n this paper, we consider continuity properties(especially, regularity, also viewed as an approximation property) for $%mathcal{p}_{0}(x)$-valued set multifunctions ($x$ being a linear,topological space), in order to obtain egoroff and lusin type theorems forset multifunctions in the vietoris hypertopology. some mathematicalapplications are established and several physical implications of thema...

Since the last twenty years an increasing interest on the fractal approach on microphysics has been observed [1-9]. The Theory of fractal space-time or the Scale Relativity Theory (SRT) [1] extends Einstein’s principle of relativity to scale transformation of resolution. The introduction of non-differentiable trajectories in physics dates back to the pioneering works of Feynman, who demonstrate...

A reconciliation of gravitation and electromagnetism has eluded physics for neearly a century. It is argued here that this is because both quantum physics and classical physics are set in differentiable space time manifolds with point particles. Once we consider extended particles as in Quantum Superstring theory, and the consequential underlying Non-Commutative geometry, then a reconciliation ...

We present a differentiable formulation of rigid-body contact dynamics for objects and robots represented as compositions convex primitives. Classical physics engines rely on non-differentiable collision detection modules. More recent optimization-based approaches simulating between primitives bilevel that separates simulation. These latest are unreliable in realistic simulation scenarios becau...

We describe how to apply adjoint sensitivity methods backward Monte-Carlo schemes arising from simulations of particles passing through matter. Relying on this, we demonstrate derivative based techniques for solving inverse problems such systems without approximations underlying transport dynamics. are implementing those algorithms various scenarios within a general purpose differentiable progr...

The problem of the efficient design material microstructures exhibiting desired properties spans a variety engineering and science applications. ability to rapidly generate that exhibit user-specified property distributions can transform iterative process traditional microstructure-sensitive design. We reformulate microstructure using constrained generative adversarial network (GAN) model. This...

The essential role played by differentiable structures in physics is reviewed in light of recent mathematical discoveries that topologically trivial space-time models, especially the simplest one, R , possess a rich multiplicity of such structures, no two of which are diffeomorphic to each other and thus to the standard one. This means that physics has available to it a new panoply of structure...

Abstract Data-driven turbulence modeling is experiencing a surge in interest following algorithmic and hardware developments the data sciences. We discuss an approach using differentiable physics paradigm that combines known with machine learning to develop closure models for Burgers’ turbulence. consider one-dimensional Burgers system as prototypical test problem unresolved terms advection-dom...

We review recent developments in differential topology with special concern for their possible significance to physical theories, especially general relativity. In particular we are concerned here with the discovery of the existence of non-standard (“fake” or “exotic”) differentiable structures on topologically simple manifolds such as S, Rand S ×R. Because of the technical difficulties involve...