Differential Models for Sandpile Growth
|
|
- Suzan Freeman
- 5 years ago
- Views:
Transcription
1 Differential Models for Sandpile Growth Piermarco Cannarsa University of Rome Tor Vergata (Italy) LABORATOIRE JACQUES-LOUIS LIONS Universite Pierre et Marie Curie (Paris VI) April 3rd, 2009
2 Outline
3 a description from wikipedia granular material is a conglomeration of discrete solid, macroscopic particles characterized by a loss of energy whenever particles interact constituents must be large enough such that they are not subject to thermal motion fluctuations lower size limit for grains is about 1 µm upper size limit undefined (icebergs may be regarded as granular materials)
4 examples of granular materials Figure: coffee, plastic materials, sugar, pills, land, fresh snow... and sand
5 common features of interest at the microscopic level classical mechanics suffices to describe behaviour at the mesoscopic and macroscopic level new phenomena appear which are hard to understand with classical mechanics understanding the macroscopic behaviour of granular matter is of interest to physics as well to engineering, chemstry, drug industries,... these materials are largely present in nature: a good description of the motion of dunes, avalanches and so on may be of great help to environmental policies
6 Brazil nut effect largest particles end up on surface when a granular material containing a mixture of objects of different sizes is shaken serious interest for manufacturing once mixture has been created it is undesirable for different particle types to segregate several factors determine severity of the effect (the sizes and densities of the particles, the pressure of any gas between the particles, and the shape of the container)
7 sand a naturally occurring granular material composed of finely divided rock and mineral particles ranging in diameter from 0.06 to 2 millimeters most common constituent of sand is silica (silicon dioxide, or SiO2), usually in the form of quartz, resistant to weathering composition is highly variable, depending on the local rock sources and conditions
8 sand + wind = the beauty of dunes Sand is transported by wind and water and deposited in the form of beaches, dunes...
9 first east-west crossing of Libyan Desert (1932) founder and first commander of British Army s Long Range Desert Group during World War II a pioneer of desert exploration Figure: Ralph Alger Bagnold (3 April May 1990)
10 an influential book Figure: The Physics of Blown Sand and Desert Dunes (1941) laid foundations for research on sand transport by wind used by NASA in studying sand dunes on Mars
11 the table problem granular matter poured by a source onto a table forms a pile of a certain maximal slope falls from the table after reaching the boundary
12 mathematical models Different models have been proposed by physicists to study granular matter discrete models (cellular automata) statistical mechanics models (particle models) continuous models (partial differential equations) variational models double layer models
13 variational models (1996)
14 Prigozhin proved analysis of variational model well-posedness f L 2 (Ω) & u 0 K 0 comparison f 1 f 2 & u 1 0 u2 0 = u 1 u 2 equivalence with u t = div(v u) + f in R + Ω u 1, u < 1 v = 0 in R + Ω u = 0 on Ω, u(0, ) = u 0 in Ω model admits rolling matter only at critical slope
15 double layer models interactions between two layers also at sub-critical slopes Figure: P.-G. de Gennes (Nobel Laureate in Physics, 1991)
16 BCRE model BCRE, Boutreux and de Gennes, Hadeler and Kuttler (1994,... ) f v(x, t) u(x, t) x Ω
17 BCRE model à la Hadeler and Kuttler v t = div(v u) (1 u )v + f u t = (1 u )v u, v 0 u 1 in Ω maximal slope normalized to 1 u(, t) Ω = 0 sand falls down from the boundary
