2-universal positive definite integral quinary quadratic forms

In this paper we prove that every positive definite n-ary integral quadratic form with 12 < n < 13 (respectively 14 ≦ n ≤ 20) that can be represented by a sum of squares of integral linear forms is represented by a sum of 2 · 3n + n + 6 (respectively 3 · 4n + n + 3) squares. We also prove that every positive definite 7-ary integral quadratic form that can be represented by a sum of squares is represented by a sum of 25 squares.

For each positive integer n n , let g Z ( n ) g_\mathbb {Z}(n) be the smallest integer such that if an integral quadratic form in n n variables can be written as a sum of squares of integral linear forms, then it can be written as a sum of g Z ( n ) g_\mathbb {Z}(n) squares of integral linear forms. We show that every positive definite integral quadratic form is equivalent to what we call a balanced Hermite–Korkin–Zolotarev-reduced form and use it to show that the growth of g Z ( n ) g_\mathbb {Z}(n) is at most an exponential of n \sqrt {n} . Our result improves the best known upper bound on g Z ( n ) g_\mathbb {Z}(n) which is on the order of an exponential of n n . We also define an analogous number g O ∗ ( n ) g_{\mathcal O}^*(n) for writing Hermitian forms over the ring of integers O \mathcal O of an imaginary quadratic field as sums of norms of integral linear forms, and when the class number of the imaginary quadratic field is 1 1 , we show that the growth of g O ∗ ( n ) g_{\mathcal O}^*(n) is at most an exponential of n \sqrt {n} . We also improve on results of both Conway and Sloane and Kim and Oh on s s -integrable lattices.

In this article, we establish the general asymptotic behaviour of the Fourier coefficients of the Hecke eigenforms supported at integers represented by a primitive integral positive-definite binary quadratic form of fixed discriminant $D < 0$ under the assumption that the class number $h(D)=1$. As a consequence, we also obtain a quantitative result for the number of sign changes of the sequence of normalised Fourier coefficients $\lambda_{f}(n)$ of Hecke eigenforms $f$ over the indices, represented by a primitive integral positive definite binary quadratic form of fixed discriminant $D < 0$ when class number $h(D)=1$; in the interval $(X,2X]$ for sufficiently large $X$. Moreover, under the assumption of the Lindel{\"o}f hypothesis, the above-said sequence has at least $X^{\frac{1}{2}-\epsilon}$ many sign changes.

Polynomial sums over automorphs of a positive definite binary quadratic form

Let q q be a power of an odd prime, and F q [ T ] \mathbb {F}_q[T] be the ring of polynomials over a finite field F q \mathbb {F}_q of q q elements. A quadratic form f f over F q [ T ] \mathbb {F}_q[T] is said to be regular if f f globally represents all polynomials that are represented by the genus of f f . In this paper, we study definite regular quadratic forms over F q [ T ] \mathbb {F}_q[T] . It is shown that for a fixed q q , there are only finitely many equivalence classes of regular definite primitive quadratic forms over F q [ T ] \mathbb {F}_q[T] , regardless of the number of variables. Characterizations of those which are universal are also given.

In this note, we give an elementary proof of the following classical fact. Any positive definite ternary quadratic form over the rational numbers fails to represent infinitely many positive integers. For any ternary quadratic form (positive definite or indefinite), our method constructs certain congruence classes whose elements, up to a square factor, are the only elements not represented over the rational numbers by that form. In the case of a positive definite ternary form, we show that these classes are non-empty. This shows that the minimum number of variables in a positive definite quadratic form representing all positive integers is four. Our proof is very elementary and only uses quadratic reciprocity of Gauss.

In this paper we generalize results of the third paper of this series. As a corollary we can show the following: Let Li (1 ≤ i ≤ n) be a positive definite quadratic form which is equivalent to one of Cartan matrices of Lie algebras of type An (n ≥ 2), Dn (n ≥ 4), E6, E7, E8 and assume that is positive definite quadratic forms and satisfies that rk Mt ≥ 2 and implies rk K or rk L = 1. Then we have n = m and Lt is equivalent to a constant multiple of Ms(i) for some permutation s. Therefore we get the uniqueness of decompositions with respect to tensor products in this case.

