Elementary Fractal Geometry. 2. Carpets Involving Irrational Rotations

Self-similar sets with the open set condition, the linear objects of fractal geometry, have been considered mainly for crystallographic data. Here we introduce new symmetry classes in the plane, based on rotation by irrational angles. Examples without characteristic directions, with strong connectedness and small complexity, were found in a computer-assisted search. They are surprising since the rotations are given by rational matrices, and the proof of the open set condition usually requires integer data. We develop a classification of self-similar sets by symmetry class and algebraic numbers. Examples are given for various quadratic number fields.

ON STARK’S CLASS NUMBER CONJECTURE AND THE GENERALISED BRAUER–SIEGEL CONJECTURE

Stark conjectured that for any $h\in \Bbb {N}$ , there are only finitely many CM-fields with class number h. Let $\mathcal {C}$ be the class of number fields L for which L has an almost normal subfield K such that $L/K$ has solvable Galois closure. We prove Stark’s conjecture for $L\in \mathcal {C}$ of degree greater than or equal to 6. Moreover, we show that the generalised Brauer–Siegel conjecture is true for asymptotically good towers of number fields $L\in \mathcal {C}$ and asymptotically bad families of $L\in \mathcal {C}$ .

Machine Learning Based Scholarship and Credit Pre-Assessment System

The researcher explained the implementation process of finding the scholarship for the students by using machine learning supervised learning algorithm i.e. Naïve Bayes algorithm. Addition to this it includes a small description of naïve bayes classifier which used to be used through the authors. It explains the significance of training facts set and trying out information set in Machine mastering techniques. Machine learning nowadays becomes plenty used technique in the field of IT industry. It is a very effective instrument and technique for many quite a number fields such as education, IT and even in enterprise industry. In this paper, the researcher attempt to find computerized end result reputation of scholarships of college students by way of using naïve bayes classifier algorithm primarily based on the scholar educational performance, conversation skills, greedy power, IHS, income, time management, regularity etc. A scholarship offers a strength and self assurance to a student. It also boosts the performance of students indirectly. Usually scholarships are furnished by governments or authorities organizations. It is very essential for students to recognize their personal potentiality early in their educational profession so that they faster its growth, receiving attention from an employer or corporation helps college students take this step. Students can apply for scholarships primarily based on the eligibility criteria (such as caste category, annual income, etc). The scholarship will be issued based on merit, student performance and career specific. Different schemes of scholarships are provided for the students based on distinct eligibility criteria. By the use of a naïve bayes classifier, the researcher acquired a end result with accuracy of 96.7% and error of 3.3%. The repute of scholarship students was once displayed in the form of yes or no.

Solving Diophantine Equations by Factoring in Number Fields

In this text we provide an introduction to algebraic number theory and show its applications in solving certain difficult diophantine equations. We begin with a quick summary of the theory of quadratic residues, before diving into a select few areas of algebraic number theory. Our article is accompanied by a couple of worked problems and exercises for the reader to tackle on their own.

Campana points and powerful values of norm forms

We give an asymptotic formula for the number of weak Campana points of bounded height on a family of orbifolds associated to norm forms for Galois extensions of number fields. From this formula we derive an asymptotic for the number of elements with m-full norm over a given Galois extension of $$\mathbb {Q}$$ Q . We also provide an asymptotic for Campana points on these orbifolds which illustrates the vast difference between the two notions, and we compare this to the Manin-type conjecture of Pieropan, Smeets, Tanimoto and Várilly-Alvarado.

Galois covers of ℙ1 and number fields with large class groups

For each finite subgroup [Formula: see text] of [Formula: see text], and for each integer [Formula: see text] coprime to [Formula: see text], we construct explicitly infinitely many Galois extensions of [Formula: see text] with group [Formula: see text] and whose ideal class group has [Formula: see text]-rank at least [Formula: see text]. This gives new [Formula: see text]-rank records for class groups of number fields.

AN EFFECTIVE ANALYTIC FORMULA FOR THE NUMBER OF DISTINCT IRREDUCIBLE FACTORS OF A POLYNOMIAL

We obtain an effective analytic formula, with explicit constants, for the number of distinct irreducible factors of a polynomial $f \in \mathbb {Z}[x]$ . We use an explicit version of Mertens’ theorem for number fields to estimate a related sum over rational primes. For a given $f \in \mathbb {Z}[x]$ , our result yields a finite list of primes that certifies the number of distinct irreducible factors of f.