18 crush course on distance function... distance function d K (x) = min y K y x any closed K R n Ω R n oriented distance from Ω C 2 d(x) = d R n \Ω(x) d Ω (x) d( ) Lipschitz Lip(d) 1 d C 2 ( Ω + ρb) d(x) proj Ω (x) = { x} d(x) = x x x x singular set of d( ) in Ω Σ := {x Ω : d(x)} cut locus Σ connected compact dim H (Σ) n 1 normal distance to Σ τ(x) = min { t 0 : x + t d(x) Σ }
19 ... at a glance Ω Σ τ(x) d(x) x Γ d(x) x Figure: distance function d and normal distance τ
20 focal points Σ = Σ Γ for x Ω \ Σ κ i (x) 1 i n 1 i-th principal curvature of Ω at x = proj Ω (x) d(x)κ i (x) 1 x Γ d(x)κ i (x) = 1 for some i x Ω \ Σ n 1 D 2 κ d(x 0 ) = i (x 0 ) 1 κ i (x 0 )d(x 0 ) e i(x 0 ) e i (x 0 ) i=1 e i orthonormal unit vectors d(x)
21 regularity of normal distance easy τ continuous Ω optimal regularity Ω C 2,1 = τ Ω Lipschitz Itoh and Tanaka (2001) Ω C Li and Nirenberg (2005) Ω C 2,1 observe τ Ω Lipschitz = τ locally Lipschitz Ω \ Σ
22 τ Lipschitz in Ω? NO global regularity? Figure: Lipschitz regularity fails at focal points (x, τ 1 ) ( τ 0, 1 ) M x 2/3 2 2
23 Hölder continuity of normal distance Theorem (C, Cardaliaguet, Giorgieri: 2007) Ω R 2 Ω analytic } = α [2/3, 1] : τ C 0,α (Ω) Moreover Γ \ Σ = (e.g. Ω = disk) α = 1 Γ \ Σ (e.g. Ω = parabola) α < 1
24 asymptotics of variational model { u t (t, ) f, φ u(t, ) L 2 (Ω) 0 φ K 0 t 0 a.e. u(0, ) = u 0 K 0 := {u W 1, (Ω) : u 1, u Ω = 0} f 0 u 0 (x) u(t, x) d(x) u 0 (x) u (x) d(x) x Ω Theorem (C, Cardaliaguet, Sinestrari: 2009) u (x) = max{u 0 (x), u f (x)} u f (x) = max [d(y) y x ] + y spt(f ) x Ω x Ω
25 u f = max y spt(f ) [d(y) y x ] + smallest u 0 : Lip(u) 1 & u f d on spt(f ) u f d in Ω u f d in Ω Σ spt(f ) f u f Ω Σ
26 convergence in finite time suppose r > 0 : f r on B r (x) x Σ then Σ spt(f ) u f d u d Theorem (C, Cardaliaguet, Sinestrari: 2009) T > 0 such that u 0 K 0 u(t, ) = d t T
27 numerics by S. Finzi Vita 0.5 (HK) N=101 x=0.01 t=0.005 supp(f)=(0,1) it=6993 Tmax = (HK) N=101 x=0.01 t=0.005 supp(f)=(0,0.4) it=6879 Tmax = (HK) N=101 x=0.01 t= supp(f)=(0,1) it=25610 Tmax = Figure: evolution of u for different source supports (BCRE)
28 stationary system equilibria of both variational and BCRE models u t = div(v u) + f v t = div(v u) (1 u )v + f u 1 u t = (1 u )v u < 1 v = 0 u 1 u, v 0 u = 0 on Ω u = 0 on Ω satisfy stationary div(v u) = f in Ω, u 1 = 0 on {v > 0} u 1 u, v 0 in Ω u = 0 on Ω of interest in its own right
29 references Hadeler, Kuttler (1999) n = 1 C, Cardaliaguet (2004) n = 2 f and v contunuous C, Cardaliaguet, Crasta, Giorgieri (2005) n 2 f and v contunuous C, Cardaliaguet, Sinestrari (2009) f and v integrable
30 a representation formula (n = 2) (d, v) smooth equilibrium solution v = 0 on Σ v =? x Ω \ Σ 0 < t < τ(x) d dt = d(x) {}}{ v(x + t d(x)) = v(x + t d(x), d(x + t d(x)) = v(x + t d(x)) d(x + t d(x)) f (x + t d(x)) }{{} κ(x) = 1 (d(x)+t)κ(x) V (t) := v(x + t d(x)) satisfies V (τ(x)) = 0 and v(x) = V (t) τ(x) 0 κ(x) V (t) + f (x + t d(x)) = 0 1 (d(x) + t)κ(x) f (x + t d(x)) 1 (d(x) + t)κ(x) dt 1 d(x)κ(x)
31 v f (x) = τ(x) 0 f (x + t d(x)) support of v f 1 (d(x) + t)κ(x) dt 1 d(x)κ(x) spt(f ) Σ Ω spt(v f ) κ > 0 κ = 0
32 definition of solution (u, v) W 1, 0 (Ω) L 1 (Ω) equilibrium solution of table problem u, v 0 and u 1 a.e. φ Cc (Ω) v(x) Du(x), Dφ(x) dx = f (x)φ(x)dx Ω Ω v(x)( Du(x) 2 1)dx = 0 Ω
33 existence Ω R n Ω C 2 f 0 f L 1 (Ω) s f (y + sν(y)) n 1 i=1 (1 sκ i(y)) in L 1 ([0, τ(y)]) for H n 1 -a.e. y Ω v f (x) = τ(x) 0 n 1 f (x + t d(x)) i=1 1 (d(x) + t)κ i (x) 1 d(x)κ i (x) Theorem (d, v f ) equilibrium solution of table problem additional facts f L v f (y + tdd(y)) 0 as t τ(y) false for f unbounded: Ω = B 1 R 2 f (x) = 1 x d(x) = 1 x, Σ = {0}, k(x) 1, τ(x) = x, v f (x) 1 d(x) = u f (x) for every x spt(v f ) dt
