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Athanasios Sourmelidis

Abstract

In this paper, we prove a discrete analogue of Voronin’s early finite-dimensional approximation result with respect to terms from a given Beatty sequence and make use of Taylor approximation in order to derive a weak universality statement.

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Ramesh Sirisetti and G. Jogarao

Abstract

In this paper, the concept of relative complementation in almost distributive lattice is generalized. We obtain several properties on the sets of weak relative complement elements. We prove a sufficient condition for a weakly relatively complemented almost distributive lattice with dense elements to become a generalized stone almost distributive lattice.

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Arpita Das and Pratima Panigrahi

Abstract

The R-graph R(G) of a graph G is the graph obtained from G by intro- ducing a new vertex ue for each e ∈ E(G) and making ue adjacent to both the end vertices of e. In this paper, we determine the adjacency, Lapla- cian and signless Laplacian spectra of R-vertex join and R-edge join of a connected regular graph with an arbitrary regular graph in terms of their eigenvalues. Moreover, applying these results we construct some non-regular A-cospectral, L-cospectral and Q-cospectral graphs, and find the number of spanning trees.

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Akbar Rezaei and Arsham Borumand Saeid

Abstract

Hilbert algebras are important tools for certain investigations in algebraic logic since they can be considered as fragments of any propositional logic containing a logical connective implication and the constant 1 which is considered as the logical value “true” and as a generalization of this was defined the notion of g-Hilbert algebra. In this paper, we investigate the relationship between g-Hilbert algebras, gi-algebras, implication gruopoid and BE-algebras. In fact, we show that every g-Hilbert algebra is a self distributive BE-algebras and conversely. We show cannot remove the condition self distributivity. Therefore we show that any self distributive commutative BE-algebras is a gi-algebra and any gi-algebra is strong and transitive if and only if it is a commutative BE-algebra. We prove that the MV -algebra is equivalent to the bounded commutative BE-algebra.

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Martine Queffélec

Abstract

We intend to unroll the surprizing properties of the Thue-Morse sequence with a harmonic analysis point of view, and mention in passing some related open questions.

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Florian Pausinger and Alev Topuzoğlu

Abstract

A permuted van der Corput sequence Sbσ in base b is a one-dimensional, infinite sequence of real numbers in the interval [0, 1), generation of which involves a permutation σ of the set {0, 1,..., b − 1}. These sequences are known to have low discrepancy DN, i.e. t(Sbσ):=limsupNDN(Sbσ)/logN is finite. Restricting to prime bases p we present two families of generating permutations. We describe their elements as polynomials over finite fields 𝔽p in an explicit way. We use this characterization to obtain bounds for t(Spσ) for permutations σ in these families. We determine the best permutations in our first family and show that all permutations of the second family improve the distribution behavior of classical van der Corput sequences in the sense that t(Spσ)<t(Spid).

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Anshika Srivastava, Ram Krishna Pandey and Om Prakash

Abstract

This paper concerns the problem of determining or estimating the maximal upper density of the sets of nonnegative integers S whose elements do not differ by an element of a given set M of positive integers. We find some exact values and some bounds for the maximal density when the elements of M are generalized Fibonacci numbers of odd order. The generalized Fibonacci sequence of order r is a generalization of the well known Fibonacci sequence, where instead of starting with two predetermined terms, we start with r predetermined terms and each term afterwards is the sum of r preceding terms. We also derive some new properties of the generalized Fibonacci sequence of order r. Furthermore, we discuss some related coloring parameters of distance graphs generated by the set M.

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Mario Neumüller and Friedrich Pillichshammer

Abstract

The star discrepancy DN*(𝒫) is a quantitative measure for the irregularity of distribution of a finite point set 𝒫 in the multi-dimensional unit cube which is intimately related to the integration error of quasi-Monte Carlo algorithms. It is known that for every integer N ≥ 2 there are point sets 𝒫 in [0, 1)d with |𝒫| = N and DN*(𝒫)=O((logN)d-1/N). However, for small N compared to the dimension d this asymptotically excellent bound is useless (e.g., for N ≤ ed −1).

In 2001 it has been shown by Heinrich, Novak, Wasilkowski and Woźniakowski that for every integer N ≥ 2there exist point sets 𝒫 in [0, 1)d with |𝒫| = N and DN*(𝒫)Cd/N. Although not optimal in an asymptotic sense in N, this upper bound has a much better (and even optimal) dependence on the dimension d.

Unfortunately the result by Heinrich et al. and also later variants thereof by other authors are pure existence results and until now no explicit construction of point sets with the above properties is known. Quite recently Löbbe studied lacunary subsequences of Kronecker’s (n α)-sequence and showed a metrical discrepancy bound of the form Cd(logd)/N with implied absolute constant C> 0 independent of N and d.

In this paper we show a corresponding result for digital Kronecker sequences, which are a non-archimedean analog of classical Kronecker sequences.

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Nareupanat Lekkoksung and Prakit Jampachon

Abstract

A generalized hypersubstitution of type τ = (ni)i∈I is a mapping σ which maps every operation symbol fi to the term σ (fi) and may not preserve arity. It is the main tool to study strong hyperidentities that are used to classify varieties into collections called strong hypervarieties. Each generalized hypersubstitution can be extended to a mapping σ̂ on the set of all terms of type τ. A binary operation on HypG(τ), the set of all generalized hypersubstitutions of type τ, can be defined by using this extension. The set HypG(τ) together with such a binary operation forms a monoid, where a hypersubstitution σid, which maps fi to fi(x1, . . . , xn₁ ) for every i ∈ I, is the neutral element of this monoid. A weak projection generalized hypersubstitution of type τ is a generalized hypersubstitution of type τ which maps at least one of the operation symbols to a variable. In semigroup theory, the various types of its elements are widely considered. In this paper, we present the characterizations of idempotent weak projection generalized hypersubstitutions of type (m, n) and give some sufficient conditions for a weak projection generalized hypersubstitution of type (m, n) to be regular, where m, n ≥ 1.

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Marapureddy Murali Krishna Rao

Abstract

In this paper, we introduce the notion of k-ideal, m−k ideal, prime ideal, maximal ideal, filter, irreducible ideal, strongly irreducible ideal in ordered Γ-semirings, study the properties of ideals in ordered Γ-semirings and the relations between them. We characterize m − k ideals using derivation of ordered Γ-semirings and prove that every ideal in a mono regular ordered Γ-semiring is a prime ideal and field ordered Γ-semiring is simple.