scholarly journals Modular forms of degree n and representation by quadratic forms IV

1987 ◽  
Vol 107 ◽  
pp. 25-47
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Quadratic Form ◽  
Finite Index ◽  
Modular Forms ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Prime Divisors ◽  
Positive Definite Quadratic Form ◽  
Finite Set ◽  
Quadratic Lattice

Let M be a quadratic lattice with positive definite quadratic form over the ring of rational integers, M’ a submodule of finite index, S a finite set of primes containing all prime divisors of 2[M: M’] and such that Mp is unimodular for p ∉ S. In [2] we showed that there is a constant c such that for every lattice N with positive definite quadratic form and every collection (fp)p∊s of isometries fp: NP → MP there is an isometry f: N → M satisfyingf ≡ fp mod M′p for every p |[M: M],f(Np) is private in Mp for every p ∉ S,provided the minimum of N ≥ c and rank M ≥ 3 rank N + 3.

scholarly journals The minimum and the primitive representation of positive definite quadratic forms II

1996 ◽  
Vol 141 ◽  
pp. 1-27 ◽  
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Positive Definite Quadratic Form ◽  
Primitive Representation ◽  
Quadratic Lattices

We are concerned with representation of positive definite quadratic forms by a positive definite quadratic form. Let us consider the following assertion Am, n : Let M, N be positive definite quadratic lattices over Z with rank(M) = m and rank(N) = n respectively. We assume that the localization Mp is represented by Np for every prime p, that is there is an isometry from Mp to Np. Then there exists a constant c(N) dependent only on N so that M is represented by N if min(M) > c(N), where min(M) denotes the least positive number represented by M.

The use of Gaussian (exponential quadratic) wave functions in molecular problems - I. General formulae for the evaluation of integrals

1960 ◽  
Vol 258 (1294) ◽  
pp. 412-420 ◽  
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Electronic Energy ◽  
Positive Definite ◽  
Wave Functions ◽  
The Other ◽  
Variation Method ◽  
Positive Definite Quadratic Form ◽  
Correlated Electron ◽  
Energy Integrals

It is shown that for wave functions of the form ψ = ∑ k ⁡ C k exp ⁡ ( − Q k ) , , where Q k is any positive definite quadratic form in the Cartesian co-ordinates of n particles and C k a constant, all integrals required for the calculation of the electronic energy of molecules by the variation method can be readily evaluated. The potential energy integrals are reduced to quadratures and the other integrals are expressed in closed form. In the general case the quadratic forms Q k determine many-electron functions formed from correlated electron orbitals of ellipsoidal symmetry and with variable centres. The results are easily extended to wave functions of the form ψ = ∑ k ⁡ P k exp ⁡ ( − Q k ) , , where P k is a polynomial.

scholarly journals Fourier Optimization and Quadratic Forms

2021 ◽  
Author(s):  
Andrés Chirre ◽  
Oscar E Quesada-Herrera
Keyword(s):  
Quadratic Form ◽  
Fourier Analysis ◽  
Quadratic Forms ◽  
Error Term ◽  
Positive Definite ◽  
Partial Sums ◽  
Type Result ◽  
Positive Definite Quadratic Form ◽  
Short Intervals ◽  
Representations Of Integers

Abstract We prove several results about integers represented by positive definite quadratic forms, using a Fourier analysis approach. In particular, for an integer $\ell\ge 1$, we improve the error term in the partial sums of the number of representations of integers that are a multiple of $\ell$. This allows us to obtain unconditional Brun–Titchmarsh-type results in short intervals and a conditional Cramér-type result on the maximum gap between primes represented by a given positive definite quadratic form.

scholarly journals Tensor products of positive definite quadratic forms, VII

1984 ◽  
Vol 96 ◽  
pp. 133-137 ◽  
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Quadratic Form ◽  
Lie Algebras ◽  
Quadratic Forms ◽  
Tensor Products ◽  
Positive Definite ◽  
Positive Definite Quadratic Form ◽  
Constant Multiple ◽  
The Third ◽  
Cartan Matrices

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.

