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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
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.
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 𝑀𝑘 be the 𝑘-th Mulatu number. Let 𝑟, 𝑠 be non-zero integers with 𝑟 ≥ 1 and 𝑠 ∈ {−1, 1}, let {𝑈𝑛}𝑛≥0 be the generalized Lucas sequence and {𝑉𝑛}𝑛≥0 its companion given respectively by 𝑈𝑛+2 = 𝑟𝑈𝑛+1 + 𝑠𝑈𝑛 and 𝑉𝑛+2 = 𝑟𝑉𝑛+1 + 𝑠𝑉𝑛, with 𝑈0 = 0, 𝑈1 = 1, 𝑉0 = 2, 𝑉1 = 𝑟. In this paper, we give effective bounds for the solutions of the following Diophantine equations 𝑀𝑘 = 𝑈𝓁𝑈𝑚𝑈𝑛 and 𝑀𝑘 = 𝑉𝓁𝑉𝑚𝑉𝑛, 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 (𝑥1, … , 𝑥𝑛) ∈ 𝐸𝑛 such that ║𝑥1║ = … = ║𝑥𝑛║ = 1 and for the multilinear operator norm it holds ‖𝑇 ‖ = |𝑇 (𝑥1, … , 𝑥𝑛)|.
Let 0 ≤ 𝜃 ≤
In this note, we characterize all norm-peak multilinear forms on
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 exp (𝑛, F), is the maximum number of edges in an 𝑛-vertex F -free planar graph.
Let Θ𝑘 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 exp (𝑛, Θ6) ≤ 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 exp (𝑛, Θ6) ≤ 18𝑛/−48𝑛/7 and then we demonstrate the existence of infinitely many positive integer 𝑛 and an 𝑛-vertex Θ6-free planar graph attaining the bound.
Suppose that 𝑇 (𝛼, 𝛽) is an obtuse triangle with base length 1 and with base angles 𝛼 and 𝛽 (where 𝛽 > 90◦). In this note a tight lower bound of the sum of the areas of squares that can parallel cover 𝑇 (𝛼, 𝛽) is given. This result complements the previous lower bound obtained for the triangles with the interior angles at the base of the measure not greater than 90◦.
We show that every positroid of rank 𝑟 ≥ 2 has a good coline. Using the definition of the chromatic number of oriented matroid introduced by J. Nešetřil, R. Nickel, and W. Hochstättler, this shows that every orientation of a positroid of rank at least 2 is 3-colorable.
