The principal topic of this article is to extend Shanks' infrastructure ideas in real quadratic number elds to the case of real quadratic congruence function elds. In this view, this paper is intended as a \low-brow" approach to the theory of ideals and operations in the ideal class group. We summarize some basic properties of ideals and provide elementary proofs of the main results. For the pu...

Variational theories of double-porosity models can be derived by homogenization of high-contrast periodic media (see [12]). Typically, we have one or more strong phases (i.e., uniformly elliptic energies on periodic connected domains) and a weak phase with a small ellipticity constant, coupled with some lower-order term. In the simplest case of quadratic energies, this amounts to considering en...

D.H. Lehmer found a quadratic polynomial such that 326 is a primitive root for the first 206 primes represented by this polynomial. It is shown that this is related to the class number one problem and prime producing quadratics. More impressive examples in the same spirit are given using recent results on prime producing quadratics. Y. Gallot holds the current record in which 206 is being repla...

In this paper, we discuss a spectral quadratic-SDP method for the iterative resolution of fixed-order H2 and H∞ design problems. These problems can be cast as regular SDP programs with additional nonlinear equality constraints. When the inequalities are absorbed into a Lagrangian function the problem reduces to solving a sequence of SDPs with quadratic objective function for which a spectral SD...

Following what is basically Kummer’s relatively neglected approach to non-unique factorization, we determine the structure of the irreducible factorizations of an element n in the ring of integers of a number field K. Consequently, we give a combinatorial expression for the number of irreducible factorizations of n in the ring. When K is quadratic, we show in certain cases how quadratic forms c...

Important progress made this year, both in theory and in experiment, helped solving the problem of 2σ–deviation from the unitarity of the first row elements in the CKM matrix. Today we have, |V us | 2 + |V ud | 2 + |V ub | 2 − 1 = −0.0008(13), −0.0010(13), or −0.0005(13), depending on whether the q 2-dependence of the relevant K ℓ3 form factor is considered as pole-like, linear or quadratic fun...

This paper investigates the relationship between instructional effectiveness of animated vs. static diagrams and levels of learner expertise in the task domain of transforming graphs of simple linear and quadratic functions. It was demonstrated on many occasions that instructional formats that are effective for low-knowledge learners could be ineffective, or even deleterious, for high-knowledge...

Abstract. We prove two related concentration inequalities concerning the number of rational points of hyperelliptic curves over subsets of a finite field. In particular, we investigate the probability of a large discrepancy between the numbers of quadratic residues and non-residues in the image of such subsets over uniformly random hyperelliptic curves of given degrees. We find a constant proba...

The University of Manitoba Sieve Unit is used to find several values of A (> 0) such that the quadratic polynomial x + x + A will have a large asymptotic density of prime values. The Hardy-Littlewood constants which characterize this density are also evaluated.

We give asymptotic formulas for the number of biquadratic extensions of Q that admit a quadratic extension which is a Galois extension of Q with a prescribed Galois group, for example, with a Galois group isomorphic to the quaternionic group. Our approach is based on a combination of the theory of quadratic equations with some analytic tools such as the Siegel–Walfisz theorem and the double osc...