Newton’s method for solving variational inequalities is known to be locally quadratically convergent. By incorporating a line search strategy for the regularized gap function, Taji et al. (Mathematical Programming, 1993) have proposed a modification of a Newton’s method which is globally convergent and whose rate of convergence is quadratic. But the quadratic convergence has been shown only und...

In the late eighteenth century both Euler and Legendre noticed that n +n+41 is prime for n = 0, 1, 2 . . . 39, and remarked that there are few polynomials with such small degree and coefficients that give such a long string of consecutive prime values. Rabinowitsch, at the 1912 International Congress of Mathematicians [18], showed that n + n + A is prime for n = 0, 1, 2, . . . A − 2 if and only...

Pattern recognition tools: • Discriminant functions a set of linear/quadratic functions of the data; the functions are evaluated for each new observation, the observation is assigned to the class having the highest discriminant value • Gaussian mixture models trained by expectation maximization; find the class that maximizes the likelihood of the test observations (acoustic features of successi...

We determine the mean number of 3-torsion elements in the class groups of quadratic orders, where the quadratic orders are ordered by their absolute discriminants. Moreover, for a quadratic order O we distinguish between the two groups: Cl3(O), the group of ideal classes of order 3; and I3(O), the group of ideals of order 3. We determine the mean values of both |Cl3(O)| and |I3(O)|, as O ranges...

Abstract Compositional data, i.e. data including only relative information, need to be transformed prior to applying the standard discriminant analysis methods that are designed for the Euclidean space. Here it is investigated for linear, quadratic, and Fisher discriminant analysis, which of the transformations lead to invariance of the resulting discriminant rules. Moreover, it is shown that f...

It is a theorem of Kaplansky that a prime p ≡ 1 (mod 16) is representable by both or none of x2 + 32y2 and x2 + 64y2, whereas a prime p ≡ 9 (mod 16) is representable by exactly one of these binary quadratic forms. In this paper five similar theorems are proved. As an example, one theorem states that a prime p ≡ 1 (mod 20) is representable by both or none of x2 + 20y2 and x2 + 100y2, whereas a p...

Stimulus arrays are inevitably presented at different positions on the retina in visual tasks, even those that nominally require fixation. In particular, this applies to many perceptual learning tasks. We show that perceptual inference or discrimination in the face of positional variance has a structurally different quality from inference about fixed position stimuli, involving a particular, qu...

A new method for predicting internal coding exons in genomic DNA sequences has been developed. This method is based on a prediction algorithm that uses the quadratic discriminant function for multivariate statistical pattern recognition. Substantial improvements have been made (with only 9 discriminant variables) when compared with existing methods: HEXON [Solovyev, V. V., Salamov, A. A. & Lawr...

This article is a study of congruence conditions, modulo powers of two, on class number of real quadratic number fields Q(vu), for which d has at most thtee distinct prime divisors. Techniques used are those associated with Gaussian composition of binary quadratic forms. 1. Let hid) denote the class number of the quadratic field Qi\ß) and let h id) denote the number of classes of primitive bina...