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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

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In this paper, we introduce and study the class of k-strictly quasi-Fredholm linear relations on Banach spaces for nonnegative integer k. Then we investigate its robustness through perturbation by finite rank operators.

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We construct an algebra of dimension 2ℵ0 consisting only of functions which in no point possess a finite one-sided derivative. We further show that some well known nowhere differentiable functions generate algebras, which contain functions which are differentiable at some points, but where for all functions in the algebra the set of points of differentiability is quite small.

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Studia Scientiarum Mathematicarum Hungarica
Authors:
Jesús A. De Loera
,
Christopher O’Neill
, and
Chengyang Wang

In this paper, we explore affine semigroup versions of the convex geometry theorems of Helly, Tverberg, and Carathéodory. Additionally, we develop a new theory of colored affine semigroups, where the semigroup generators each receive a color and the elements of the semigroup take into account the colors used (the classical theory of affine semigroups coincides with the case in which all generators have the same color). We prove an analog of Tverberg’s theorem and colorful Helly’s theorem for semigroups, as well as a version of colorful Carathéodory’s theorem for cones. We also demonstrate that colored numerical semigroups are particularly rich by introducing a colored version of the Frobenius number.

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We prove the endomorphism conjecture for graded posets with largest Whitney number at most 4.

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Lovejoy introduced the partition function A l ¯ n as the number of 𝑙-regular overpartitions of 𝑛. Andrews defined (𝑘, 𝑖)-singular overpartitions counted by the partition function C ¯ k , i n , and pointed out that C ¯ 3 , 1 n = A 3 ¯ n . Meanwhile, Andrews derived an interesting divisibility property that C ¯ 3 , 1 9 n + 3 C ¯ 3 , 1 9 n + 6 0 (mod 3). Recently, we constructed the partition statistic 𝑟𝑙-crank of 𝑙-regular overpartitions and give combinatorial interpretations for some congruences of A l ¯ n as well as the congruences of Andrews. In this paper, we aim to prove some equalities for the 𝑟3-crank of 3-regular overpartitions.

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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.

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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 I g G = I g G ¯ when 𝐺 is the star-shaped graph. Finally, we give a necessary and sufficient condition when the generalized edge ideal of a complete graph is integrally closed.

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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.

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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.

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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.

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