# 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

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.

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

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.

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.

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.

For a lattice *L* of finite length *n*, let RCSub(*L*) be the collection consisting of the empty set and those sublattices of *L* that are closed under taking relative complements. That is, a subset *X* of *L* belongs to RCSub(*L*) if and only if *X* is join-closed, meet-closed, and whenever {*a, x, b*} *⊆ S, y* ∈ *L, x* ∧ *y* = *a*, and *x* ∨ *y* = *b*, then *y* ∈ *S*. We prove that (1) the poset RCSub(*L*) with respect to set inclusion is lattice of length *n* + 1, (2) if RCSub(*L*) is a ranked lattice and *L* is modular, then *L* is 2-distributive in András P. Huhn’s sense, and (3) if *L* is distributive, then RCSub(*L*) is a ranked lattice.

In this paper, centralizing (semi-centralizing) and commuting (semi-commuting) derivations of semirings are characterized. The action of these derivations on Lie ideals is also discussed and as a consequence, some significant results are proved. In addition, Posner’s commutativity theorem is generalized for Lie ideals of semirings and this result is also extended to the case of centralizing (semi-centralizing) derivations of prime semirings. Further, we observe that if there exists a skew-commuting (skew-centralizing) derivation *D* of *S*, then *D* = 0. It is also proved that for any two derivations *d*
_{1} and *d*
_{2} of a prime semiring *S* with char *S* ≠ 2 and *x*
^{d1}
*x*
^{d2} = 0, for all *x* ∈ *S* implies either *d*
_{1} = 0 or *d*
_{2} = 0.

We offer new properties of the special Gini mean *S*(*a, b*) = *a ^{a}*

^{/(}

^{a}^{+}

^{b}^{)}⋅

*b*

^{b}^{/(}

^{a}^{+}

^{b}^{)}, in connections with other special means of two arguments.

We treat a variation of graph domination which involves a partition (*V*
_{1}, *V*
_{2},..., *V _{k}*) of the vertex set of a graph

*G*and domination of each partition class

*V*

_{i}over distance

*d*where all vertices and edges of

*G*may be used in the domination process. Strict upper bounds and extremal graphs are presented; the results are collected in three handy tables. Further, we compare a high number of partition classes and the number of dominators needed.

Proctor and Scoppetta conjectured that

(1) there exists an infinite locally finite poset that satisfies their conditions VT and NTC but not SIS;

(2) there exists an infinite locally finite poset satisfying their conditions D3

^{-}C and D3MF but not both VT and FT; and(3) there exists an infinite locally finite poset satisfying their conditions D3

^{-}C and D3MD but not NCC.

In this note, the conjecture of Proctor and Scoppetta, which is related to *d*-complete posets, is proven.

In this paper we introduce the concept of the Hamilton triangle of a given triangle in an isotropic plane and investigate a number of important properties of this concept. We prove that the Hamilton triangle is homological with the observed triangle and with its contact and complementary triangles. We also consider some interesting statements about the relationships between the Hamilton triangle and some other significant elements of the triangle, like e.g. the Euler and the Feuerbach line, the Steiner ellipse and the tangential triangle.

We prove certain Menon-type identities associated with the subsets of the set {1, 2,..., n} and related to the functions *f, f _{k}*, Ф and Ф

*, defined and investigated by Nathanson.*

_{k}Generalizing results of Schatte [11] and Atlagh and Weber [2], in this paper we give conditions for a sequence of random variables to satisfy the almost sure central limit theorem along a given sequence of integers.

In the 1980’s the author proved lower bounds for the mean value of the modulus of the error term of the prime number theorem and other important number theoretic functions whose oscillation is in connection with the zeros of the Riemann zeta function. In the present work a general theorem is shown in a simple way which gives a lower bound for the mentioned mean value as a function of a hypothetical pole of the Mellin transform of the function. The conditions are amply satisfied for the Riemann zeta function. In such a way the results recover the earlier ones (even in a slightly sharper form). The obtained estimates are often optimal apart from a constant factor, at least under reasonable conditions as the Riemann Hypothesis. This is the case, in particular, for the error term of the prime number theorem.

In this paper we establish some Ostrowski type inequalities for double integral mean of absolutely continuous functions. An application for special means is given as well.

We prove the weak consistency of the trimmed least square estimator of the covariance parameter of an AR(1) process with stable errors.

