In this paper we give a characterization of two-weighted inequalities for maximal, singular operators and their commutators in generalized weighted Morrey spaces $\mathcal{M}^{p,\varphi}_{\omega}(\mathbb{R}^n)$. We prove the boundedness of maximal operator $M$ and maximal commutators $[M,b]$ from the spaces $\mathcal{M}^{p,\varphi_1}_{\omega_1^\delta}(\mathbb{R}^n)$ to the spaces $\mathcal{M}^{p,\varphi_2}_{\omega_2^\delta}(\mathbb{R}^n)$, where $1< p<\infty$, $0<\delta<1$ and $(\omega_1,\omega_2)\in \widetilde{A}_{p}(\mathbb{R}^n)$. We also prove the boundedness of the Calderón--Zygmund singular operators $T$ and their commutators $[b,T]$ from $\mathcal{M}^{p,\varphi_1}_{\omega_1^\delta}(\mathbb{R}^n)$ to $\mathcal{M}^{p,\varphi_2}_{\omega_2^\delta}(\mathbb{R}^n)$. Finally we give generalized weighted Morrey a priori estimates as applications of our results.

In this paper, we investigate the Ulam type stability for the $3$-dimensional quadratic functional equation $$g(a +b+c)+g(a-b)+g(a-c)+g(b-c)=3[g(a) + g(b)+g(c)]$$ in the setting of quasi-Banach space using the Brzdęk fixed point approach. The results obtained in this paper extend various previously known results in the setting of quasi-Banach space. We also give some examples to illustrate our main results.

We correct a mistake in [St] leading to erroneous formulas in Theorems 5.2 and 5.4. As an immediate corollary of a formula in [BCJ] we give a formula which relates the Hecke operators $T(p^2)\circ T(p^{2l-2})$, $T(p^{2l})$ and $T(p^{2l-4})$ and comment on it.

Let $K=\mathbb{Q}(\theta)$ be a number field generated by a complex root $\theta$ of a monic irreducible trinomial $F(x) = x^n+ax+b \in \mathbb{Z}[x]$. There is an extensive literature on monogenity of number fields defined by trinomials. For example, Gaál studied the multi-monogenity of sextic number fields defined by trinomials. Jhorar and Khanduja provide some explicit conditions on $a$, $b$ and $n$ for $(1, \theta, \ldots, \theta^{n-1})$ to be a power integral basis in $K$. But, if $\theta$ does not generate a power integral basis of $\mathbb{Z}_K$, then Jhorar's and Khanduja's results cannot answer the monogenity of $K$. In this paper, based on Newton polygon techniques, we deal with the problem of non-monogenity of $K$. More precisely, when $\theta$ does not generate a power integral basis of $\mathbb{Z}_K$, we give sufficient conditions on $n$, $a$ and $b$ for $K$ to be not monogenic. For $n\in {5, 6, 3^r, 2^k\cdot 3^r, 2^s\cdot 3^k+1}$, we give explicitly some infinite families of these number fields that are not monogenic. Finally, we illustrate our results by some computational examples.

In [2] the third author studied arithmetic properties of a class of sequences that generalize the sequence of derangements. The aim of the following paper is to disprove two conjectures stated in [2]. The first conjecture regards the set of prime divisors of their terms. The latter one is devoted to the order of magnitude of considered sequences.

Let $p$ be a prime number such that $p=2$ or $p\equiv 1\pmod 4$. Let $\varepsilon_p$ denote the fundamental unit of $\mathbb{Q}(\sqrt{p})$ and let $a$ be a positive square-free integer. In the present paper, we construct the Hilbert genus field of the real cyclic quartic fields $\mathbb{Q}(\sqrt{a\varepsilon_p\sqrt{p}})$.

Under special conditions, we prove that the set of preperiodic points for semigroups of self-morphisms of affine spaces falling on cyclotomic closures is not dense, generalising results of Ostafe and Young (2020). We also extend previous results about boundness of house and height on certain preperiodicity sets of higher dimension in semigroup dynamics.