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Mathematics and Statistics

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Motivated by the examples of Heppes and Wegner, we present several other examples of the following kind: a bounded convex region 𝐷 and a convex disk 𝐾 in the plane are described, such that every thinnest covering of 𝐷 with congruent copies of 𝐾 contains crossing pairs.

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In this paper we show that the spherical cap discrepancy of the point set given by centers of pixels in the HEALPix tessellation (short for Hierarchical, Equal Area and iso-Latitude Pixelation) of the unit 2-sphere is lower and upper bounded by order square root of the number of points, and compute explicit constants. This adds to the currently known (short) collection of explicitly constructed sets whose discrepancy converges with order 𝑁 −1/2, matching the asymptotic order for i.i.d. random point sets. We describe the HEALPix framework in more detail and give explicit formulas for the boundaries and pixel centers. We then introduce the notion of an 𝑛-convex curve and prove an upper bound on how many fundamental domains are intersected by such curves, and in particular we show that boundaries of spherical caps have this property. Lastly, we mention briefly that a jittered sampling technique works in the HEALPix framework as well.

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We apply a recent general zero density theorem of us (valid for a large class of complex functions) to improve earlier density theorems of Heath-Brown and Paul–Sankaranarayanan for Dedekind zeta functions attached to a number field 𝐾 of degree 𝑛 with 𝑛 > 2.

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Asymptotic uniform upper density, shortened as a.u.u.d., or simply upper density, is a classical notion which was first introduced by Kahane for sequences in the real line.

Syndetic sets were defined by Gottschalk and Hendlund. For a locally compact group 𝐺, a set 𝑆 ⊂ 𝐺 is syndetic, if there exists a compact subset 𝐶 ⋐ 𝐺 such that 𝑆𝐶 = 𝐺. Syndetic sets play an important role in various fields of applications of topological groups and semigroups, ergodic theory and number theory. A lemma in the book of Fürstenberg says that once a subset 𝐴 ⊂ ℤ has positive a.u.u.d., then its difference set 𝐴 − 𝐴 is syndetic.

The construction of a reasonable notion of a.u.u.d. in general locally compact Abelian groups (LCA groups for short) was not known for long, but in the late 2000’s several constructions were worked out to generalize it from the base cases of ℤ𝑑 and ℝ𝑑. With the notion available, several classical results of the Euclidean setting became accessible even in general LCA groups.

Here we work out various versions in a general locally compact Abelian group 𝐺 of the classical statement that if a set 𝑆 ⊂ 𝐺 has positive asymptotic uniform upper density, then the difference set 𝑆 − 𝑆 is syndetic.

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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.

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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.

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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.

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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, 2x] for sufficiently large x.

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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.

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Let 𝑛 ∈ ℕ. An element (x 1, … , x 𝑛) ∈ En is called a norming point of T L ( nE) if ‖x 1‖ = ⋯ = ‖xn ‖ = 1 and |T (x 1, … , xn )| = ‖T‖, where L ( nE) denotes the space of all continuous n-linear forms on E. For T L ( nE), we define

Norm(T) = {(x 1, … , x n) ∈ En ∶ (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 L (2 𝑑 (1, w)2), where 𝑑 (1, w)2 = ℝ2 with the octagonal norm of weight 0 < w < 1 endowed with x , y d * 1 , w = max x , y , x + y 1 + w .

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