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

Recently [3] we proved a general zero density theorem for a large class of functions which included among others the Riemann zeta function, Dedekind zeta functions, Dirichlet *𝐿*-functions. The goal of the present work is a (slight) improvement of this general theorem which might lead to stronger results in some applications.

^{ }

This paper serves as a kick-off to address the question of how to define and investigate the stability of bi-continuous semigroups. We will see that the mixed topology is the key concept in this framework.

^{ }

^{ }

^{ }

An endo-commutative algebra is a nonassociative algebra in which the square mapping preserves multiplication. In this paper, we give a complete classification of 2-dimensional endo-commutative straight algebras of rank one over an arbitrary non-trivial field, where a straight algebra of dimension 2 satisfies the condition that there exists an element *x* such that *x* and *x*
^{2} are linearly independent. We list all multiplication tables of the algebras up to isomorphism.

^{ }

In this paper, we consider the simultaneous sign changes of coefficients of Rankin–Selberg *L*-functions associated to two distinct Hecke eigenforms supported at positive integers represented by some certain primitive reduced integral binary quadratic form with negative discriminant *D*. We provide a quantitative result for the number of sign changes of such sequence in the interval (*x*, 2*x*] for sufficiently large *x*.

^{ }

^{ }

^{ }

In this paper, we derive several asymptotic formulas for the sum of *d*(gcd(*m,n*)), where *d*(*n*) is the divisor function and *m,n* are in Piatetski-Shapiro and Beatty sequences.

^{ }

Let *𝑛* ∈ ℕ. An element (*x*
_{1}, … , *x*
_{𝑛}) ∈ *E ^{n}
* is called a

*norming point*of

*T*∈

*) if ‖*

^{n}E*x*

_{1}‖ = ⋯ = ‖

*x*‖ = 1 and |

_{n}*T*(

*x*

_{1}, … ,

*x*)| = ‖

_{n}*T*‖, where

*) denotes the space of all continuous*

^{n}E*n*-linear forms on

*E*. For

*T*∈

*), we define*

^{n}ENorm(*T*) = {(*x*
_{1}, … , *x*
_{n}) ∈ *E ^{n}
* ∶ (

*x*

_{1}, … ,

*x*

_{n}) is a norming point of

*T*}.

Norm(*T*) is called the *norming set* of *T*. We classify Norm(*T*) for every *T* ∈ ^{2}
*𝑑*
_{∗}(1, *w*)^{2}), where *𝑑*
_{∗}(1, *w*)^{2} = ℝ^{2} with the octagonal norm of weight 0 < *w* < 1 endowed with

^{ }

^{ }

^{ }

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.

^{ }

We construct an algebra of dimension 2^{ℵ0} 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.

^{ }

^{ }

^{ }

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.

^{ }

^{ }

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

^{ }

^{ }

Lovejoy introduced the partition function _{𝑙}-crank of 𝑙-regular overpartitions and give combinatorial interpretations for some congruences of _{3}-crank of 3-regular overpartitions.

^{ }

^{ }

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 𝔹^{𝑛}. 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.

^{ }

This article studies a new class of monomial ideals associated with a simple graph 𝐺, called generalized edge ideal, denoted by 𝐼_{𝑔}(𝐺). Assuming that all the vertices 𝑥 have an exponent greater than 1 in 𝐼_{𝑔}(𝐺), we completely characterize the graph 𝐺 for which 𝐼_{𝑔}(𝐺) is integrally closed, and show that this is equivalent to 𝐼_{𝑔}(𝐺) being normal i.e., all integral powers of 𝐼_{𝑔}(𝐺) are integrally clased. We also give a necessary and sufficient condition for

^{ }

^{ }

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 𝑐𝑛^{2} edges. Here we study two refinements of 𝑞(𝐺) arising from the analogous (7, 4)-conjecture. The first is 𝑞_{𝐴}(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 𝑞_{𝐵}(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 𝑞_{𝐴}(𝐺), 𝑞_{𝐵}(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 _{𝑞}𝐴(𝐺) and _{𝑞}𝐵(𝐺), we study these parameters for some graphs.

We note that 𝑞_{𝐴}(𝐺) has already been extensively studied from various motivations. However, as far as we know the behavior of 𝑞_{𝐵}(𝐺) is studied here for the first time.

^{ }

^{ }

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.

^{ }

^{ }

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.

^{ }

Over integral domains of characteristics different from 2, we determine all the matrices

^{ }

We present generalizations of the Pinelis extension of Stolarsky’s inequality and its reverse. In particular, a new Stolarsky-type inequality is obtained. We study the properties of the linear functional related to the new Stolarsky-type inequality, and finally apply these new results in the theory of fractional integrals.

