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We prove some extension theorems for quaternionic holomorphic functions in the sense of Fueter. Starting from the existence theorem for the nonhomogeneous Cauchy-Riemann-Fueter Problem, we prove that an $\mathbb{H}$-valued function $f$ on a…

In this article we prove that every isometric copy of C(L) in C(K) is complemented if L is compact Hausdorff of finite height and K is a compact Hausdorff space satisfying the extension property, i.e., every closed subset of K admits an…

We establish a necessary and sufficient condition for the differentiability of the distance function generated by a nonempty closed set K in a real normed linear space X under a proximinality condition on K. We do not assume the uniform…

Let $\mathbb{K}$ be an uncountable field of characteristic zero and let $f$ be a function from $\mathbb{K}^n$ to $\mathbb{K}$. We show that if the restriction of $f$ to every affine plane $L\subset\mathbb{K}^n$ is regular, then $f$ is a…

We show that if $X$ is a Banach space whose dual $X^{*}$ has an equivalent locally uniformly rotund (LUR) norm, then for every open convex $U\subseteq X$, for every $\varepsilon >0$, and for every continuous and convex function $f:U…

Given a compact and H-convex subset $K$ of the Heisenberg group ${\mathbb H}$, with the origin $e$ in its interior, we are interested in finding a homogeneous H-convex function $f$ such that $f(e)=0$ and $f\bigl|_{\partial K}=1$; we will…

We consider a compact $C^\omega$ manifold $X$ and finitely many regular $C^\omega$ submanifolds $Y_1, \dots, Y_q$ of $X$, which are closed subsets in $X$, such that the union of $Y_j$'s has only normal crossings. We show that every…

Let $E$ be a Banach space and $\X$ be the closed unit ball of the dual space $E^*$. For a compact set $K$ in $E$, we prove that $K$ is polynomially convex in $E$ if and only if there exist a unital commutative Banach algebra $A$ and a…

Let D be a strictly convex domain of C^n, f_1 and f_2 be two holomorphic functions defined on a neighborhood of closure of D and set X_l={z, f_l(z)=0}, l=1,2. Suppose that X_l\cap bD is transverse for l=1 and l=2, and that X_1\cap X_2 is a…

We introduce certain functors from the category of commutative rings (and related categories) to that of $\mathbb{Z}$-algebras (not necessarily associative or commutative). One of the motivating examples is the Leavitt path algebra functor…

This article gives necessary and sufficient conditions for the dual representation of Rockafellar in (Integrals which are convex functionals. II, Pacific J. Math., 39:439--469, 1971) for integral functionals on the space of continuous…

In this very short paper, we provide a strong motivation for the study of the following problem: given a real normed space $E$, a closed, convex, unbounded set $X\subseteq E$ and a function $f:X\to X$, find suitable conditions under which,…

Let $V$ be a real finite dimensional representation of a compact Lie group $G$. It is well-known that the algebra $\mathbb R[V]^G$ of $G$-invariant polynomials on $V$ is finitely generated, say by $\sigma_1,...,\sigma_p$. Schwarz proved…

We introduce a new point of view towards Glaeser's theorem on composite $C^\infty$ functions [Ann. of Math. 1963], with respect to which we can formulate a ``$C^k$ composite function property" that is satisfied by all semiproper real…

Given commuting functions f,g, with at most a countable compact set of essential singularities, recent results for entire functions are extended to prove that Julia sets match, J(f)=J(g), in a particular case in the class K. With this…

We prove two theorems about differentiable functions on the Banach space C(K), where K is compact. (i) If C(K) admits a non-trivial function of class C^m and of bounded support, then all continuous real-valued functions on C(K) may be…

Let Y be an infinite covering space of a projective manifold M in P^N of dimension n geq 2. Let C be the intersection with M of at most n-1 generic hypersurfaces of degree d in P^N. The preimage X of C in Y is a connected submanifold. Let…

Motivated by a result of [17], we determine necessary and sufficient conditions on $F\/$ with $|E(F)| \leq n-1\/$ for which $K_n - F\/$ admits a $g$-angulation. For $|E(F)| \geq n\/$, we investigate the possibility of placing $F\/$ in…

We prove polarity duality for covering problems in Hilbert geometry. Let $G$ and $K$ be convex bodies in $\mathbb{R}^d$ where $G \subset \operatorname{int}(K)$ and $\operatorname{int}(G)$ contains the origin. Let $N^H_K(G,\alpha)$ and…

Let $X$ be a separable Banach space with a separating polynomial. We show that there exists $C\geq 1$ (depending only on $X$) such that for every Lipschitz function $f:X\rightarrow\mathbb{R}$, and every $\epsilon>0$, there exists a…