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Yasushige Watase

Summary

We formalize in the Mizar system [3], [4] basic definitions of commutative ring theory such as prime spectrum, nilradical, Jacobson radical, local ring, and semi-local ring [5], [6], then formalize proofs of some related theorems along with the first chapter of [1].

The article introduces the so-called Zariski topology. The set of all prime ideals of a commutative ring A is called the prime spectrum of A denoted by Spectrum A. A new functor Spec generates Zariski topology to make Spectrum A a topological space. A different role is given to Spec as a map from a ring morphism of commutative rings to that of topological spaces by the following manner: for a ring homomorphism h : AB, we defined (Spec h) : Spec B → Spec A by (Spec h)(𝔭) = h −1(𝔭) where 𝔭 2 Spec B.

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Ida Aichinger and Gerhard Larcher

Abstract

We study sets of bounded remainder for the billiard on the unit square. In particular, we note that every convex set S whose boundary is twice continuously differentiable with positive curvature at every point, is a bounded remainder set for almost all starting angles a and every starting point x. We show that this assertion for a large class of sets does not hold for all irrational starting angles α.

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Gamaliel Cerda-Morales

Abstract

In this paper, we give quadratic approximation of generalized Tribonacci sequence {Vn}n ≥0 defined by Vn = rVn−1 + sV n−2 + tV n−3 (n ≥ 3) and use this result to give the matrix form of the n-th power of a companion matrix of {Vn}n ≥0. Then we re-prove the cubic identity or Cassini-type formula for {Vn} n ≥0 and the Binet’s formula of the generalized Tribonacci quaternions.

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Roland Coghetto

Summary

In this article, using the Mizar system [3], [4], we define a structure [1], [6] in order to build a Pythagorean pentatonic scale and a Pythagorean heptatonic scale1 [5], [7].

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Joseph Rosenblatt and Mrinal Kanti Roychowdhury

Abstract

Quantization for a probability distribution refers to the idea of estimating a given probability by a discrete probability supported by a finite number of points. In this paper, firstly a general approach to this process is outlined using independent random variables and ergodic maps; these give asymptotically the optimal sets of n-means and the nth quantization errors for all positive integers n. Secondly two piecewise uniform distributions are considered on R: one with infinite number of pieces and one with finite number of pieces. For these two probability measures, we describe the optimal sets of n-means and the nth quantization errors for all n ∈ N. It is seen that for a uniform distribution with infinite number of pieces to determine the optimal sets of n-means for n ≥ 2 one needs to know an optimal set of (n − 1)-means, but for a uniform distribution with finite number of pieces one can directly determine the optimal sets of n-means and the nth quantization errors for all n ∈ N.

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Christian Weiss

Abstract

This paper addresses the question whether the LS-sequences constructed in [Car12] yield indeed a new family of low-discrepancy sequences. While it is well known that the case S = 0 corresponds to van der Corput sequences, we prove here that the case S = 1 can be traced back to symmetrized Kronecker sequences and moreover that for S ≥ 2 none of these two types occurs anymore. In addition, our approach allows for an improved discrepancy bound for S = 1 and L arbitrary.

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Lisa Kaltenböck and Wolfgang Stockinger

Abstract

The goal of this overview article is to give a tangible presentation of the breakthrough works in discrepancy theory [3, 5] by M. B. Levin. These works provide proofs for the exact lower discrepancy bounds of Halton’s sequence and a certain class of (t, s)-sequences. Our survey aims at highlighting the major ideas of the proofs and we discuss further implications of the employed methods. Moreover, we derive extensions of Levin’s results.

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Cécile Dartyge, Katalin Gyarmati and András Sárközy

Abstract

In Part I of this paper we studied the irregularities of distribution of binary sequences relative to short arithmetic progressions. First we introduced a quantitative measure for this property. Then we studied the typical and minimal values of this measure for binary sequences of a given length. In this paper our goal is to give constructive bounds for these minimal values.

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Małgorzata Elżbieta Hryniewicka

Abstract

In the paper, we introduce the notion of a nondistributive ring N as a generalization of the notion of an associative ring with unit, in which the addition needs not be abelian and the distributive law is replaced by n0 = 0n = 0 for every element n of N. For a nondistributive ring N, we introduce the notion of a nondistributive ring of left quotients S −1 N with respect to a multiplicatively closed set SN, and determine necessary and sufficient conditions for the existence of S −1 N.

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Martin Rehberg

Abstract

For a function f satisfying f (x) = o((log x) K), K > 0, and a sequence of numbers (qn) n, we prove by assuming several conditions on f that the sequence (αf (qn)) n≥n 0 is uniformly distributed modulo one for any nonzero real number α. This generalises some former results due to Too, Goto and Kano where instead of (qn) n the sequence of primes was considered.