^{ }

^{ }

In this paper, we consider the Feuerbach point and the Feuerbach line of a triangle in the isotropic plane, and investigate some properties of these concepts and their relationships with other elements of a triangle in the isotropic plane. We also compare these relationships in Euclidean and isotropic cases.

^{ }

^{ }

^{ }

^{ }

We define the order of the double hypergeometric series, investigate the properties of the new confluent Kampé de Fériet series, and build systems of partial differential equations that satisfy the new Kampé de Fériet series. We solve the Cauchy problem for a degenerate hyperbolic equation of the second kind with a spectral parameter using the high-order Kampé de Fériet series. Thanks to the properties of the introduced Kampé de Fériet series, it is possible to obtain a solution to the problem in explicit forms.

^{ }

Let 𝔼*
^{𝑑}
* denote the 𝑑-dimensional Euclidean space. The 𝑟-ball body generated by a given set in 𝔼

*is the intersection of balls of radius 𝑟 centered at the points of the given set. The author [Discrete Optimization 44/1 (2022), Paper No. 100539] proved the following Blaschke–Santaló-type inequalities for 𝑟-ball bodies: for all 0 <*

^{𝑑}*𝑘*<

*𝑑*and for any set of given 𝑑-dimensional volume in 𝔼

*the 𝑘-th intrinsic volume of the 𝑟-ball body generated by the set becomes maximal if the set is a ball. In this note we give a new proof showing also the uniqueness of the maximizer. Some applications and related questions are mentioned as well.*

^{𝑑}^{ }

^{ }

We discuss the outline of the shapes of graphs of χ ^{2} statistics for distributions of leading digits of irrational rotations under some conditions on *m*th convergent. We give some estimates of important coefficients *L _{k}
*’s, which determine the graphical shapes of χ

^{2}statistics. This means that the denominator

*q*of

_{m}*m*th convergent and the large partial quotient

*a*

_{m}_{+1}determine the outline of shapes of graphs, when we observe values of χ

^{2}statistics with step

*q*.

_{m}^{ }

^{ }

In this note, we introduce the concept of semi-*-IFP, the involutive version of semi-IFP, which is a generalization of quasi-*-IFP and *-reducedness of *-rings. We study the basic structure and properties of *-rings having semi-*-IFP and give results for IFPs in rings with involution. Several results and counterexamples are stated to connect the involutive versions of IFP. We discuss the conditions for the involutive IFPs to be extended into *-subrings of the ring of upper triangular matrices. In *-rings with quasi-*-IFP, it is shown that Köthe’s conjecture has a strong affirmative solution. We investigate its related properties and the relationship between *-rings with quasi-*-IFP and *-Armendariz properties.

^{ }

^{ }

In the present paper, we establish the convergence rates of the single logarithm and the iterated logarithm for martingale differences which give some further results for the open question in Stoica [6].

^{ }

^{ }

We give a full, correct proof of the following result, earlier claimed in [1]. If the Continuum Hypothesis holds then there is a coloring of the plane with countably many colors, with no monocolored right triangle.

^{ }

^{ }

^{ }

The famous Hadwiger–Nelson problem asks for the minimum number of colors needed to color the points of the Euclidean plane so that no two points unit distance apart are assigned the same color. In this note we consider a variant of the problem in Minkowski metric planes, where the unit circle is a regular polygon of even and at most 22 vertices. We present a simple lattice–sublattice coloring scheme that uses 6 colors, proving that the chromatic number of the Minkowski planes above are at most 6. This result is new for regular polygons having more than 8 vertices.

^{ }

^{ }

^{ }

^{ }

John Horton Conway stood out from many famous mathematicians for his love of games and puzzles. Among others, he is known for inventing the two-player topological games called Sprouts and Brussels Sprouts. These games start with *n* spots (*n* crosses resp.), have simple rules, last for finitely many moves, and the player who makes the last move wins. In the misère versions, the player who makes the last move loses. In this paper, we make Brussels Sprouts colored, preserving the aesthetic interest and balance of the game. In contrast to the original Sprouts, Colored Brussels Sprouts allows mathematical analysis without computer programming and has winning strategies for a large family of the number of spots.

^{ }

Given graphs *H* and *F*, the generalized Turán number ex(*n, H*, *F*) is the largest number of copies of *H* in *n*-vertex *F*-free graphs. Stability refers to the usual phenomenon that if an *n*-vertex *F*-free graph *G* contains almost ex(*n, H*, *F*) copies of *H*, then *G* is in some sense similar to some extremal graph. We obtain new stability results for generalized Turán problems and derive several new exact results.

