numerical_integration_6
37 Pages

numerical_integration_6

Course Number: PH 4390, Fall 2008

College/University: Mich Tech

Word Count: 2730

Rating:

Document Preview

Numerical Integration Monte Carlo Integration We want to Monte Carlo integrate the function 2 0 sin (x)dx = 1.0 In addition to the previous example we also compute the standard deviation, , and include it in the output. Then we want to vary (increase) the total number of MC evaluation points, N , and see how the results will change. A First Course in Computational Physics, PL Vries, Chp 4, ISBN...

Unformatted Document Excerpt
Coursehero >> Michigan >> Mich Tech >> PH 4390

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.

Integration Monte Numerical Carlo Integration We want to Monte Carlo integrate the function 2 0 sin (x)dx = 1.0 In addition to the previous example we also compute the standard deviation, , and include it in the output. Then we want to vary (increase) the total number of MC evaluation points, N , and see how the results will change. A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.1/30 Numerical Integration Monte Carlo Integration A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.2/30 Numerical Integration Monte Carlo Integration A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.2/30 Numerical Integration Monte Carlo Integration A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.2/30 Numerical Integration Monte Carlo Integration A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.2/30 Numerical Integration Monte Carlo Integration MC converges very slowly ! We need to go to very large values of N to get an overlap of the condence interval with the expected result. The standard deviation becomes smaller with increasing N. A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.2/30 Numerical Integration Monte Carlo Integration The error in a Monte Carlo integration is fundamentally different to that discussed in previous numerical integration methods. When we compare it to the trapezoidal rule, we recall that there the error was found to be proportional to h2 . Thus when we decrease h in the trapezoidal scheme h like h we have actually halved the error. 2 h Making h smaller as however means making N 2 larger like N 2N . A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.3/30 Numerical Integration Monte Carlo Integration In contrast to the trapezoidal rule, the error in the Mone Carlo scheme is a probabilistic error. Including the calculation of allows us to say that our estimate is 68.3 % of the time within one standard deviation of the correct answer. Adding more points means that the average will improve following this 68.3 % criterion. Nevertheless in the case of a multidimensional integral, the error will remain to decrease according to the probabilistic nature of the averaging process and not according to the dimensionality of the integral ! A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.4/30 Numerical Integration Monte Carlo Integration Exactly this independence of dimensionality of the integration makes Mone Carlo integration an interesting technique for many problems. Considering again the trapezoidal analogue we would have to increase N by a factor of 2 for a two-dimensional integral in order to halve the error ( 2N in x-dimension and 2N in y -dimension ). In general we had to increase N in the trapezoidal technique as 2N for each additional dimension, thus some d-dimensional integral would require an increase d of N by a factor of 2 2 to halve the error. A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.5/30 Numerical Integration Monte Carlo Integration We have already seen that the convergence of Mone Carlo integration is slow and goes as N . So to halve the error in MC schemes we had to increase N by a factor of 4. But since this happens regardless of the dimensionality of the integration, Monte Carlo becomes superior to e.g. the trapezoidal rule for dimensions greater than d = 4. So here is the actual strength of MC: faster convergence than any other scheme for high-dimensional integrals. A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.6/30 Numerical Integration Monte Carlo Integration Multidimensional integrals are frequently encountered in many areas of Physics, especially in Statistical Mechanics. For example the expectation value of some macroscopic property u is given as phase space integral, i.e. u= R R ue kT dx3N dv 3N e kT dx3N dv 3N E E where dx3N , dv 3N means integration over the 3 components of position coordinates and velocity coordinates of each particle N . Already for a smaller value of N such a process is virtually impossible to all but the MC method. A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.7/30 Numerical Integration Monte Carlo Integration As an example we want to carry out a multidemensional integration using the Monte Carlo method. Let us try to solve the 9 dimensional integral 1 1 1 .... 0 dax day daz dbx dby dbz dcx dcy dcz 0 0 (a+b)c We again want to monitor the evolution of the standard deviation with increasing numbers of evaluation points N . A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.8/30 Numerical Integration Monte Carlo Integration A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.9/30 Numerical Integration Monte Carlo Integration A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.9/30 Numerical Integration Monte Carlo Integration A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.9/30 Numerical Integration Monte Carlo Integration A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.9/30 Numerical Integration Monte Carlo Simulations Monte Carlo methods are also a common way to describe random processes. One of the earliest applications was to study scattering neturons in nuclear reactors. In particular, how much shielding is necessary to stop these neutrons. Many problems in physics are similar in spirit to random movement. Random processes are said to be stochastic . A rather well known example is the drunken sailor problem: After an extended bar tour and lots of partying the sailor is supposed to get back to the ship. But his choices of going into a certain direction happen arbitrarily and random. The question is how far will he have traveled after having taken N steps ? A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.10/30 Numerical Integration Monte Carlo Simulations Simulation of the random walk can be done on a square grid. A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.11/30 Numerical Integration Monte Carlo Simulations A path similar to the one shown on the previous slide can be generated with the help of random numbers. Say the probability to move in any of the four directions is equal. A random number shall be created and if it is between 0 and 0.25 the movement shall be towards north. If the random number is between 0.25 and 0.5 the direction to move forward shall be east and so on... Of course, a single path wont tell us much and we need a number of pathes (repeat the simulation a number of times) to build up a distribution. After that we can state, how far the sailor is likely to