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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
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.
Let 𝑇 be a tree, a vertex of degree one is called a leaf. The set of all leaves of 𝑇 is denoted by Leaf(𝑇). The subtree 𝑇 − Leaf(𝑇) of 𝑇 is called the stem of 𝑇 and denoted by Stem(𝑇). A tree 𝑇 is called a caterpillar if Stem(𝑇) is a path. In this paper, we give two sufficient conditions for a connected graph to have a spanning tree whose stem is a caterpillar. We also give some examples to show that these conditions are sharp.
We revisit the problem of property testing for convex position for point sets in ℝ𝑑. Our results draw from previous ideas of Czumaj, Sohler, and Ziegler (2000). First, their testing algorithm is redesigned and its analysis is revised for correctness. Second, its functionality is expanded by (i) exhibiting both negative and positive certificates along with the convexity determination, and (ii) significantly extending the input range for moderate and higher dimensions.
The behavior of the randomized tester on input set 𝑃 ⊂ ℝ𝑑 is as follows: (i) if 𝑃 is in convex position, it accepts; (ii) if 𝑃 is far from convex position, with probability at least 2/3, it rejects and outputs a (𝑑 +2)-point witness of non-convexity as a negative certificate; (iii) if 𝑃 is close to convex position, with probability at least 2/3, it accepts and outputs a subset in convex position that is a suitable approximation of the largest subset in convex position. The algorithm examines a sublinear number of points and runs in subquadratic time for every fixed dimension 𝑑.
We prove zero density theorems for Dedekind zeta functions in the vicinity of the line Re s = 1, improving an earlier result of W. Staś.
A positive integer
Let (𝑃𝑛)𝑛≥0 and (𝑄𝑛)𝑛≥0 be the Pell and Pell–Lucas sequences. Let 𝑏 be a positive integer such that 𝑏 ≥ 2. In this paper, we prove that the following two Diophantine equations 𝑃𝑛 = 𝑏𝑑𝑃𝑚 + 𝑄𝑘 and 𝑃𝑛 = 𝑏𝑑𝑄𝑚 + 𝑃𝑘 with 𝑑, the number of digits of 𝑃𝑘 or 𝑄𝑘 in base 𝑏, have only finitely many solutions in nonnegative integers (𝑚, 𝑛, 𝑘, 𝑏, 𝑑). Also, we explicitly determine these solutions in cases 2 ≤ 𝑏 ≤ 10.
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 𝟐𝑛 ⊕ 𝟏 can be characterized by the property of not having a *-homomorphism onto 𝟐𝑖 ⊕ 𝟏 for 1 < 𝑖 < 𝑛.
In this article, their question from 1971 is answered.
This paper introduces and examines the concept of a *-Rickart *-ring, and proves that every Rickart *-ring is also a *-Rickart *-ring. A necessary and sufficient condition for a *-Rickart *-ring to be a Rickart *-ring is also provided. The relationship between *-Rickart *-rings and *-Baer *-rings is investigated, and several properties of *-Rickart *-rings are presented. The paper demonstrates that the property of *-Rickart extends to both the center and *-corners of a *-ring, and investigates the extension of a *-Rickart *-ring to its polynomial *-ring. Additionally, *-Rickart *-rings with descending chain condition on *-biideals are studied, and all *-Rickart (*-Baer) *-rings with finitely many elements are classified.
