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

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 𝑆^{𝑑−1}, where a zone is the intersection of an origin symmetric Euclidean plank with 𝑆^{𝑑−1}. We prove that, for sufficiently large 𝑛, it is possible to arrange 𝑛 congruent zones of suitable width on 𝑆^{𝑑−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 𝑆^{𝑑−1} by 𝑛 congruent zones such that each point of 𝑆^{𝑑−1} belongs to at most 𝐴_{𝑑} ln 𝑛 zones, where the 𝐴_{𝑑} 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 𝑆^{𝑑−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.

We study translative arrangements of centrally symmetric convex domains in the plane (resp., of congruent balls in the Euclidean 3-space) that neither pack nor cover. We define their soft density depending on a soft parameter and prove that the largest soft density for soft translative packings of a centrally symmetric convex domain with 3-fold rotational symmetry and given soft parameter is obtained for a proper soft lattice packing. Furthermore, we show that among the soft lattice packings of congruent soft balls with given soft parameter the soft density is locally maximal for the corresponding face centered cubic (FCC) lattice.

Let 𝑀_{𝑘} be the 𝑘-th Mulatu number. Let 𝑟, 𝑠 be non-zero integers with 𝑟 ≥ 1 and 𝑠 ∈ {−1, 1}, let {𝑈_{𝑛}}_{𝑛≥0} be the generalized Lucas sequence and {𝑉_{𝑛}}_{𝑛≥0} its companion given respectively by 𝑈_{𝑛+2} = 𝑟𝑈_{𝑛+1} + 𝑠𝑈_{𝑛} and 𝑉_{𝑛+2} = 𝑟𝑉_{𝑛+1} + 𝑠𝑉_{𝑛}, with 𝑈_{0} = 0, 𝑈_{1} = 1, 𝑉_{0} = 2, 𝑉_{1} = 𝑟. In this paper, we give effective bounds for the solutions of the following Diophantine equations 𝑀_{𝑘} = 𝑈_{𝓁}𝑈_{𝑚}𝑈_{𝑛} and 𝑀_{𝑘} = 𝑉_{𝓁}𝑉_{𝑚}𝑉_{𝑛}, where 𝓁, 𝑚, 𝑛 and 𝑘 are nonnegative integers and 𝓁 ≤ 𝑚 ≤ 𝑛. Then, we explicitly solve the above Diophantine equations for the Fibonacci, Pell, Balancing sequences and their companions respectively.

Let 𝑛 ≥ 2. A continuous 𝑛-linear form 𝑇 on a Banach space 𝐸 is called *norm-peak* if there is a unique (𝑥_{1}, … , 𝑥_{𝑛}) ∈ 𝐸^{𝑛} such that ║𝑥_{1}║ = … = ║𝑥_{𝑛}║ = 1 and for the multilinear operator norm it holds ‖𝑇 ‖ = |𝑇 (𝑥_{1}, … , 𝑥_{𝑛})|.

Let 0 ≤ 𝜃 ≤ ^{2} with the rotated supremum norm ‖(𝑥, 𝑦)‖_{(∞,𝜃)} = max {|𝑥 cos 𝜃 + 𝑦 sin 𝜃|, |𝑥 sin 𝜃 − 𝑦 cos 𝜃|}.

In this note, we characterize all norm-peak multilinear forms on ^{2} with the 𝓁_{𝑝}-norm for 𝑝 = 1, ∞.

In this paper we introduce a construction for a weighted CW complex (and the associated lattice cohomology) corresponding to partially ordered sets with some additional structure. This is a generalization of the construction seen in [4] where we started from a system of subspaces of a given vector space. We then proceed to prove some basic properties of this construction that are in many ways analogous to those seen in the case of subspaces, but some aspects of the construction result in complexities not present in that scenario.

Let *F* be a nonempty family of graphs. A graph 𝐺 is called *F* -*free* if it contains no graph from *F* as a subgraph. For a positive integer 𝑛, the *planar Turán number* of *F*, denoted by ex_{p} (𝑛, *F*), is the maximum number of edges in an 𝑛-vertex *F* -free planar graph.

Let Θ_{𝑘} be the family of Theta graphs on 𝑘 ≥ 4 vertices, that is, graphs obtained by joining a pair of non-consecutive of a 𝑘-cycle with an edge. Lan, Shi and Song determined an upper bound ex_{p} (𝑛, Θ_{6}) ≤ 18𝑛/7−36𝑛/7, but for large 𝑛, they did not verify that the bound is sharp. In this paper, we improve their bound by proving ex_{p} (𝑛, Θ_{6}) ≤ 18𝑛/−48𝑛/7 and then we demonstrate the existence of infinitely many positive integer 𝑛 and an 𝑛-vertex Θ_{6}-free planar graph attaining the bound.