# Register now to access 7 million high quality study materials (What's Course Hero?) Course Hero is the premier provider of high quality online educational resources. With millions of study documents, online tutors, digital flashcards and free courseware, Course Hero is helping students learn more efficiently and effectively. Whether you're interested in exploring new subjects or mastering key topics for your next exam, Course Hero has the tools you need to achieve your goals.

2 Pages

### sol3

Course: AM 224, Fall 2009
School: Sanford-Brown Institute
Rating:

Word Count: 545

#### Document Preview

HW 1 PDE, 3 solutions 7, p.163. Suppose g is C 1 . The Hopf-Lax formula implies Dg(y) L x-y t , at a inverse Lagrangian point y. By problem 6, this is is equivalent to x-y H(Dg(y)), t which implies y B(x, Rt). 8,p.163. The Hamiltonian is H(p) = |p|2 , thus L = H = |q|2 /4. The Hopf-Lax formula says u(x, t) = infn yR |x - y|2 + u0 (y) . 4t If y Rn \E, the right hand side is +. Thus, it suffices to consider...

Register Now

#### Unformatted Document Excerpt

Coursehero >> Georgia >> Sanford-Brown Institute >> AM 224

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
HW 1 PDE, 3 solutions 7, p.163. Suppose g is C 1 . The Hopf-Lax formula implies Dg(y) L x-y t , at a inverse Lagrangian point y. By problem 6, this is is equivalent to x-y H(Dg(y)), t which implies y B(x, Rt). 8,p.163. The Hamiltonian is H(p) = |p|2 , thus L = H = |q|2 /4. The Hopf-Lax formula says u(x, t) = infn yR |x - y|2 + u0 (y) . 4t If y Rn \E, the right hand side is +. Thus, it suffices to consider the infimum over E where u0 (y) = 0. Hence, u(x, t) = inf yE |x - y|2 1 = dist(x, E)2 . 4t 4t 13, p. 291. We compute |Du(x)| = so that B(0,1) 1 1 1 , log (1 + 1/|x|) 1 + |x| |x| 1 0 |Du(x)|n dx = n 1 1 1 dr n log(1 + 1/r) 1 + r r log 2 1 dt tn after the change of variables t = log(1 + 1/r). 18, p. 291. Let u() denote the Fourier transform of u(x). By the Fourier ^ inversion formula u L Cn u L1 , which is controlled by the H s norm, ^ Rn |^| d u Rn 1 d (1 + ||2 )s 1/2 Rn (1 + ||2 )s |^|2 d u Hs 1/2 . The first term is finite if s > n/2, the second is u by Parseval's equality. 2 2,p.563. This is hard work. The problem requires that you absorb Kruzkov's proof for conservations laws, and adapt it to viscosity solutions. While this is a worthy goal, I am not going to do it here. The proof may be found in V.2, Crandall and Lions, Trans. AMS, Vol. 277, (1983), pp.1-42. 3,p.563. Suppose u - v has a global maximum at (x0 , t0 ). Without loss of generality, we may suppose , u(x0 t0 ) = v(x0 , t0 ). Define e = u - u 2 , = 4e , v (x, t) = v(x, t) + (|x - x0 |2 + |t - t0 |2 ). Observe that if |x - x0 |2 + |t - t0 |2 = r 2 with r 2 we have 2 e - r 2 e - = -3e . (u - v )(x, t) = (u - u)(x, t) + (v - v )(x, t) + (u - v)(x, t) On the other hand, (u - v )(x0 , t0 ) = u - u(x0 , t0 ) -e . Therefore, u - v has a global maximum (x , t ) in the ball B((x0 , t0 ), ). At this point, u = vt and Du = Dv and since t u + H(Du , x ) = aij Dij u 0, t we have the inequality Now let 0 using 0, (x , t ) (x0 , t0 ), vt vt , DV DV . vt + H(Dv , x ) 0. 4, p. 563. (a) It will suffice to check the definition of viscosity solution at x0 = 0. Suppose u - v has a maximum at 0. Then, (u - v)(x) (u - v)(0) for all x (-1, 1) which implies 1 - |x| - v(0) + v (0)x + o(x) 1 - v(0). First suppose x &g...

Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

Sanford-Brown Institute - AM - 223
1PDE, HW 5 solutionsProblem 1. Change variables to p = |x|2 /4t. Then 0 n 1 1 2 e|x| /4t dt = n/2 |x|2n ep p 2 2 dp n/2 (4t) 4 0 1 1 |x|2n 2 n = n/2 |x|2n 1 = n/2 |x|2n = . 2 n2 n (n 2) 4 4Here the -function identity (z + 1) = z(z) has been
Sanford-Brown Institute - CS - 19
cs019 Homework Solution and AnnouncementsSeptember 18, 20081Dots and BoxesDots and Boxes has been extended to Monday due to some omissions of the project requirements in the initial handout: 1. You must turn in a README file with your code. Th
Sanford-Brown Institute - EN - 193
Camera Network Calibration from Dynamic SilhouettesSudipta N. Sinha Marc Pollefeys Leonard McMillan. Dept. of Computer Science, University of North Carolina at Chapel Hill. ssinha, marc, mcmillan @cs.unc.eduAbstractIn this paper we present an aut
Sanford-Brown Institute - EN - 193
Calibration of a Multicamera NetworkPatrick T. Baker Center for Automation Research University of Maryland College Park, MD 20742 Yiannis Aloimonos Center for Automation Research University of Maryland College Park, MD 20742for a particular network
Sanford-Brown Institute - PH - 161
Phys 1610 Homework 2 solution, Fall 2007
Sanford-Brown Institute - PH - 2620
REPORTSReversible Unfolding of Individual Titin Immunoglobulin Domains by AFMMatthias Rief, Mathias Gautel, Filipp Oesterhelt, Julio M. Fernandez, Hermann E. Gaub*Single-molecule atomic force microscopy (AFM) was used to investigate the mechanica
Sanford-Brown Institute - PH - 2620
Vol 440|2 March 2006|doi:10.1038/nature04525ARTICLESOptimal isotope labelling for NMR protein structure determinations Masatsune Kainosho1, Takuya Torizawa1, Yuki Iwashita1, Tsutomu Terauchi1, Akira Mei Ono1 &amp; Peter Guntert2Nuclear-magnetic-reso
Sanford-Brown Institute - M - 42
SOLAR ACTIVITY AND CLIMATE CHANGES OF THE EARTH. V. A. Alexeev, Institute of Geochemistry and Analytical Chemistry, Russian Academy of Sciences, Moscow 119991 Russia; e-mail: aval@icp.ac.ru The solar radiation is the fundamental source of energy that
Sanford-Brown Institute - EC - 266
Econometrica, Vol. 72, No. 1 (January, 2004), 219255HIGHER ORDER PROPERTIES OF GMM AND GENERALIZED EMPIRICAL LIKELIHOOD ESTIMATORS BY WHITNEY K. NEWEY AND RICHARD J. SMITH1In an effort to improve the small sample properties of generalized method o
Sanford-Brown Institute - EC - 111
Name_Economics 111: Intermediate Microeconomics Spring 2005 Midterm 2 Answer KeyYou have 1 hour and 20 minutes. Only clarifying questions are allowed. Do not cheat. Do not panic. Enjoy the exam. Questions 1 to 5 are multiple choice. Circle the co
Sanford-Brown Institute - CS - 159
CS159 Introduction to Computational ComplexityThe VLSI Model IIThe VLSI ModelArchitectural Model:Chips realize FSMs Wires are rectilinear. Wires have bounded width and separation . Gates can be binary or non-binary. Gates/memory cells occupy are
Sanford-Brown Institute - CSCI - 2560
CS256 Applied Theory of ComputationVLSI Model V John E SavageOverviewDerivation of lower bounds on planar circuit size. Area-time lower bounds for functions computated by VLSI chips.Lect 28 VLSI Model VCS256 @John E Savage2Area-Time Compu
Sanford-Brown Institute - CS - 159
