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We apply weak smooth bump functions to construct smooth surjections with derivatives of rank-1 and smooth bump functions whose derivatives are surjections in infinite-dimensional Banach spaces. We also construct weak smooth bump functions whose derivatives have empty interior.
We improve a recent splitting theorem of Domański and Vogt by showing that in the category of PLS-spaces each exact sequence 0 -> F -> X -> 0 splits whenever E is isomorphic to a subspace of D' and F is isomorphic to a quotient of D', the space of distributions on some open subse of [R^d].
We give a simple relation between the relative extremal function and the pluricomplex Green function. Using this relation, we give a new proof that the multipole pluricomplex Green function has the product property g[Omega_1 x Omega_2] = max{g[Omega_1],g[Omega_2]} for any domains [Omega_1 is a subset of C^n] and [Omega_2 is a subset of C^m].
We prove that the operation conv of taking a convex hull is continuous but is not Lipschitz continuous with respect to the radial metric for the class of star sets at 0 with 0 in interiors.
The notion of an absolute approximate retract for a class Q of topological spaces (or an AAR(Q)-space) generalizes the concept of an absolute retract for the class Q. For many classes Q, it is shown that AAR(Q)-spaces are preserved under retraction mappings and that a fully normal AAR(Q)-space X must be contractible and can be expressed as a product of finite-dimensional compacta if and only if X...
Some relations between the modified version of the orthogonal Cauchy equation and the following more general equation f(x + y) = g ([...]) [f(x) + f(y)] are obtained. Further the Cauchy equation with the right-hand side multiplied by some constant is considered. This equation is assumed for all x,y satisfying the equality [...] = [alpha]. Finally solutions of this cooditional equation in the case...
We consider three types of geometries of circles (Moebius plane, Laguerre plane and Minkowski plane, cf. [4) with respect to so-called multicentral automorphisms. An automorphism [phi] of any geometry of circles is central if it has a fix point P and [phi] becomes a central collineation in the derived projective plane M(P). For any central automorphism [phi] we try to establish the whole set of points...
Let v be the weight on the disc and M[phi] be the pointwise multiplication operator, M[phi](f) = [phi]f, on the weighted Banach space of analytic functions H[...](D) on the disc with the sup-norms. We characterize when M[phi] : H[...] --> H[...] is an isomorphism into for weights tending exponentially to zero at the boundary. In particular, the result holds for v(z) = exp[...] or v(z) = exp [....
We construct general examples of affine normal varieties X which have a hypersurface H [is a subset of] X, such that the variety X [...] H is not affine. We also show, that if X is an affine normal surface and H is a curve in X, then X [...] H is affine, too.
The notion of bundle convergence for sequences of operators in a von Neumann algebra A equipped with a faithful and normal state phi as well as for sequences of vectors in their [L_2]-spaces were introduced by Hensz, Jajte and Paszkiewicz in 1996 as an appropriate substitute for almost everywhere convergence in the commutative setting. First, we prove that if (B_k : k = l, 2,...) is a sequence in...
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