^{ }

Let *T* be a tree. The *reducible stem* of *T* is the smallest subtree that contains all branch vertices of *T*. In this paper, we first use a new technique of Gould and Shull [5] to state a new short proof for a result of Kano et al. [10] on the spanning tree with a bounded number of leaves in a claw-free graph. After that, we use a similar idea to prove a sharp sufficient condition for a claw-free graph having a spanning tree whose reducible stem has few leaves.

^{ }

Let *n* ∈ ℕ. An element (*x*
_{1}, … , *x _{n}
*) ∈

*E*is called a

^{n}*norming point*of

*n*-linear forms on

*E*. For

Norm(*T*) is called the *norming set* of *T*.

Let

In this paper, we classify Norm(*T*) for every

^{ }

This article indicates another set-theoretic formula, solely in terms of union and intersection, for the set of the limits of any given sequence (net, in general) in an arbitrary *T*
_{1} space; this representation in particular gives a new characterization of a *T*
_{1} space.

^{ }

^{ }

We give all solutions of completely multiplicative functions ƒ , g, for which the equation *Ag*(*n* + 1) = *B*ƒ (*n*) + *C* holds for every *n* ∈ ℕ. We also study the equation *G*(*p* + 1) = *F*(*p* − 1) + *D* and we prove some results concerning it.

^{ }

We consider a graph whose vertices are legally colored using *k* colors and ask if the graph contains a *k*-clique. As it turns out this very special type of *k*-clique problem is in an intimate connection with constructing schedules. The practicality this clique search based construction of schedules is checked by carrying out numerical experiments.

^{ }

Assume that *A _{j}
*,

*j*∈ {1, … ,

*m*} are positive definite matrices of order

*n*. In this paper we prove among others that, if 0 <

*l I*≤

_{n}*A*,

_{j}*j*∈ {1, … ,

*m*} in the operator order, for some positive constant

*l*, and

*I*is the unity matrix of order

_{n}*n*, then

where *Pk* ≥ 0 for *k* ϵ {1, …, *m*} and

^{ }

The evolute of a conic in the pseudo-Euclidean plane is the locus of centers of all its osculating circles. It’s a curve of order six and class four in general case. In this paper we discuss and compute the order and class of evolutes of different types of conics. We will highlight those cases that have no analogy in the Euclidean plane.

^{ }

^{ }

^{ }

Fast [12] is credited with pioneering the field of statistical convergence. This topic has been researched in many spaces such as topological spaces, cone metric spaces, and so on (see, for example [19, 21]). A cone metric space was proposed by Huang and Zhang [17]. The primary distinction between a cone metric and a metric is that a cone metric is valued in an ordered Banach space. Li et al. [21] investigated the definitions of statistical convergence and statistical boundedness of a sequence in a cone metric space. Recently, Sakaoğlu and Yurdakadim [29] have introduced the concepts of quasi-statistical convergence. The notion of quasi I-statistical convergence for triple and multiple index sequences in cone metric spaces on topological vector spaces is introduced in this study, and we also examine certain theorems connected to quasi I-statistically convergent multiple sequences. Finally, we will provide some findings based on these theorems.

^{ }

^{ }

For a graph *G*, we define the lower bipartite number LB(*G*) as the minimum order of a maximal induced bipartite subgraph of *G*. We study the parameter, and the related parameter bipartite domination, providing bounds both in general graphs and in some graph families. For example, we show that there are arbitrarily large 4-connected planar graphs *G* with LB(*G*) = 4 but a 5-connected planar graph has linear LB(*G*). We also show that if *G* is a maximal outerplanar graph of order *n*, then LB(*G*) lies between (*n* + 2)/3 and 2 *n*/3, and these bounds are sharp.

^{ }

^{ }

The motions of a bar structure consisting of two congruent tetrahedra are investigated, whose edges in their basic position are the face diagonals of a rectangular parallelepiped. The constraint of the motion is the following: the originally intersecting edges have to remain coplanar. All finite motions of our bar structure are determined. This generalizes our earlier work, where we did the same for the case when the rectangular parallelepiped was a cube. At the end of the paper we point out three further possibilities to generalize the question about the cube, and give for them examples of finite motions.

^{ }

^{ }

We study the path behavior of the symmetric walk on some special comb-type subsets of ℤ^{2} which are obtained from ℤ^{2} by generalizing the comb having finitely many horizontal lines instead of one.

^{ }

^{ }

In a typical maximum clique search algorithm when optimality testing is inconclusive a forking takes place. The instance is divided into smaller ones. This is the branching step of the procedure. In order to ensure a balanced work load for the processors for parallel algorithms it is essential that the resulting smaller problems are do not overly vary in difficulty. The so-called splitting partitions of the nodes of the given graph were introduced earlier to meliorate this problem. The paper proposes a splitting partition of the edges for the same purpose. In the lack of available theoretical tools we assess the practical feasibility of constructing suboptimal splitting edge partitions by carrying out numerical experiments. While working with splitting partitions we have realized that they can be utilized as preconditioning tools preliminary to a large scale clique search. The paper will discuss this new found role of the splitting edge partitions as well.