CSCI 1590 Intro to Computational ComplexityFormula Size John E. SavageBrown UniversityMarch 5, 2008John E. Savage (Brown University)CSCI 1590 Intro to Computational ComplexityMarch 5, 20081 / 13Summary1Review2Application of Neci
Sanford-Brown Institute - CSCI - 2560
CS256 Applied Theory of ComputationCircuit Complexity III John E SavageOverviewIndirect storage access function Neciporuks formula size lower bound Krapchenkos formula size lower bound Monotone function The path elimination methodLecture 11 Circ
Sanford-Brown Institute - CS - 256
Programming with Stores is NP-hardARRAY PROGRAMMING (AP)CS256: Applied Theory of Computation Lecture 29 Data Storage in Nanoarrays IInstance: (W,k) where W is an n m array over {0,1} and k is an integer. Answer: Yes if there exists a set of at
Sanford-Brown Institute - CS - 149
CS149Introduction to Combinatorial OptimizationHomework 5Due: 4:00 pm, Thr, Oct. 16thProblem 1a) Apply the Simplex algorithm to solve the problem max x1 s.t. 2x1 3x1 x1 +3x2 +2x2 2x2 3x2 x1 , x2 10 10 10 0.b) Graph the feasible region
Sanford-Brown Institute - CS - 015
TREES Definitions, Terminology, and Properties Binary Trees Search Trees: Improving Search Speed Traversing Binary Search TreesSearching in a List When we search for an element in a sorted linked list, we have to check consecutive nodes to fin
Sanford-Brown Institute - CS - 168
Internet Protocol Goal: Glue lower-level networks together Wasnt that the goal of switching?Network 1 (Ethernet) H7 R3 H8H1H2H3Network 2 (Ethernet) R1Network 4 (point-to-point)R2 H4 Network 3 (FDDI)H5H6H1 TCP IP ETH ETH IP FDDI F
Sanford-Brown Institute - CS - 190
-Requirements for Awedio (pronounced aw - dee - oh)-Description/Background-a DJ/programmer and i were discussing the possibility of creating amore universal interface to sound to coincide with the tide of digitalmedia as a replacement for ph
Sanford-Brown Institute - CS - 190
Colin Hartnett (cphartne)Ego and _The Mythical Man-Month_In an ideal situation, the stratfication of teams Brooks lays out, suchas the distinct separation into architects and implementors and thedivision of implementors using the surgical team m
Michigan - SPP - 638
As an advocacy organization, Act Up was invited to be here to voice our opinion on the effect of the AIDS crisis on the millions of men, women, and children afflicted with the disease. For three days we have listened, debated, and argued on the futur
Michigan - WHOLEISSUE - 2004
Monitoring the Endangered Species Act: Revisiting the Eastern North Pacific Gray WhaleAndrew C. Keller &amp; Leah R. GerberEcology, Evolution and Environmental Sciences Arizona State University College &amp; University Dr. Tempe, AZ 85287-1501AbstractTh
Michigan - X - 04
X04 Input/Output UtilitiesChapter X04 Input/Output Utilities Contents1 Scope of the Chapter 2 Background to the Problems 2.1 Input/Output on Parallel Machines . . . . . . 2.2 Output from NAG Parallel Library Routines 2.3 Matrix Output Routines .
Michigan - D - 01
D01 QuadratureD01DAFP NAG Parallel Library Routine DocumentNote: Before using this routine, please read the Users' Note for your implementation to check for implementation-dependent details. You are advised to enclose any calls to NAG Parallel Li
Michigan - X - 04
