Abstract
In this article we deal with the special matrices used in the analysis of the Boolean quadratic forms. We define basic notation and terminology, like Boolean independence or Boolean ...cumulants, which are useful to formulate the matrix problem. We check for which matrices appropriately constructed quadratic forms are Boolean independent. We show that the size of the matrix matters in this problem. We present the methods useful to find matrices for which corresponding Boolean quadratic forms are or are not Boolean independent.
An integral quadratic lattice is called indefinite
k
-universal if it represents all integral quadratic lattices of rank
k
for a given positive integer
k
. For
k
≥
3
, we prove that the indefinite
k
...-universal property satisfies the local–global principle over number fields. For
k
=
2
, we show that a number field
F
admits an integral quadratic lattice which is locally 2-universal but not indefinite 2-universal if and only if the class number of
F
is even. Moreover, there are only finitely many classes of such lattices over
F
. For
k
=
1
, we prove that
F
admits a classic integral lattice which is locally classic 1-universal but not classic indefinite 1-universal if and only if
F
has a quadratic unramified extension where all dyadic primes of
F
split completely. In this case, there are infinitely many classes of such lattices over
F
. All quadratic fields with this property are determined.
For a positive integer m, a (positive definite integral) quadratic form is called primitively m-universal if it primitively represents all quadratic forms of rank m. It was proved in 9 that there are ...exactly 107 equivalence classes of primitively 1-universal quaternary quadratic forms. In this article, we prove that the minimal rank of primitively 2-universal quadratic forms is six, and there are exactly 201 equivalence classes of primitively 2-universal senary quadratic forms.
In this work we study a class of nonlocal quadratic forms given by Ej(u,v)=12∫RN∫RN(u(x)−u(y))(v(x)−v(y))j(x−y)dxdy,where j:RN→0,∞ is a measurable even function with min{1,|⋅|2}j∈L1(RN). Assuming ...merely j∉L1(RN), we show local compactness of the embedding Dj(RN)↪L2(RN), where Dj(RN) denotes the space of functions u∈L2(RN) with Ej(u,u)<∞. Using this local compactness, we establish an alternative which allows to distinguish vanishing and nonvanishing of bounded sequences in Dj(RN). As an application, we show the existence of maximizers for a class of integral functionals defined on the unit sphere in Dj(RN). Our main results extend to cylindrical unbounded sets of the type Ω=U×Rk, where U⊂RN−k is open and bounded. Finally, we note that a Poincaré inequality associated with Ej holds for unbounded domains of this type, thereby extending the corresponding result in Felsinger et al. (2015) for bounded domains.
Let
n
⩾ 2 be an integer. We give necessary and sufficient conditions for an integral quadratic form over dyadic local fields to be
n
-universal by using invariants from Beli’s theory of bases of norm ...generators. Also, we provide a minimal set for testing
n
-universal quadratic forms over dyadic local fields, as an analogue of Bhargava and Hanke’s 290-theorem (or Conway and Schneeberger’s 15-theorem) on universal quadratic forms with integer coefficients.
Let Fm denote the set of positive-definite primitive integral quadratic forms in m variables. Let f,g∈Fm. In this paper we introduce a new concept, namely that of g being derivable from f. This ...concept is based on a certain theta function identity being valid. A consequence of this concept is that if g is derivable from f then the representation number of g can be given in terms of that of f. Many examples are given, especially for diagonal ternary quadratic forms.
PDF
