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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
In this article, we use the idea of “negation” to construct new unit distributions, i.e., continuous distributions with support equal to the unit interval [0, 1]. A notable feature of these distributions is that they have opposite shape properties to the unit distributions from which they are derived; “opposite” in the sense that, from a graphical point of view, a certain horizontal symmetry is operated. We then examine the main properties of these negation-type distributions, including distributional functions, moments, and entropy measures. Finally, concrete examples are described, namely the negation-type power distribution, the negation-type [0, 1]-truncated exponential distribution, the negation-type truncated [0, 1]-sine distribution, the negation-type [0, 1]-truncated Lomax distribution, the negation-type Kumaraswamy distribution, and the negation-type beta distribution. Some of their properties are studied, also with the help of graphics that highlight their original modeling behavior. After the analysis, it appears that the negation-type Kumaraswamy distribution stands out from the others by combining simplicity with a high degree of flexibility, in a sense completing the famous Kumaraswamy distribution. Overall, our results enrich the panel of unit distributions available in the literature with an innovative approach.
High proved the following theorem. If the intersections of any two congruent copies of a plane convex body are centrally symmetric, then this body is a circle. In our paper we extend the theorem of High to the sphere and the hyperbolic plane. Let us have in 𝑆2, ℝ2 or 𝐻2 a pair of convex bodies (for 𝑆2 different from 𝑆2), such that the intersections of any congruent copies of them are centrally symmetric. Then our bodies are congruent circles. If the intersections of any congruent copies of them are axially symmetric, then our bodies are (incongruent) circles. Let us have in 𝑆2, ℝ2 or 𝐻2 proper closed convex subsets 𝐾, 𝐿 with interior points, such that the numbers of the connected components of the boundaries of 𝐾 and 𝐿 are finite. If the intersections of any congruent copies of 𝐾 and 𝐿 are centrally symmetric, then 𝐾 and 𝐿 are congruent circles, or, for ℝ2, parallel strips. For ℝ2 we exactly describe all pairs of such subsets 𝐾, 𝐿, whose any congruent copies have an intersection with axial symmetry (there are five cases).
This paper is mainly about direct summand right ideals of nearrings with 𝐷𝐶𝐶𝑁 which cannot be expressed as a non-trivial direct sum. A fairly natural condition (Φ-irreducibility) makes it possible to study these right ideals in reasonable depth. It turns out they are either very ring like or right ideals (called shares) controlling considerable nearring structure. The two cases are studied in some detail. A surprising feature of the last section is that, with weak hypercentrality present, the nearring is a unique finite direct sum of these right ideals if, and only if, all such right ideals are ideals.
In the present paper we aim to calculate with the exclusive use of real methods, an atypical harmonic series with a weight 4 structure, featuring the harmonic number of the kind 𝐻2𝑘. Very simple relations and neat results are considered for the evaluation of the main series.
This article describes a general analytical derivation of the Fuss’ relation for bicentric polygons with an odd number of vertices. In particular, we derive the Fuss’ relations for the bicentric tridecagon and the bicentric pentadecagon.
We improve the lower bound on the translative covering density of tetrahedra found by Y. Li, M. Fu and Y. Zhang. Our method improves the bound from 1.00122 to 1.0075, but also shows the existence of similar lower density bounds for any polyhedron which has a face without opposite parallel face or edge.
In this paper, we consider the asymptotic behaviour of the expectation of the number of vertices of a uniform random spherical disc-polygon. This provides a connection between the corresponding results in spherical convexity, and in Euclidean spindle-convexity, where the expectation tends to the same constant. We also extend the result to a more general case, where the random points generating the uniform random disc-polygon are chosen from spherical convex disc with smooth boundary.
We prove existence of Helly numbers for crystals and for cut-and-project sets with convex windows. Also we show that for a two-dimensional crystal consisting of 𝑘 copies of a single lattice the Helly number does not exceed 𝑘 + 6.
Consider an arrangement of 𝑛 congruent zones on the 𝑑-dimensional unit sphere 𝑆𝑑−1, where a zone is the intersection of an origin symmetric Euclidean plank with 𝑆𝑑−1. We prove that, for sufficiently large 𝑛, it is possible to arrange 𝑛 congruent zones of suitable width on 𝑆𝑑−1 such that no point belongs to more than a constant number of zones, where the constant depends only on the dimension and the width of the zones. Furthermore, we also show that it is possible to cover 𝑆𝑑−1 by 𝑛 congruent zones such that each point of 𝑆𝑑−1 belongs to at most 𝐴𝑑 ln 𝑛 zones, where the 𝐴𝑑 is a constant that depends only on 𝑑. This extends the corresponding 3-dimensional result of Frankl, Nagy and Naszódi [8]. Moreover, we also examine coverings of 𝑆𝑑−1 with congruent zones under the condition that each point of the sphere belongs to the interior of at most 𝑑 − 1 zones.
In this note we introduce a pseudometric on closed convex planar curves based on distances between normal lines and show its basic properties. Then we use this pseudometric to give a shorter proof of the theorem by Pinchasi that the sum of perimeters of 𝑘 convex planar bodies with disjoint interiors contained in a convex body of perimeter 𝑝 and diameter 𝑑 is not greater than 𝑝 + 2(𝑘 − 1)𝑑.