34 (quasi-)uniqueness Theorem (u, v) W 1, 0 (Ω) L 1 (Ω) stationary solution iff v = v f a.e. u 1 and u f u d v unique ( v f ) u unique only if Σ spt(f ) u determined on {u f = d} spt(v f ) {u f = d}
35 numerics by S. Finzi Vita different dynamical models yield may converge to different solutions (with same rolling layer!) 0.5 (HK) N=51 x=0.02 t=0.01 supp(f)=(0,1) itstep=100 Tmax = (HK) N=51 x=0.02 t=0.01 supp(f)=(0,0.4) itstep=100 Tmax = (HK) N=51 x=0.02 t= supp(f)=(0,1) itstep=100 Tmax = Figure: equilibrium solutions u and (u + v) compared with d and u f
36 Figure: source does not cover ridge, BCRE versus variational model
37 related results optimal mass transport Evans, Feldman, Gariepy (1997) Feldman (1999) constrained problems in calculus of variations Cellina, Perrotta (1998) Bouchitté, Buttazzo (2001) Crasta, Malusa (2007) anisotropic geometries superconductivity Prigozhin (1998) Crasta, Malusa (2006) table problem with walls Crasta, Finzi Vita (2008)
38 pictures at an exhibition Figure: Apriamo la Mente, Roma 2007 Merci de votre attention
A Semi-Lagrangian Scheme for the Open Table Problem in Granular Matter Theory
A Semi-Lagrangian Scheme for the Open Table Problem in Granular Matter Theory M. Falcone and S. Finzi Vita Abstract We introduce and analyze a new scheme for the approximation of the two-layer model proposed
More informationA variational approach to the macroscopic. Electrodynamics of hard superconductors
A variational approach to the macroscopic electrodynamics of hard superconductors Dept. of Mathematics, Univ. of Rome La Sapienza Torino, July 4, 2006, joint meeting U.M.I. S.M.F. Joint work with Annalisa
More informationSur le tas de Sable. Noureddine Igbida (LAMFA CNRS-UMR 6140, Université de Picardie Jules Sur le Verne, tas de SableAmiens)
Sur le tas de Sable Noureddine Igbida LAMFA CNRS-UMR 6140, Université de Picardie Jules Verne, 80000 Amiens Nancy, 4 Mars 2008 Nancy, 4 Mars 2008 1 / 26 Une source de sable qui tourne : résultat d une
More informationMotivation Power curvature flow Large exponent limit Analogues & applications. Qing Liu. Fukuoka University. Joint work with Prof.
On Large Exponent Behavior of Power Curvature Flow Arising in Image Processing Qing Liu Fukuoka University Joint work with Prof. Naoki Yamada Mathematics and Phenomena in Miyazaki 2017 University of Miyazaki
More informationBoundary value problems for the infinity Laplacian. regularity and geometric results
: regularity and geometric results based on joint works with Graziano Crasta, Roma La Sapienza Calculus of Variations and Its Applications - Lisboa, December 2015 on the occasion of Luísa Mascarenhas 65th
More informationOn a weighted total variation minimization problem
On a weighted total variation minimization problem Guillaume Carlier CEREMADE Université Paris Dauphine carlier@ceremade.dauphine.fr Myriam Comte Laboratoire Jacques-Louis Lions, Université Pierre et Marie
More informationOn the stability of filament flows and Schrödinger maps
On the stability of filament flows and Schrödinger maps Robert L. Jerrard 1 Didier Smets 2 1 Department of Mathematics University of Toronto 2 Laboratoire Jacques-Louis Lions Université Pierre et Marie
More informationContinuum Model of Avalanches in Granular Media
Continuum Model of Avalanches in Granular Media David Chen May 13, 2010 Abstract A continuum description of avalanches in granular systems is presented. The model is based on hydrodynamic equations coupled
More informationMinimal time mean field games
based on joint works with Samer Dweik and Filippo Santambrogio PGMO Days 2017 Session Mean Field Games and applications EDF Lab Paris-Saclay November 14th, 2017 LMO, Université Paris-Sud Université Paris-Saclay
More informationAN INTRODUCTION TO VARIFOLDS DANIEL WESER. These notes are from a talk given in the Junior Analysis seminar at UT Austin on April 27th, 2018.
