1 When patterns occur in the form of groups generated by the same source, distinctions between sources can be exploited to improve accuracy. We present a method for exploiting such ‘style’ consistency using a quadratic discriminant. We show that under reasonable assumptions on the feature and class distributions, the estimation of style parameters is simple and accurate.

We prove that the Shimizu L-function of a real quadratic field is obtained from a (Lorentzian) spectral triple on a noncommutative torus with real multiplication, as an adiabatic limit of the Dirac operator on a 3-dimensional solvmanifold. The Dirac operator on this 3dimensional geometry gives, via the Connes-Landi isospectral deformations, a spectral triple for the noncommutative tori obtained...

In this paper we describe an algorithm for computing the rank of an elliptic curve defined over a real quadratic field of class number one. This algorithm extends the one originally described by Birch and Swinnerton-Dyer for curves over Q. Several examples are included.

This paper presents despeckling and information extraction using non-quadratic regularization. The novelty of this paper is that instead of the Gaussian prior model a Gauss-Markov random field model is chosen, because it can efficiently model textures in the images. The iterative procedure consist of noise-free image and texture parameter. The experimental results show that the proposed method ...

We prove that the Shimizu L-function of a real quadratic field is obtained from a (Lorentzian) spectral triple on a noncommutative torus with real multiplication, as an adiabatic limit of the Dirac operator on a 3-dimensional solvmanifold. The Dirac operator on this 3-dimensional geometry gives, via the Connes–Landi isospectral deformations, a spectral triple for the noncommutative tori obtaine...

Let E be a real quadratic field with discriminant d 6≡ 0 (mod p) where p is an odd prime. For ρ = ±1 we determine ∏ 0<c<d, ( d c )=ρ ( p−1 bpc/dc ) modulo p2 in terms of a Lucas sequence, the fundamental unit and the class number of E.

Let E be an elliptic curve over Q attached to a newform f of weight two on Γ0(N). Let K be a real quadratic field, and let p||N be a prime of multiplicative reduction for E which is inert in K, so that the p-adic completion Kp of K is the quadratic unramified extension of Qp. Subject to the condition that all the primes dividing M := N/p are split in K, the article [Dar] proposes an analytic co...

Let p be an odd prime with p ⌘ 1 (mod 4) and " = (t + upp)/2 > 1 be the fundamental unit of the real quadratic field K = Q(pp) over the rationals. The Ankeny-Artin-Chowla conjecture asserts that p u, which still remains unsolved. In this paper, we investigate various kinds of congruences equivalent to its negation p | u by making use of Dirichlet’s class number formula, the products of quadrati...

This note is a detailed explanation of Shanks–Atkin NUCOMP— composition and reduction carried out “simultaneously”—for all quadratic fields, that is, including real quadratic fields. That explanation incidentally deals with various “exercises” left for confirmation by the reader in standard texts. Extensive testing in both the numerical and function field cases by Michael J Jacobson, Jr, report...

We present algorithms for the computation of extreme binary Humbert forms in real quadratic number fields. With these algorithms we are able to compute extreme Humbert forms for the number fields Q( √ 13) and Q( √ 17). Finally we compute the Hermite-Humbert constant for the number field Q( √ 13).