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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
Since Henrik Strietz’s 1975 paper proving that the lattice Part(𝑛) of all partitions of an 𝑛-element finite set is four-generated, more than half a dozen papers have been devoted to four-element generating sets of this lattice. We prove that each element of Part(𝑛) with height one or two (in particular, each atom) belongs to a four-element generating set. Furthermore, our construction leads to a concise and easy proof of a 1996 result of the author stating that the lattice of partitions of a countably infinite set is four-generated as a complete lattice. In a recent paper “Generating Boolean lattices by few elements and exchanging session keys”, see https://doi.org/10.30755/NSJOM.16637, the author establishes a connection between cryptography and small generating sets of some lattices, including Part(𝑛). Hence, it is worth pointing out that by combining a construction given here with a recent paper by the author, “Four-element generating sets with block count width at most two in partition lattices”, available at https://tinyurl.com/czg-4gw2, we obtain many four-element generating sets of Part(𝑛).
Based on two involutions and a bijection, we combinatorially determine the difference between the number of 𝓁-regular partitions of 𝑛 into an even number of parts and into an odd number of parts for all positive integers 𝑛 and 𝓁 > 1, which extends two recent results due to Ballantine and Merca. As an application, we provide a combinatorial proof of Hickerson’s identity on the number of partitions into an even and odd number of parts.
Let 𝐺 be a connected 𝐾1,5-free graph with 𝑛 vertices. In this paper, we study some optimal sufficient conditions for a connected 𝐾1,5-free graph to have a spanning tree with few leaves and branch vertices in total. In particular, we first prove that if 𝜎5(𝐺) ≥ 𝑛 − 2, then 𝐺 contains a spanning tree with at most seven leaves and branch vertices. After that, we show all graphs 𝐺 which have no spanning tree with at most seven leaves and branch vertices and 𝜎5(𝐺) = 𝑛 − 3.
In this paper, we study controlled fusion frame in tensor product of Hilbert spaces and discuss some of its properties. We describe the resolution of the identity operator on a tensor product of Hilbert spaces using the theory of controlled fusion frame. Finally, we discuss alternative dual with the help of controlled fusion frame in tensor product of Hilbert spaces.
Alfréd Rényi, the founding director of the Mathematical Institute of the Hungarian Academy of Sciences was the first mathematician who proved a density theorem for the zeros of Dirichlet’s 𝐿-functions with variable moduli. This was based on a refinement of the large sieve of Linnik, developed by Rényi himself. He used this to show a weaker form of the binary Goldbach conjecture. His density theorem was the first forerunner of the famous Bombieri–Vinogradov theorem. We give a simple alternative proof of a weaker form of the Bombieri–Vinogradov theorem, based only on classical facts about 𝐿-functions (including Siegel’s theorem) and a simple but ingenious idea of Halász, but without using any form of the large sieve.
In the present paper, we study the asymptotic properties of an exponential-type operator which was recently constructed. It is connected with 𝑝(𝑥) = 𝑥4/3. The main result is a pointwise complete asymptotic expansion valid for locally smooth functions. All coefficients are derived and explicitly given.
Following previous observations on 𝑞-Appell and 𝑞-Lauricella functions, the purpose of this article is to find canonical 𝑞-difference equations for the four intermediate 𝑞-Lauricella functions
We prove that, when 𝑛 goes to infinity, Kostant’s problem has negative answer for almost all simple highest weight modules in the principal block of the BGG category O for the Lie algebra sl𝑛(ℂ).
In this article, we introduce a non-negative integer-valued function that measures the obstruction for converting topological isotopy between two Legendrian knots into a Legendrian isotopy. We refer to this function as the Cost function. We show that the Cost function induces a metric on the set of topologically isotopic Legendrian knots. Hence, the set of topologically isotopic Legendrian knots can be seen as a graph with path-metric given by the Cost function. Legendrian simple knot types are shown to be characterized using the Cost function. We also get a quantitative version of Fuchs–Tabachnikov’s Theorem that says any two Legendrian knots in (𝕊3, 𝜉𝑠𝑡𝑑) in the same topological knot type become Legendrian isotopic after sufficiently many stabilizations [8]. We compute the Cost function for Legendrian simple knots (for example torus knots) and we note the behavior of Cost function for twist knots and cables of torus knots (some of which are Legendrian non-simple). We also construct examples of Legendrian representatives of 2-bridge knots and compute the Cost between them. Further, we investigate the behavior of the Cost function under the connect sum operation. We conclude with some questions about the Cost function, its relation with the standard contact structure, and the topological knot type.
Recent results have provided important functional generalizations, extensions and improvements of the Hardy and Levinson integral inequalities. However, they require some assumptions on the main functions, such as monotonicity or convexity assumptions, which remain somewhat restrictive. In this article, we propose two new ideas of functional generalizations, one based on a series expansion approach and the other on an integral approach. Both achieve the goal of offering adaptable generalizations and extensions of the Hardy and Levinson integral inequalities. They are formulated in two different general theorems, which are proved in detail. Several examples of new integral inequalities are derived.