AN INTRODUCTION TO VARIFOLDS DANIEL WESER These notes are from a talk given in the Junior Analysis seminar at UT Austin on April 27th, 2018. 1 Introduction Why varifolds? A varifold is a measure-theoretic
More informationBehaviour of Lipschitz functions on negligible sets. Non-differentiability in R. Outline
Behaviour of Lipschitz functions on negligible sets G. Alberti 1 M. Csörnyei 2 D. Preiss 3 1 Università di Pisa 2 University College London 3 University of Warwick Lars Ahlfors Centennial Celebration Helsinki,
More informationContinuous dependence estimates for the ergodic problem with an application to homogenization
Continuous dependence estimates for the ergodic problem with an application to homogenization Claudio Marchi Bayreuth, September 12 th, 2013 C. Marchi (Università di Padova) Continuous dependence Bayreuth,
More informationModelling our sense of smell
Modelling our sense of smell by Carlos Conca What are olfactory cilia? The nasal epithelium (mucous) is the part of the nose which traps smells and communicates them to the brain. The microscopic olfactory
More informationMASS MOVEMENTS, WIND, AND GLACIERS
Date Period Name MASS MOVEMENTS, WIND, AND GLACIERS SECTION.1 Mass Movements In your textbook, read about mass movements. Use each of the terms below just once to complete the passage. avalanche creep
More informationOn the infinity Laplace operator
On the infinity Laplace operator Petri Juutinen Köln, July 2008 The infinity Laplace equation Gunnar Aronsson (1960 s): variational problems of the form S(u, Ω) = ess sup H (x, u(x), Du(x)). (1) x Ω The
More informationMass Movements, Wind, and Glaciers
Mass Movements,, and Glaciers SECTION 8.1 Mass Movement at Earth s Surface In your textbook, read about mass movement. Use each of the terms below just once to complete the passage. avalanche creep landslide
More informationHow the Cookie Crumbles
How the Cookie Crumbles Data Analysis for Experimental Granular Materials Research Victoria V.C. Winbow Dr. Rena Zieve UC-Davis REU 2005 Abstract: Granular materials are characterized as a conglomeration
More informationUna aproximación no local de un modelo para la formación de pilas de arena
Cabo de Gata-2007 p. 1/2 Una aproximación no local de un modelo para la formación de pilas de arena F. Andreu, J.M. Mazón, J. Rossi and J. Toledo Cabo de Gata-2007 p. 2/2 OUTLINE The sandpile model of
More informationSIMULATION OF A 2D GRANULAR COLUMN COLLAPSE ON A RIGID BED
1 SIMULATION OF A 2D GRANULAR COLUMN COLLAPSE ON A RIGID BED WITH LATERAL FRICTIONAL EFFECTS High slope results and comparison with experimental data Nathan Martin1, Ioan Ionescu2, Anne Mangeney1,3 François
More informationPratice Surface Processes Test
1. The cross section below shows the movement of wind-driven sand particles that strike a partly exposed basalt cobble located at the surface of a windy desert. Which cross section best represents the
More informationSoil Mechanics. Chapter # 1. Prepared By Mr. Ashok Kumar Lecturer in Civil Engineering Gpes Meham Rohtak INTRODUCTION TO SOIL MECHANICS AND ITS TYPES
Soil Mechanics Chapter # 1 INTRODUCTION TO SOIL MECHANICS AND ITS TYPES Prepared By Mr. Ashok Kumar Lecturer in Civil Engineering Gpes Meham Rohtak Chapter Outlines Introduction to Soil Mechanics, Soil
More informationWave operators with non-lipschitz coefficients: energy and observability estimates
Wave operators with non-lipschitz coefficients: energy and observability estimates Institut de Mathématiques de Jussieu-Paris Rive Gauche UNIVERSITÉ PARIS DIDEROT PARIS 7 JOURNÉE JEUNES CONTRÔLEURS 2014
More informationGeology 229 Engineering Geology. Lecture 7. Rocks and Concrete as Engineering Material (West, Ch. 6)
Geology 229 Engineering Geology Lecture 7 Rocks and Concrete as Engineering Material (West, Ch. 6) Outline of this Lecture 1. Rock mass properties Weakness planes control rock mass strength; Rock textures;
More informationBoundary value problems for the infinity Laplacian. regularity and geometric results
: regularity and geometric results based on joint works with Graziano Crasta, Roma La Sapienza Calculus of variations, optimal transportation, and geometric measure theory: from theory to applications
More informationRegularity of competitive equilibria in a production economy with externalities
Regularity of competitive equilibria in a production economy with externalities Elena del Mercato Vincenzo Platino Paris School of Economics - Université Paris 1 Panthéon Sorbonne QED-Jamboree Copenhagen,
More informationAnticipation guide # 3
Wind Anticipation guide # 3 Creep is a type of mass movement that happens slowly over many years Oxidation is a type of physical weathering A delta is a depositional feature that occurs with glaciers The
More informationFrequency functions, monotonicity formulas, and the thin obstacle problem
Frequency functions, monotonicity formulas, and the thin obstacle problem IMA - University of Minnesota March 4, 2013 Thank you for the invitation! In this talk we will present an overview of the parabolic
More informationTopology of the set of singularities of a solution of the Hamilton-Jacobi Equation
Topology of the set of singularities of a solution of the Hamilton-Jacobi Equation Albert Fathi IAS Princeton March 15, 2016 In this lecture, a singularity for a locally Lipschitz real valued function
More informationRocks. Sedimentary Rocks. Before You Read. Read to Learn
chapter 3 Rocks section 4 Sedimentary Rocks What You ll Learn how sedimentary rocks form how sedimentary rocks are classified Before You Read Imagine you are stacking slices of bread, one on top of the
More informationChapter 9 Notes: Ice and Glaciers, Wind and Deserts
Chapter 9 Notes: Ice and Glaciers, Wind and Deserts *Glaciers and Glacial Features glacier is a mass of ice that moves over land under its own weight through the action of gravity Glacier Formation must
More informationExistence of viscosity solutions for a nonlocal equation modelling polymer
Existence of viscosity solutions for a nonlocal modelling Institut de Recherche Mathématique de Rennes CNRS UMR 6625 INSA de Rennes, France Joint work with P. Cardaliaguet (Univ. Paris-Dauphine) and A.
