We introduce a notion of rectifiability modeled on Carnot groups. Precisely, we say that a subset E of a Carnot group M and N is a subgroup of M , we say E is Nrectifiable if it is the Lipschitz image of a positive measure subset of N . First, we discuss the implications of N-rectifiability, where N is a Carnot group (not merely a subgroup of a Carnot group), which include N-approximability and...

We study multi-parameter Carnot-Carathéodory balls, generalizing results due to Nagel, Stein, and Wainger in the single parameter setting. The main technical result is seen as a uniform version of the theorem of Frobenius. In addition, we study maximal functions associated to certain multi-parameter families of Carnot-Carathéodory balls.

AND Abstract. Graded nilpotent Lie groups, or Carnot Groups are to subRiemannian geometry as Euclidean spaces are to Riemannian geometry. They are the metric tangent cones for this geometry. Hoping that the analogy between subRiemannian and Riemannian geometry is a strong one, one might conjecture that the subRiemannian geo-desic ow on any Carnot group is completely integrable. We prove this co...

15.1. The First Law of Thermodynamics • Define the first law of thermodynamics. • Describe how conservation of energy relates to the first law of thermodynamics. • Identify instances of the first law of thermodynamics working in everyday situations, including biological metabolism. • Calculate changes in the internal energy of a system, after accounting for heat transfer and work done. 15.2. Th...

Thermostatics of Carnot engines has been extended by more recent research based on endo-reversible model. Our model assumes exo-reversibility but endo-irreversibility to determine new upper-bound thermomechanical conversion. We propose a functional expression entropy production related cycle transformation durations. This approach analyses the energy, and power consequences. introduce concept a...

We have devised a thermodynamic model of cortical neurodynamics expressed at the classical level by neural networks and at the quantum level by dissipative quantum field theory. Our model is based on features in the spatial images of cortical activity newly revealed by high-density electrode arrays. We have incorporated the mechanism and necessity for so-called dark energy in knowledge retrieva...

A dilatation structure encodes the approximate self-similarity of a metric space. A metric space (X, d) which admits a strong dilatation structure (definition 2.2) has a metric tangent space at any point x ∈ X (theorem 4.1), and any such metric tangent space has an algebraic structure of a conical group (theorem 4.2). Particular examples of conical groups are Carnot groups: these are simply con...