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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 ℤ^{𝑑} and ℝ^{𝑑}. 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.

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*, 2*x*] 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}) ∈ *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

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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.

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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.

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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.