Browse Our Mathematics and Statistics Journals

**Mathematics and statistics journals publish papers on the theory and application of mathematics, statistics, and probability. Most mathematics journals have a broad scope that encompasses most mathematical fields. These commonly include logic and foundations, algebra and number theory, analysis (including differential equations, functional analysis and operator theory), geometry, topology, combinatorics, probability and statistics, numerical analysis and computation theory, mathematical physics, etc.**

# Mathematics and Statistics

A positive integer _{๐} โฃ 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 โค_{+} 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.

Let (๐_{๐})_{๐โฅ0} and (๐_{๐})_{๐โฅ0} be the Pell and PellโLucas sequences. Let ๐ be a positive integer such that ๐ โฅ 2. In this paper, we prove that the following two Diophantine equations ๐_{๐} = ๐^{๐}๐_{๐} + ๐_{๐} and ๐_{๐} = ๐^{๐}๐_{๐} + ๐_{๐} with ๐, the number of digits of ๐_{๐} or ๐_{๐} in base ๐, have only finitely many solutions in nonnegative integers (๐, ๐, ๐, ๐, ๐). Also, we explicitly determine these solutions in cases 2 โค ๐ โค 10.

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^{๐} โ ๐ can be characterized by the property of not having a *-homomorphism onto ๐^{๐} โ ๐ for 1 < *๐* < *๐*.

In this article, their question from 1971 is answered.

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.

Very recently, the authors in [5] proposed the exponential-type operator connected with

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

A classical result of Dowker (Bull. Amer. Math. Soc. 50: 120-122, 1944) states that for any plane convex body ๐พ, the areas of the maximum (resp. minimum) area convex ๐-gons inscribed (resp. circumscribed) in ๐พ 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 ๐ถ is the unit disk of a normed plane, then the same properties hold for the area of ๐ถ-๐-gons circumscribed about a ๐ถ-convex disk ๐พ and for the perimeters of ๐ถ-๐-gons inscribed or circumscribed about a ๐ถ-convex disk ๐พ, but for a typical origin-symmetric convex disk ๐ถ with respect to Hausdorff distance, there is a ๐ถ-convex disk ๐พ such that the sequence of the areas of the maximum area ๐ถ-๐-gons inscribed in ๐พ 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 ๐ถ.

A ๐-graph with ๐ edges and ๐ vertices is defined as an ๐ ร ๐ matrix with entries from {0, โฆ , ๐}, such that each row of the matrix (called a ๐-edge) contains exactly two nonzero entries. If ๐ป is a ๐-graph, then ๐ป is said to contain an ๐ -copy of the ordinary graph ๐น, if a set ๐ of ๐-edges can be selected from ๐ป such that their intersection graph is isomorphic to ๐น, and for any vertex ๐ฃ of ๐ and any two incident edges ๐, ๐ โ ๐ the sum of the entries of ๐ and ๐ is at least ๐ . The extremal number ex(๐, ๐น, ๐, ๐ ) is defined as the maximal number of edges in an ๐-vertex ๐-graph such that it does not contain contain an ๐ -copy of the forbidden graph ๐น.

In the present paper, we reduce the problem of finding ex(๐, ๐น, ๐, ๐ + 1) for even ๐ to the case ๐ = 2, and determine the asymptotics of ex(๐, ๐ถ_{2๐+1}, ๐, ๐ + 1).

For positive integers ๐, ๐, ๐ with ๐ > ๐ , the set-coloring Ramsey number ๐
(๐; ๐, ๐ ) is the minimum ๐ such that if every edge of the complete graph ๐พ_{๐} receives a set of ๐ colors from a palette of ๐ colors, then there is guaranteed to be a monochromatic clique on ๐ vertices, that is, a subset of ๐ vertices where all of the edges between them receive a common color. In particular, the case ๐ = 1 corresponds to the classical multicolor Ramsey number. We prove general upper and lower bounds on ๐
(๐; ๐, ๐ ) which imply that ๐
(๐; ๐, ๐ ) = 2^{ฮ(๐๐)} if ๐ /๐ is bounded away from 0 and 1. The upper bound extends an old result of Erdลs and Szemerรฉdi, who treated the case ๐ = ๐ โ 1, while the lower bound exploits a connection to error-correcting codes. We also study the analogous problem for hypergraphs.