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Christophe Reutenauer and Laurent Vuillon

Abstract

We state a new formula to compute the Markoff numbers using iterated palindromic closure and the Thue-Morse substitution. The main theorem shows that for each Markoff number m, there exists a word v ∈ {a, b}∗ such that m − 2 is equal to the length of the iterated palindromic closure of the iterated antipalindromic closure of the word av. This construction gives a new recursive construction of the Markoff numbers by the lengths of the words involved in the palindromic closure. This construction interpolates between the Fibonacci numbers and the Pell numbers.

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Marapureddy Murali Krishna Rao

Abstract

In this paper we study the properties of structures of the semigroup (M,+) and the Γ-semigroup M of Γ -semiring M and regular Γ-semiring M satisfying the identity a + aαb = a or aαb + a = a or a + aαb + b = a or a + 1 = 1, for all a ∈ M, α ∈ Γ. We also study the properties of Γ-semiring with unity 1 which is also an additive identity.

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Marcelo Fiore

Abstract

We consider discrete enriched abstract clones and provide two constructions investigating their representation as discrete enriched clones of operations on objects in concrete enriched categories over the enriching category. Our first construction embeds a discrete enriched abstract clone into the concrete discrete enriched clone of operations on an object in the enriching category. Our second construction refines the given embedding by introducing a monoid action and restricting attention to the concrete discrete enriched clone of its equivariant operations. As in the classical theory of abstract clones, our main focus is on discrete enriched abstract clones with finite arities. However, we also consider discrete enriched abstract clones with countable arities to show that the representation theory of the former is conceptually explained by that of the latter.

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Carlo Sanna

Abstract

For any real number s, let σs be the generalized divisor function, i.e., the arithmetic function defined by σs(n) := ∑d|n ds, for all positive integers n. We prove that for any r > 1 the topological closure of σ−r(N+) is the union of a finite number of pairwise disjoint closed intervals I1, . . . , I. Moreover, for k = 1, . . . , ℓ, we show that the set of positive integers n such that σ−r(n) ∈ Ik has a positive rational asymptotic density dk. In fact, we provide a method to give exact closed form expressions for I1, . . . , I and d1, . . . , d, assuming to know r with sufficient precision. As an example, we show that for r = 2 it results ℓ = 3, I1 = [1, π2/9], I2 = [10/9, π2/8], I3 = [5/4, π2/6], d1 = 1/3, d2 = 1/6, and d3 = 1/2.

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Ravi Kumar Bandaru

Abstract

In this paper, the notion of a QI-algebra is introduced which is a generalization of a BI-algebra and there are studied its properties. We considered ideals, congruence kernels in a QI-algebra and characterized congruence kernels whenever a QI-algebra is right distributive.

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Angsuman Das

Abstract

In this short paper, we characterize the positive integers n for which intersection graph of ideals of ℤn is perfect.

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Farrukh Jamal, M. H. Tahir, Morad Alizadeh and M. A. Nasir

Abstract

Generalizing distributions is important for applied statisticians and recent literature has suggested several ways of extending well-known distributions. We propose a new class of distributions called the Marshall-Olkin Burr X family, which yields exible shapes for its density such as symmetrical, left-skewed, right-skewed and reversed-J shaped, and can have increasing, decreasing,constant, bathtub and upside-down bathtub hazard rates shaped. Some of its structural properties including quantile and generating functions, ordinary and incomplete moments, and mean deviations are obtained. One special model of this family, the Marshall- Olkin-Burr-Lomax distribution, is investigated in details. We also derive the density of the order statistics. The model parameters are estimated by the maximum likelihood method. For illustrative purposes, three applications to real life data are presented.

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Shahabaddin Ebrahimi Atani and Fatemeh Esmaeili Khalil Saraei

Abstract

Let L be a complete lattice. In a manner analogous to a commutative ring, we introduce and investigate the L-fuzzy multiplication modules over a commutative ring with non-zero identity. The basic properties of the prime L-fuzzy submodules of L-fuzzy multiplication modules are characterized.

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Yoshiyuki Kitaoka

Abstract

Let f(x) be a monic polynomial in Z[x] with roots α1, . . ., αn. We point out the importance of linear relations among 1, α1, . . . , αn over rationals with respect to the distribution of local roots of f modulo a prime. We formulate it as a conjectural uniform distribution in some sense, which elucidates data in previous papers.

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Hendrik Jager

Abstract

Denote by Θ12, · · · the sequence of approximation coefficients of Minkowski’s diagonal continued fraction expansion of a real irrational number x. For almost all x this is a uniformly distributed sequence in the interval [0, 1/2 ]. The average distance between two consecutive terms of this sequence and their correlation coefficient are explicitly calculated and it is shown why these two values are close to 1/6 and 0, respectively, the corresponding values for a random sequence in [0, 1/2].