More information1. Which type of climate has the greatest amount of rock weathering caused by frost action? A) a wet climate in which temperatures remain below
1. Which type of climate has the greatest amount of rock weathering caused by frost action? A) a wet climate in which temperatures remain below freezing B) a wet climate in which temperatures alternate
More informationEOLIAN PROCESSES & LANDFORMS
EOLIAN PROCESSES & LANDFORMS Wind can be an effective geomorphic agent under conditions of sparse vegetation & abundant unconsolidated sediment egs. hot & cold deserts, beaches & coastal regions, glacial
More informationGRANULAR MEDIA. Between Fluid and Solid
GRANULAR MEDIA Between Fluid and Solid Sand, rice, sugar, snow,cement...although ubiquitous in our daily lives, granular media still challenge engineers and fascinate researchers. This book provides the
More informationDraw a picture of an erupting volcano and label using the following words/phrases: magma; lava; cools slowly; cools quickly; intrusive; extrusive
Lesson 3.2a NOTES: Igneous Rocks (Unlock) Essential Question: How are igneous rocks described? Learning Target: I can describe how igneous rocks are formed and classified Igneous Rock How does igneous
More informationVariational coarse graining of lattice systems
Variational coarse graining of lattice systems Andrea Braides (Roma Tor Vergata, Italy) Workshop Development and Analysis of Multiscale Methods IMA, Minneapolis, November 5, 2008 Analysis of complex lattice
More informationRome - May 12th Université Paris-Diderot - Laboratoire Jacques-Louis Lions. Mean field games equations with quadratic
Université Paris-Diderot - Laboratoire Jacques-Louis Lions Rome - May 12th 2011 Hamiltonian MFG Hamiltonian on the domain [0, T ] Ω, Ω standing for (0, 1) d : (HJB) (K) t u + σ2 2 u + 1 2 u 2 = f (x, m)
More informationAnalyse d un Modèle de tas de Sable
Analyse d un Modèle de tas de Sable Noureddine Igbida LAMFA CNRS-UMR 6140, Université de Picardie Jules Verne, 80000 Amiens Premier congrès de la SM2A, Rabat 5-8 Février 2008 de Sable Premier congrés de
More informationEarth s crust is made mostly of Igneous rocks. There are 3 main types of Sedimentary Rocks: 1. Clastic 2. Chemical 3. Organic
Sedimentary Rocks Earth s crust is made mostly of Igneous rocks. But, most rocks on Earth s s surface are Sedimentary Rocks.. (75%) Sedimentary Rocks Sedimentary rocks are rocks that are made of broken-down
More informationMATH 2250 Final Exam Solutions
MATH 225 Final Exam Solutions Tuesday, April 29, 28, 6: 8:PM Write your name and ID number at the top of this page. Show all your work. You may refer to one double-sided sheet of notes during the exam
More informationWeathering, Erosion, and Deposition Guided Notes
1. Weathering, Erosion, and Deposition 2. Outline Section 1: Weathering Section 2: Erosion Section 3: Deposition Section 4: Case Study Weathering, Erosion, and Deposition Guided Notes 3. Section 1: Weathering
More informationSmall energy regularity for a fractional Ginzburg-Landau system
Small energy regularity for a fractional Ginzburg-Landau system Yannick Sire University Aix-Marseille Work in progress with Vincent Millot (Univ. Paris 7) The fractional Ginzburg-Landau system We are interest
More informationExistence of minimizers for the pure displacement problem in nonlinear elasticity
Existence of minimizers for the pure displacement problem in nonlinear elasticity Cristinel Mardare Université Pierre et Marie Curie - Paris 6, Laboratoire Jacques-Louis Lions, Paris, F-75005 France Abstract
More informationAvalanche Segregation of Granular Media. James Koeppe and Michael Enz
Avalanche Segregation of Granular Media James Koeppe and Michael Enz School of Physics and Astronomy University of Minnesota - Twin Cities 116 Church St. SE Minneapolis, MN 55455 Faculty Advisor: James
More informationSedimentary Rocks. All sedimentary rocks begin to form when existing rocks are broken down into sediments Sediments are mainly weathered debris
Rocks! Objectives Describe the major processes involved in the formation of sedimentary rock Distinguish between clastic sedimentary rocks and chemical sedimentary rocks Identify the features that are
More informationHyperbolic Systems of Conservation Laws
Hyperbolic Systems of Conservation Laws III - Uniqueness and continuous dependence and viscous approximations Alberto Bressan Mathematics Department, Penn State University http://www.math.psu.edu/bressan/
More informationcore mantle crust the center of the Earth the middle layer of the Earth made up of molten (melted) rock
core the center of the Earth mantle the middle layer of the Earth made up of molten (melted) rock crust the surface layer of the Earth that includes the continents and oceans 1 continental drift the theory
More informationNumerical Approximation of L 1 Monge-Kantorovich Problems
Rome March 30, 2007 p. 1 Numerical Approximation of L 1 Monge-Kantorovich Problems Leonid Prigozhin Ben-Gurion University, Israel joint work with J.W. Barrett Imperial College, UK Rome March 30, 2007 p.
