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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.

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.

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*.

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.

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

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.

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.

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.

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.

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