following type:Let x0, y0 be any real numbers. Then for certain functions f(x, y), there exist numbers x, y such that1.1 x ≡ x0, y ≡ y0 (mod 1),and1.2 .The first result of this type, but with replaced by min , was given by Barnes (3) for the case when the function is an indefinite binary quadratic form. A generalisation of this was proved by elementary geometry by K. Rogers (6).

On a conjecture of Watson

1983 ◽  
Vol 94 (1) ◽  
pp. 9-22 ◽  
Author(s):  
Madhu Raka
Keyword(s):  
Quadratic Form ◽  
Classical Result ◽  
Preceding Paper ◽  
Real Numbers ◽  
Linear Forms ◽  
Indefinite Quadratic Form ◽  
Indefinite Quadratic ◽  
Image Position

Let Qn be a real indefinite quadratic form in n variables x1, x2,…, xn, of determinant D ≠ 0 and of type (r, s), 0 < r < n, n = r + s. Let σ denote the signature of Qn so that σ = r − s. It is known (see e.g. Blaney(4)) that, given any real numbers c1 c2, …, cn, there exists a constant C depending upon n and σ only such that the inequalityhas a solution in integers x1, x2, …, xn. Let Cr, s denote the infimum of all such constants. Clearly Cr, s = Cs, r, so we need consider non-negative signatures only. For n = 2, C1, 1 = ¼ follows from a classical result of Minkowski on the product of two linear forms. When n = 3, Davenport (5) proved that C2, 1 = 27/100. For all n and σ = 0, Birch (3) proved that Cr, r = ¼. In 1962, Watson(18) determined the values of Cr, s for all n ≥ 21 and for all signatures σ. He proved thatWatson also conjectured that (1·2) holds for all n ≥ 4. Dumir(6) proved Watson's conjecture for n = 4. For n = 5, it was proved by Hans-Gill and Madhu Raka(7, 8). The author (12) has proved the conjecture for σ = 1 and all n. In the preceding paper (13) we proved that C5, 1 = 1. In this paper we prove Watson's conjecture for σ = 2, 3 and 4.

Asymmetric inequalities for non-homogeneous ternary quadratic forms

1967 ◽  
Vol 63 (2) ◽  
pp. 291-303 ◽  
Author(s):  
Vishwa Chander Dumir
Keyword(s):  
Quadratic Forms ◽  
Real Numbers ◽  
Linear Forms ◽  
Ternary Quadratic Forms ◽  
Image Position

A well-known theorem of Minkowski on the product of two linear forms states that ifare two linear forms with real coefficients and determinant Δ = |αδ − βγ| ≠ 0, then given any real numbers c1, c2 we can find integers x, y such that

scholarly journals Theorems Relating to Quadratic Forms and their Discriminant Matrices

1953 ◽  
Vol 10 (1) ◽  
pp. 13-15
Author(s):  
S. Vajda
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Degrees Of Freedom ◽  
Normal Population ◽  
Royal Statistical Society ◽  
Unit Variance ◽  
The Matrix ◽  
Chi Squared ◽  
Image Position ◽  
Latent Roots

In a paper read before the Research Branch of the Royal Statistical Society (Ref. 1, p. 150) the following case was considered:Let the expression be given; introduce, for c, a linear form in and obtainIf the yi are sample values from a normal population with unit variance, then it is known (Ref. 2) that (1) is distributed as where zi varies as chi-squared with one degree of freedom and the li are the latent roots of the matrix of the quadratic form. If these latent roots are f times unity and n—f times zero, then this reduces to a chi-squared distribution with f degrees of freedom.

Integral Bases for Quadratic Forms

1963 ◽  
Vol 15 ◽  
pp. 412-421 ◽  
Author(s):  
J. H. H. Chalk
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Integer Variables ◽  
Inhomogeneous Minimum ◽  
Integral Bases ◽  
Indefinite Quadratic Form ◽  
Homogeneous Minimum ◽  
Indefinite Quadratic ◽  
Image Position

Letbe an indefinite quadratic form in the integer variables x1, . . . , xn with real coefficients of determinant D = ||ars||(n) ≠ 0. The homogeneous minimum MH(Qn) and the inhomogeneous minimum MI(Qn) of Qn(x) are defined as follows :

scholarly journals Almost sure convergence of quadratic forms in random variables

1973 ◽  
Vol 15 (3) ◽  
pp. 257-258
Author(s):  
V. K. Rohatgi
Keyword(s):  
Quadratic Forms ◽  
Random Variables ◽  
Almost Sure Convergence ◽  
Real Numbers ◽  
Image Position

Let X1, X2, … be a sequence of random variables and let {ajk}, j,k = 1, 2, …, be a matrix of real numbers. Write We establish the following result.

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