X04 Input/Output UtilitiesX04BDFP NAG Parallel Library Routine DocumentNote: Before using this routine, please read the Users' Note for your implementation to check for implementation-dependent details. You are advised to enclose any calls to NAG
Michigan - X - 04
X04 Input/Output UtilitiesX04BVFP NAG Parallel Library Routine DocumentNote: Before using this routine, please read the Users Note for your implementation to check for implementation-dependent details. You are advised to enclose any calls to NAG
Michigan - X - 04
X04 Input/Output UtilitiesX04BRFP NAG Parallel Library Routine DocumentNote: Before using this routine, please read the Users Note for your implementation to check for implementation-dependent details. You are advised to enclose any calls to NAG
Michigan - X - 04
X04 Input/Output UtilitiesX04YAFP NAG Parallel Library Routine DocumentNote: Before using this routine, please read the Users' Note for your implementation to check for implementation-dependent details. You are advised to enclose any calls to NAG
Michigan - Z - 01
Z01 Library UtilitiesZ01BGFP NAG Parallel Library Routine DocumentNote: Before using this routine, please read the Users' Note for your implementation to check for implementation-dependent details. You are advised to enclose any calls to NAG Para
Michigan - D - 01
D01 QuadratureD01AUFP NAG Parallel Library Routine DocumentNote: Before using this routine, please read the Users' Note for your implementation to check for implementation-dependent details. You are advised to enclose any calls to NAG Parallel Li
Michigan - Z - 01
Z01 Library UtilitiesZ01AAFP NAG Parallel Library Routine DocumentNote: Before using this routine, please read the Users' Note for your implementation to check for implementation-dependent details.1DescriptionZ01AAFP defines a logical proce
Michigan - X - 04
X04 Input/Output UtilitiesX04BCFP NAG Parallel Library Routine DocumentNote: Before using this routine, please read the Users Note for your implementation to check for implementation-dependent details. You are advised to enclose any calls to NAG
Michigan - Z - 01
Z01 Library UtilitiesZ01ABFP NAG Parallel Library Routine DocumentNote: Before using this routine, please read the Users' Note for your implementation to check for implementation-dependent details.1DescriptionZ01ABFP undefines a logical pro
Michigan - Z - 01
Z01 Library UtilitiesZ01BBFP NAG Parallel Library Routine DocumentNote: Before using this routine, please read the Users' Note for your implementation to check for implementation-dependent details. You are advised to enclose any calls to NAG Para
Michigan - Z - 01
Z01 Library UtilitiesZ01BAFP NAG Parallel Library Routine DocumentNote: Before using this routine, please read the Users Note for your implementation to check for implementation-dependent details. You are advised to enclose any calls to NAG Paral
Michigan - APPLE - 2
] PROARC V1.0 [ _ The PROdos ARChival Utility for 5.25 floppy disks and Files. Programmed by The Freebooter 5/29/87 Software Encryption Analysts of South Texas Description: P
Michigan - ENG - 225
225.031 syllabus, 1 English 225.031 Argumentative Writing 4:00-5:30 221 DENN Winter 2003 Instructor: Rob Cosgrove Office: 3043 Tisch Hall Mailbox: 3161 Angell Hall E-Mail: rcosgrov@umich.edu Office Hours: TBACourse Description The purpose of this c
Michigan - PHYSICS - 406
