We introduce the notion of \emph{primitive arc} of a curve defined over a field $k$ and study criterions for the existence of such objects in terms of the geometry of the curve. We prove that this notion provides a criterion which determines when the normalization of an irreductible curve singularity $(X,x)$ induces an isomorphism between the formal neighborhoods of the associated arc schemes at the constant arc $x$ and its lifting $\bar x$ to the normalization $\bar X$. We also show that the existence of a primitive arc at $x\in X$ is equivalent to the smoothness of the analytically irreducible curve $X$ at $x$. In this end, we interpret this notion in terms of the formal deformations of the constant arc $x$ in the associated arc scheme.

In this paper, we find coefficient estimates by a new method making use of the Faber polynomial expansions for a comprehensive subclass of analytic bi-univalent functions, which is defined by subordinations in the open unit disk. The coefficient bounds presented in this paper would generalize and improve some recent works appeared in the literature.

We show that the subset of the disk algebra of the functions that are not in some Dales-Davie algebra is algebrable. In other words, the set $\Big\{f\in\mathcal{A}(D)\,:\,\sum_{n=0}^{\infty}\dfrac{\|f^{(n)}\|}{n!}=+\infty\Big\}$ is shown to be algebrable.

In this paper we first investigate linear extendability of an isometric embedding $T:\mathcal U \longrightarrow \mathcal Y$ from an open subset $\mathcal U$ of a real Banach space $\mathcal X$ into a real Banach space $\mathcal Y$ in the case where $\mathcal Y$ is either the space $C_\Bbb R(K)$ of continuous real-valued functions on a compact space $K$, or is a strictly convex Banach space. Then we obtain similar results for the case where $\mathcal Y$ is an arbitrary real Banach space and $T:\mathcal U \longrightarrow \mathcal Y$ is an isometry whose range satisfies some additional conditions.

In this paper we study common fixed point properties of non-linear actions of semi-topological semigroups on non-void weak* compact convex sets in dual Banach spaces. Among other things, we derive from our main result Theorem 1, the existence of a common fixed point property for semigroups of non-expansive mappings acting on non-empty weakly compact convex sets, generalizing a result of Hsu [13], Mitchell [25].

For a Banach algebra ${\cal A}$, we introduce and investigate weak amenability of ${\cal A}$ with respect to a character. We give some necessary conditions for the weak amenability of $\cal A$ with respect to a character and describe a class of Banach algebras that are not weakly amenable with respect to characters. Finally, we give examples of Banach algebras which are weakly amenable with respect to characters but neither weakly amenable nor amenable with respect to characters.

We prove that if $\mathcal{D}$ is a non-trivial non-symmetric $2$-$(v,k,2)$ design admitting a flag-transitive point-primitive automorphism group $G$ with $Soc(G)=A_{n}$ for $n\geq5$, then $\mathcal{D}$ is a $2$-$(6,3,2)$ or $2$-$(10,4,2)$ design.

We give a formula for a Frobenius vector of a Gorenstein simplicial affine semigroup $S$, and when the semigroup is Cohen-Macaulay we give an algorithm computing the set of minimal Frobenius vectors of $S$ for a special class of semigroups.

In this paper, we study two optimization problems where solutions on a dense set yield global solution. We study these problems for spaces of Bochner integrable functions and for spaces of continuous functions. The first one deals with expressing the length of a vector as a sum of the distance to a best approximation and minimal best approximation and the second one relates to approximating a subsequence of a minimizing sequence with a sequence of proximinal vectors.

Let $X$ be a perfect compact plane set, $n\in \mathbb{N}$ and $D^n(X)$ be the algebra of complex-valued functions on $X$ with continuous $n$-th derivative. In this paper we study weighted composition operators on algebras $D^n(X)$. We give a necessary and sufficient condition for these operators to be compact. As a consequence, we characterize power compact composition operators on these algebras. Then we determine the spectra of Riesz weighted composition operators on these algebras.

It is known that there is not a two dimensional linear space in $\mathbb R^\mathbb R$ every non-zero element of which is an injective function. Here, we generalize this result to arbitrarily large dimensions. We also study the convolution of non-differentiable functions which gives, as a result, a differentiable function. In this latter case, we are able to show the existence of linear spaces of the largest possible dimension formed by functions enjoying the previous property. By doing this we provide both positive and negative results to the recent field of lineability. Some open questions are also provided.

We study the buildings in which parallelism of residues is an equivalence relation. If the building admits a group action, we describe how parallel residues are related to residues with equal stabilizers. This permits to retrieve the fact that in a Coxeter group or in a graph product, intersections of parabolic subgroups are parabolic.