Agostino Prastaro | Searcharxiv

arXiv2012

Quantum Extended Crystal Super Pde's

We generalize our geometric theory on extended crystal PDE's and their stability, to the category $\mathfrak{Q}_S$ of quantum supermanifolds. By using algebraic topologic techniques, obstructions to the existence of global quantum smooth solutions for such equations are obtained. Applications are given to encode quantum dynamics of nuclear nuclides, identified with graviton-quark-gluon plasmas, and study their stability. We prove that such quantum dynamical systems are encoded by suitable quantum extended crystal Yang-Mills super PDE's. In this way stable nuclear-charged plasmas and nuclides are characterized as suitable stable quantum solutions of such quantum Yang-Mills super PDE's. An existence theorem of local and global solutions with mass-gap, is given for quantum super Yang-Mills PDE's, $\hat{(YM)}$, by identifying a suitable constraint, $\hat{(Higgs)}\subset \hat{(YM)}$, {\em Higgs quantum super PDE}, bounded by a quantum super partial differential relation $\hat{(Goldstone)}\subset \hat{(YM)}$, {\em quantum Goldstone-boundary}. A global solution $V\subset\hat{(YM)}$, crossing the quantum Goldstone-boundary acquires (or loses) mass. Stability properties of such solutions are characterized.

math.AT↗

arXiv2011

Extended Crystal PDE's

In this paper we show that between PDE's and crystallographic groups there is an unforeseen relation. In fact we prove that integral bordism groups of PDE's can be considered extensions of crystallographic subgroups. In this respect we can consider PDE's as {\em extended crystals}. Then an algebraic-topological obstruction ({\em crystal obstruction}), characterizing existence of global smooth solutions for smooth boundary value problems, is obtained. Applications of this new theory to the Ricci-flow equation and Navier-Stokes equation are given that solve some well-known fundamental problems. These results, are also extended to singular PDE's, introducing ({\em extended crystal singular PDE's}). An application to singular MHD-PDE's, is given following some our previous results on such equations, and showing existence of (finite times stable smooth) global solutions crossing critical nuclear energy production zone.

math.AT↗