Physics 406 1. a)Homework #104/13/988.96 .gm cm3Z29A63.54 massT300 .KConsider one mole of copper:63.54 .gm 6 m3N atomsNAVmass V = 7.0915 10 na N atoms VConcentration of atoms: Mass of a single atom:28 n a = 8
Michigan - PHYSICS - 406
Physics 406 1. T := 300 K Etranslational := 1 2Homework #39/30/05These atmospheric molecules have three translational degrees of freedom. mass v2According to the Equipartition Theorem: 1 Etranslational := 3 k T 2 Etranslational = 6.2
Michigan - PHYSICS - 305
Physics 305 April 26, 2002Name_ Final ExamPlease show all of your work (formulas used and intermediate steps) on these pages. Students showing the most work will receive the most credit. All you will need for this exam are a pencil, a calculator,
Michigan - PHYSICS - 150
Physics 150 May 16, 2008Name_ Exam #1Instructions: There are 10 multiple choice questions (worth 2 points each), and 2 work problems (worth a total of 40 points). You must answer all questions to receive full credit. You must show all equations a
Michigan - EECS - 595
Tutorial: 1) Compile with java &gt; 1.4 Javac EmailClient.java 2) Run the application Java EmailClient 3) Go to File Menu, select Open A. Select a data file i. Data file is a collection of email messages in the following XML like format. --&lt;EMAIL&gt; &lt;SUBJ
Michigan - EECS - 595
AutoSchedulerAn Email add-on simulation for automatic scheduling of email messages.Abstract:As a student in a graduate school, I get a number of emails about talks, seminars, meetings and such date-oriented messages. AutoScheduler is a simulation
Sanford-Brown Institute - MICRO - 36
Microsymposium 36, MS022, 2002PROPOSING A HIGH VOLATILE CONTENT IN THE EQUATORIAL LAYERED DEPOSITS INCLUDING THE MEDUSAE FOSSAE FORMATION, MARS. E. R. Fuller and J. W. Head, III, Planetary Geosciences Group, Department of Geological Sciences, Brown
Sanford-Brown Institute - MICRO - 36
Microsymposium 36, MS060, 2002CLASSIFICATION OF MARTIAN LACUSTRINE TARGETS FOR MEX-HRSC. H. Lahtela, J. Raitala, M. Aittola and V.-P. Kostama, Astronomy Division, Department of Physical Sciences, University of Oulu, P.O. BOX 3000, FIN-90014 Univers
Sanford-Brown Institute - MICRO - 36
Microsymposium 36, MS103, 2002MOUNTAIN GLACIERS ON MARS?: WESTERN ARSIA MONS FAN-SHAPED DEPOSIT SMOOTH FACIES AS ROCK GLACIERS: James W. Head1 and David R. Marchant2, 1Dept. Geol. Sci., Brown Univ., Providence, RI 02912 USA, 2Dept. Earth Sciences,
Sanford-Brown Institute - MICRO - 36
Microsymposium 36, MS072, 2002KARA REVISITED. T. hman 1, D. Badjukov 2, J. Raitala 3 &amp; C. Lorenz 2, 1Institute of Geosciences, Department of Geology, P.O. Box 3000, FIN-90014 University of Oulu, Finland, &lt;teemu.ohman@oulu.fi&gt;; 2V.I. Vernadsky Insti
Sanford-Brown Institute - MICRO - 36
Microsymposium 36, MS050, 2002IR FLASHES INDUCED BY METEOROID IMPACTS ONTO THE PLUTO'S SURFACE. I.B.Kosarev, I.V.Nemtchinov. Institute for Dynamics of Geospheres, Russian Academy of Sciences,Leninsky pr., 38, bld.6, Moscow, 119334, kosarev@idg.chp
Sanford-Brown Institute - MICRO - 36
Microsymposium 36, MS086, 2002COLD BASED GLACIAL ORIGIN OF THE PAVONIS MONS LOBATE DEPOSITS. D. E. Shean 1 and J. W. Head 2, 1Brown University, David_Shean@Brown.edu, 2Brown University, James_Head@Brown.eduIntroduction: Each of the three Tharsis M
Sanford-Brown Institute - MICRO - 36
