We employ a classical idea of Ehrenpreis, together with a new algebraic result, to give a new proof that regular functions of several quaternionic variables cannot have compact singularities. As a byproduct we characterize those inhomogeneous Cauchy{ Fueter systems which admit solutions on convex sets. Our method readily extends to the case of monogenic functions on Cliiord Algebras. We nally s...

‎Let $X$ be a real normed  space, then  $C(subseteq X)$  is  functionally  convex  (briefly, $F$-convex), if  $T(C)subseteq Bbb R $ is  convex for all bounded linear transformations $Tin B(X,R)$; and $K(subseteq X)$  is  functionally   closed (briefly, $F$-closed), if  $T(K)subseteq Bbb R $ is  closed  for all bounded linear transformations $Tin B(X,R)$. We improve the    Krein-Milman theorem  ...

In this paper, we investigate the concept of topological stationary for locally compact semigroups. In [4], T. Mitchell proved that a semigroup S is right stationary if and only if m(S) has a left Invariant mean. In this case, the set of values ?(f) where ? runs over all left invariant means on m(S) coincides with the set of constants in the weak* closed convex hull of right translates of f. Th...

For a set S in a Banach space, we denote by dim(S) its covering dimension [1, p. 42]. Recall that, when S is a convex set, the covering dimension of S coincides with the algebraic dimension of S, this latter being understood as ∞ if it is not finite [1, p. 57]. Also, S and conv(S) will denote the closure and the convex hull of S, respectively. In [3], we proved what follows. dim({x ∈ X : Φ(x) =...

Let $C_0(alpha)$ denote the class of concave univalent functions defined in the open unit disk $mathbb{D}$. Each function $f in C_{0}(alpha)$ maps the unit disk $mathbb{D}$ onto the complement of an unbounded convex set. In this paper, we study the mapping properties of this class under integral operators.

We characterize in a reflexive Banach space all the closed convex sets containing no lines for which the condition ensures the closedness of the algebraic difference for all closed convex sets . We also answer a closely related problem: determine all the pairs , of closed convex sets containing no lines such that the algebraic difference of any sufficiently small uniform perturbation of and rem...

We study the algebraic boundary of a convex semi-algebraic set via duality in convex and algebraic geometry. We generalise the correspondence of facets of a polytope with the vertices of the dual polytope to general semi-algebraic convex sets. In this case, exceptional families of extreme points might exist and we characterise them semi-algebraically. We also give a strategy for computing a com...

1.1 Definitions We say a set S ⊆ Rd is convex if for any two points x,x′ ∈ S, the line segment conv{x,x′} := {(1−α)x+αx′ : α ∈ [0, 1]} between x and x′ (also called the convex hull of {x,x′}) is contained in S. Overloading terms, we say a function f : S → R is convex if its epigraph epi(f) := {(x, t) ∈ S × R : f(x) ≤ t} is a convex set (in Rd × R). Proposition 1. A function f : S → R is convex ...

The following result of convex analysis is well–known [2]: If the function f : X → [−∞, +∞] is convex and some x0 ∈ core (dom f) satisfies f(x0) > −∞, then f never takes the value −∞. From a corresponding theorem for convex functions with values in semi–linear spaces a variety of results is deduced, among them the mentioned theorem, a theorem of Deutsch and Singer on the single–valuedness of co...

in this paper, we define the notions of fuzzy congruence relations and fuzzy convex subalgebras on a commutative residuated lattice and we obtain some related results. in particular, we will show that there exists a one to one correspondence between the set of all fuzzy congruence relations and the set of all fuzzy convex subalgebras on a commutative residuated lattice. then we study fuzzy...