^{ }

^{ }

We prove that for any collection *F* of *n* ≥ 2 pairwise disjoint compact convex sets in the plane there is a pair of sets *A* and *B* in *F* such that any line that separates *A* from *B* separates either *A* or *B* from a subcollection of *F* with at least *n*/18 sets.

^{ }

In this paper, we study the existence of positive solutions for a system of nonlinear fractional differential equations. The results are based upon the fixed-point theorem of cone expansion and compression type due to Krasnosel’skill. Moreover, Our results generalize and include some known results.

^{ }

^{ }

^{ }

Criteria for a diffeomorphism of a smooth manifold *M* to be lifted to a linear automorphism of a given real vector bundle *p : V → M*, are stated. Examples are included and the metric and complex vector-bundle cases are also considered.

^{ }

^{ }

Let *X* be an irreducible complex projective variety of dimension *n* ≥ 1. Let D be a Cartier divisor on *X* such that *H ^{i}(X, O_{X} (mD))* = 0 for

*m*> 0 and for all

*i*> 0, then it is not true in general that

*D*is a nef divisor (cf. [4]). Also, in general, effective divisors on smooth surfaces are not necessarily nef (they are nef provided they are semiample). In this article, we show that, if

*X*is a smooth surface of general type and

*C*is a smooth hyperplane section of it, then for any non-zero effective divisor

*D*on

*X*satisfying

*H*= 0 for all

^{1}(X, O_{X}(mD))*m > C.K*,

_{X}*D*is a nef divisor.

^{ }

^{ }

In this paper, we introduce the notion of a Gel’fand Γ-semiring and discuss the various characterization of simple, *k*-ideal, strong ideal, *t*-small elements and additively cancellative elements of a Gel’fand Γ-semiring *R*, and prove that the set of additively cancellative elements, set of all *t*-small elements of *R* and set of all maximal ideal of *R* are strong ideals. Further, let *R* be a simple Gel’fand Γ-semiring and 1 ≠ *t* ∈ *R*. Let *M* be the set of all maximal left (right) ideals of *R*. Then an element *x* of *R* is *t*-small if and only if it belongs to every maximal one sided left (right)ideal of *R* containing *t*.

^{ }

For a continuous and positive function *w*(λ), *λ >
* 0 and

*μ*a positive measure on (0, ∞) we consider the following

*integral transform*

where the integral is assumed to exist for *t* > 0.

We show among others that *D*(*w, μ*) is operator convex on (0, ∞). From this we derive that, if *f* : [0, ∞) → **R** is an operator monotone function on [0, ∞), then the function [*f*(0) -*f*(*t*)] *t*
^{-1} is operator convex on (0, ∞). Also, if *f* : [0, ∞) → **R** is an operator convex function on [0, ∞), then the function

under some natural assumptions for the positive operators A and B are given. Examples for power, exponential and logarithmic functions are also provided.

^{ }

Problem 2 of Welsh’s 1976 text *Matroid Theory*, asking for criteria telling when two families of sets have a common transversal, is solved.

Another unsolved problem in the text *Matroid Theory*, on whether the “join” of two non-decreasing submodular functions is submodular, is answered in the negative. This resolves an issue first raised by Pym and Perfect in 1970.

^{ }

^{ }

With distributed computing and mobile applications becoming ever more prevalent, synchronizing diverging replicas of the same data is a common problem. Reconciliation – bringing two replicas of the same data structure as close as possible without overriding local changes – is investigated in an algebraic model. Our approach is to consider two sequences of simple commands that describe the changes in the replicas compared to the original structure, and then determine the maximal subsequences of each that can be propagated to the other. The proposed command set is shown to be functionally complete, and an update detection algorithm is presented which produces a command sequence transforming the original data structure into the replica while traversing both simultaneously. Syntactical characterization is provided in terms of a rewriting system for semantically equivalent command sequences. Algebraic properties of sequence pairs that are applicable to the same data structure are investigated. Based on these results the reconciliation problem is shown to have a unique maximal solution. In addition, syntactical properties of the maximal solution allow for an efficient algorithm that produces it.

^{ }

^{ }

^{ }

This manuscript deals with the global existence and asymptotic behavior of solutions for a Kirchhoff beam equation with internal damping. The existence of solutions is obtained by using the Faedo-Galerkin method. Exponential stability is proved by applying Nakao’s theorem.