In this work, we provide a characterization result for lower semicontinuity of surface energies defined on piecewise rigid functions, i.e., functions which are affine Caccioppoli partition where the derivative in each component is skew symmetric matrix. This achieved by means an integral condition, called BD-ellipticity , spirit BV -ellipticity Ambrosio and Braides [5] . By specific examples sh...

A cornerstone of current–density functional theory (CDFT) in its paramagnetic formulation is proven. After a brief outline the mathematical structure CDFT, lower semicontinuity and expectation-valuedness CDFT constrained-search proven, meaning that there always minimizing density matrix universal functional. These results place framework on same footing as standard DFT.

We give a categorical treatment, in the spirit of Baez and Fritz, of relative entropy for probability distributions defined on standard Borel spaces. We define a category called SbStat suitable for reasoning about statistical inference on standard Borel spaces. We define relative entropy as a functor into Lawvere’s category [0,∞] and we show convexity, lower semicontinuity and uniqueness.

In the pointfree topological context of locales and frames, real functions on a locale L are represented as localic morphisms S(L)→ L(R) (i.e. frame homomorphisms L(R) → S(L)) where S(L) stands for the frame of all sublocales of L and L(R) denotes the frame of reals. This is reminiscent of dealing with (not necessarily continuous) real functions X → R as with continuous functions D(X)→ R where ...