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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 β„€+ 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 MΓΆbius function are preserved in this process.

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Let (𝑃𝑛)𝑛β‰₯0 and (𝑄𝑛)𝑛β‰₯0 be the Pell and Pell–Lucas sequences. Let 𝑏 be a positive integer such that 𝑏 β‰₯ 2. In this paper, we prove that the following two Diophantine equations 𝑃𝑛 = π‘π‘‘π‘ƒπ‘š + π‘„π‘˜ and 𝑃𝑛 = π‘π‘‘π‘„π‘š + π‘ƒπ‘˜ with 𝑑, the number of digits of π‘ƒπ‘˜ or π‘„π‘˜ in base 𝑏, have only finitely many solutions in nonnegative integers (π‘š, 𝑛, π‘˜, 𝑏, 𝑑). Also, we explicitly determine these solutions in cases 2 ≀ 𝑏 ≀ 10.

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GrΓ€tzer 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 πŸπ‘› βŠ• 𝟏 can be characterized by the property of not having a *-homomorphism onto πŸπ‘– βŠ• 𝟏 for 1 < 𝑖 < 𝑛.

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 𝑝-th order moment; 𝑝 ∈ β„• βˆͺ {0}. Further, we establish the simultaneous approximation for the operator under consideration.

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A bi-cyclic 4-polytope in ℝ4 was introduced by Z. Smilansky as the convex hull of evenly spaced points on a generalized trigonometric moment curve in ℝ4. We present combinatorial geometric conditions that yield the face lattices of a class of such 4-polytopes.

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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 𝐢 is the unit disk of a normed plane, then the same properties hold for the area of 𝐢-𝑛-gons circumscribed about a 𝐢-convex disk 𝐾 and for the perimeters of 𝐢-𝑛-gons inscribed or circumscribed about a 𝐢-convex disk 𝐾, but for a typical origin-symmetric convex disk 𝐢 with respect to Hausdorff distance, there is a 𝐢-convex disk 𝐾 such that the sequence of the areas of the maximum area 𝐢-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 𝐢.

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Studia Scientiarum Mathematicarum Hungarica
Authors:
KoppΓ‘ny Encz
,
MΓ‘rton Marits
,
Benedek VΓ‘li
, and
MΓ‘tΓ© Weisz

A π‘ž-graph with 𝑒 edges and 𝑛 vertices is defined as an 𝑒 Γ— 𝑛 matrix with entries from {0, … , π‘ž}, such that each row of the matrix (called a π‘ž-edge) contains exactly two nonzero entries. If 𝐻 is a π‘ž-graph, then 𝐻 is said to contain an 𝑠-copy of the ordinary graph 𝐹, if a set 𝑆 of π‘ž-edges can be selected from 𝐻 such that their intersection graph is isomorphic to 𝐹, and for any vertex 𝑣 of 𝑆 and any two incident edges 𝑒, 𝑓 ∈ 𝑆 the sum of the entries of 𝑒 and 𝑓 is at least 𝑠. The extremal number ex(𝑛, 𝐹, π‘ž, 𝑠) is defined as the maximal number of edges in an 𝑛-vertex π‘ž-graph such that it does not contain contain an 𝑠-copy of the forbidden graph 𝐹.

In the present paper, we reduce the problem of finding ex(𝑛, 𝐹, π‘ž, π‘ž + 1) for even π‘ž to the case π‘ž = 2, and determine the asymptotics of ex(𝑛, 𝐢2π‘˜+1, π‘ž, π‘ž + 1).

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Studia Scientiarum Mathematicarum Hungarica
Authors:
David Conlon
,
Jacob Fox
,
Xiaoyu He
,
Dhruv Mubayi
,
Andrew Suk
, and
Jacques VerstraΓ«te

For positive integers 𝑛, π‘Ÿ, 𝑠 with π‘Ÿ > 𝑠, the set-coloring Ramsey number 𝑅(𝑛; π‘Ÿ, 𝑠) is the minimum 𝑁 such that if every edge of the complete graph 𝐾𝑁 receives a set of 𝑠 colors from a palette of π‘Ÿ colors, then there is guaranteed to be a monochromatic clique on 𝑛 vertices, that is, a subset of 𝑛 vertices where all of the edges between them receive a common color. In particular, the case 𝑠 = 1 corresponds to the classical multicolor Ramsey number. We prove general upper and lower bounds on 𝑅(𝑛; π‘Ÿ, 𝑠) which imply that 𝑅(𝑛; π‘Ÿ, 𝑠) = 2Θ(π‘›π‘Ÿ) if 𝑠/π‘Ÿ is bounded away from 0 and 1. The upper bound extends an old result of ErdΕ‘s and SzemerΓ©di, who treated the case 𝑠 = π‘Ÿ βˆ’ 1, while the lower bound exploits a connection to error-correcting codes. We also study the analogous problem for hypergraphs.

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