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A positive integer d = โˆ i = 1 r p i d i is said to be an exponential divisor or an e-divisor of n = โˆ i = 1 r p i n i > 1 if ๐‘‘๐‘– โˆฃ ๐‘›๐‘– for all prime divisors ๐‘๐‘– of ๐‘›. In addition, 1 is an e-divisor of 1. It is easy to see that โ„ค+ is a poset under the e-divisibility relation. Utilizing this observation we show that e-convolution of arithmetical functions is an example of the convolution of incidence functions of posets. We also note that the identity, units and the Mรถbius function are preserved in this process.

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Let (๐‘ƒ๐‘›)๐‘›โ‰ฅ0 and (๐‘„๐‘›)๐‘›โ‰ฅ0 be the Pell and Pellโ€“Lucas sequences. Let ๐‘ be a positive integer such that ๐‘ โ‰ฅ 2. In this paper, we prove that the following two Diophantine equations ๐‘ƒ๐‘› = ๐‘๐‘‘๐‘ƒ๐‘š + ๐‘„๐‘˜ and ๐‘ƒ๐‘› = ๐‘๐‘‘๐‘„๐‘š + ๐‘ƒ๐‘˜ with ๐‘‘, the number of digits of ๐‘ƒ๐‘˜ or ๐‘„๐‘˜ in base ๐‘, have only finitely many solutions in nonnegative integers (๐‘š, ๐‘›, ๐‘˜, ๐‘, ๐‘‘). Also, we explicitly determine these solutions in cases 2 โ‰ค ๐‘ โ‰ค 10.

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Grรคtzer and Lakser asked in the 1971 Transactions of the American Mathematical Society if the pseudocomplemented distributive lattices in the amalgamation class of the subvariety generated by ๐Ÿ๐‘› โŠ• ๐Ÿ can be characterized by the property of not having a *-homomorphism onto ๐Ÿ๐‘– โŠ• ๐Ÿ for 1 < ๐‘– < ๐‘›.

In this article, their question from 1971 is answered.

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Mathematica Pannonica
Authors:
Muhammad T. Tajuddin
,
Usama A. Aburawash
, and
Muhammad Saad

This paper introduces and examines the concept of a *-Rickart *-ring, and proves that every Rickart *-ring is also a *-Rickart *-ring. A necessary and sufficient condition for a *-Rickart *-ring to be a Rickart *-ring is also provided. The relationship between *-Rickart *-rings and *-Baer *-rings is investigated, and several properties of *-Rickart *-rings are presented. The paper demonstrates that the property of *-Rickart extends to both the center and *-corners of a *-ring, and investigates the extension of a *-Rickart *-ring to its polynomial *-ring. Additionally, *-Rickart *-rings with descending chain condition on *-biideals are studied, and all *-Rickart (*-Baer) *-rings with finitely many elements are classified.

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Very recently, the authors in [5] proposed the exponential-type operator connected with x 4 3 and studied its convergence estimates. In the present research, we extend the study and obtain the general form of its ๐‘-th order moment; ๐‘ โˆˆ โ„• โˆช {0}. Further, we establish the simultaneous approximation for the operator under consideration.

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A classical result of Dowker (Bull. Amer. Math. Soc. 50: 120-122, 1944) states that for any plane convex body ๐พ, the areas of the maximum (resp. minimum) area convex ๐‘›-gons inscribed (resp. circumscribed) in ๐พ is a concave (resp. convex) sequence. It is known that this theorem remains true if we replace area by perimeter, or convex ๐‘›-gons by disk-๐‘›-gons, obtained as the intersection of ๐‘› closed Euclidean unit disks. It has been proved recently that if ๐ถ is the unit disk of a normed plane, then the same properties hold for the area of ๐ถ-๐‘›-gons circumscribed about a ๐ถ-convex disk ๐พ and for the perimeters of ๐ถ-๐‘›-gons inscribed or circumscribed about a ๐ถ-convex disk ๐พ, but for a typical origin-symmetric convex disk ๐ถ with respect to Hausdorff distance, there is a ๐ถ-convex disk ๐พ such that the sequence of the areas of the maximum area ๐ถ-๐‘›-gons inscribed in ๐พ is not concave. The aim of this paper is to investigate this question if we replace the topology induced by Hausdorff distance with a topology induced by the surface area measure of the boundary of ๐ถ.

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In this paper, some basic characterizations of a weighted Bloch space with the differentiable strictly positive weight ๐œ” on the unit disc are given, including the growth, the higher order or free derivative descriptions, and integral characterizations of functions in the space.

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We present examples of multiplicative semigroups of positive reals (Beurlingโ€™s generalized integers) with gaps bounded from below.

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In this paper, we propose some new positive linear approximation operators, which are obtained from a composition of certain integral type operators with certain discrete operators. It turns out that the new operators can be expressed in discrete form. We provide representations for their coefficients. Furthermore, we study their approximation properties and determine their moment generating functions, which may be useful in finding several other convergence results in different settings.

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