We present recent results on the computation of quadratic function fields with high 3-rank. Using a generalization of a method of Belabas on cubic field tabulation and a theorem of Hasse, we compute quadratic function fields with 3-rank ≥ 1, of imaginary or unusual discriminant D, for a fixed |D| = q. We present numerical data for quadratic function fields over F5, F7, F11 and F13 with deg(D) ≤...

The quadratic discriminant function (QDF) derived from the multivariate Gaussian distribution is effective for classification in many pattern recognition tasks. In particular, a variant of QDF, called MQDF, has achieved great success and is widely recognized as the state-of-the-art method in character recognition. However, when the number of training samples is small, covariance estimation invo...

For many statistical pattern recognition methods, distributions of sample vectors are assumed to be normal, and the quadratic discriminant function derived from the probability density function of multivariate normal distribution is used for classification. However, the computational cost is O(n) for n-dimensional vectors. Moreover, if there are not enough training sample patterns, covariance m...

Proof. The discriminant of x + 1 is D = 256 = 2. We have x + 1 ≡ (x + 1) (mod 2). Let p be an odd prime (so p D), and suppose the irreducible factors of x + 1 have degrees n1, n2, . . . , nk. By Corollary 41, the Galois group of x + 1 contains an element with cycle structure (n1, n2, . . . , nk). Since the Galois group of x +1 over Q is the Klein 4-group, in which every element has order dividi...

Quadratic discriminant analysis is a common tool for classification, but estimation of the Gaussian parameters can be ill-posed. This paper contains theoretical and algorithmic contributions to Bayesian estimation for quadratic discriminant analysis. A distribution-based Bayesian classifier is derived using information geometry. Using a calculus of variations approach to define a functional Bre...

We show that the theorem of Ellenberg and Venkatesh on representation of integral quadratic forms by integral positive definite quadratic forms is valid under weaker conditions on the represented form. In the article [5] Ellenberg and Venkatesh prove that for any integral positive definite quadratic form f in n variables there is a constant C(f) such that integral quadratic forms g of square fr...

For handwriting recognition integrating segmentation and cluss$icution, the underlying classifier is desired to give both high uccuruc~ and resistance to outlil3-S. In a previous evaluation study, the modified quadratic discriminunt function (MQDF) proposed by Kimuru et al. was shown to be superior in out&r rejection but inferior in cluss$icution uccuruc~ us compared to neural cluss$ers. This p...

Fisher proposed a linear discriminant function (Fisher’s LDF). From 1971, we analysed electrocardiogram (ECG) data in order to develop the diagnostic logic between normal and abnormal symptoms by Fisher’s LDF and a quadratic discriminant function (QDF). Our four years research was inferior to the decision tree logic developed by the medical doctor. After this experience, we discriminated many d...