More informationLecture No 2 Degenerate Diffusion Free boundary problems
Lecture No 2 Degenerate Diffusion Free boundary problems Columbia University IAS summer program June, 2009 Outline We will discuss non-linear parabolic equations of slow diffusion. Our model is the porous
More informationGeology and Geography. for grain size analysis. Trend surface, a least-squares fit method, is used.
1 Milton Recolizado St. ID: 0032477 Math 308 A Project Fall 2001 Geology and Geography Abstract In the areas of geology and geography, linear algebra can be applied. Linear models are often used for modeling
More informationTHE CAHN-HILLIARD EQUATION WITH A LOGARITHMIC POTENTIAL AND DYNAMIC BOUNDARY CONDITIONS
THE CAHN-HILLIARD EQUATION WITH A LOGARITHMIC POTENTIAL AND DYNAMIC BOUNDARY CONDITIONS Alain Miranville Université de Poitiers, France Collaborators : L. Cherfils, G. Gilardi, G.R. Goldstein, G. Schimperna,
More informationFigure 1 The map shows the top view of a meandering stream as it enters a lake. At which points along the stream are erosion and deposition dominant?
1. In which type of climate does chemical weathering usually occur most rapidly? 1. hot and dry 3. cold and dry 2. hot and wet 4. cold and wet 2. Figure 1 The map shows the top view of a meandering stream
More informationSediment and sedimentary rocks Sediment
Sediment and sedimentary rocks Sediment From sediments to sedimentary rocks (transportation, deposition, preservation and lithification) Types of sedimentary rocks (clastic, chemical and organic) Sedimentary
More informationPinning and depinning of interfaces in random media
Pinning and depinning of interfaces in random media Patrick Dondl joint work with Nicolas Dirr and Michael Scheutzow March 17, 2011 at Université d Orléans signal under crossed polarizers is measured and
More informationPhysical Geology, 15/e
Lecture Outlines Physical Geology, 15/e Plummer, Carlson & Hammersley Deserts & Wind Action Physical Geology 15/e, Chapter 13 Deserts Desert any arid region that receives less than 25 cm of precipitation
More informationc) metamorphosis d) rock transformation a) melting and cooling b) heat and pressure a) igneous rock b) sedimentary rock
Quizizz Rocks and Soil Name : Class : Date : 1. The process where rocks is transformed from one type to another is called a) rock cycle b) water cycle c) metamorphosis d) rock transformation 2. How are
More informationIsodiametric problem in Carnot groups
Conference Geometric Measure Theory Université Paris Diderot, 12th-14th September 2012 Isodiametric inequality in R n Isodiametric inequality: where ω n = L n (B(0, 1)). L n (A) 2 n ω n (diam A) n Isodiametric
More informationL.O: SLOWING STREAMS DEPOSIT (SORT) SEDIMENT HORIZONTALLY BY SIZE.
L.O: SLOWING STREAMS DEPOSIT (SORT) SEDIMENT HORIZONTALLY BY SIZE. 1. Base your answer to the following question on the profile shown below, which shows the pattern of horizontal sorting produced at a
More informationUNIT SEVEN: Earth s Water. Chapter 21 Water and Solutions. Chapter 22 Water Systems. Chapter 23 How Water Shapes the Land
UNIT SEVEN: Earth s Water Chapter 21 Water and Solutions Chapter 22 Water Systems Chapter 23 How Water Shapes the Land Chapter Twenty-Three: How Water Shapes the Land 23.1 Weathering and Erosion 23.2
More informationKinematic segregation of granular mixtures in sandpiles
Eur. Phys. J. B 7, 271 276 (1999) THE EUROPEAN PHYSICAL JOURNAL B c EDP Sciences Società Italiana di Fisica Springer-Verlag 1999 Kinematic segregation of granular mixtures in sandpiles H.A. Makse a Laboratoire
More informationAsymptotic behavior of the degenerate p Laplacian equation on bounded domains
Asymptotic behavior of the degenerate p Laplacian equation on bounded domains Diana Stan Instituto de Ciencias Matematicas (CSIC), Madrid, Spain UAM, September 19, 2011 Diana Stan (ICMAT & UAM) Nonlinear
More informationErosion and Deposition
CHAPTER 3 LESSON 2 Erosion and Deposition Landforms Shaped by Water and Wind Key Concepts What are the stages of stream development? How do water erosion and deposition change Earth s surface? How do wind
More informationTemperature and Heat. Ken Intriligator s week 4 lectures, Oct 21, 2013
Temperature and Heat Ken Intriligator s week 4 lectures, Oct 21, 2013 This week s subjects: Temperature and Heat chapter of text. All sections. Thermal properties of Matter chapter. Omit the section there
More informationsort examples of weathering into categories of biological, chemical, and physical;
Key Question How are rocks and minerals weathered? Learning Goals sort examples of weathering into categories of biological, chemical, and physical; observe and describe physical and chemical changes in
More informationSedimentary Rocks, our most Valuable Rocks. Or, what you will probably find when you are outdoors exploring.
