Quotients and Inverse Limits of Spaces of Orderings

1979 ◽  
Vol 31 (3) ◽  
pp. 604-616 ◽  
Author(s):  
Murray A. Marshall
Keyword(s):  
Quadratic Forms ◽  
Sums Of Squares ◽  
Inverse Limits ◽  
Witt Ring ◽  
Spaces Of Orderings ◽  
Image Position

A connection between the theory of quadratic forms defined over a given field F, and the space XF of all orderings of F is developed by A. Pfister in [12]. XF can be viewed as a set of characters acting on the group F×/ΣF×2, where ΣF×2 denotes the subgroup of F× consisting of sums of squares. Namely, each ordering P ∈ XF can be identified with the characterdefined byIt follows from Pfister's result that the Witt ring of F modulo its radical is completely determined by the pair (XF, F×/ΣF×2).

scholarly journals Abstract theory of semiorderings

2005 ◽  
Vol 72 (2) ◽  
pp. 225-250
Author(s):  
Thomas C. Craven ◽  
Tara L. Smith
Keyword(s):  
Quadratic Forms ◽  
Algebraic Theory ◽  
Covering Number ◽  
Finitely Generated ◽  
Abstract Theory ◽  
Witt Ring ◽  
Witt Rings ◽  
Covering Numbers ◽  
Spaces Of Orderings

Marshall's abstract theory of spaces of orderings is a powerful tool in the algebraic theory of quadratic forms. We develop an abstract theory for semiorderings, developing a notion of a space of semiorderings which is a prespace of orderings. It is shown how to construct all finitely generated spaces of semiorderings. The morphisms between such spaces are studied, generalising the extension of valuations for fields into this context. An important invariant for studying Witt rings is the covering number of a preordering. Covering numbers are defined for abstract preorderings and related to other invariants of the Witt ring.

Quadratic forms between spheres and the non-existence of sums of squares formulae

1986 ◽  
Vol 100 (3) ◽  
pp. 493-504 ◽  
Author(s):  
Paul Y. H. Yiu
Keyword(s):  
Quadratic Forms ◽  
Sums Of Squares ◽  
Bilinear Forms ◽  
Bilinear Maps ◽  
Bilinear Map ◽  
Image Position

Hurwitz [6] posed in 1898 the problem of determining, for given integers r and s, the least integer n, denoted by r s, for which there exists an [r, s, n] formula, namely a sums of squares formula of the typewhere are bilinear forms with real coefficients in and . Such an [r, s, n] formula is equivalent to a normed bilinear map satisfying . We shall, therefore, speak of sums of squares formulae and normed bilinear maps interchangeably.

scholarly journals Annihilating polynomials for quadratic forms

2001 ◽  
Vol 27 (7) ◽  
pp. 449-455 ◽  
Author(s):  
David W. Lewis
Keyword(s):  
Quadratic Forms ◽  
Witt Ring ◽  
Short Survey

This is a short survey of the main known results concerning annihilating polynomials for the Witt ring of quadratic forms over a field.

Ternary Quadratic Forms and Eta Quotients

2015 ◽  
Vol 58 (4) ◽  
pp. 858-868 ◽  
Author(s):  
Kenneth S. Williams
Keyword(s):  
Quadratic Forms ◽  
Fourier Coefficients ◽  
Arithmetic Progressions ◽  
Dedekind Eta Function ◽  
Ternary Quadratic Forms ◽  
Sum Formula ◽  
Image Position ◽  
Positive Integers

AbstractLet denote the Dedekind eta function. We use a recent productto- sum formula in conjunction with conditions for the non-representability of integers by certain ternary quadratic forms to give explicitly ten eta quotientssuch that the Fourier coefficients c(n) vanish for all positive integers n in each of infinitely many non-overlapping arithmetic progressions. For example, we show that if we have c(n) = 0 for all n in each of the arithmetic progressions

Decomposition of Witt Rings

1982 ◽  
Vol 34 (6) ◽  
pp. 1276-1302 ◽  
Author(s):  
Andrew B. Carson ◽  
Murray A. Marshall
Keyword(s):  
Quadratic Forms ◽  
Classification Problem ◽  
Bilinear Forms ◽  
Interesting Problem ◽  
Witt Ring ◽  
Witt Rings ◽  
Finite Case ◽  
Characteristic 2 ◽  
Definition Of

We take the definition of a Witt ring to be that given in [13], i.e., it is what is called a strongly representational Witt ring in [8]. The classical example is obtained by considering quadratic forms over a field of characteristic ≠ 2 [17], but Witt rings also arise in studying quadratic forms or symmetric bilinear forms over more general types of rings [5,7, 8, 9]. An interesting problem in the theory is that of classifying Witt rings in case the associated group G is finite. The reduced case, i.e., the case where the nilradical is trivial, is better understood. In particular, the above classification problem is completely solved in this case [4, 12, or 13, Corollary 6.25]. Thus, the emphasis here is on the non-reduced case. Although some of the results given here do not require |G| < ∞, they do require some finiteness assumption. Certainly, the main goal here is to understand the finite case, and in this sense this paper is a continuation of work started by the second author in [13, Chapter 5].

