# 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.

3 Pages

### lecture17

Course: CSCI 4011, Fall 2008
School: Minnesota
Rating:

Word Count: 1255

#### Document Preview

LIMITATIONS INHERENT OF COMPUTER PROGRAMS CSci 4011 ((lambda (x) (list x (list (quote quote) x))) (quote (lambda (x) (list x (list (quote quote) x))))) Print the next sentence twice, the second time in quotes. Print the next sentence twice, the second time in quotes. Can we make a TM that does the same? WARNING: HEADACHES AHEAD SELF-REFERENCE UNDECIDABILITY: There are problems that no computer can solve...

Register Now

#### Unformatted Document Excerpt

Coursehero >> Minnesota >> Minnesota >> CSCI 4011

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.
LIMITATIONS INHERENT OF COMPUTER PROGRAMS CSci 4011 ((lambda (x) (list x (list (quote quote) x))) (quote (lambda (x) (list x (list (quote quote) x))))) Print the next sentence twice, the second time in quotes. Print the next sentence twice, the second time in quotes. Can we make a TM that does the same? WARNING: HEADACHES AHEAD SELF-REFERENCE UNDECIDABILITY: There are problems that no computer can solve INCOMPLETENESS: There are true theorems that have no proof Theorem: There is a computable q : * *, where for any string w, q(w) is the description of a TM Pw that on any input, prints out w and then accepts s w Q Pw w A TM THAT MAKES CANNIBALS s M CM PM M M(M) A TM THAT PRINTS ITSELF s w PCM CM PCM CM SELF Run the cannibal maker on itself! THE RECURSION THEOREM Theorem: Let T be a Turing machine that computes a function t : * * *. There is a Turing machine R that computes a function r : * *, where for every w r(w) = t(R, w) (a,b) T R t(a,b) w t(R,w) Proof: (a,b) s T t(a,b) M s CM P!M" M M(!M") PM , w M,w CM M,s M M(M,s) s Proof: (a,b) T t(a,b) M,w CM s P!M", w !M,s" M M(!M",s) PCMT w PCMT CM CM T T , w t(R,w) t(PCMT,s) A Turing machine can obtain its own description and then go on to compute with it 1. Design TM t(M,w) that assumes M is code for t(w) 2. Use recursion theorem to get TM R(w) = t(R,w) Theorem: ATM is undecidable Proof (using the recursion theorem): Assume H decides ATM Construct machine B such that on input w: 1. Obtains, via the recursion theorem, its own description B 2. Runs H on B,w and flips the output Running B on input w does the opposite of what H says it should! INCOMPLETENESS There are true statements that have no proof GREAT LOGICIANS OF THE TWENTIETH CENTURY James T. Kirk Kurt Gdel This sentence is a lie. This sentence has no proof. Like Gdel, we will build a formal statement that is true but unprovable STATEMENTS We want to express mathematical statements, eg: If x < y and y < z then x < z 8 x,y,z [ (x < y) (y < z) ! (x < z) ] There are infinitely many prime numbers 8 q 9 p 8 x,y [ p > q (x,y > 1 ! xy p) ] STATEMENTS We can make a formal grammar for statements: = { , , , [, ], (, ), 8, 9, x, R, ;} S!Q[F] Q ! 