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

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

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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.

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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.

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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.

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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.*

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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*.

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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.

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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].

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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.

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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.

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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.

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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.

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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.

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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

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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.

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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.

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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.

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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

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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*.

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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.

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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.

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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.

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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.

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We consider hypersphere x = x(*u, v, w*) in the four dimensional Euclidean space. We calculate the Gauss map, and the curvatures of it. Moreover, we compute the second Laplace-Beltrami operator the hypersphere satisfying Δ^{II}x = *A*x, where *A* ϵ *Mat* (4,4).

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In this paper, we show a Marcinkiewicz type interpolation theorem for Orlicz spaces. As an application, we obtain an existence result for a parabolic equation in divergence form.

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Extending Blaschke and Lebesgue’s classical result in the Euclidean plane, it has been recently proved in spherical and the hyperbolic cases, as well, that Reuleaux triangles have the minimal area among convex domains of constant width *D*. We prove an essentially optimal stability version of this statement in each of the three types of surfaces of constant curvature. In addition, we summarize the fundamental properties of convex bodies of constant width in spaces of constant curvature, and provide a characterization in the hyperbolic case in terms of horospheres.

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Let P be a set of n points in general position in the plane. Let *R* be a set of points disjoint from P such that for every *x, y € P* the line through *x* and *y* contains a point in *R*. We show that if *c* in the plane, then *P* has a special property with respect to the natural group structure on *c*. That is, *P* is contained in a coset of a subgroup *H* of c of cardinality at most |*R*|.

We use the same approach to show a similar result in the case where each of *B* and *G* is a set of n points in general position in the plane and every line through a point in *B* and a point in *G* passes through a point in *R*. This provides a partial answer to a problem of Karasev.

The bound

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Let *E, G* be Fréchet spaces and *F* be a complete locally convex space. It is observed that the existence of a continuous linear not almost bounded operator *T* on *E* into *F* factoring through *G* causes the existence of a common nuclear Köthe subspace of the triple (*E, G, F*). If, in addition, *F* has the property (*y*), then (*E, G, F*) has a common nuclear Köthe quotient.

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In this paper we study the sum *n*, and {*n _{p}
*} is a sequence of integers indexed by primes. Under certain assumptions we show that the aforementioned sum is

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In this paper we derive new inequalities involving the generalized Hardy operator. The obtained results generalized known inequalities involving the Hardy operator. We also get new inequalities involving the classical Hardy–Hilbert inequality.

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The bipartite domination number of a graph is the minimum size of a dominating set that induces a bipartite subgraph. In this paper we initiate the study of this parameter, especially bounds involving the order, the ordinary domination number, and the chromatic number. For example, we show for an isolate-free graph that the bipartite domination number equals the domination number if the graph has maximum degree at most 3; and is at most half the order if the graph is regular, 4-colorable, or has maximum degree at most 5.

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This study proposes a new family of continuous distributions, called the Gudermannian generated family of distributions, based on the Gudermannian function. The statistical properties, including moments, incomplete moments and generating functions, are studied in detail. Simulation studies are performed to discuss and evaluate the maximum likelihood estimations of the model parameters. The regression model of the proposed family considering the heteroscedastic structure of the scale parameter is defined. Three applications on real data sets are implemented to convince the readers in favour of the proposed models.

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Let [ · ] be the fioor function. In this paper, we show that when 1 < c < 37/36, then every sufficiently large positive integer *N* can be represented in the form

where p_{1}, p_{2}, p_{3} are primes close to squares.

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In this article, we study a family of subgraphs of the Farey graph, denoted as *ℱ _{N}
* for every

*N*∈ ℕ. We show that

*ℱ*is connected if and only if

_{N}*N*is either equal to one or a prime power. We introduce a class of continued fractions referred to as

*ℱ*-continued fractions for each

_{N}*N*> 1. We establish a relation between

*ℱ*-continued fractions and certain paths from infinity in the graph

_{N}*ℱ*. Using this correspondence, we discuss the existence and uniqueness of

_{N}*ℱ*-continued fraction expansions of real numbers.

_{N}^{ }

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Given a finite point set *P* in the plane, a subset S⊆P is called an *island* in *P* if conv(S) ⋂ *P = S*. We say that S ⊂ *P* is a *visible island* if the points in S are pairwise visible and S is an island in P. The famous Big-line Big-clique Conjecture states that for any *k ≥* 3 and *l* ≥ 4, there is an integer *n = n(k, l*), such that every finite set of at least *n* points in the plane contains *l* collinear points or *k* pairwise visible points. In this paper, we show that this conjecture is false for visible islands, by replacing each point in a Horton set by a triple of collinear points. Hence, there are arbitrarily large finite point sets in the plane with no 4 collinear members and no visible island of size 13.

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In this article, we define the notion of a generalized open book of a *n*-manifold over the *k*−sphere *S ^{k}
*,

*k < n*. We discuss Lefschetz open book embeddings of Lefschetz open books of closed oriented 4-manifolds into the trivial open book over

*S*of the 7−sphere

^{2}*S*. If X is the double of a bounded achiral Lefschetz fibration over

^{7}*D*, then

^{2}*X*naturally admits a Lefschetz open book given by the bounded achiral Lefschetz fibration. We show that this natural Lefschetz open book of

*X*admits a Lefschetz open book embedding into the trivial open book over

*S*of the 7−sphere

^{2}*S*.

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We show that if a non-degenerate PL map *f* : *N* → *M* lifts to a topological embedding in *N ^{n}
* →

*M*≥

^{m}, m*n*, lifts to a topological embedding in

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This short note deals with polynomial interpolation of complex numbers verifying a Lipschitz condition, performed on consecutive points of a given sequence in the plane. We are interested in those sequences which provide a bound of the error at the first uninterpolated point, depending only on its distance to the last interpolated one.