Sedimentary Rocks, our most Valuable Rocks Or, what you will probably find when you are outdoors exploring. Sedimentary rocks give us evidence to earth s earlier history. We look at processes happening
More informationLaboratory 5. Sedimentary Rocks
Laboratory 5. Sedimentary Rocks The two primary types of sediment are chemical and detrital. Sediment becomes lithified into sedimentary rocks by cementation and compaction. Chemical sedimentconsists of
More informationDynamical Systems & Lyapunov Stability
Dynamical Systems & Lyapunov Stability Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University Outline Ordinary Differential Equations Existence & uniqueness Continuous dependence
More informationWeathering, Erosion, & Deposition Lab Packet
Weathering, Erosion, & Deposition Lab Packet Name Hour Grade /50 *Copper-Bearing Rocks and Iron Rocks need to be started on Tuesday or Wednesday. Freezing is done at home and will need to be completed
More informationLab 7: Sedimentary Structures
Name: Lab 7: Sedimentary Structures Sedimentary rocks account for a negligibly small fraction of Earth s mass, yet they are commonly encountered because the processes that form them are ubiquitous in the
More informationClastic Sedimentary Rocks
Clastic Sedimentary Rocks Alessandro Grippo, Ph.D. Alternating sandstones and mudstones in Miocene turbidites Camaggiore di Firenzuola, Firenze, Italy Alessandro Grippo review Mechanical weathering creates
More informationTD M1 EDP 2018 no 2 Elliptic equations: regularity, maximum principle
TD M EDP 08 no Elliptic equations: regularity, maximum principle Estimates in the sup-norm I Let be an open bounded subset of R d of class C. Let A = (a ij ) be a symmetric matrix of functions of class
More informationA Mathematical Trivium
A Mathematical Trivium V.I. Arnold 1991 1. Sketch the graph of the derivative and the graph of the integral of a function given by a freehand graph. 2. Find the limit lim x 0 sin tan x tan sin x arcsin
More informationSedimentary Structures
Sedimentary Structures irection of transport cards A5 cards with a sedimentary structure diagram on each are held up one at a time for students to work whether the direction is to the left or right. Interpreting
More informationHyperbolic conservation laws and applications Schedule and Abstracts
Hyperbolic conservation laws and applications Schedule and Abstracts The Graduate Center, CUNY 365 Fifth Avenue New York, NY 10016 Science Center, Room 4102 Thursday, April 26th, 2012 9:30am till 5:30pm
More informationIsoperimetric inequalities and cavity interactions
Laboratoire Jacques-Louis Lions Université Pierre et Marie Curie - Paris 6, CNRS May 17, 011 Motivation [Gent & Lindley 59] [Lazzeri & Bucknall 95 Dijkstra & Gaymans 93] [Petrinic et al. 06] Internal rupture
More informationUNIT 4 SEDIMENTARY ROCKS
UNIT 4 SEDIMENTARY ROCKS WHAT ARE SEDIMENTS Sediments are loose Earth materials (unconsolidated materials) such as sand which are transported by the action of water, wind, glacial ice and gravity. These
More informationSand in Forensic Geology. Modified from a PowerPoint presentation by J. Crelling, Southern Illinois University
Sand in Forensic Geology Modified from a PowerPoint presentation by J. Crelling, Southern Illinois University Characterizing Properties of Sand Remember that sand is actually a size of sediment Characterizing
More informationFiltered scheme and error estimate for first order Hamilton-Jacobi equations
and error estimate for first order Hamilton-Jacobi equations Olivier Bokanowski 1 Maurizio Falcone 2 2 1 Laboratoire Jacques-Louis Lions, Université Paris-Diderot (Paris 7) 2 SAPIENZA - Università di Roma
More informationThe Curious Case of Soft Matter Ranjini Bandyopadhyay Raman Research Institute
The Curious Case of Soft Matter Ranjini Bandyopadhyay Raman Research Institute ranjini@rri.res.in. My research interests: structure, dynamics, phase behavior and flow of soft materials Panta rei: everything
More informationGIOVANNI COMI AND MONICA TORRES
ONE-SIDED APPROXIMATION OF SETS OF FINITE PERIMETER GIOVANNI COMI AND MONICA TORRES Abstract. In this note we present a new proof of a one-sided approximation of sets of finite perimeter introduced in
More informationShape optimization under convexity constraint
Shape optimization under convexity constraint Jimmy LAMBOLEY University Paris-Dauphine with D. Bucur, G. Carlier, I. Fragalà, W. Gangbo, E. Harrell, A. Henrot, A. Novruzi, M. Pierre 07/10/2013, Ottawa
More informationStrong uniqueness for stochastic evolution equations with possibly unbounded measurable drift term
1 Strong uniqueness for stochastic evolution equations with possibly unbounded measurable drift term Enrico Priola Torino (Italy) Joint work with G. Da Prato, F. Flandoli and M. Röckner Stochastic Processes
More informationParticle Distribution Analysis of Simulated Granular Avalanches. Abstract
Particle Distribution Analysis of Simulated Granular Avalanches Abstract Initial results of two-dimensional, bimodal distribution of grains undergoing avalanches suggests there does not exist a relationship
