For a product preserving gauge bundle functor on vector bundles, we present some lifts of smooth functions that are constant or linear on fibers, and some lifts of projectable vector fields that are vector bundle morphisms.

Properties of Lipschitz and d.c. surfaces of finite codimension in a Banach space and properties of generated $\sigma$-ideals are studied. These $\sigma$-ideals naturally appear in the differentiation theory and in the abstract approximation theory. Using these properties, we improve an unpublished result of M. Heisler which gives an alternative proof of a result of D. Preiss on singular points of convex functions.

We find conditions for a smooth nonlinear map f: U → V between open subsets of Hilbert or Banach spaces to be locally convex in the sense that for some c and each positive ɛ < c the image f(B ɛ(x)) of each ɛ-ball B ɛ(x) ⊂ U is convex. We give a lower bound on c via the second order Lipschitz constant Lip2(f), the Lipschitz-open constant Lipo(f) of f, and the 2-convexity number conv2(X) of the Banach space X.

We consider a Schrödinger-type differential expression $H_V={\nabla }^{*}\nabla +V$, where $\nabla$ is a $C^{\infty }$-bounded Hermitian connection on a Hermitian vector bundle $E$ of bounded geometry over a manifold of bounded geometry $(M,g)$ with metric $g$ and positive $C^{\infty }$-bounded measure $d\mu$, and $V$ is a locally integrable section of the bundle of endomorphisms of $E$. We give a sufficient condition for $m$-sectoriality of a realization of $H_V$ in $L^2\left(E\right)$. In the proof we use generalized Kato’s inequality as well as a result on the positivity of $u\in L^2\left(M\right)$ satisfying the...

We prove that four manifolds diffeomorphic on the complement of a point have the same Donaldson invariants.

The paper explains the notion of projectively equivariant quantization. It gives a sketch of Martin Bordemann's proof of the existence of projectively equivariant quantization on arbitrary manifolds.

Motivati dall'analisi asintotica dei vortici nella teoria di Chern-Simons-Higgs, si studia l'equazione $$-\mathrm{\Delta }u=\lambda \left(\frac{e^u}{{\int }_{\mathrm{\Omega }}{e}^{u}dx}-\frac{1}{\left|\mathrm{\Omega }\right|}\right),u\in H^1\left(\mathrm{\Omega }\right)$$ dove $\mathrm{\Omega }={\mathbb{R}}^2/{\mathbb{Z}}^2$ é il toro piatto bidimensionale. In contrasto con l'analogo problema di Dirichlet, si dimostra che per $\lambda \in \left(8\pi ,4{\pi }^2\right)$ l'equazione ammette una soluzione non banale. Tale soluzione cattura il carattere bidimensionale dell'equazione, nel senso che, per tali valori di $\lambda$, l'equazione non può ammettere soluzioni (periodiche) non banali dipendenti da una sola variabile (vedi [10]).

Some properties and applications of natural vector bundle morphisms $T_A\circ {⨂}_s^q\to {⨂}_s^q\circ T_A$ over $i{d}_{T_A}$ are presented.

We deduce that all natural operators of the type of the Helmholtz map from the variational calculus in fibered manifolds are the constant multiples of the Helmholtz operator.

We discuss possible topological configurations of nodal sets, in particular the number of their components, for spherical harmonics on $S^2$. We also construct a solution of the equation $\Delta u=u$ in ${ℝ}^2$ that has only two nodal domains. This equation arises in the study of high energy eigenfunctions.