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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 𝑝-th order moment; 𝑝 ∈ ℕ ∪ {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 𝐾, the areas of the maximum (resp. minimum) area convex 𝑛-gons inscribed (resp. circumscribed) in 𝐾 is a concave (resp. convex) sequence. It is known that this theorem remains true if we replace area by perimeter, or convex 𝑛-gons by disk-𝑛-gons, obtained as the intersection of 𝑛 closed Euclidean unit disks. It has been proved recently that if 𝐶 is the unit disk of a normed plane, then the same properties hold for the area of 𝐶-𝑛-gons circumscribed about a 𝐶-convex disk 𝐾 and for the perimeters of 𝐶-𝑛-gons inscribed or circumscribed about a 𝐶-convex disk 𝐾, but for a typical origin-symmetric convex disk 𝐶 with respect to Hausdorff distance, there is a 𝐶-convex disk 𝐾 such that the sequence of the areas of the maximum area 𝐶-𝑛-gons inscribed in 𝐾 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 𝐶.

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In this paper, some basic characterizations of a weighted Bloch space with the differentiable strictly positive weight 𝜔 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 (Beurling’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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Let 𝑓 be a normalized primitive cusp form of even integral weight for the full modular group Γ = 𝑆𝐿(2, ℤ). In this paper, we investigate upper bounds for the error terms related to the average behavior of Fourier coefficients 𝜆𝑓 ⊗𝑓 ⊗⋯⊗𝑙𝑓 (𝑛) of 𝑙-fold product 𝐿-functions, where 𝑓 ⊗ 𝑓 ⊗ ⋯ ⊗𝑙 𝑓 denotes the 𝑙-fold product of 𝑓. These results improves and generalizes the recent developments of Venkatasubbareddy and Sankaranarayanan [41]. We also provide some other similar results related to the error terms of general product 𝐿-functions.

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We apply a recent general zero density theorem of us (valid for a large class of complex functions) to improve earlier density theorems of Heath-Brown and Paul–Sankaranarayanan for Dedekind zeta functions attached to a number field 𝐾 of degree 𝑛 with 𝑛 > 2.

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Asymptotic uniform upper density, shortened as a.u.u.d., or simply upper density, is a classical notion which was first introduced by Kahane for sequences in the real line.

Syndetic sets were defined by Gottschalk and Hendlund. For a locally compact group 𝐺, a set 𝑆 ⊂ 𝐺 is syndetic, if there exists a compact subset 𝐶 ⋐ 𝐺 such that 𝑆𝐶 = 𝐺. Syndetic sets play an important role in various fields of applications of topological groups and semigroups, ergodic theory and number theory. A lemma in the book of Fürstenberg says that once a subset 𝐴 ⊂ ℤ has positive a.u.u.d., then its difference set 𝐴 − 𝐴 is syndetic.

The construction of a reasonable notion of a.u.u.d. in general locally compact Abelian groups (LCA groups for short) was not known for long, but in the late 2000’s several constructions were worked out to generalize it from the base cases of ℤ𝑑 and ℝ𝑑. With the notion available, several classical results of the Euclidean setting became accessible even in general LCA groups.

Here we work out various versions in a general locally compact Abelian group 𝐺 of the classical statement that if a set 𝑆 ⊂ 𝐺 has positive asymptotic uniform upper density, then the difference set 𝑆 − 𝑆 is syndetic.

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Recently [3] we proved a general zero density theorem for a large class of functions which included among others the Riemann zeta function, Dedekind zeta functions, Dirichlet 𝐿-functions. The goal of the present work is a (slight) improvement of this general theorem which might lead to stronger results in some applications.

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