On inequalities for powers of linear operators and for quadratic forms

1981 ◽  
Vol 89 (1-2) ◽  
pp. 25-50 ◽  
Author(s):  
Vũ Qúôc Phóng
Keyword(s):  
Quadratic Forms ◽  
Linear Operators ◽  
Dissipative Operator ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Dense Domain ◽  
The Best Possible Constant ◽  
Bilinear Functional ◽  
Image Position ◽  
Necessary And Sufficient

SynopsisLetHbe a Hilbert space in which a symmetric operatorSwith a dense domainDsis given and letShave a finite deficiency index (r, s). This paper contains a necessary and sufficient condition for validity of the following inequalities of Kolmogorov typeand a method for calculating the best possible constantsCn,m(S).Moreover, let φ be a symmetric bilinear functional with a dense domainDφsuch thatDs⊂Dφand φ(f, g) = (Sf, g) for allf∈Ds,g∈Dφ. A necessary and sufficient condition for validity of the inequalityas well as a method for calculating the best possible constantKare obtained. Then an analogous approach is worked out in order to obtain the best possible additive inequalities of the formThe paper is concluded by establishing the best possible constants in the inequalitieswhereTis an arbitrary dissipative operator. The theorems are extensions of the results of Ju. I. Ljubič, W. N. Everitt, and T. Kato.

ANOVA Sums of Squares as Quadratic Forms

1992 ◽  
Vol 46 (4) ◽  
pp. 288 ◽  
Author(s):  
Sutaip L. C. Saw
Keyword(s):  
Quadratic Forms ◽  
Sums Of Squares

scholarly journals Pairs of quadratic forms modulo one

1993 ◽  
Vol 35 (1) ◽  
pp. 51-61
Author(s):  
R. C. Baker ◽  
J. Brüdern
Keyword(s):  
Natural Number ◽  
Quadratic Forms ◽  
Image Position ◽  
Real Quadratic

Let s be a natural number, s ≥ 2. We seek a positive number λ(s) with the following property:Let ε > 0. Let Q1(x1, …, xs), Q2(x1, …, xs) be real quadratic forms, then for N > C1(s, ε) we havefor some integers n1, …, ns,

Finite Spaces of Signatures

1989 ◽  
Vol 41 (5) ◽  
pp. 808-829 ◽  
Author(s):  
Victoria Powers
Keyword(s):  
Quadratic Forms ◽  
Local Rings ◽  
Character Group ◽  
Witt Rings ◽  
Ordered Field ◽  
Spaces Of Orderings ◽  
Ordered Fields ◽  
Skew Fields ◽  
Finite Spaces ◽  
Field Setting

Marshall's Spaces of Orderings are an abstract setting for the reduced theory of quadratic forms and Witt rings. A Space of Orderings consists of an abelian group of exponent 2 and a subset of the character group which satisfies certain axioms. The axioms are modeled on the case where the group is an ordered field modulo the sums of squares of the field and the subset of the character group is the set of orders on the field. There are other examples, arising from ordered semi-local rings [4, p. 321], ordered skew fields [2, p. 92], and planar ternary rings [3]. In [4], Marshall showed that a Space of Orderings in which the group is finite arises from an ordered field. In further papers Marshall used these abstract techniques to provide new, more elegant proofs of results known for ordered fields, and to prove theorems previously unknown in the field setting.

On the region defined by | xy | ≤ 1, x2 + y2 ≤ t

1953 ◽  
Vol 49 (1) ◽  
pp. 63-71 ◽  
Author(s):  
Kathleen Ollerenshaw
Keyword(s):  
Quadratic Forms ◽  
Positive Definite ◽  
Binary Quadratic Forms ◽  
Image Position

In this paper I find the critical lattices of the region Kt defined by the inequalities | xy | ≤ 1, x2 + y2 ≤ t (t > 0). I deduce in §9 the arithmetic minima of a pair of binary quadratic formswhere f1 is indefinite, f2 is positive definite, and f1, f2 are harmonically related, that is to sayand

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