A simple four-moment approximation to the distribution of a positive definite quadratic form, with applications to testing

2022 ◽  
Keyword(s):  
Quadratic Form ◽  
Positive Definite ◽  
Positive Definite Quadratic Form ◽  
Moment Approximation

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.

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 The minimum and the primitive representation of positive definite quadratic forms II

1996 ◽  
Vol 141 ◽  
pp. 1-27 ◽  
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Positive Definite Quadratic Form ◽  
Primitive Representation ◽  
Quadratic Lattices

We are concerned with representation of positive definite quadratic forms by a positive definite quadratic form. Let us consider the following assertion Am, n : Let M, N be positive definite quadratic lattices over Z with rank(M) = m and rank(N) = n respectively. We assume that the localization Mp is represented by Np for every prime p, that is there is an isometry from Mp to Np. Then there exists a constant c(N) dependent only on N so that M is represented by N if min(M) > c(N), where min(M) denotes the least positive number represented by M.

scholarly journals ACTION OF HECKE OPERATORS ON SIEGEL THETA SERIES I

2006 ◽  
Vol 02 (02) ◽  
pp. 169-186 ◽  
Author(s):  
LYNNE H. WALLING
Keyword(s):  
Quadratic Form ◽  
Linear Combination ◽  
Commutation Relation ◽  
Theta Series ◽  
Positive Definite ◽  
Hecke Operators ◽  
Elementary Functions ◽  
Positive Definite Quadratic Form

We apply the Hecke operators T(p) and [Formula: see text] to a degree n theta series attached to a rank 2k ℤ-lattice L, n ≤ k, equipped with a positive definite quadratic form in the case that L/pL is hyperbolic. We show that the image of the theta series under these Hecke operators can be realized as a sum of theta series attached to certain closely related lattices, thereby generalizing the Eichler Commutation Relation (similar to some work of Freitag and of Yoshida). We then show that the average theta series (averaging over isometry classes in a given genus) is an eigenform for these operators. We show the eigenvalue for T(p) is ∊(k - n, n), and the eigenvalue for T′j(p2) (a specific linear combination of T0(p2),…,Tj(p2)) is pj(k-n)+j(j-1)/2β(n,j)∊(k-j,j) where β(*,*), ∊(*,*) are elementary functions (defined below).

On Eutactic Forms

1977 ◽  
Vol 29 (5) ◽  
pp. 1040-1054 ◽  
Author(s):  
Avner Ash
Keyword(s):  
Quadratic Form ◽  
Symmetric Matrix ◽  
Positive Definite ◽  
Column Vector ◽  
Positive Definite Quadratic Form ◽  
Minimal Vectors

Let (aij) = A be a positive definite n × n symmetric matrix with real entries. To it corresponds a positive definite quadratic form ƒ on Rn: ƒ(x) = txAx = ∑ aijXiXj for x any column vector in Rn. The set of values ƒ(y) for y in Zn — {0} has a minimum m (A) > 0 and the number of “minimal vectors“ y1, … , yr in Zn for which ƒ(yi) = m (A) is finite. By definition, ƒ and A are called eutactic if and only if there are positive numbers s1 ,… , sr such that

scholarly journals A class of D-extreme Minkowski-reduced forms

1981 ◽  
Vol 31 (3) ◽  
pp. 269-275 ◽  
Author(s):  
D. W. Trenerry
Keyword(s):  
Quadratic Form ◽  
Local Minimum ◽  
Positive Definite ◽  
Positive Definite Quadratic Form ◽  
Extreme Form ◽  
Reduced Forms

AbstractBarnes (1978, 1979) introduced the concept of a -extreme form, which is a Minkowski-reduced positive definite quadratic form having prescribed diagonal coefficients α1, α2, …, αn and providing a local minimum of the determinant of the form over all such forms. Here a class of forms which are -extreme for all α and all n is described.

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.

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