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## Incidence Functions of the Exponential Divisor Poset

Mathematica Pannonica
Author:
Pentti Haukkanen

A positive integer $d = ∏ i = 1 r p i d i$ is said to be an exponential divisor or an e-divisor of $n = ∏ i = 1 r p i n i > 1$ if 𝑑𝑖 ∣ 𝑛𝑖 for all prime divisors 𝑝𝑖 of 𝑛. In addition, 1 is an e-divisor of 1. It is easy to see that ℤ+ 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 Möbius function are preserved in this process.

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## On Mixed 𝐵-Concatenations of Pell and Pell–Lucas Numbers which are Pell Numbers

Mathematica Pannonica
Authors:
and
Marija Bliznac Trebješanin

Let (𝑃𝑛)𝑛≥0 and (𝑄𝑛)𝑛≥0 be the Pell and Pell–Lucas sequences. Let 𝑏 be a positive integer such that 𝑏 ≥ 2. In this paper, we prove that the following two Diophantine equations 𝑃𝑛 = 𝑏𝑑𝑃𝑚 + 𝑄𝑘 and 𝑃𝑛 = 𝑏𝑑𝑄𝑚 + 𝑃𝑘 with 𝑑, the number of digits of 𝑃𝑘 or 𝑄𝑘 in base 𝑏, have only finitely many solutions in nonnegative integers (𝑚, 𝑛, 𝑘, 𝑏, 𝑑). Also, we explicitly determine these solutions in cases 2 ≤ 𝑏 ≤ 10.

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## Answer to a 1971 Question of Grätzer and Lakser on Pseudocomplemented Lattices

Mathematica Pannonica
Author:
Jonathan David Farley

Grätzer 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 𝟐𝑛 ⊕ 𝟏 can be characterized by the property of not having a *-homomorphism onto 𝟐𝑖 ⊕ 𝟏 for 1 < 𝑖 < 𝑛.

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## Notice of Self-Retraction and Replacement: A 1971 Question of Grätzer and Lakser on Pseudocomplemented Lattices

Mathematica Pannonica
Authors:
Jonathan David Farley
and
Dominic van der Zypen
Open access

## *-Rickart Property for Rings with Involution

Mathematica Pannonica
Authors:
,
Usama A. Aburawash
, and

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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## Simultaneous Approximation for an Exponential Operator Connected with x4/3

Mathematica Pannonica
Authors:
Anjali
and
Vijay Gupta

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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## On a Dowker-Type Problem for Convex Disks with Almost Constant Curvature

Studia Scientiarum Mathematicarum Hungarica
Authors:
Bushra Basit
and
Zsolt Lángi

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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## The Bloch Spaces with Differentiable Strictly Positive Weights

Mathematica Pannonica
Authors:
Ding Nan
and
Hasi Wulan

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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## Beurling-Integers with Lacunarity

Mathematica Pannonica
Author:
Imre Z. Ruzsa

We present examples of multiplicative semigroups of positive reals (Beurling’s generalized integers) with gaps bounded from below.

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## Construction of New Operators by Composition of Integral-Type Operators and Discrete Operators

Mathematica Pannonica
Authors:
Ulrich Abel
and
Vijay Gupta

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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