Energy measure closability for Dirichlet forms

We consider symmetric Dirichlet forms on locally compact and non-locally compact spaces and provide an elementary proof for their closability with respect to energy dominant measures. We also discuss how to use known potential theoretic results to furnish an alternative proof of this theorem

Vector analysis for Dirichlet forms and quasilinear PDE and SPDE on metric measure spaces

Starting with a regular symmetric Dirichlet form on a locally compact separable metric space $X$, our paper studies elements of vector analysis, $L_p$-spaces of vector fields and related Sobolev spaces. These tools are then employed to obtain existence and uniqueness results for some quasilinear elliptic PDE and SPDE in variational form on $X$ by standard methods. For many of our results locality is not assumed, but most interesting applications involve local regular Dirichlet forms on fractal spaces such as nested fractals and Sierpinski carpets

Metrics and spectral triples for Dirichlet and resistance forms

The article deals with intrinsic metrics, Dirac operators and spectral triples induced by regular Dirichlet and resistance forms. We show, in particular, that if a local resistance form is given and the space is compact in resistance metric, then the intrinsic metric yields a geodesic space. Given a regular Dirichlet form, we consider Dirac operators within the framework of differential 1-forms proposed by Cipriani and Sauvageot, and comment on its spectral properties. If the Dirichlet form admits a carr\'e operator and the generator has discrete spectrum, then we can construct a related spectral triple, and in the compact and strongly local case the associated Connes distance coincides with the intrinsic metric. We finally give a description of the intrinsic metric in terms of vector fields

A mono‐substituted silicon(II) cation – A crystalline “supersilylene”

Mono‐coordinated silicon(II) cations are predicted to be reactive ambiphiles, combining the typically high Lewis acidity of silicon cations with nucleophilicity due to the presence of an electron pair at the same atomic centre. Here, a carbazole‐derived scaffold was used to isolate salts with a mono‐coordinated silicon(II) cation, [RSi]$^{+}$ (R=bulky carbazolyl substituent), by halide abstraction from a base‐free halosilylene, RSiI, with Ag[Al(O$^{t}$Bu$^{F}$)$_{4}$]. Despite the bulk of the carbazolyl moiety, the silylenylium cation [RSi]$^{+}$ retains high reactivity. It was shown to react with an amine to form three bonds at the silicon atom in one reaction which conforms with the notion of a “supersilylene”. The resulting silylium cation [RSi(H)NR′$_{2}$]$^{+}$ (in the formal oxidation state Si$^{IV}$) obtained by oxidative addition of an NH bond at [RSi]$^{+}$ is even more acidic than the silylenylium cation (Si$^{II}$) due to the absence of a lone pair of electrons the silicon atom