Class Number Two for Real Quadratic Fields of Richaud-Degert Type

2000 Mathematics Subject Classification: Primary: 11D09, 11A55, 11C08, 11R11, 11R29; Secondary: 11R65, 11S40; 11R09.This paper contains proofs of conjectures made in [16] on class number 2 and what this author has dubbed the Euler-Rabinowitsch polynomial for real quadratic fields. As well, we complete the list of Richaud-Degert types given in [16] and show how the behaviour of the Euler-Rabinowitsch polynomials and certain continued fraction expansions come into play in the complete determination of the class number 2 problem for such types. For some values the determination is unconditional, and for others, the wide Richaud-Degert types, the determination is conditional on the generalized Riemann hypothesis (GRH)

The rational canonical form of a matrix

ABSTRACT. The purpose of this paper is to provide an efficient algorithmic means of determining the rational canonical form of a matrix using computational symbolic algebraic manipulation packages, and is in fact the practical implementation of a classical mathematical method

On permutation polynomials over finite fields

A polynomial f over a finite field F is called a permutation polynomial if the mapping F→F defined by f is one-to-one. In this paper we consider the problem of characterizing permutation polynomials; that is, we seek conditions on the coefficients of a polynomial which are necessary and sufficient for it to represent a permutation. We also give some results bearing on a conjecture of Carlitz which says essentially that for any even integer m, the cardinality of finite fields admitting permutation polynomials of degree m is bounded

Mitigating Gender Bias in Machine Learning Data Sets

Artificial Intelligence has the capacity to amplify and perpetuate societal biases and presents profound ethical implications for society. Gender bias has been identified in the context of employment advertising and recruitment tools, due to their reliance on underlying language processing and recommendation algorithms. Attempts to address such issues have involved testing learned associations, integrating concepts of fairness to machine learning and performing more rigorous analysis of training data. Mitigating bias when algorithms are trained on textual data is particularly challenging given the complex way gender ideology is embedded in language. This paper proposes a framework for the identification of gender bias in training data for machine learning.The work draws upon gender theory and sociolinguistics to systematically indicate levels of bias in textual training data and associated neural word embedding models, thus highlighting pathways for both removing bias from training data and critically assessing its impact.Comment: 10 pages, 5 figures, 5 Tables, Presented as Bias2020 workshop (as part of the ECIR Conference) - http://bias.disim.univaq.i

On well-rounded ideal lattices - II

We study well-rounded lattices which come from ideals in quadratic number fields, generalizing some recent results of the first author with K. Petersen. In particular, we give a characterization of ideal well-rounded lattices in the plane and show that a positive proportion of real and imaginary quadratic number fields contains ideals giving rise to well-rounded lattices.Comment: 13 pages; to appear in the International Journal of Number Theor

Cryptography Adapted to the New European Area of Higher Education

9 pages, no figures.-- Contributed to: Computational Science – ICCS 2008: 8th International Conference, Kraków, Poland, June 23-25, 2008, Proceedings, Part II.A new experience for teaching Cryptography to engineering students is shown. The aim is to give them a better understanding of secure and cryptographic algorithms by using Maple software, in a graduate-level course. In this paper we discuss how to structure, define, and implement a web-based course as a part of the traditional classes, according to the convergence of the European Higher Education Project. The proposed course facilitates the use of new Information and Communication Technologies.This work has been supported by Ministerio de Industria, Turismo y Comercio (Spain) in collaboration with Telef¶onica I+D (Project SEGUR@) with reference CENIT-2007 2004.Peer reviewe

A solid-phase enzyme-linked assay for vitamin B 12

A new solid-phase enzyme-linked competitive binding assay for vitamin B 12 (cyanocobalamin) is described. The assay is based on the competition between analyte B 12 molecules and a glucose-6-phosphate dehydrogenase-vitamin B 12 conjugate for a limited number of R-protein binding sites immobilized on sepharose particles. After appropriate incubation and washing steps, the enzyme activity bound to the solid-phase is inversely related to the concentration of B 12 in the sample. Under optimized conditions, the method can detect B 12 in the range of 3×10 −10 −1×10 −8 M (using 100 μ l sample) with high selectivity over other biological molecules.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41626/1/604_2005_Article_BF01197285.pd