The ultrapower *T** of an arbitrary ordered set *T* is introduced as an infinitesimal extension of *T*. It is obtained as the set of equivalence classes of the sequences in *T*, where the corresponding relation is generated by a free ultrafilter on the set of natural numbers. It is established that *T** always satisfies Cantor’s property, while one can give the necessary and sufficient conditions for *T* so that *T** would be complete or it would fulfill the open completeness property, respectively. Namely, the density of the original set determines the open completeness of the extension, while independently, the completeness of *T** is determined by the cardinality of *T*.

We prove a theorem on the preservation of inequalities between functions of a special form after differentiation on an ellipse. In particular, we obtain generalizations of the Duffin–Schaeffer inequality and the Vidensky inequality for the first and second derivatives of algebraic polynomials to an ellipse.

In this paper we work out a Riemann–von Mangoldt type formula for the summatory function

A congruence is defined for a matroid. This leads to suitable versions of the algebraic isomorphism theorems for matroids. As an application of the congruence theory for matroids, a version of Birkhoff’s Theorem for matroids is given which shows that every nontrivial matroid is a subdirect product of subdirectly irreducible matroids.

Let (*M*, [*g*]) be a Weyl manifold and TM be its tangent bundle equipped with Riemannian *g*−natural metrics which are linear combinations of Sasaki, horizontal and vertical lifts of the base metric with constant coefficients. The aim of this paper is to construct a Weyl structure on TM and to show that TM cannot be Einstein-Weyl even if (*M, g*) is fiat.

We give all functions ƒ , E: ℕ → ℂ which satisfy the relation

for every a, b, c ∈ ℕ, where h ≥ 0 is an integers and K is a complex number. If n cannot be written as a^{2} + b^{2} + c^{2} + h for suitable a, b, c ∈ ℕ, then ƒ (n) is not determined. This is more complicated if we assume that ƒ and E are multiplicative functions.

In this article, we study a fractional control problem that models the maximization of the profit obtained by exploiting a certain resource whose dynamics are governed by the fractional logistic equation. Due to the singularity of this problem, we develop different resolution techniques, both for the classical case and for the fractional case. We perform several numerical simulations to make a comparison between both cases.

The main aim of this paper is to prove that the nonnegativity of the Riesz’s logarithmic kernels with respect to the Walsh– Kaczmarz system fails to hold.

Binary groups are a meaningful step up from non-associative rings and nearrings. It makes sense to study them in terms of their nearrings of zero-fixing polynomial maps. As this involves algebras of a more specialized nature these are looked into in sections three and four. One of the main theorems of this paper occurs in section five where it is shown that a binary group *V* is a *P*
_{0}(*V*) ring module if, and only if, it is a rather restricted form of non-associative ring. Properties of these non-associative rings (called terminal rings) are investigated in sections six and seven. The finite case is of special interest since here terminal rings of odd order really are quite restricted. Sections eight to thirteen are taken up with the study of terminal rings of order *p*
^{n} (*p* an odd prime and *n* ≥ 1 an integer ≤ 7).

Column-row products have zero determinant over any commutative ring. In this paper we discuss the converse. For domains, we show that this yields a characterization of pre-Schreier rings, and for rings with zero divisors we show that reduced pre-Schreier rings have this property.

Finally, for the rings of integers modulo *n*, we determine the 2x2 matrices which are (or not) full and their numbers.

For a continuous and positive function w(λ), λ > 0 and μ a positive measure on (0, ∞) we consider the following*monotonic integral transform*

where the integral is assumed to exist for*T* a positive operator on a complex Hilbert space*H*. We show among others that, if β ≥ A, B ≥ α > 0, and 0 < δ ≤ (B − A)^{2} ≤ Δ for some constants α, β, δ, Δ, then

and

where

Applications for power function and logarithm are also provided.

Let ƒ be analytic in the unit disk B and normalized so that ƒ (z) = z + a_{2}z^{2} + a_{3}z^{3} +܁܁܁. In this paper, we give upper bounds of the Hankel determinant of second order for the classes of starlike functions of order α, Ozaki close-to-convex functions and two other classes of analytic functions. Some of the estimates are sharp.

The authors have studied the curvature of the focal conic in the isotropic plane and the form of the circle of curvature at its points has been obtained. Hereby, we discuss several properties of such circles of curvature at the points of a parabola in the isotropic plane.

Let *k* ≥ 1. A *Sperner k-family* is a maximum-sized subset of a finite poset that contains no chain with *k* + 1 elements. In 1976 Greene and Kleitman defined a lattice-ordering on the set *S _{k}*(

*P*) of Sperner

*k*-families of a fifinite poset

*P*and posed the problem: “Characterize and interpret the join- and meet-irreducible elements of

*S*(

_{k}*P*),” adding, “This has apparently not been done even for the case

*k*= 1.”

