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2.
Let K be a number field, and let W be a subspace of KN, N?1. Let V1,…,VM be subspaces of KN of dimension less than dimension of W. We prove the existence of a point of small height in , providing an explicit upper bound on the height of such a point in terms of heights of W and V1,…,VM. Our main tool is a counting estimate we prove for the number of points of a subspace of KN inside of an adelic cube. As corollaries to our main result we derive an explicit bound on the height of a nonvanishing point for a decomposable form and an effective subspace extension lemma.  相似文献   

3.
Let F(x1,…,xn) be a nonsingular indefinite quadratic form, n=3 or 4. For n=4, suppose the determinant of F is a square. Results are obtained on the number of solutions of
F(x1,…,xn)=0  相似文献   

4.
Let a, b, c, d be given nonnegative integers with a,d?1. Using Chebyshev?s inequalities for the function π(x) and some results concerning arithmetic progressions of prime numbers, we study the Diophantine equation
  相似文献   

5.
This paper studies the existence of the uniformly minimum risk unbiased (UMRU) estimators of parameters in a class of linear models with an error vector having multivariate normal distribution or t-distribution, which include the growth curve model, the extended growth curve model, the seemingly unrelated regression equations model, the variance components model, and so on. The necessary and sufficient existence conditions are established for UMRU estimators of the estimable linear functions of regression coefficients under convex losses and matrix losses, respectively. Under the (extended) growth curve model and the seemingly unrelated regression equations model with normality assumption, the conclusions given in the literature can be derived by applying the general results in this paper. For the variance components model, the necessary and sufficient existence conditions are reduced as terse forms.  相似文献   

6.
Let p>3 be a prime, u,v,dZ, gcd(u,v)=1, p?u2dv2 and , where is the Legendre symbol. In the paper we mainly determine the value of by expressing p in terms of appropriate binary quadratic forms. As applications, for we obtain a general criterion for and a criterion for εd to be a cubic residue of p, where εd is the fundamental unit of the quadratic field . We also give a general criterion for , where {Un} is the Lucas sequence defined by U0=0, U1=1 and Un+1=PUnQUn−1 (n?1). Furthermore, we establish a general result to illustrate the connections between cubic congruences and binary quadratic forms.  相似文献   

7.
Both a general and a diagonal u-invariant for forms of higher degree are defined, generalizing the u-invariant of quadratic forms. We give a survey of both old and new results on these u-invariants.  相似文献   

8.
For a normally distributed random matrix Y with mean zero and general covariance matrix ΣY and for a symmetric matrix W, necessary and sufficient conditions are derived for the Wishartness of YWY.  相似文献   

9.
For a class of multivariate skew normal distributions, the noncentral skew chi-square distribution is studied. The necessary and sufficient conditions under which a sequence of quadratic forms is generalized noncentral skew chi-square distributed random variables are obtained. Several examples are given to illustrate the results.  相似文献   

10.
Let M?5. For any odd prime power q and any prime ??q, we show that there are at least pairwise coprime DFq[T] which are square-free and of odd degree ?M, such that ? does not divide the class number of the complex quadratic functions fields .  相似文献   

11.
For a normal random matrix Y with mean zero, necessary and sufficient conditions are obtained for YWkY to be Wishart-Laplace distributed and {YWkY} to be independent, where each Wk is assumed to be symmetric rather than nonnegative definite.  相似文献   

12.
Notions of linear sufficiency and quadratic sufficiency are of interest to some authors. In this paper, the problem of nonnegative quadratic estimation for βHβ+hσ2 is discussed in a general linear model and its transformed model. The notion of quadratic sufficiency is considered in the sense of generality, and the corresponding necessary and sufficient conditions for the transformation to be quadratically sufficient are investigated. As a direct consequence, the result on (ordinary) quadratic sufficiency is obtained. In addition, we pose a practical problem and extend a special situation to the multivariate case. Moreover, a simulated example is conducted, and applications to a model with compound symmetric covariance matrix are given. Finally, we derive a remark which indicates that our main results could be extended further to the quasi-normal case.  相似文献   