In this note we prove an elementary result concerning the distribution of the positive integers which are represented by a positive-definite integral binary quadratic form Theorem 1. Let f(X, Y) = aX ~ + b X Y + c y2 be a positive-definite integral binary quadratic form of discriminant - A (= b 2 - 4ac = mi , there exist integers x and y such that n < ax 2 q- bxy + cy 2 < n + 2ml/~A1/4n ~m + m i . P r 0 0 f. Replacing the form f by an equivalent form we may suppose that m~ = c. We define integers x and y by =F(4cn~l/2q I!4cn-Ax2)l/2-bx l x L\~) /' Y= 2c -+1.

Root systems and inflations of non-negative quasi-Cartan matrices

Refining the notion of regularity introduced by Dickson, an integral quadratic form is said to be spinor regular if it represents all integers represented by its spinor genus. Examples of positive definite primitive integral ternary quadratic forms which have this property are presented, and it is proved that there exist only finitely many equivalence classes containing such forms.

Classifications and representations are two main topics in the theory of quadratic forms. In this paper, we consider these topics of ternary quadratic forms. For a given squarefree integer N N , first we give the classification of positive definite ternary quadratic forms of level 4 N 4N explicitly. Second, we give explicit formulas of the weighted sum of representations over each class in every genus of ternary quadratic forms of level 4 N 4N , which are involved with modified Hurwitz class number. In the proof of the main results, we use the relations among ternary quadratic forms, quaternion algebras, and Jacobi forms. As a corollary, we get the formula for the class number of positive ternary quadratic forms of level 4 N 4N . As applications, we derive an explicit base of Eisenstein series space of modular forms of weight 3 / 2 3/2 and level 4 N 4N , and give new proofs of some interesting identities involving representation number of ternary quadratic forms.

We investigate the existence of a weak solution u ∈ W 1,p 0 (Ω) to the degenerate quasi-linear Dirichlet boundary value problem (P) -Δ p u = λ 1 |u| p-2 u + f(x) in Ω; u = 0 on ∂Ω. It is assumed that 1 < p < ∞, p ¬= 2, Ω is a bounded domain in R N , f ∈ L∞ (Q) is a given function, and the number λ 1 stands for the first (smallest) eigenvalue of the positive p-Laplacian -Ap, where Δ p u ≡ div(|⊇u| p-2 ⊇u). The eigenvalue λ 1 being simple, let φ 1 denote the eigenfunction associated with λ 1 . We show the existence of a solution for problem (P) when f satisfies the orthogonality condition ⊃ Ω fφ 1 dx = 0, in which case the set of solutions is bounded in C 1 (Ω) provided p ¬= 2 and f ≢ 0 in Ω. A key role in our proofs is played by a second-order Taylor formula in its integral form (near φ 1 ) which contains certain Gâteaux derivatives and a positive semidefinite quadratic form. This quadratic form compensates for lack of coercivity in the energy functional corresponding to problem (P). When combined with well-known regularity results, its positive semidefiniteness renders a priori estimates from which existence and boundedness (for p ¬= 2) follow.

In this paper the researchers are presenting an upper bound for the distribution function of quadratic forms in normal vector with mean zero and positive definite covariance matrix. They also will show that the new upper bound is more precise than the one introduced by Okamoto [4] and the one introduced by Siddiqui [5]. Theoretical Error bounds for both, the new and Okamoto upper bounds are derived in this paper. For larger number of terms in any given positive definite quadratic form, a rough and easier upper bound is suggested.

The positive definite homogeneous multivariate forms play an important role in the automatic control and medical imaging, and the definiteness of the forms can be identified by special structured tensors. In this paper, we first state the equivalence between the positive definite multivariate forms and the corresponding tensors and account for the links between the positive definite tensors with H-tensors. Then based on diagonal dominance, some iterative criterions are presented to test H-tensors. Furthermore, we establish new iterative schemes for testing the positive definite multivariate homogeneous forms. The efficiency and validity of new methods are illustrated by numerical examples.

SUMMARY This paper is concerned with the characterization of the variance covariance matrix for a normal family in which certain quadratic forms are independent and follow cχ 2 distributions. It is also proved that the necessary and sufficient condition for any quadratic or linear form to be distributed independently of a sum of a finite number of non-negative definite quadratic forms is that it is distributed independently of each one of them.