A Note on Quadratic Forms and the u-invariant

1974 ◽  
Vol 26 (5) ◽  
pp. 1242-1244 ◽  
Author(s):  
Roger Ware
Keyword(s):  
Finite Number ◽  
Quadratic Forms ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Real Field ◽  
Witt Ring ◽  
Necessary And Sufficient

The u-invariant of a field F, u = u(F), is defined to be the maximum of the dimensions of anisotropic quadratic forms over F. If F is a non-formally real field with a finite number q of square classes then it is known that u ≦ q. The purpose of this note is to give some necessary and sufficient conditions for equality in terms of the structure of the Witt ring of F.

On multilattice groups. II

1966 ◽  
Vol 62 (2) ◽  
pp. 149-164 ◽  
Author(s):  
D. B. Mcalister
Keyword(s):  
Finite Number ◽  
Sufficient Conditions ◽  
Positive Element ◽  
Necessary And Sufficient Conditions ◽  
Countable Number ◽  
Direct Sums ◽  
Lattice Group ◽  
Ordered Groups ◽  
Group A ◽  
Necessary And Sufficient

Conrad ((2)), has shown that any lattice group which obeys (C.F.) each strictly positive element exceeds at most a finite number of pairwise orthogonal elements may be constructed, from a family of simply ordered groups, by carrying out, alternately, the operations of forming finite direct sums and lexico extensions, at most a countable number of times. The main result of this paper, Theorem 3.1, gives necessary and sufficient conditions for a multilattice group, which obeys (ℋ*), to be isomorphic to a multilattice group which is constructed from a family of almost ordered groups, by carrying out, alternately, the operations of forming arbitrary direct sums and lexico extensions, any number of times; we call such a group a lexico sum of the almost ordered groups.

scholarly journals Pointwise stabilization of the Poisson integral for the diffusion type equations with inertia

2016 ◽  
Vol 8 (2) ◽  
pp. 279-283
Author(s):  
H.P. Malytska ◽  
I.V. Burtnyak
Keyword(s):  
Finite Number ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Poisson Integral ◽  
Diffusion Type ◽  
Necessary And Sufficient ◽  
Parabolic Degeneracy

In this paper we consider the pointwise stabilization of the Poisson integral for the diffusion type equations with inertia in the case of finite number of parabolic degeneracy groups. We establish necessary and sufficient conditions of this stabilization for a class of bounded measurable initial functions.

On the Preservation of Root Numbers and the Behavior of Weil Characters under Reciprocity Equivalence

1997 ◽  
Vol 40 (4) ◽  
pp. 402-415
Author(s):  
Jenna P. Carpenter
Keyword(s):  
Number Field ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Witt Ring ◽  
Root Numbers ◽  
Local Root ◽  
Necessary And Sufficient ◽  
Selection Of ◽  
Local Root Numbers

AbstractThis paper studies how the local root numbers and the Weil additive characters of the Witt ring of a number field behave under reciprocity equivalence. Given a reciprocity equivalence between two fields, at each place we define a local square class which vanishes if and only if the local root numbers are preserved. Thus this local square class serves as a local obstruction to the preservation of local root numbers. We establish a set of necessary and sufficient conditions for a selection of local square classes (one at each place) to represent a global square class. Then, given a reciprocity equivalence that has a finite wild set, we use these conditions to show that the local square classes combine to give a global square class which serves as a global obstruction to the preservation of all root numbers. Lastly, we use these results to study the behavior of Weil characters under reciprocity equivalence.

The Brascamp–Lieb Polyhedron

2010 ◽  
Vol 62 (4) ◽  
pp. 870-888 ◽  
Author(s):  
Stefán Ingi Valdimarsson
Keyword(s):  
Finite Number ◽  
Related Problem ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Maps ◽  
Rank 2 ◽  
Necessary And Sufficient ◽  
Concise Description

AbstractA set of necessary and sufficient conditions for the Brascamp–Lieb inequality to hold has recently been found by Bennett, Carbery, Christ, and Tao. We present an analysis of these conditions. This analysis allows us to give a concise description of the set where the inequality holds in the case where each of the linear maps involved has co-rank 1. This complements the result of Barthe concerning the case where the linear maps all have rank 1. Pushing our analysis further, we describe the case where the maps have either rank 1 or rank 2.A separate but related problem is to give a list of the finite number of conditions necessary and sufficient for the Brascamp–Lieb inequality to hold. We present an algorithm which generates such a list.

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