9X Q | 8X Q | 9X | 8X F ! A | F F | F F |F | (F) A ! RA | R(L) X ! xX | x L ! X | X; L A statement is a string from this grammar with: no free variables: if xi appears in F, it appears in Q well-formed relations: each appearance of Ri has the same number of arguments. STATEMENTS 8x1 9x2 9x3 [ R1(x1) R2(x1;x2;x3) ] 9x1 8x2 [ R1(x1;x2) R1(x2;x1) ] 8x [ R(x) ] NOT STATEMENTS R1(x) R2(x) 9x1 [ R(x1; x2) ] 9x1 8x2 [ R(x1) R(x1;x2) ] TRUTH & MODELS A model M is a k-tuple (U, P1, , Pk) where: U is the universe of values each xi can have P1 Pk are relations over U M assigns meaning or truth to a statement: Ri(u1,u2,) is true iff (u1,u2) 2 Pi 8xS is true iff for every x 2 U, S is true 9xS is true iff for some x 2 U, S is true. ,, have the usual meanings. EXAMPLE Let M = (R, {(x,y,z) : x+y=z}). xy Then: [ R(y,y,x) ] is true. yx [ R(y,y,x) ] is false. 9y 8x [ R(y,x) ] is ill-formed. EXAMPLE Let M = (N, {(x,y) : x y}, {(x,y,z) : z = x y} ). Then: 8x 8y [ R1(y,x) R1(x,y) ] is true. 9y 8x [ R1(x,y) ] is false. 9y 8x [ R1(y,x,x) ] is ill-formed. LANGUAGE & THEORY Let M = (U, P1, , Pk) be a model. The language of M, L(M) is the set of well-formed statements under M. The theory of M, Th(M) is the set of true statements in L(M). PROOFS are sequences of statements that logically follow from each other. For example S follows from T in first-order logic if: S is a tautology; or S is an axiom; or S = x T ( where x T); or S and T () ( ) ); etc A sequence S1Sm where for each i, (Sj<i, Si) 2 or (, Si) 2 , is a proof of Sm in logic . In general, a logic is a decidable subset of S S Given a logic and a model M, define the set Provable(M) = { S 2 L(M): S has a proof in } Theorem: Provable(M) is Turing recognizable. Proof: Since a proof is just a string, we can check if S is provable by checking all possible proofs: def is_provable(S): for in {0,1,00,01,10,11,100,}: (S1,,Sm) = parse_as_stmt_list() S0 = None is_proof = True for i in {1,,m}: follows[i] = S[i] 2 L(M) and (S[:i],S[i]) 2 . is_proof = is_proof and follows[i] if (is_proof and (S[m] == S)): return True INCOMPLETENESS I Theorem: For every TM M and string w, there is a computable formula M,w(x) 2 L(N,+,) such that 9x M,w(x) is true iff M(w) accepts. Proof: We make a sentence that says x encodes an accepting computation history of M on w: (x mod b = q0w) i[ (u = x div bi mod b ) (v = x div b(i+1) mod b) (u Mv) ] j [ x div bj mod b = qaccept ] INCOMPLETENESS II Theorem: For every TM M and string w, there is a computable formula M,w(x)L(,+,) such that x M,w(x) is true iff M(w) accepts. Corollary: Th(N, +, ) is undecidable. Proof: Suppose Th(,+,) was ...

