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Given a topological space 〈X, T〉 ∈M, an elementary submodel of set theory, we defineX Mto beX ∩M with the topology generated by {U ∩M : U ∈T ∩M}. We prove that it is undecidable whetherX Mhomeomorphic toω 1 impliesX =X M,yet it is true in ZFC that ifX Mis homeomorphic to the long line, thenX =X M.The former result generalizes to other cardinals of uncountable confinality while the latter generalizes to connected, locally compact, locally hereditarily LindelöfT 2 spaces.

We useL 2 methods to show that if a group with a presentation of deficiency one is an extension ofZ by a finitely generated normal subgroup then the 2-complex corresponding to any presentation of optimal deficiency is aspherical and to prove a converse of the Cheeger-Gromov-Gottlieb theorem relating Euler characteristic and asphericity. These results are applied to the Whitehead conjecture, 4-manifolds and 2-knot groups.

We prove the existence of a positive solution for the problem $${\rm{\gamma}}{{\rm{\Delta}}^2}u - m\left(u \right){\rm{\Delta}}u = \mu a\left(x \right){u^q} + b\left(x \right){u^p},\,\,{\rm{in}}\,{\rm{\Omega ,}}\,\,\,\,\,u = {\rm{\gamma \Delta}}u = 0,\,\,{\rm{on}}\,\,\partial {\rm{\Omega ,}}$$ where Ω ⊂ ℝN is a bounded smooth domain, γ ∈ {0, 1},0 < q > 1 < p, m is weakly continuous in $${H^2}\left({\rm{\Omega}} \right) \cap H_0^1\left({\rm{\Omega}} \right),a \in {L^\infty}\left({\rm{\Omega}} \right)$$ is nonnegative and b is a bounded potential which can change sign. The solution is obtained via a sub-supersolution approach when the parameter µ > 0 is small.

Let $${\{ {f_{\lambda ;j}}\} _{\lambda \in V;1 \leqslant j \leqslant k}}$$ be families of holomorphic functions in the open unit disk $${\text{D}} \subset {\Bbb C}$$ ⊂ ℂ depending holomorphically on a parameter λ ∈ V ⊂ ℂ n . We establish a Rolle type theorem for the generalized multiplicity (called cyclicity) of zeros of the family of univariate holomorphic functions $${\left\{ {\sum\nolimits_{j = 1}^k {{f_{\lambda ;j}}} } \right\}_{\lambda \in V}}$$ at 0 ∈ D. As a corollary, we estimate the cyclicity of the family of generalized exponential polynomials, that is, the family of entire functions of the form $$\sum\nolimits_{k = 1}^m {{P_k}(z){e^{{Q_k}(z)}}} $$ , z ∈ ℂ, where P k and Q k are holomorphic polynomials of degrees p and q, respectively, parameterized by vectors of coefficients of P k and Q k .

We construct a homogeneous subspace of 2ω whose complement is dense in 2ω and rigid. Using the same method, assuming Martin’s Axiom, we also construct a countable dense homogeneous subspace of 2ω whose complement is dense in 2ω and rigid.

LetX be a (not necessarily closed) subspace of the dual spaceB * of a separable Banach spaceB. LetX 1 denote the set of all weak * limits of sequences inX. DefineX a , for every ordinal numbera, by the inductive rule:X a = (U b < a X b ) 1 .There is always a countable ordinala such thatX a is the weak * closure ofX; the first sucha is called theorder ofX inB * . LetE be a closed subset of a locally compact abelian group. LetPM(E) be the set of pseudomeasures, andM(E) the set of measures, whose supports are contained inE. The setE obeys synthesis if and only ifM(E) is weak * dense inPM(E). Varopoulos constructed an example in which the order ofM(E) is 2. The authors construct, for every countable ordinala, a setE inR that obeys synthesis, and such that the order ofM(E) inPM(E) isa.