Irregular behaviour of class numbers and Euler-Kronecker constants of cyclotomic fields: the log log log devil at play

Kummer (1851) and, many years later, Ihara (2005) both posed conjectures on invariants related to the cyclotomic field $\mathbb Q(\zeta_q)$ with $q$ a prime. Kummer's conjecture concerns the asymptotic behaviour of the first factor of the class number of $\mathbb Q(\zeta_q)$ and Ihara's the positivity of the Euler-Kronecker constant of $\mathbb Q(\zeta_q)$ (the ratio of the constant and the residue of the Laurent series of the Dedekind zeta function $\zeta_{\mathbb Q(\zeta_q)}(s)$ at $s=1$). If certain standard conjectures in analytic number theory hold true, then one can show that both conjectures are true for a set of primes of natural density 1, but false in general. Responsible for this are irregularities in the distribution of the primes. With this survey we hope to convince the reader that the apparently dissimilar mathematical objects studied by Kummer and Ihara actually display a very similar behaviour.Comment: 20 pages, 1 figure, survey, to appear in `Irregularities in the Distribution of Prime Numbers - Research Inspired by Maier's Matrix Method', Eds. J. Pintz and M. Th. Rassia

Open Problems on Central Simple Algebras

We provide a survey of past research and a list of open problems regarding central simple algebras and the Brauer group over a field, intended both for experts and for beginners.Comment: v2 has some small revisions to the text. Some items are re-numbered, compared to v

Polyamide-Scorpion Cyclam Lexitropsins Selectively Bind AT-Rich DNA Independently of the Nature of the Coordinated Metal

Cyclam was attached to 1-, 2- and 3-pyrrole lexitropsins for the first time through a synthetically facile copper-catalyzed “click” reaction. The corresponding copper and zinc complexes were synthesized and characterized. The ligand and its complexes bound AT-rich DNA selectively over GC-rich DNA, and the thermodynamic profile of the binding was evaluated by isothermal titration calorimetry. The metal, encapsulated in a scorpion azamacrocyclic complex, did not affect the binding, which was dominated by the organic tail

Hybrid learning scheme for data mining applications

Classification of large datasets is a challenging task in Data Mining. In the current work, we propose a novel method that compresses the data and classifies the test data directly in its compressed form. The work forms a hybrid learning approach integrating the activities of data abstraction, frequent item generation, compression, classification and use of rough sets

Hybrid learning scheme for data mining applications

Classification of large datasets is a challenging task in Data Mining. In the current work, we propose a novel method that compresses the data and classifies the test data directly in its compressed form. The work forms a hybrid learning approach integrating the activities of data abstraction, frequent item generation, compression, classification and use of rough sets

Classification of run-length encoded binary data

In classification of binary featured data, distance computation is carried out by considering each feature. We represent the given binary data as run-length encoded data. This would lead to a compact or compressed representation of data. Further, we propose an algorithm to directly compute the Manhattan distance between two such binary encoded patterns.We show that classification of data in such compressed form would improve the computation time by a factor of 5 on large handwritten data. The scheme is useful in large data clustering and classification which depend on distance measures

Adaptive Boosting with Leader Based Learners for Classification of Large Handwritten Data

Boosting is a general method for improving the accuracy of a learning algorithm. AdaBoost, short form for adaptive boosting method, consists of repeated use of a weak or a base learning algorithm to find corresponding weak hypothesis by adapting to the error rates of the individual weak hypotheses. A large, complex handwritten data is under study. A repeated use of weak learner on the huge data results in large amount of processing time. In view of this, instead of using the entire training data for learning, we propose to use only prototypes. Further, in the current work, the base learner consists of a nearest neighbour classifier that employs prototypes generated using "leader" clustering algorithm. The leader algorithm is a single pass algorithm and is linear in terms of time as well as computation complexity. The prototype set alone is used as training data. In the process of developing an algorithm, domain knowledge of the Handwritten data, which is under study, is made use of. With the fusion of clustering, prototype selection, AdaBoost and Nearest Neighbour classifier, a very high classification accuracy, which is better than reported earlier on the considered data, is obtained in less number of iterations. The procedure integrates clustering outcome in terms of prototypes with boosting

Hybrid Learning Scheme for Data Mining Applications

Classification of large datasets is a challenging task in data mining. In the current work, we propose a novel method that compresses the data and classifies the test data directly in its compressed form. The work forms a hybrid learning approach integrating the activities of data abstraction, frequent item generation, compression, classification and use of rough sets