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:

Neumont - CSC - 1885
Supreme Court of Canada Ings v. The Bank of P.E.I., 11 S.C.R. 265 Date: 1885-06-23 John Ings (Defendant) Appellant; and The President, Directors And Company of the Bank of Prince Edward Island (Plaintiffs) Respondents.1885: February 24, 25; 1885: Ju
Minnesota - CSCI - 3003
CSci 3003: Introduction to Computing in Biology Lab Assignment #620 points Assigned: 3/25/09 Due: Tuesday, 4/7/09 (before midnight)Goals of this lab: Practice using Matlab to load/analyze data. Learn about statistical analyses of gene expressio
Minnesota - CSCI - 3003
Introduction to Matlab Today: Introduction to Matlab environment Variables and expressions Loading/viewing your data Matrices and common matrix tasks Next week: Spring break! Following week Introduction to probability/statistics More matlab: F
Minnesota - CSCI - 3921
CSCI 3921, CLASS 22: INTELLECTUAL PROPERTY=I. INTRODUCTIONII. WHY IS THIS AN IMPORTANT AREA?III. GOALSIV. TYPES OF INTELLECTUAL PROPERTY PROTECTIONV. SURVEYVI. ANNOUNCEMENTS==I. INTRODUCTION-A bill was recently introduced into t
Minnesota - CSCI - 3081
:CSci 3081 L4.1: Software Specification and DesignUse CasesProf. Eric Van WykSpring 2008c Eric Van WykCSci 3081: Program Design and Development - Spring 2008Page 1 / 9Software Specification and DesignUse Cases:Use CasesA scenario i
Naval Academy - EE - 228
Introduction to Electrical Engineering EE228 Laboratory Exercise Underdamped, Critically Damped and Overdamped Responses ReferencesFundamentals of Electric Circuits, Alexander and Sadiku, Chapter 8.Prelab1. Design a parallel RLC circuit with the
Naval Academy - EE - 228
COSMOS: Complete Online Solutions Manual Organization SystemEE228Spring 2007Problem Set #17Chapter 14, Solution 13.G () =(1 10)(1 + j) 1 + j = 2 ( j) (10 + j) ( j) 2 (1 + j 10)G dB = -20 + 20 log10 1 + j 40 log10 j 20 log10 1 + j 10 =
Minnesota - CSCI - 5511
CSci 5511 2nd Midterm Fall 2008 CSci 5511 Fall 2008 1Answer key for 2nd Midterm ExamTuesday November 18 75 minutes = 75 points1. 15 points Show the backed-up values for all the nodes in the following game tree and show the branches that are prune
Minnesota - CSCI - 5471
Minnesota - CSCI - 1001
INTELLECTUAL PROPERTY BASICS NOTES=Here are a few notes on intellectual property basics. The book discussescopyright, so the purpose of these notes is to provide a little moregeneral background.Basis in the US Constitution-U.S. Constitution
Minnesota - CSCI - 3921
SOLUTION TO 3/25 IN-CLASS FAIR USE EXERCISE=(1)The use was commercial, involved creative works, and involved theentire (or almost entire) work. So these all argue against fairuse. The only factor for fair use is the market for the movies is no
Minnesota - CSCI - 1001
Minnesota - CSCI - 5221
How to use project VMsFirst Method:1) Log into a Linux ITLabs machine in EE/CS 2-170 or EE/CS 4-250or Lind Hall 24. (Use the -X forwarding, e.g. ssh -X johnsmith@cs2170-12.itlabs.umn.edu).Or ssh directly into itclus28.itlabs.umn.edu2) Load
Minnesota - CSCI - 1001
CSCI 1001 CLASS 4: ETHICS AND COMPUTING=-KEY QUESTIONS-1) Why are ethics important?2) What techniques do ethicists use when analyzing moral decisions?--WHERE THIS MATERIAL APPEARS IN THE TEXT AND CLASS ASSIGNMENTS-This material isn't in
Minnesota - CSCI - 1001
&lt;?xml version=&quot;1.0&quot; encoding=&quot;UTF-8&quot;?&gt;&lt;Error&gt;&lt;Code&gt;NoSuchKey&lt;/Code&gt;&lt;Message&gt;The specified key does not exist.&lt;/Message&gt;&lt;Key&gt;b1eacbc216212dcaaa465347152b4071b5255440.txt&lt;/Key&gt;&lt;RequestId&gt;7FEEC2B4B31CC5E0&lt;/RequestId&gt;&lt;HostId&gt;bzsAY+tg2GFEC7H1bqIWJT45o+To
Minnesota - CSCI - 5801
Requirement #: Unique # Requirements Type:Use Case:Introduction: A short description about what this requirement is for Rationale: Why is the requirement here? Author: Source: Who came up with the requirement.Inputs: The inputs needed to perfor
UCLA - STATISTICS - 100
