Quaestiones Mathematicae

On a weak law of large numbers and L^p-convergence for general random variables ¹

2025-05-29T15:25:43+00:00

Let {X_n, n ≥ 1} be a sequence of random variables with partial sums S_n = X₁ + · · · + X_n for every n ≥ 1. For the independent identically distributed random variables, the following Kolmogorov-Feller theorem provides a necessary and sufficient condition for the weak law of large numbers to hold.

Some characterizations of ω-balanced topological groups with a q-point ¹

2025-05-29T15:31:16+00:00

In this paper, we study some characterizations of q-spaces, strict q-spaces and strong q-spaces under ω-balanced topological groups. First we characterize a topological group G to be ω-balanced and a q-space whenever for each open neighborhood O of the identity in G, there is a countably compact invariant subgroup H which is of countable character in G, such that H ⊆ O and the canonical quotient mapping p : G → G/H is quasi-perfect and the quotient group G/H is metrizable. Secondly, we characterize G to be ω-balanced and a strict q-space, replacing the condition of being a q-space with an appropriate condition which is designed for the so-called “strict q-spaces”. Finally, we characterize G to be ω-balanced and a strong q-space, illustrating an additional condition to be replaced to that of q-space, or strict q-space. As applications of our three main results, we found new characterizations of the ω-narrow topological groups.

On a variant of extremal disconnectedness in locales and spaces

2025-05-29T15:37:41+00:00

We study a variant of extremal disconnectedness in frames, defined by requiring that any two disjoint open sublocales should be separable by disjoint open sublocales each of which is the interior of some zero-sublocale. In spaces, this notion was recently introduced in [1] in order to study some properties of frames of z-ideals of rings of continuous functions. The point-free approach adopted here brings to the fore some topological results not considered in [1], such as that a dense z-embedded subspace of a Tychonoff space has this property if and only if the space containing it has the property. In [1], this was proved only for the embedding X ⊆ βX. We give a ring-theoretic characterization by defining a property of f-rings in a transparent manner which shows why the property should be defined the way we do. The property is that if two annihilator ideals meet trivially, then the annihilator of each of the annihilator ideals contains a positive element such that the sum of the two elements is a non-zerodivisor.

Powers as Fibonacci sums

2025-05-29T15:42:34+00:00

No abstract.

Proofs of three conjectured internal congruences modulo 32 for Schur-type overpartitions ¹

2025-05-29T15:47:16+00:00

Let S(n) denote the number of overpartitions of Schur-type. Recently, Chern, da Silva and Sellers proved many congruences modulo 8 and 16 for S(n). At the end of their paper, they also posed a conjecture on internal congruences modulo 32 for S(n). In this paper, we confirm their conjecture by using theta function identities and the (p, k)-parameterization of theta functions given by Alaca and Williams. In particular, we show that for any integer j with 0 ≤ j ≤ 31, there are infinitely many integers u_j such that S(u_j ) ≡ j (mod 32).

Interior vertices and edges in integer partitions

2025-05-29T15:51:41+00:00

This paper characterises the shape of the Young diagram associated with integer partitions in terms of two newly invented statistics which are defined in the text: namely interior vertices and horizontal edges. Generating functions and associated asymptotic results are developed for both interior vertices and horizontal edges.

On elliptic curves assigned to a pair of right triangles with an equal hypotenuse

2025-05-29T15:55:36+00:00

Consider a pair of right triangles with an equal hypotenuse. This turns out to solve the diophantine system of equations a² + b² = c² + d² = e² in integers. To this system we associate a family of elliptic curves with defining equation y² = (x−a² )(x−b²)(x−c² )+a²b²c² . We show that there exists a subfamily of rank ≥ 3 over Q(m, n, k, ℓ) and obtain a subfamily of rank (exactly) four over Q(k) and determine a set of its free generators. Besides, we show there exist infinitely many elliptic curves of rank ≥ 5 parameterized by a rank five quartic elliptic curve. We also find a few particular examples with higher ranks. The families we construct have ℤ/2ℤ torsion subgroups in general. The previous work [13] has obtained similar Pythagorean quadruplet elliptic curve families in two parameters with rank ≥ 3. (Recall that by a Pythagorean quadruplet (a, b, c, d), we mean an integer solution to the quadratic equation a² + b² = c² + d² .)

Maker-Breaker Domination game played on corona products of graphs

2025-05-29T17:17:53+00:00

In the Maker-Breaker domination game, Dominator and Staller play on a graph G by taking turns in which each player selects a not yet played vertex of G. Dominator’s goal is to select all the vertices in a dominating set, while Staller aims to prevent this from happening. In this paper, the game is investigated on corona products of graphs. Its outcome is determined as a function of the outcome of the game on the second factor. Staller-Maker-Breaker domination numbers are determined for arbitrary corona products, while Maker-Breaker domination numbers of corona products are bounded from both sides. All the bounds presented are demonstrated to be sharp. Corona products as well as general graphs with small (Staller-)Maker-Breaker domination numbers are described.

Multiplication operators on the weighted Sobolev disk algebra ¹

2025-05-29T17:37:14+00:00

Let D be the unit disk in the complex plane C. For α > −1, the weighted Sobolev disk algebra SA(D, dA_α) consists of all analytic functions in the weighted Sobolev space W^2,2 (D, dA_α). In this paper, we prove that the multiplication operator M_zn is similar to M_zm on SA(D, dA_α) if and only if n = m, where n, m are positive integers. Then we characterize when a bounded operator P on SA(D, dA_α) belongs to the commutant A′ (M_zn ) of M_zn by using the matrix representation of P. In addition, we compute the exact norm of M_z on SA(D, dA_α). Finally, we prove that on the unweighted Sobolev disk algebra SA(D) the restrictions of M_zn to different invariant subspaces z^kSA(D) (k ≥ 1) are not unitarily equivalent, and the restrictions of M_zn (n ≥ 2) to different invariant subspaces S_j (0 ≤ j < n) are also not unitarily equivalent.

Lipschitz closed injective hull ideals and Lipschitz interpolative ideals

2025-05-29T17:41:23+00:00

In this paper we present two constructions: the Lipschitz closed injective hull ideals and the Lipschitz interpolative ideals of a Lipschitz operator ideal. These procedures aim to construct new Lipschitz operator ideals between pointed metric spaces and Banach spaces. The new ideals are characterized by specific criteria that determine whether a Lipschitz operator belongs to them, using summability properties and interpolation formulas.

The extended Frobenius problem for Lucas series incremented by a Lucas number

2025-05-29T17:45:04+00:00

We study the extended Frobenius problem for sequences of the form {l_a} ∪ {l_a + l_n}_n∈N and {l_a + l_n}_n∈N, where {l_n}_n∈N is the Lucas series and l_a is a Lucas number. As a consequence, we show that the families of numerical semigroups associated to both sequences satisfy the Wilf’s conjecture.