493 research outputs found
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 , our paper studies elements of vector analysis,
-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 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(OBu)]. 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′] (in the formal oxidation state Si) obtained by oxidative addition of an NH bond at [RSi] is even more acidic than the silylenylium cation (Si) due to the absence of a lone pair of electrons the silicon atom
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