University of California, Los Angeles Department of Statistics Statistics 100A Instructor: Nicolas ChristouNormal distribution The normal distribution is the most important distribution. It describes well the distribution of random variables that
Stanford - EE - 374
EE374 Inference in Graphical ModelsLecture 9 - 2/8/2007Correlation MethodsLecturer: Andrea Montanari Scribe: Arash Asadpour and Dominic DiPalantino1 Wrapping up from last time about Monte Carlo Markov chains: we proved that for &lt; k , the Metr
Minnesota - CSCI - 5211
Authorized licensed use limited to: University of Minnesota. Downloaded on October 28, 2008 at 14:32 from IEEE Xplore. Restrictions apply.Authorized licensed use limited to: University of Minnesota. Downloaded on October 28, 2008 at 14:32 from IEEE
Stanford - PSYCH - 221
LM75 Digital Temperature Sensor and Thermal Watchdog with Two-Wire InterfaceOctober 2005LM75 Digital Temperature Sensor and Thermal Watchdog with Two-Wire InterfaceGeneral DescriptionThe LM75 is a temperature sensor, Delta-Sigma analog-todigita
UCLA - EE - 206
4) a) C is one of the hidden terminals from A and E because those nodes cannot hear that C is transmitting and may attempt to initiate connections with their respective neighbors thus causing collisions. b) B is the exposed terminal. Collisions ha
Allan Hancock College - COMP - 3201
COMP3201 Computer Graphics Assignment 2: 3D BeginningsAssignment Due: electronic submission by 6pm on Wednesday 12 April 2006Task:Construct a 3D environment for the prairie dog.To understand how to create a 3D scene in Computer Graphics. In pa
East Los Angeles College - ENVI - 0310
u\$ B!e u\$ e u ze u ze SShDSHV\$ u #g YDPD u x w } d h d k d d u ! ddi g6h !k ~6Yd!gy 6d6 idiYVm q q \$ \$ \$ tzWDV\$ u #g q \$ q B!D(zD\$ u x u s q o m h k i d ke d { d d { h
Neumont - CSC - 1948
Supreme Court of Canada Berwick v. Canada Trust Co., [1948] S.C.R. 151 Date: 1948-03-23 Alexander Raymond Berwick (Defendant) Appelant; and The Canada Trust Company (Plaintiff) Respondent1948: February 27; 1948: March 23. Present:-Kerwin, Rand, Kell
UCLA - ESS - 265
enA imAging: seeing The inVisibLeenA imaging: seeing the invisiblePontus C. Brandt, Donald G. Mitchell, Edmond C. Roelof, Stamatios M. Krimigis, Christopher P. Paranicas, Barry H. Mauk, Joachim Saur, and Robert DeMajistreIn what follows, we de
Minnesota - D - 4110
MMinnesota Geological SurveyINNESOTA1872GQuaternary Glacial GeologyThe Quaternary Period began about 2 million years ago. It is divided into the Pleistocene Epoch (2 million to 10,000 years ago) and the Holocene Epoch (10,000 years ago
Neumont - CSC - 1948
Supreme Court of Canada Berwick v. Canada Trust Co., [1948] S.C.R. 151 Date: 1948-03-23 Alexander Raymond Berwick (Defendant) Appelant; and The Canada Trust Company (Plaintiff) Respondent1948: February 27; 1948: March 23. Present:-Kerwin, Rand, Kell
Neumont - CSC - 1885
Supreme Court of Canada The Queen v. The Bank of Nova Scotia, 11 S.C.R. 1 Date: 1885-06-26 Her Majesty the Queen Appellant; and The Bank of Nova Scotia Respondents.1885: February 23; 1885: June 26. Present:Sir W. J. Ritchie, C.J., and Strong, Fourni
Cal Poly - ELE - 6306
Test de Systmes lectroniques Cours ELE6306Abdelhakim KhouasDpartement de Gnie lectrique cole Polytechnique de MontralCourriel : akhouas@polymtl.ca Site WEB : www.cours.polymtl.ca/ele6306 Consultation : Lundi de 9h-10h et Jeudi 10h-12h Local : D-62
Cal Poly - ELE - 6306
Chapitre 5 Gnration Automatique des Vecteurs de Test ExercicesAbdelhakim KhouasDpartement de Gnie lectrique cole Polytechnique de MontralExercice 11Exercice 2! # ! &quot; % ! #' \$( * ' \$( +' \$( ' \$( &amp; ) ) ) )# ! &amp;\$ &quot;2Exercice 3, !' ! #
Cal Poly - ELE - 6306
Plan de coursELE6306 - Test de systmes lectroniquesDpartement gnie lectrique3 Crdits (3 - 1,5 - 4,5)http:/www.cours.polymtl.ca/ele6306I.INFORMATIONS GNRALESELE6306 Test de systmes lectroniques Automne 2006 Cours : Mardi 13h45-16h35, local C-
Cal Poly - ELE - 6306
