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Mathematics and Statistics
The Hilbert metric between two points 𝑥, 𝑦 in a bounded convex domain 𝐺 is defined as the logarithm of the cross-ratio 𝑥, 𝑦 and the intersection points of the Euclidean line passing through the points 𝑥, 𝑦 and the boundary of the domain. Here, we study this metric in the case of the unit ball 𝔹𝑛. We present an identity between the Hilbert metric and the hyperbolic metric, give several inequalities for the Hilbert metric, and results related to the inclusion properties of the balls defined in the Hilbert metric. Furthermore, we study the distortion of the Hilbert metric under conformal and quasiregular mappings.
This article studies a new class of monomial ideals associated with a simple graph 𝐺, called generalized edge ideal, denoted by 𝐼𝑔(𝐺). Assuming that all the vertices 𝑥 have an exponent greater than 1 in 𝐼𝑔(𝐺), we completely characterize the graph 𝐺 for which 𝐼𝑔(𝐺) is integrally closed, and show that this is equivalent to 𝐼𝑔(𝐺) being normal i.e., all integral powers of 𝐼𝑔(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 𝑐𝑛2 edges. Here we study two refinements of 𝑞(𝐺) arising from the analogous (7, 4)-conjecture. The first is 𝑞𝐴(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 𝑞𝐵(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 𝑞𝐴(𝐺), 𝑞𝐵(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 𝑞𝐴(𝐺) and 𝑞𝐵(𝐺), we study these parameters for some graphs.
We note that 𝑞𝐴(𝐺) has already been extensively studied from various motivations. However, as far as we know the behavior of 𝑞𝐵(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.
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 𝑑 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 < 𝑘 < 𝑑 and for any set of given 𝑑-dimensional volume in 𝔼 𝑑 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.
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 mth convergent. We give some estimates of important coefficients Lk ’s, which determine the graphical shapes of χ2 statistics. This means that the denominator qm of mth convergent and the large partial quotient am +1 determine the outline of shapes of graphs, when we observe values of χ 2 statistics with step qm .
