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

The isodiametric inequality is derived from the isoperimetric inequality trough a variational principle, establishing that balls maximize the perimeter among convex sets with fixed diameter. This principle brings also quantitative improvements to the isodiametric inequality, shown to be sharp by explicit nearly optimal sets.

Quantitative stability in the isodiametric inequality via the isoperimetric inequality

We consider the isoperimetric problem in \(\mathbb R^2\) with density. We show that, if the density is \(\mathrm{C}^{0,\alpha }\), then the boundary of any isoperimetric set is of class \(\mathrm{C}^{1,\frac{\alpha }{3-2\alpha }}\). This improves the previously known regularity.

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Regularity of isoperimetric sets in \mathbb R^2 with density

Given a continuous, injective function φ defined on the boundary of a planar open set Ω, we consider the problem of minimizing the total variation among all the BV homeomorphisms on Ω coinciding with φ on the boundary. We find the explicit value of this infimum in the model case when Ω is a rectangle. We also present two important consequences of this result: first, whatever the domain Ω is, the infimum above remains the same also if one restricts himself to consider only W 1,1 homeomorphisms. Second, any BV homeomorphism can be approximated in the strict BV sense with piecewise affine homeomorphisms and with diffeomorphisms.

On the planar minimal BV extension problem

In 1938 Herman Auerbach published a paper where he showed a deep connection between the solutions of the Ulam problem of floating bodies and a class of sets studied by Zindler, that are the planar sets whose bisecting chords have all the same length. In the same paper he conjectured that among Zindler sets the one with minimal area, as well as with maximal perimeter, is given by the so-called “Auerbach triangle”. We prove here that his conjecture was true.

https://www.ems-ph.org/journals/show_pdf.php?issn=1435-9855&vol=13&iss=6&rank=4

On a conjecture by Auerbach

Denoting by \({\varepsilon\subseteq\mathbb{R}^2}\) the set of the pairs \({(\lambda_1(\Omega),\,\lambda_2(\Omega))}\) for all the open sets \({\Omega\subseteq\mathbb{R}^N}\) with unit measure, and by \({\Theta\subseteq\mathbb{R}^N}\) the union of two disjoint balls of half measure, we give an elementary proof of the fact that \({\partial\varepsilon}\) has horizontal tangent at its lowest point \({(\lambda_1(\Theta),\,\lambda_2(\Theta))}\) .

https://hal.archives-ouvertes.fr/hal-00653903/document

Aldo Pratelli

Papers

Quantitative stability in the isodiametric inequality via the isoperimetric inequality

Regularity of isoperimetric sets in \mathbb R^2 with density

On the planar minimal BV extension problem

On a conjecture by Auerbach

On the boundary of the attainable set of the dirichlet spectrum