Plan de coursELE6306 - Test de systmes lectroniquesDpartement gnie lectrique3 Crdits (3 - 1,5 - 4,5)http:/www.cours.polymtl.ca/ele6306I.INFORMATIONS GNRALESELE6306 Test de systmes lectroniques Automne 2007 Cours : Mardi 9h30-12h20, local A-4
Cal Poly - ELE - 6306
TestdunRseausurpuce gnrique (genericNetworkonChip)Prsent par: David Bafumba-Lokilo &amp; Silvio ForneraPlandeprsentation Introduction Le circuit Hypothses Architecture et mthode de test Simulations Conclusion QuestionsIntroduction|Circuit|Hy
Cal Poly - ELE - 6306
ELE6306 : Test de systmes lectroniques Projet de cours VHDL-AMS :Un Atout pour la Conception des Systmes Microlectroniques Analogiques Numriques. Abdelmajid Iguermia ; Boujemaa NbaheddaProfesseur : A. KhouasDpartement de gnie lectrique cole Polytec
Cal Poly - ELE - 6306
ELE6306 : Test de systmes lectroniquesProjet de coursTechniques de Test pour les Convertisseurs Analogiques/Numriques (CAN)Mhammed HAMINE &amp; Iliasse BENAMRANEProfesseur : A. KHOUAScole Polytechnique de MontralDpartement de gnie lectrique Autom
Cal Poly - ELE - 6306
1Rduction de la consommation de puissance durant le test des circuits munis de chaines de scanKhalid El Amrani 1322403 ELE-6306, Test de systmes lectroniques Dpartement de gnie lectrique, cole Polytechnique, Montral (Qc)Abstract-la consommation
Maryland - E - 812
Fall 2003 BMGT 650 Section 0101 MARKETING MANAGEMENT Instructor: Prof. Venkatesh (Venky) Shankar Office: 3307 Marketing Department, Van Munching Hall, College Park Telephone: (301) 405-2175 (W) Email: vshankar@rhsmith.umd.edu www address: http:/www.v
UCLA - CHEM - 156
Chem 156 Homework Set #7 Due Wed 3/13/021. Given a polymer with N = 1000 and the spacing between two successive protomers of 5.0 . a. Derive an expression for the end-to-end distance of a polymer of this type. b. Using the expression derived in 1a,
CSU Channel Islands - ECE - 275
ECE 275B Homework #2 1) (15 pts.) At a temperature of T=300 K, what is the frequency at which there is the maximum black-body radiation energy per frequency range d? What is the wavelength at which there is the maximum blackbody energy per wavelength
CSU Channel Islands - LIB - 0708
LAUC- I 2007 ~ 2008 Executive Board and Committees Roster, Charges, and Implementation Tasks Executive Board Last Updated: 12 June 2008 LM Chair: Linda Murphy (IV.5.) a. Represent LAUC-I in its communications with the University Librarian and library
CSU Channel Islands - LIB - 0506
Dear colleagues, The LAUC-I Executive Board and Program Committee invite you to attend our next Lunch with LAUC-I on The Anteater Chronicles! Annie Mar and Jackie Dooley will present an overview of Anteater Chronicles: The UC Irvine Story, the UCI hi
Stanford - ME - 218
ME218a Midterm Exam Due by 5pm on 10/28/94Name:_ I Certify that I have taken this examination in compliance with the Stanford University Honor Code. _ #1_ #2_ #3_ #4_ #5__ #6_ #7_ #8_ #9_ #10_ #11_ Total_Midterm Examination for ME 218aDue by 5:0
Stanford - B - 132
B132, Room 102 Area Hazard Analysis (AHA) Title: B132, Room 102 AHA Location (Bldg &amp; Rm): B132, room 102 Date: August 4, 2008This Area Hazard Analysis (AHA) is a process that is used to evaluate the area to 1) determine the hazards that may be pres
Stanford - B - 116
B116 Area Hazard Analysis (AHA)Title: B116 AHA Location (Bldg &amp; Rm): B116 Date: August 4, 2008This Area Hazard Analysis (AHA) is a process that is used to evaluate the area to 1) determine the hazards that may be present 2) determine appropriate c
Penn State - T - 597
FDR at Gettysburg Tom Benson April 19971D R A F TFDR at Gettysburg: The New Deal and the Rhetoric of Public MemoryThomas W. BensonChapter for Rhetoric and Public Memory, edited by Stephen Browne and David Henry for Michigan State University
Neumont - INU - 1030
1INU 1030 COURS 7 ASPECTS LGAUX DE LA PRSERVATION NUMRIQUEI. LA VALSE DES LOIS TOUCHANT AUX ARCHIVES ET AUX DOCUMENTS LECTRONIQUESNIVEAU INTERNATIONAL Exemple dorganismes UNESCO CIA (Conseil international des archives) OCDE (Organisation de coo
