We introduce a novel approach towards Harnack’s inequality in the context of spaces of homogeneous type. This approach, based on the socalled critical density property and doubling properties for weights, avoids the explicit use of covering lemmas and BMO.

This paper is a sequel to [7], where a local version of ZFC, LZFC, was introduced and examined and transitive models of ZFC with properties that resemble large cardinal properties, namely Mahlo and Π1-indescribable models, were considered. By analogy we refer to such models as “large models”, and the properties in question as “large model properties”. Continuing here in the same spirit we consi...

Internet has undeniably benefited human society as a whole and become the fulcrum of every sector across globe. Cyberspace, borderless world, thrown various challenges which need to be maneuvered with care caution future belongs internet information age. The research article projects light on problems protection of- copyright in Infringement cyberspace very ordinary. Copyright can infringed thr...

Deep learning and graph-based models have gained popularity in various life science applications such as property modeling, achieving state-of-the-art performance. However, the quantification of prediction uncertainty these is less studied not applied low dataset size regime, which characterizes many chemical engineering-related molecular properties. In this work, we apply two to model critical...

We present a syntactic method for proving observational equivalences in reduction systems. The method is based on establishing a weak diamond property for critical pairs. It has been used successfully in proofs on the observational equivalence theories of var and .