FINITENESS RESULTS FOR REGULAR DEFINITE TERNARY QUADRATIC FORMS OVER ${\mathbb Q}(\sqrt{5})$

2007 ◽  
Vol 03 (04) ◽  
pp. 541-556 ◽  
Author(s):  
WAI KIU CHAN ◽  
A. G. EARNEST ◽  
MARIA INES ICAZA ◽  
JI YOUNG KIM
Keyword(s):  
Quadratic Form ◽  
Number Field ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Integral Quadratic Form ◽  
Ring Of Integers ◽  
Ternary Quadratic Forms ◽  
Totally Positive ◽  
Finiteness Result

Let 𝔬 be the ring of integers in a number field. An integral quadratic form over 𝔬 is called regular if it represents all integers in 𝔬 that are represented by its genus. In [13,14] Watson proved that there are only finitely many inequivalent positive definite primitive integral regular ternary quadratic forms over ℤ. In this paper, we generalize Watson's result to totally positive regular ternary quadratic forms over [Formula: see text]. We also show that the same finiteness result holds for totally positive definite spinor regular ternary quadratic forms over [Formula: see text], and thus extends the corresponding finiteness results for spinor regular quadratic forms over ℤ obtained in [1,3].

scholarly journals New Estimates of the Singular Series Corresponding to Positive Quaternary Quadratic Forms

2006 ◽  
Vol 13 (4) ◽  
pp. 687-691
Author(s):  
Guram Gogishvili
Keyword(s):  
Quadratic Form ◽  
General Result ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Precise Estimate ◽  
Definite Integral ◽  
Singular Series ◽  
Prime Factors ◽  
Quaternary Quadratic Form ◽  
Best Estimates

Abstract Let 𝑚 ∈ ℕ, 𝑓 be a positive definite, integral, primitive, quaternary quadratic form of the determinant 𝑑 and let ρ(𝑓,𝑚) be the corresponding singular series. When studying the best estimates for ρ(𝑓,𝑚) with respect to 𝑑 and 𝑚 we proved in [Gogishvili, Trudy Tbiliss. Univ. 346: 72–77, 2004] that where 𝑏(𝑘) is the product of distinct prime factors of 16𝑘 if 𝑘 ≠ 1 and 𝑏(𝑘) = 3 if 𝑘 = 1. The present paper proves a more precise estimate where 𝑑 = 𝑑0𝑑1, if 𝑝 > 2; 𝑕(2) ⩾ –4. The last estimate for ρ(𝑓,𝑚) as a general result for quaternary quadratic forms of the above-mentioned type is unimprovable in a certain sense.

scholarly journals Note On Extreme Forms

1955 ◽  
Vol 7 ◽  
pp. 150-154 ◽  
Author(s):  
E. S. Barnes
Keyword(s):  
Quadratic Form ◽  
Small Variation ◽  
Positive Definite ◽  
Positive Definite Quadratic Form

Letƒ(x1, … ,xn) = Σaijxixjbe a positive definite quadratic form of determinantD= |aij|, and letMbe the minimum offfor integralx1, … ,xnnot all zero. The formƒis said to beextremeif the ratioMn/Ddoes not increase when the coefficients aijoffsuffer any sufficiently small variation.

II.—On Fitting Polynomials to Data with Weighted and Correlated Errors

1935 ◽  
Vol 54 ◽  
pp. 12-16 ◽  
Author(s):  
A. C. Aitken
Keyword(s):  
Quadratic Form ◽  
Least Squares ◽  
Positive Definite ◽  
Correlated Errors ◽  
Positive Definite Quadratic Form ◽  
Matrix Notation ◽  
Image Position

This paper concludes the study of fitting polynomials by Least Squares, treated in two previous papers. The problem being concerned with the minimum of a positive definite quadratic form, it makes for conciseness to use matrix notation. We shall therefore adopt the following conventions :—The n values of the variable x, of the data u0, u1, …, un−1, of certain polynomials qr(x) entering into the solution, and so on, will be regarded compositely as vectors. They will be imagined as having their components or elements disposed in column array, but when written in full will be written horizontally, to save space, enclosed by curled brackets. Row vectors, when written out in full, will be enclosed by square brackets. In the shorter notation we shall write, for example, u, x for column vectors, u′, x′ for the row vectors obtained by transposition. The vectors occurring in the problem will be the following:—

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