scholarly journals On spinor exceptional representations

1982 ◽  
Vol 87 ◽  
pp. 247-260 ◽  
Author(s):  
J. W. Benham ◽  
J. S. Hsia
Keyword(s):  
Quadratic Form ◽  
Real Numbers ◽  
The Real ◽  
Integer Coefficients ◽  
Spinor Genus ◽  
Exceptional Representations

Let f(x1 …, xm) be a quadratic form with integer coefficients and c ∈ Z. If f(x) = c has a solution over the real numbers and if f(x) ≡ c (mod N) is soluble for every modulus N, then at least some form h in the genus of f represents c. If m ≧ 4 one may further conclude that h belongs to the spinor genus of f. This does not hold when m = 3.

On manifolds containing a submanifold whose complement is contractible

1961 ◽  
Vol 57 (3) ◽  
pp. 507-515
Author(s):  
G. M. Kelly
Keyword(s):  
Quadratic Form ◽  
Critical Point ◽  
Unit Sphere ◽  
Real Function ◽  
Critical Point Theory ◽  
Compact Manifold ◽  
Point Theory ◽  
Critical Levels ◽  
Real Numbers ◽  
The Real

The problem discussed here arose in the course of some reflections on the critical point theory of Lusternik and Schnirelmann (4). In (4) it is shown how it is possible to associate, with a suitably differentifiable real-valued function f defined on a compact manifold M, a set of real numbers λ1 ≤ λ2 ≤ … λc, which are critical levels of f and which in certain respects are analogous to, and indeed generalizations of, the eigenvalues of a quadratic form. The number c depends on M and is called the category of M. If Rn is Euclidean n-space, Sn the unit sphere of Rn+1, and Pn the real projective n-space obtained from Sn by identifying opposite points, then a quadratic form φ in the (n + 1) coordinates of Rn+1 defines a real function on Sn and, by passage to the quotient, on Pn. Pn has category n + 1, and the numbers λ in this case are just the eigenvalues of the quadratic form.

Quadratic forms over Quadratic Extensions of Fields with Two Quaternion Algebras

1979 ◽  
Vol 31 (5) ◽  
pp. 1047-1058 ◽  
Author(s):  
Craig M. Cordes ◽  
John R. Ramsey
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Square Root ◽  
Real Numbers ◽  
Quaternion Algebras ◽  
The Real ◽  
Quadratic Extensions

In this paper, we analyze what happens with respect to quadratic forms when a square root is adjoined to a field F which has exactly two quaternion algebras. There are many such fields—the real numbers and finite extensions of the p-adic numbers being two familiar examples. For general quadratic extensions, there are many unanswered questions concerning the quadratic form structure, but for these special fields we can clear up most of them.It is assumed char F ≠ 2 and K = F (√a) where a ∊ Ḟ – Ḟ2. Ḟ denotes the non-zero elements of F. Generally the letters a, b, c, … and α, β, … refer to elements from Ḟ and x, y, z, … come from .

Shuffling of Linear Orders

1995 ◽  
Vol 38 (2) ◽  
pp. 223-229
Author(s):  
John Lindsay Orr
Keyword(s):  
Ordered Set ◽  
Linear Orders ◽  
Real Numbers ◽  
The Real ◽  
Linearly Ordered Set

AbstractA linearly ordered set A is said to shuffle into another linearly ordered set B if there is an order preserving surjection A —> B such that the preimage of each member of a cofinite subset of B has an arbitrary pre-defined finite cardinality. We show that every countable linearly ordered set shuffles into itself. This leads to consequences on transformations of subsets of the real numbers by order preserving maps.

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).

scholarly journals A question of van den Dries and a theorem of Lipshitz and Robinson; Not everything is standard

2007 ◽  
Vol 72 (1) ◽  
pp. 119-122 ◽  
Author(s):  
Ehud Hrushovski ◽  
Ya'acov Peterzil
Keyword(s):  
Real Numbers ◽  
The Real ◽  
First Order ◽  
Real Closed Fields ◽  
Minimal Structure ◽  
Closed Fields ◽  
New Construction ◽  
The Relationship

AbstractWe use a new construction of an o-minimal structure, due to Lipshitz and Robinson, to answer a question of van den Dries regarding the relationship between arbitrary o-minimal expansions of real closed fields and structures over the real numbers. We write a first order sentence which is true in the Lipshitz-Robinson structure but fails in any possible interpretation over the field of real numbers.

On expressible sets and p-adic numbers

2011 ◽  
Vol 54 (2) ◽  
pp. 411-422
Author(s):  
Jaroslav Hančl ◽  
Radhakrishnan Nair ◽  
Simona Pulcerova ◽  
Jan Šustek
Keyword(s):  
Haar Measure ◽  
Real Numbers ◽  
The Real ◽  
Expressible Set

AbstractContinuing earlier studies over the real numbers, we study the expressible set of a sequence A = (an)n≥1 of p-adic numbers, which we define to be the set EpA = {∑n≥1ancn: cn ∈ ℕ}. We show that in certain circumstances we can calculate the Haar measure of EpA exactly. It turns out that our results extend to sequences of matrices with p-adic entries, so this is the setting in which we work.

Models of the Real Numbers

2020 ◽  
pp. 203-221
Author(s):  
Lorenz Halbeisen ◽  
Regula Krapf
Keyword(s):  
Real Numbers ◽  
The Real

The Real Numbers

2020 ◽  
pp. 29-54
Author(s):  
Daniel W. Cunningham
Keyword(s):  
Real Numbers ◽  
The Real
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