Let us consider the following two norms in the vector space \,$\mathcal{P}$ \,of all complex polynomials: $$ \|p\|_{D_{r}}:= \sup\{|p(z)|: |z|<r\}, \text{ and } \|p\|_{1}:=\sum_{i=0}^{n}|a_{i}|, $$ where \,$p(z) = \sum_{i=0}^{n} a_{i} z^{i}$. In this note we show that, if $0 <\varepsilon < \varepsilon' < 1 < r < r'$, then $$ \|\cdot\|_{D_{\varepsilon}}\prec\|\cdot\|_{D_{\varepsilon'}}\prec\|\cdot\|_{D_{1}}\prec \|\cdot\|_{1}\prec \|\cdot\|_{D_{r}}\prec \|\cdot\|_{D_{r'}}, $$ where \,$\prec$ \,represents the natural (strict) partial order in their corresponding induced topologies.

We establish the existence of a solution for the following system of differential equations \begin{equation*} \label{system}\begin{cases}x'(t) = f (t, x(t)), & x(t_0) = x^*,\\ y'(t) = g (t, y(t)), & y(t_0) = x^{**}, \end{cases} \end{equation*} in the space of all bounded and continuous real functions on $[0,+\infty[$. We use best proximity point methods and measure of noncompactness theory under suitable assumptions on $f$ and $g$. Some new best proximity point theorems play a key role in the above result.

We show that, in contrast to the real analytic case, quasianalytic ultradifferentiability can never be tested in lower dimensions. Our results are based on a construction due to Jaffe.

This paper shows that the group of auto-homeomorphisms of a topological group can be endowed with a topology so that the resulting topological group plays, for topological groups, the role of the group of automorphisms of a group: it represents the internal actions on the given topological group.

A space $X$ is {\it monotonically star $\sigma$-compact} if one assigns to each open cover $\mathcal U$ of $X$ a subspace $s(\mathcal U)\subseteq X$, called a kernel, such that $s(\mathcal U)$ is a $\sigma$-compact subset of $X$, and $st(s(\mathcal U),\mathcal U)=X$, and if $\mathcal V$ refines $\mathcal U$ then $s(\mathcal U)\subseteq s(\mathcal V)$, where $st(s(\mathcal U),\mathcal U)=\bigcup\{U\in \mathcal U:U\cap s(\mathcal U)\neq\emptyset\}.$ In this paper, we investigate the relationship between monotonically star $\sigma$-compact spaces and related spaces, and also study topological properties of monotonically star $\sigma$-compact spaces.

Each hypersurface of a nearly Kähler manifold is naturally equipped with two tensor fields of $(1,1)$-type, namely the shape operator $A$ and the induced almost contact structure $\phi$. In this paper, we show that, in the homogeneous nearly Kähler $\mathbb{S}^6$ a hypersurface satisfies the condition $A\phi+\phi A=0$ if and only if it is totally geodesic; moreover, similar as for the non-flat complex space forms, the homogeneous nearly Kähler manifold $\mathbb{S}^3\times\mathbb{S}^3$ does not admit a hypersurface that satisfies the condition $A\phi+\phi A=0$.

This papers examines the general differential equation \[y''(z)+a(z) y'(z)+ b(z)y(z)=0\] in the unit disk of the complex plane, and finds conditions on the analytic functions $a$ and $b$ that ensures the solutions are Janowski starlike. Also studied is Janowski convexity of solutions to \[z (1-z)y''(z)+ a(z) y'(z)+ \alpha y(z) =0,\] where $\alpha$ is a given constant. Janowski starlikeness and Janowski convexity encompass various widely studied classes of classical starlikeness and convexity. As application, we give convexity and starlikeness geometric description of solutions to differential equations related to several important special functions.

In this paper, we study new contractive conditions which are strong enough to generate fixed points but which do not force the map to be continuous at fixed points. In this context, we give new results on the fixed-circle problem. We investigate some applications to complex-valued metric spaces and to discontinuous activation functions in real and complex valued neural networks.

We prove decidability results on the existence of constant subsequences of uniformly recurrent morphic sequences along arithmetic progressions. We use spectral properties of the subshifts they generate to give a first algorithm deciding whether, given $p\in \mathbb{N}$, there exists such a constant subsequence along an arithmetic progression of common difference $p$. In the special case of uniformly recurrent automatic sequences we explicitly describe the sets of such $p$ by means of automata.

In this paper we introduce a new class of rings whose elements are a sum of a central and a unit element, namely a ring $R$ is called $CU$ if each element $a\in R$ has a decomposition $a = c + u$ where $c$ is central and $u$ is unit. One of the main results in this paper is that if $F$ is a field which is not isomorphic to $\Bbb Z_2$, then $M_2(F)$ is a $CU$ ring. This implies, in particular, that any square matrix over a field which is not isomorphic to $\Bbb Z_2$ is the sum of a central matrix and a unit matrix.

We prove that the class of additive perfectly everywhere surjective functions contains (with the exception of the zero function) a vector space of maximal possible dimension ($2^\cont$). Additionally, we show under the assumption of regularity of $\cont$ that the family of additive everywhere surjective functions that are not strongly everywhere surjective contains (with the exception of the zero function) a vector space of dimension $\cont^+$.