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

This article studies a new class of monomial ideals associated with a simple graph 𝐺, called generalized edge ideal, denoted by 𝐼_{𝑔}(𝐺). Assuming that all the vertices 𝑥 have an exponent greater than 1 in 𝐼_{𝑔}(𝐺), we completely characterize the graph 𝐺 for which 𝐼_{𝑔}(𝐺) is integrally closed, and show that this is equivalent to 𝐼_{𝑔}(𝐺) being normal i.e., all integral powers of 𝐼_{𝑔}(𝐺) are integrally clased. We also give a necessary and sufficient condition for

A proper edge coloring of a graph 𝐺 is *strong* if the union of any two color classes does not contain a path with three edges (i.e. the color classes are *induced matchings*). The *strong chromatic index* 𝑞(𝐺) is the smallest number of colors needed for a strong coloring of 𝐺. One form of the famous (6, 3)-theorem of Ruzsa and Szemerédi (solving the (6, 3)-conjecture of Brown–Erdős–Sós) states that 𝑞(𝐺) cannot be linear in 𝑛 for a graph 𝐺 with 𝑛 vertices and 𝑐𝑛^{2} edges. Here we study two refinements of 𝑞(𝐺) arising from the analogous (7, 4)-conjecture. The first is 𝑞_{𝐴}(𝐺), the smallest number of colors needed for a proper edge coloring of 𝐺 such that the union of any two color classes does not contain a path or cycle with four edges, we call it an *A-coloring*. The second is 𝑞_{𝐵}(𝐺), the smallest number of colors needed for a proper edge coloring of 𝐺 such that all four-cycles are colored with four different colors, we call it a *B-coloring*. These notions lead to two stronger and one equivalent form of the (7, 4)-conjecture in terms of 𝑞_{𝐴}(𝐺), 𝑞_{𝐵}(𝐺) where 𝐺 is a balanced bipartite graph. Since these are questions about graphs, perhaps they will be easier to handle than the original ^{special}(7, 4)-conjecture. In order to understand the behavior of _{𝑞}𝐴(𝐺) and _{𝑞}𝐵(𝐺), we study these parameters for some graphs.

We note that 𝑞_{𝐴}(𝐺) has already been extensively studied from various motivations. However, as far as we know the behavior of 𝑞_{𝐵}(𝐺) is studied here for the first time.

We study a natural set of refinements of the Ehrhart series of a closed polytope, first considered by Chapoton. We compute the refined series in full generality for a simplex of dimension 𝑑, a cross-polytope of dimension 𝑑, respectively a hypercube of dimension 𝑑 ≤ 3, using commutative algebra. We deduce summation formulae for products of 𝑞-integers with different arguments, generalizing a classical identity due to MacMahon and Carlitz. We also present a characterisation of a certain refined Eulerian polynomial in algebraic terms.

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

In this article, this question is answered.

Over integral domains of characteristics different from 2, we determine all the matrices

We present generalizations of the Pinelis extension of Stolarsky’s inequality and its reverse. In particular, a new Stolarsky-type inequality is obtained. We study the properties of the linear functional related to the new Stolarsky-type inequality, and finally apply these new results in the theory of fractional integrals.

In this paper, we consider the Feuerbach point and the Feuerbach line of a triangle in the isotropic plane, and investigate some properties of these concepts and their relationships with other elements of a triangle in the isotropic plane. We also compare these relationships in Euclidean and isotropic cases.

We define the order of the double hypergeometric series, investigate the properties of the new confluent Kampé de Fériet series, and build systems of partial differential equations that satisfy the new Kampé de Fériet series. We solve the Cauchy problem for a degenerate hyperbolic equation of the second kind with a spectral parameter using the high-order Kampé de Fériet series. Thanks to the properties of the introduced Kampé de Fériet series, it is possible to obtain a solution to the problem in explicit forms.

Let 𝔼*
^{𝑑}
* denote the 𝑑-dimensional Euclidean space. The 𝑟-ball body generated by a given set in 𝔼

*is the intersection of balls of radius 𝑟 centered at the points of the given set. The author [Discrete Optimization 44/1 (2022), Paper No. 100539] proved the following Blaschke–Santaló-type inequalities for 𝑟-ball bodies: for all 0 <*

^{𝑑}*𝑘*<

*𝑑*and for any set of given 𝑑-dimensional volume in 𝔼

*the 𝑘-th intrinsic volume of the 𝑟-ball body generated by the set becomes maximal if the set is a ball. In this note we give a new proof showing also the uniqueness of the maximizer. Some applications and related questions are mentioned as well.*

^{𝑑}We discuss the outline of the shapes of graphs of χ ^{2} statistics for distributions of leading digits of irrational rotations under some conditions on *m*th convergent. We give some estimates of important coefficients *L _{k}
*’s, which determine the graphical shapes of χ

^{2}statistics. This means that the denominator

*q*of

_{m}*m*th convergent and the large partial quotient

*a*

_{m}_{+1}determine the outline of shapes of graphs, when we observe values of χ

^{2}statistics with step

*q*.

_{m}