In this article, the case *k* = 1 is done.

A linear operator on a Hilbert space *S* is shown to be densely defined and closed if and only if

In a more general setup, we can consider relations instead of operators and we prove that in this situation a similar result holds. We give a necessary and sufficient condition for a linear relation to be densely defined and self-adjoint.

Let *X* be a topological space. For any positive integer *n*, we consider the *n*-fold symmetric product of *X*, ℱ* _{n}*(

*X*), consisting of all nonempty subsets of

*X*with at most

*n*points; and for a given function

*ƒ*:

*X*→

*X*, we consider the induced functions ℱ

*(*

_{n}*ƒ*): ℱ

*(*

_{n}*X*) → ℱ

*(*

_{n}*X*). Let

*M*be one of the following classes of functions: exact, transitive, ℤ-transitive, ℤ

_{+}-transitive, mixing, weakly mixing, chaotic, turbulent, strongly transitive, totally transitive, orbit-transitive, strictly orbit-transitive, ω-transitive, minimal,

*I N, T T*

_{++}, semi-open and irreducible. In this paper we study the relationship between the following statements:

*ƒ*∈

*M*and ℱ

*(*

_{n}*ƒ*) ∈

*M*.

Infinite matroids have been defined by Reinhard Diestel and coauthors in such a way that this class is (together with the finite matroids) closed under dualization and taking minors. On the other hand, Andreas Dress introduced a theory of matroids with coefficients in a fuzzy ring which is – from a combinatorial point of view – less general, because within this theory every circuit has a finite intersection with every cocircuit. Within the present paper, we extend the theory of matroids with coefficients to more general classes of matroids, if the underlying fuzzy ring has certain properties to be specified.

In many clique search algorithms well coloring of the nodes is employed to find an upper bound of the clique number of the given graph. In an earlier work a non-traditional edge coloring scheme was proposed to get upper bounds that are typically better than the one provided by the well coloring of the nodes. In this paper we will show that the same scheme for well coloring of the edges can be used to find lower bounds for the clique number of the given graph. In order to assess the performance of the procedure we carried out numerical experiments.

This paper solves an enumerative problem which arises naturally in the context of Pascal’s hexagram. We prove that a general Desargues configuration in the plane is associated to *six* conical sextuples via the theorems of Pascal and Kirkman. Moreover, the Galois group associated to this problem is isomorphic to the symmetric group on six letters.

The purpose of this paper is to study the principal fibre bundle (*P*, *M*, *G*, *π*
_{p} ) with Lie group *G*, where M admits Lorentzian almost paracontact structure (*Ø*, *ξ*
_{p}, η_{p}, *g*) satisfying certain condtions on (1, 1) tensor field *J*, indeed possesses an almost product structure on the principal fibre bundle. In the later sections, we have defined trilinear frame bundle and have proved that the trilinear frame bundle is the principal bundle and have proved in Theorem 5.1 that the Jacobian map *π*
^{*} is the isomorphism.

Many combinatorial optimization problems can be expressed in terms of zero-one linear programs. For the maximum clique problem the so-called edge reformulation is applied most commonly. Two less frequently used LP equivalents are the independent set and edge covering set reformulations. The number of the constraints (as a function of the number of vertices of the ground graph) is asymptotically quadratic in the edge and the edge covering set LP reformulations and it is exponential in the independent set reformulation, respectively. F. D. Croce and R. Tadei proposed an approach in which the number of the constraints is equal to the number of the vertices. In this paper we are looking for possible tighter variants of these linear programs.

## Abstract

In 1975 C. F. Chen and C. H. Hsiao established a new procedure to solve initial value problems of systems of linear differential equations with constant coefficients by Walsh polynomials approach. However, they did not deal with the analysis of the proposed numerical solution. In a previous article we study this procedure in case of one equation with the techniques that the theory of dyadic harmonic analysis provides us. In this paper we extend these results through the introduction of a new procedure to solve initial value problems of differential equations with not necessarily constant coefficients.

