Related papers: Entropy for Monge-Amp\`ere Measures in the Prescri…

Let $\phi: \Bbb R^n \mapsto \Bbb R$ be a strictly convex and smooth function, and $\mu= \text{det}\,D^2 \phi$ be the Monge-Amp\`ere measure generated by $\phi.$ For $x\in \Bbb R^n$ and $t>0$, let $S(x,t):=\{y\in \Bbb R^n:…

We consider mixed Monge-Amp\`ere products of quasiplurisubharmonic functions with analytic singularities, and show that such products may be regularized as explicit one parameter limits of mixed Monge-Amp\`ere products of smooth functions,…

We consider suspension flows with continuous roof function over the full shift $\Sigma$ on a finite alphabet. For any positive entropy subshift of finite type $Y \subset \Sigma$, we explictly construct a roof function such that the…

We consider the complex Monge-Amp\'{e}re equation on complete K\"{a}hler manifolds with cusp singularity along a divisor when the right hand side $F$ has rather weak regularity. We proved that when the right hand side $F$ is in some…

Finite energy pluripotential theory accommodates the variational theory of equations of complex Monge-Amp\`ere type arising in K\"ahler geometry. Recently it has been discovered that many of the potential spaces involved have a rich metric…

We consider the entropy of sums of independent discrete random variables, in analogy with Shannon's Entropy Power Inequality, where equality holds for normals. In our case, infinite divisibility suggests that equality should hold for…

We prove a complete realization theorem for multifractal entropy spectra of continuous potentials on a broad class of dynamical systems. More precisely, for every $H>0$ and every continuous concave function on a compact interval with…

We investigate the uniqueness for the Monge-Amp\`{e}re type equation \begin{equation} \label{eq-abstract} det(u_{ij}+\delta_{ij}u)_{i,j=1}^{n-1}=G(u),\ \ \ \ \ \ \ (*)\end{equation}on $S^{n-1}$, where $u$ is the restriction of the support…

We show that the non pluripolar product of positive currents is a bimeromorphic invariant. Under some natural assumptions, we show that the (weighted) energy associated to big cohomology classes are also bimeromorphic invariants. We compare…

In this work we consider companion conservation laws to general systems of conservation laws. We investigate sufficient regularity for weak solutions to satisfy companion laws, assuming the fluxes to be $C^{1,\gamma}$, $0<\gamma<1$,…

We prove a geodesic distance formula for quasi-psh functions with finite entropy, extending results by Chen and Darvas. We work with big and nef cohomology classes: a key result we establish is the convexity of the $K$-energy in this…

Let $(X,\omega)$ be a compact K\"ahler manifold. We introduce and study the largest set $DMA(X,\omega)$ of $\omega$-plurisubharmonic (psh) functions on which the complex Monge-Amp\`ere operator is well defined. It is much larger than the…

This paper studies the behavior of the entropy numbers of classes of functions with bounded integral norms from a given finite dimensional linear subspace. Upper bounds of these entropy numbers in the uniform norm are obtained and applied…

We study a generalization of the Fr\'echet mean on metric spaces, which we call $\phi$-means. Our generalization is indexed by a convex function $\phi$. We find necessary and sufficient conditions for $\phi$-means to be finite and provide a…

We investigate the finite $p$-energy classes $E_p$ of quaternionic plurisubharmonic functions of Cegrell type. We also construct an example to show that the optimal constant in the energy estimate is strictly bigger than $1$ for $p>0$,…

Let $\Omega$ be a bounded, pseudoconvex domain of $\mathbb C^n$ satisfying the "$f$-Property". The $f$-Property is a consequence of the geometric "type" of the boundary; it holds for all pseudoconvex domains of finite type but may also…

We prove that $\mathcal{C}^{1,\alpha}$ solutions to the Monge-Amp\`ere system in dimension $d$ and codimension $k= d_*-d+1$, where $d_*$ denotes the Janet dimension, are dense in the space of continuous functions, for every H\"older…

The principles of energy, symmetry, entropy, and causality conservation are discussed in a "Tetrahedron Model" of a conceptually complete "T.O.E.:" "The charges of matter are the symmetry debts of light" (Noether's Theorem). Symmetry debts…

We study the ratio of the entropy to the total energy in conformal field theories at finite temperature. For the free field realizations of {\cal N}=4 super Yang-Mills theory in D=4 and the (2,0) tensor multiplet in D=6, the ratio is…

Information plays an important role in our understanding of the physical world. We hence propose an entropic measure of information for any physical theory that admits systems, states and measurements. In the quantum and classical world,…