More informationSobolev Spaces. Chapter Hölder spaces
Chapter 2 Sobolev Spaces Sobolev spaces turn out often to be the proper setting in which to apply ideas of functional analysis to get information concerning partial differential equations. Here, we collect
More informationHomogenization and Multiscale Modeling
Ralph E. Showalter http://www.math.oregonstate.edu/people/view/show Department of Mathematics Oregon State University Multiscale Summer School, August, 2008 DOE 98089 Modeling, Analysis, and Simulation
More informationC E C U R R I C U L U M I E N S C B L E I T A. i N T E G R A T I N G A R T S i n O N A T I D U C B L I P U. Student Learning Objectives:
We athering E Q U I T A B L E S C I E N C E C U R R I C U L U M Lesson 1 i N T E G R A T I N G A R T S i n P U B L I C E D U C A T I O N NGSS Science Standard: 4-ESS1-1 Identify evidence from patterns
More information1 Shoreline Erosion and Deposition
CHAPTER 12 1 Shoreline Erosion and Deposition SECTION Agents of Erosion and Deposition BEFORE YOU READ After you read this section, you should be able to answer these questions: What is a shoreline? How
More informationA BEACH IS A BEACH. Or Is It? Hawaii. St. Croix, US Virgin Islands
A BEACH IS A BEACH Or Is It? Pt. Reyes, California Western Florida Hawaii AGI What is a beach? Eastern Maine A beach is a strip of shoreline washed by waves and tides. Crane Key, Florida Bay St. Croix,
More informationContents: 1. Minimization. 2. The theorem of Lions-Stampacchia for variational inequalities. 3. Γ -Convergence. 4. Duality mapping.
Minimization Contents: 1. Minimization. 2. The theorem of Lions-Stampacchia for variational inequalities. 3. Γ -Convergence. 4. Duality mapping. 1 Minimization A Topological Result. Let S be a topological
More informationThe Rocky Road Game. Sedimentary Rock. Igneous Rock. Start. Metamorphic Rock. Finish. Zone of Transportation. Weathering Way.
Sedimentary Rock Deposition Depot Zone of Transportation Transported: Advance 3 Weathering Way The Rocky Road Game Uplift: Advance 5 Lithification Lane Crystallization Crossway Submerge Detour take the
More informationFrom nonlocal to local Cahn-Hilliard equation. Stefano Melchionna Helene Ranetbauer Lara Trussardi. Uni Wien (Austria) September 18, 2018
From nonlocal to local Cahn-Hilliard equation Stefano Melchionna Helene Ranetbauer Lara Trussardi Uni Wien (Austria) September 18, 2018 SFB P D ME S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal
More informationIgneous, Metamorphic & Sedimentary. Chapter 5 & Chapter 6
Igneous, Metamorphic & Sedimentary Chapter 5 & Chapter 6 Section 5.1 What are Igneous Rocks? Compare and contrast intrusive and extrusive igneous rocks. Describe the composition of magma Discuss the factors
More informationDry granular flows: gas, liquid or solid?
Dry granular flows: gas, liquid or solid? Figure 1: Forterre & Pouliquen, Annu. Rev. Fluid Mechanics, 2008 1 Characterizing size and size distribution Grains are not uniform (size, shape, ) Statistical
More informationHomogenization limit for electrical conduction in biological tissues in the radio-frequency range
Homogenization limit for electrical conduction in biological tissues in the radio-frequency range Micol Amar a,1 Daniele Andreucci a,2 Paolo Bisegna b,2 Roberto Gianni a,2 a Dipartimento di Metodi e Modelli
More informationScience EOG Review: Landforms
Mathematician Science EOG Review: Landforms Vocabulary Definition Term canyon deep, large, V- shaped valley formed by a river over millions of years of erosion; sometimes called gorges (example: Linville
More informationExamining the Terrestrial Planets (Chapter 20)
GEOLOGY 306 Laboratory Instructor: TERRY J. BOROUGHS NAME: Examining the Terrestrial Planets (Chapter 20) For this assignment you will require: a calculator, colored pencils, a metric ruler, and your geology
More informationName: Class: Date: Multiple Choice Identify the letter of the choice that best completes the statement or answers the question.
Name: Class: Date: geology ch 7 test 2008 Multiple Choice Identify the letter of the choice that best completes the statement or answers the question. 1. Which of the following is true about ice sheets?
More informationAn introduction to the physics of. Granular Matter. Devaraj van der Meer
An introduction to the physics of Granular Matter Devaraj van der Meer GRANULAR MATTER is everywhere: in nature: beach, soil, snow, desert, mountains, sea floor, Saturn s rings, asteroids... in industry:
More informationThe Agents of Erosion
The Agents of Erosion 1. Erosion & Deposition 2. Water 3. Wind 4. Ice California Science Project 1 1. Erosion and Deposition Erosion is the physical removal and transport of material by mobile agents such
More informationSobolev spaces. May 18
Sobolev spaces May 18 2015 1 Weak derivatives The purpose of these notes is to give a very basic introduction to Sobolev spaces. More extensive treatments can e.g. be found in the classical references
More information