Microsymposium 36, MS105, 2002MOUNTAIN GLACIERS ON MARS? CHARACTERIZATION OF WESTERN ARSIA MONS FANSHAPED DEPOSITS USING MGS DATA: James W. Head 1 and David R. Marchant2, 1Dept. Geol. Sci., Brown Univ., Providence, RI 02912 USA, 2Dept. Earth Scienc
Sanford-Brown Institute - MICRO - 36
Microsymposium 36, MS037, 2002AUTOMATED MAKING THE MAP OF ISIDISS BASIN. J.A.Iluhina and J.F.Rodionova. Sternberg State Astronomical Institute, jeanna@sai.msu.ru , 2Moscow University The plain of Isidis was choosen for landing of the spacecraft of
Sanford-Brown Institute - MICRO - 36
Microsymposium 36, MS002, 2002ABLATION OF METEORITES V.A. Alexeev, Vernadsky Institute of Geochemistry and Analytical Chemistry, Russian Academy of Sciences, 117975, Moscow, Russia (aval@icp.ac.ru) The interaction of galactic cosmic-ray particles w
Sanford-Brown Institute - MICRO - 36
Microsymposium 36, MS001, 2002ANALYSIS OF VENUSIAN CORONAE, ARACHNOIDS AND NOVAE ON THE BASIS OF THEIR GEOLOGICAL ENVIRONMENT. M. Aittola and V.-P. Kostama, 1 Astronomy, Dept. of Physical Sciences, Univ. of Oulu, P.O. BOX 3000, FIN-90014, Finland (
Sanford-Brown Institute - MICRO - 36
Microsymposium 36, MS073, 2002LANDSAT TM CLASSIFICATION OF THE ARCTIC KARA IMPACT CRATER TUNDRA, RUSSIA. K. Ojala 1, D. Badjukov 2, J. Raitala1(jouko.raitala@oulu.fi), T. hman 1 and C. Lorenz2, 1Physics Dept., Univ. of Oulu, Finland; 2Vernadsky Ins
Sanford-Brown Institute - MICRO - 36
Microsymposium 36, MS063, 2002SEASONAL VARIATIONS OF SUBSURFACE HYDROGEN AS SEEN BY HIGH ENERGY NEUTRON DETECTOR, MARS ODISSEY. M. L. Litvak1, I. G. Mitrofanov1, A.S. Kozyrev1, A.B. Sanin1, W. Boynton 2, C. Shinohara2, D. Hamara2, R. S. Saunders3; 1
Sanford-Brown Institute - MICRO - 36
Microsymposium 36, MS051, 2002DISTRIBUTION AND CLASSIFICATION OF VOLCANO-TECTONIC FEATURES OF VENUS. V.-P. Kostama 1 and M. Aittola 1, 1Astronomy, Dept. of Physical Sciences, University of Oulu, P.O. BOX 3000, FIN90014, Finland (&lt;petri.kostama@oulu
Sanford-Brown Institute - MICRO - 36
Microsymposium 36, MS062, 2002DISTRIBUTION OF CHEMICALLY BOUND WATER IN SURFACE LAYER OF MARS BASED ON DATA ACQUIRED BY HIGH ENERGY NEUTRON SPECTROMETER, MARS ODYSSEY. M. L. Litvak1, A. T. Basilevsky 2, I. G. Mitrofanov 1, W. Boynton 3, R. S. Saunde
Michigan - APPLE - 2
From caen!spool.mu.edu!howland.reston.ans.net!pipex!uunet!news2.uunet.ca!ionews.io.org!r-node.io.org!nobody Sun Oct 10 13:27:16 EDT 1993Article: 5632 of comp.binaries.apple2Path: caen!spool.mu.edu!howland.reston.ans.net!pipex!uunet!news2.uunet.ca!i
Sanford-Brown Institute - EC - 218
A Supergame-Theoretic Model of Price Wars during BoomsBy Julio Rotemberg and Garth SalonerMacro motivation: What happens with prices when demand increases? Can fiscal policy affect aggregate production?Repeated Oligopoly: Prices may decrease whe
Sanford-Brown Institute - EC - 218
Repeated Games under Imperfect MonitoringIn the basic repeated games model deviations are observed allowing for punishments. But in many real life situations players do not have perfect observability of other players' actions.Example: Tacit collu
Sanford-Brown Institute - CSCI - 1780
ASSIGNMENT 3: Client-Server GameOut: 2/21/02; Due: 3/7/02 Programming Parallel and Distributed Systems Computer Science 178, Spring 2002 Steven P. ReissOBJECTIVEThe purpose of this assignment is to let you develop and play with a client-server ap