## Abstract

Fejes Tóth [] studied approximations of smooth surfaces in three-space by piecewise flat triangular meshes with a given number of vertices on the surface that are optimal with respect to Hausdorff distance. He proves that this Hausdorff distance decreases inversely proportional with the number of vertices of the approximating mesh if the surface is convex. He also claims that this Hausdorff distance is inversely proportional to the *square* of the number of vertices for a specific non-convex surface, namely a one-sheeted hyperboloid of revolution bounded by two congruent circles. We refute this claim, and show that the asymptotic behavior of the Hausdorff distance is linear, that is the same as for convex surfaces.

## Abstract

Let *m* ≠ 0, ±1 and *n* ≥ 2 be integers. The ring of algebraic integers of the pure fields of type *n* = 2, 3,4. It is well known that for *n* = 2, an integral basis of the pure quadratic fields can be given parametrically, by using the remainder of the square-free part of m modulo 4. Such characterisation of an integral basis also exists for cubic and quartic pure fields, but for higher degree pure fields there are only results for special cases.

In this paper we explicitly give an integral basis of the field *n*.

## Abstract

In , a universal linear algebraic model was proposed for describing homogeneous conformal geometries, such as the spherical, Euclidean, hyperbolic, Minkowski, anti-de Sitter and Galilei planes (). This formalism was independent from the underlying field, providing an extension and general approach to other fields, such as finite fields. Some steps were taken even for the characteristic 2 case.

In this article, we undertake the study of the characteristic 2 case in more detail. In particular, the concept of virtual quadratic spaces is used (), and a similar result is achieved for finite fields of characteristic 2 as for other fields. Some differences from the non-characteristic 2 case are also pointed out.

We present an algorithm to compute the primary decomposition of a submodule *N* of the free module ℤ[*x*
_{1},...,*x*
_{n}]^{m}. For this purpose we use algorithms for primary decomposition of ideals in the polynomial ring over the integers. The idea is to compute first the minimal associated primes of *N*, i.e. the minimal associated primes of the ideal Ann (ℤ[*x*
_{1},...,*x*
_{n}]^{m}/*N*) in ℤ[*x*
_{1},...,*x*
_{n}] and then compute the primary components using pseudo-primary decomposition and extraction, following the ideas of Shimoyama-Yokoyama. The algorithms are implemented in Singular.

Let *A*
_{1},...,*A*
_{N} and *B*
_{1},...,*B*
_{M} be two sequences of events and let *ν*
_{N}(*A*) and *ν*
_{M}(*B*) be the number of those *A*
_{i} and *B*
_{j}, respectively, that occur. Based on multivariate Lagrange interpolation, we give a method that yields linear bounds in terms of *S*
_{k,t}, *k*+*t* ≤ *m* on the distribution of the vector (*ν*
_{N}(*A*), *ν*
_{M}(*B*)). For the same value of *m*, several inequalities can be generated and all of them are best bounds for some values of *S*
_{k,t}. Known bivariate Bonferroni-type inequalities are reconstructed and new inequalities are generated, too.

M. Giusti’s classification of the simple complete intersection singularities is characterized in terms of invariants. This is a basis for the implementation of a classifier in the computer algebra system Singular.

Let *S* be a set of *n* points distributed uniformly and independently in a convex, bounded set in the plane. A four-gon is called empty if it contains no points of *S* in its interior. We show that the expected number of empty non-convex four-gons with vertices from *S* is 12*n*
^{2}log*n* + *o*(*n*
^{2}log*n*) and the expected number of empty convex four-gons with vertices from *S* is Θ(*n*
^{2}).

This paper attempts an exposition of the connection between valuation theory and hyperstructure theory. In this regards, by considering the notion of totally ordered canonical hypergroup we define a hypervaluation of a hyperfield onto a totally ordered canonical hypergroup and obtain some related basic results.

We provide sufficient conditions for a mapping acting between two Banach spaces to be a diffeomorphism. We get local diffeomorhism by standard method while in making it global we employ a critical point theory and a duality mapping. We provide application to integro-differential initial value problem for which we get differentiable dependence on parameters.

We obtain new lower and upper bounds for probabilities of unions of events. These bounds are sharp. They are stronger than earlier ones. General bounds may be applied in arbitrary measurable spaces. We have improved the method that has been introduced in previous papers. We derive new generalizations of the first and second parts of the Borel-Cantelli lemma.

It is proved that there exists an *NI* ring *R* over which the polynomial ring *R*[*x*] is not an *NLI* ring. This answers an open question of Qu and Wei (*Stud. Sci. Math. Hung.*, **51(2)**, 2014) in the negative. Moreover a sufficient condition of *R*[*x*] to be an *NLI* ring is included for an *NLI* ring *R*.