Allan Hancock College - PHYS - 1160
Solution to PHYS1160 Tutorial Sheet 7, 20032.If a person with wet hands grasps two conductors, and has a resistance of 1000 ohms, how large a potential difference would be required to produce a 10 mA current that may freeze the hands to the condu
Minnesota - ELED - 1010
TOPIC: What wejed/02KnowWhat weWant to knowHow to find what wewant to knowWhat weLearnedWhat weKnowWhat weWant to knowWhat weLearned
Minnesota - KAIS - 0099
Northside Home Fund Cluster LocationsOLIVER AVE N53RD AVE N53RD AVE NLYNDAKNOX AVE N52ND AVE NLE AVE51ST AVE NShingle Creek6TH ST NLOGAN AVE NSh50TH AVE NLind - BohanonALDRICH AVE NNorthside Home Fund ClustersWVINCE
Neumont - CSC - 1955
Supreme Court of Canada Thompson v. Fraser, [1955] S.C.R. 419 Date: 1955-04-26 Henry A. Thompson (Defendant) Appellant; and David Frederick Fraser (Plaintiff) Respondent.1955: February 8, 9; 1955: April 26. Present: Taschereau, Estey, Locke, Cartwri
Allan Hancock College - AUST - 1000
AUST 1000 1/2005 Fiona Nicoll Anzac Day In no unreal sense it was on the 25th of April, 1915, that the consciousness of Australian nationhood was born - Charles Edwin Woodrow Bean, official World War One historian and war correspondent. The Anzac Myt
Neumont - CSC - 1956
Supreme Court of Canada Roberts v. The Queen, [1957] S.C.R. 28 Date: 1956-12-21 Llewellyn M. Roberts And George B. Bagwell (Suppliants) Appellants; and Her Majesty The Queen (Respondent) Respondent.1956: October 16, 17, 18, 19; 1956: December 21. Pr
Minnesota - MEREV - 001
Solutions to Homework 4FM 5001 Preparation for Financial Mathematics1-1 Compute 4 1 5 0 9 1 5 6 4 3 2 3 2 4 4 3 0 0 0 2 0 4 0 1 1(4) + 5(1) + 0(2) 9(3) 1(0) + 5(0) + (0) 9(0) 1(0) + 5(2) + 0(4) 9(1) 5(0) 6(0) + 4(0) + 3(0) 5(0) 6
Neumont - CSC - 1961
Supreme Court of Canada Great Eastern Oil and Import Company Limited and Angus Oakley v. F.E. Best Motor Accessories Company Limited, [1962] S.C.R. 118 Date: 1961-12-15 Great Eastern Oil and Import Company Limited and Angus Oakley (Defendants) Appell
Minnesota - MATH - 1372
MATH 1372 Exam 2 study guide The second midterm exam will be held Thursday, October 30, at your choice of 5 - 6 p.m. or 6:10 - 7:10 p.m. in the following rooms: Brenner, Lecture 010: Fraser 101 Mosher, Lecture 020: Fraser 102 If you take the exam a
Minnesota - MATH - 2374
1. Find both first-order partial derivatives of both the functions f (x, y) = e3y sin(x) g(x, y) = xy 2 ln(3x)2. Now find the four second order partial derivatives of f (x, y) and g(x, y).3. A function f (x, y) is harmonic if it satisfies the Lap
Minnesota - MATH - 4428
4428 Spring 09, Project 2: Constrained optimization, Lagrange multipliers, numerics, linear programming, integer programing (no late homework) Due Wednesday February 25 Instructions: You can use a computer algebra system for computing derivatives, so
Minnesota - MATH - 8583
Math 8583, Fall 2003 Solutions for Final Exam 1 (Kelvin transformation). Let u be a harmonic function in an open set Rn , n 1. Then the function u (x) := |x|2n u(|x|2 x) is harmonic in := {x Rn : |x|2 x }. Proof. This statement is a special cas
Stanford - CS - 0708
CS140 - Winter 2008Past &amp; PresentCS 140: Operating Systems Lecture 25: Network LayerMendel RosenblumLast time: pushing bits from with hardware Link layer. How to: encode bits on wire, parse bits into packets, arbitrate between senders, name r
Minnesota - MATH - 8583
Math 8583: Theory of Partial Dierential Equations: Fall 2007 Homework Assignment 1 (due on Wednesday, October 3, till 10:10 am) 50 points are distributed between 5 problems, 10 points each. 1. Show that Laplaces equation is invariant with respect to
Penn State - MOL - 108
Michael Lin NY Times Journal #2 1/21/03 What If Biosphere 2 Worked? (Columbia Is Sharply Cutting Money for Biosphere Project) The leaves filtered the suns bright yellow light as it struck the ground in broken patterns. Steve blinked as he slowly retu