Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These

Phd by thesis

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Convex Analysisの二,三の進展について

Couplings and changes of variables.- Three examples of coupling techniques.- The founding fathers of optimal transport.- Qualitative description of optimal transport.- Basic properties.- Cyclical monotonicity and Kantorovich duality.- The Wasserstein distances.- Displacement interpolation.- The Monge-Mather shortening principle.- Solution of the Monge problem I: global approach.- Solution of the Monge problem II: Local approach.- The Jacobian equation.- Smoothness.- Qualitative picture.- Optimal transport and Riemannian geometry.- Ricci curvature.- Otto calculus.- Displacement convexity I.- Displacement convexity II.- Volume control.- Density control and local regularity.- Infinitesimal displacement convexity.- Isoperimetric-type inequalities.- Concentration inequalities.- Gradient flows I.- Gradient flows II: Qualitative properties.- Gradient flows III: Functional inequalities.- Synthetic treatment of Ricci curvature.- Analytic and synthetic points of view.- Convergence of metric-measure spaces.- Stability of optimal transport.- Weak Ricci curvature bounds I: Definition and Stability.- Weak Ricci curvature bounds II: Geometric and analytic properties.

Optimal Transport: Old and New

Preface.- Primal and Dual Problems.- One-Dimensional Issues.- L^1 and L^infinity Theory.- Minimal Flows.- Wasserstein Spaces.- Numerical Methods.- Functionals over Probabilities.- Gradient Flows.- Exercises.- References.- Index.

Optimal Transport for Applied Mathematicians : Calculus of Variations, PDEs, and Modeling

Optimal Transport: A Semi-Discrete Approach

It is well-known that duality in the Monge–Kantorovich transport problem holds true provided that the cost function c : X × Y → [0, ∞] is lower semi-continuous or finitely valued, but it may fail otherwise. We present a suitable notion of rectification cr of the cost c, so that the Monge-Kantorovich duality holds true replacing c by cr. In particular, passing from c to cr only changes the value of the primal Monge–Kantorovich problem. Finally, the rectified function cr is lower semi-continuous as soon as X and Y are endowed with proper topologies, thus emphasizing the role of lower semi-continuity in the duality-theory of optimal transport.

Duality for rectified cost functions

We prove that, given a planar bi-Lipschitz homeomorphism $u$ defined on the boundary of the unit square, it is possible to extend it to a function $v$ of the whole square, in such a way that $v$ is still bi-Lipschitz. In particular, denoting by $L$ and $\widetilde L$ the bi-Lipschitz constants of $u$ and $v$, with our construction one has $\widetilde L \leq C L^4$ (being $C$ an explicit geometrical constant). The same result was proved in 1980 by Tukia (see \cite{Tukia}), using a completely different argument, but without any estimate on the constant $\widetilde L$. In particular, the function $v$ can be taken either smooth or (countably) piecewise affine.

A planar bi-Lipschitz extension Theorem

We study the optimal sets $\Omega^\ast\subset\mathbb{R}^d$ for spectral functionals $F\big(\lambda_1(\Omega),\dots,\lambda_p(\Omega)\big)$, which are bi-Lipschitz with respect to each of the eigenvalues $\lambda_1(\Omega),\dots,\lambda_p(\Omega)$ of the Dirichlet Laplacian on $\Omega$, a prototype being the problem $$ \min{\big\{\lambda_1(\Omega)+\dots+ \lambda_p(\Omega)\;:\;\Omega\subset\mathbb{R}^d,\ |\Omega|=1\big\}}. $$ We prove the Lipschitz regularity of the eigenfunctions $u_1,\dots,u_p$ of the Dirichlet Laplacian on the optimal set $\Omega^*$ and, as a corollary, we deduce that $\Omega^*$ is open. For functionals depending only on a generic subset of the spectrum, as for example $\lambda_k(\Omega)$ or $\lambda_{k_1}(\Omega)+\dots+\lambda_{k_p}(\Omega)$, our result proves only the existence of a Lipschitz continuous eigenfunction in correspondence to each of the eigenvalues involved.

Lipschitz regularity of the eigenfunctions on optimal domains

Davide Buoso: "A few spectral optimization problems for elliptic equations and systems".- Giuseppe Buttazzo: "Optimization problems involving the first Dirichlet eigenvalue and the torsional rigidity".- Michel Delfour: "Metrics spaces of shapes and geometries from set parametrized functions".- Gianpaolo Leonardi.- Dario Mazzoleni.- Al-hassem Nayam: "Asymptotics of higher codimensional shapes and related problems".- Alberto Paganini: "Approximate Shape Gradients for Interface Problems".- Volker Schulz: "Towards a Lagrange-Newton Approach for PDE Constrained Shape Optimization".- Jan Sokolowski: "Shape optimization for nonlinear PDEs".- Kevin Sturm: "On the existence and identification of the shape derivative for PDE constrained problems".- Michiel Van den Berg.- Bozhidar Velichkov.- Alessandro Zilio: "Optimal regularity results for a partition problem involving".

Aldo Pratelli

Papers

Duality for rectified cost functions

A planar bi-Lipschitz extension Theorem

Lipschitz regularity of the eigenfunctions on optimal domains

New trends in shape optimization

The ε - εβ property, the boundedness of isoperimetric sets in ℝ N with density, and some applications