13.
Let L, N and M be positive definite integral \({\mathbb{Z}}\) -lattices. In this paper, we show some relation between the weighted sum of representations of L and N by gen(M) and the weighted sum of extensions of \(\tilde M_{\tilde \sigma}\) in the gen(M σ) via N η when M is even and gcd(dL, dM) =  1. As a consequence of the particular case when M is even unimodular, we recapture the Böcherer formula (13) in (Böcherer, Maths Z 183:21–46, 1983) for the relation of the Fourier coefficients between Eisenstein series and Jacobi–Eisenstein series.  相似文献   

14.
Let F be a field of characteristic distinct from 2, L=F(d) a quadratic field extension. Let further f and g be quadratic forms over L considered as polynomials in n variables, Mf, Mg their matrices. We say that the pair (f,g) is a k-pair if there exist SGLn(L) such that all the entries of the k×k upper-left corner of the matrices SMfSt and SMgSt are in F. We give certain criteria to determine whether a given pair (f,g) is a k-pair. We consider the transfer corL(t)/F(t) determined by the F(t)-linear map s:L(t)F(t) with s(1)=0, s(d)=1, and prove that if dimcorL(t)/F(t)(f+tg)an2(n?k), then (f,g) is a [k+12]-pair. If, additionally, the form f+tg does not have a totally isotropic subspace of dimension p+1 over L(t), we show that (f,g) is a (k?2p)-pair. In particular, if the form f+tg is anisotropic, and dimcorL(t)/F(t)(f+tg)an2(n?k), then (f,g) is a k-pair.  相似文献   

15.
Let be a prime. Let a,bZ with p?a(a2+b2). In the paper we mainly determine by assuming p=c2+d2 or p=Ax2+2Bxy+Cy2 with ACB2=a2+b2. As an application we obtain simple criteria for εD to be a quadratic residue , where D>1 is a squarefree integer such that D is a quadratic residue of p, εD is the fundamental unit of the quadratic field with negative norm. We also establish the congruences for and obtain a general criterion for p|U(p−1)/4, where {Un} is the Lucas sequence defined by U0=0, U1=1 and Un+1=bUn+k2Un−1(n?1).  相似文献   

16.
We give two congruence properties of Hermitian modular forms of degree 2 over and . The one is a congruence criterion for Hermitian modular forms which is generalization of Sturm?s theorem. Another is the well-definedness of the p-adic weight for Hermitian modular forms.  相似文献   

17.
We prove Witt’s cancelation and extension theorems for Galois Ring valued quadratic forms. The proof is based on the properties of the invariant I, previously defined by the authors, that classifies, together with the type of the corresponding bilinear form (alternating or not), nonsingular Galois Ring valued quadratic forms. Our results extend the Witt’s theorem for mod four valued quadratic forms. On the other hand, the known relation between the invariant I and the Arf invariant of an ordinary quadratic form (if the associated nonsingular bilinear form is alternating) is extended to the nonalternating case by explaining the invariant I in terms of Clifford algebras.  相似文献   

18.
We establish character sum bounds of the form
  相似文献   

19.
The radical of a field consists of all nonzero elements that are represented by every binary quadratic form representing 1. Here, the radical is studied in relation to local–global principles, and further in its behavior under quadratic field extensions. In particular, an example of a quadratic field extension is constructed where the natural analogue to the square-class exact sequence for the radical fails to be exact. This disproves a conjecture of Kijima and Nishi.  相似文献   

20.
In this article weakly isotropic quadratic forms over a (formally) real field are studied. Conditions on the field are given which imply that every weakly isotropic form over that field has a weakly isotropic subform of small dimension. Fields over which every quadratic form can be decomposed into an orthogonal sum of a strongly anisotropic form and a torsion form are characterized in different ways.  相似文献   

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