travel. A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.12/30 Numerical Integration Monte Carlo Simulations The histogram for a MC simulation of the Drunken Sailor Scenario of N = 100 steps. A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.13/30 Numerical Integration Monte Carlo Simulations The drunken sailor example has some relevance to real physics problems. For example the mobility of an atom within the lattice points of a crystal could be described this way. Or also, the problem of diffusion could be addressed with MC simulations. In the case of diffusion, we know that at room temperature molecules in the air have velocities the on order of hundreds of meters per second. But if a bottle of perfume is opened at one end of the classroom, it takes several minutes to perceive the aroma at the other end of the classroom. Why ? A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.14/30 Numerical Integration Monte Carlo Simulations The explanation of the limited progress of the aroma of the perfume is that the molecules move around very much like the drunken sailor. There will be a large number of collisions with other molecules and each of these collisions will change the direction of the moving particle so that there will be little net gain in actually traveled distance. If we want to simulate that process we need to allow for random movement in all different directions. So we need a method to obtain uniformly distributed directional movements on the unit sphere. A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.15/30 Numerical Integration Monte Carlo Simulations We can get the requested uniform distribution of solid angles by rst selecting from a uniform distribution of random variables on the interval [0, 2] followed by choosing a second random variable g from the interval [1, 1] and derive as = arccos (g) In order to simulate the diffusion process we would let the molecule move some given distance before some collision would occur. The collision would result in a random change of direction of continued movement. A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.16/30 Numerical Integration Monte Carlo Simulations Of course the distance of unperturbed traveling is another critical component and should not be taken as a constant. But as a rst approximation we keep this as a xed constant and say it is the mean free path. So the scenario looks as follows: a molecule travels a mean free path distance, collides with another molecule, which leads to scattering in a random direction, travels another mean free path distance, and so on... We can simulate an example of real diffusion with a modied version of our drunken sailor MC program. A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.17/30 Numerical Integration Monte Carlo Simulations For example, let us consider the diffusion of an aromatic molecule in air. The velocity of the molecule shall be 500 m and the mean free path shall be = 1m. We s want to calculate the distance < d > a molecule moves in one second. Our average shall include 100 trial runs of a single aromatic moecule. We nd a net distance traveled much smaller than the 500m that theoretically were possible. Our results show some clear accumulation around 19m. The importance of MC methods is not so much related to exact results but more to the fact that they can yield qualitatively valid results. A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.18/30 Numerical Integration Monte Carlo Simulations The histogram for the diffusion of some particle as obtained from MC simulation. A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.19/30 Numerical Integration Monte Carlo Simulations Although the results seem to be statistically solid, they still are physically invalid. It is true that the net displacement is considerably less than the free displacement. However, the estimated 19m per second still seem to be too much. From experience we would estimate the diffusion time for the molecules of a perfume traveling for about 20m to be on the order of minutes and not just one second. We could rerun the MC simulation with a different value for the mean free path. A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.20/30 Numerical Integration Monte Carlo Simulations We might have actually found something more general. The net displacement in units of the mean free path length after 500 collisions. It would be interesting to see if there is a relationship between the magnitude of the net displacement and the number of collisions. One simple way of investigating whether there is some relation between the number of collisions and the net displacement is to modify our MC program and monitor the average net displacement as a function of the number of collisions. A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.21/30 Numerical Integration Monte Carlo Simulations If we do the mentioned modied MC simulation we nd the following behaviour ( 1000 different paths ). A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.22/30 Numerical Integration Monte Carlo Simulations The obtained plot is very smooth ! We could argue that there should be a relationship of the kind <d> Nq and the interesting question is what value do we get for the exponent q . A simple plot on logarithmic axes would yield a value of q 1. 2 We therefore have discovered that the displacement is proportional to the square root of N . A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.23/30 Numerical Integration Uniform Charge Distribution Let us assume we have a square region in the xy -plane, such that 1 x 1 and 1 y 1. In this square region there shall be a charge distribution . The charge shall be distributed uniformly. We are interested in the electrostatic potential at the point (xp , yp ). y (xp,yp) x A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.24/30 Numerical Integration Uniform Charge Distribution Since the charge is distributed uniformly every innitesimal volume element will contribute the same fraction of source charge and we can get the net effect via integration. y (xp,yp) x A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.25/30 Numerical Integration Uniform Charge Distribution Thus the electrostatic potential is (xp , yp ) = 40 1 1 1 1 dxdy (xp x)2 +(yp y)2 40 and we may take the prefactor y to be 1. (xp,yp) x A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.26/30 Numerical Integration Uniform Charge Distribution On the other hand we could make use of the set of orthogonal functions to expand the electrostatical potential, i.e. (r, ) = i=0 i (r)Pi (cos ) where Pi denotes Legendre polynomials in the unnormalized form. The coefcients i (r) cover the radial dependence, while the Legendre functions will do the angular dependence . A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.27/30 Numerical Integration Uniform Charge Distribution The initial unnormalized Legendre polynomials are P0 (x) P1 (x) P2 (x) P3 (x) P4 (x) P5 (x) = 1 = x 3x2 1 = 2 3 3x = 5x 2 35x4 30x2 +3 = 8 3 5 = 63x 70x +15x 8 A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.28/30 Numerical Integration Uniform Charge Distribution The orthogonality condition for these unnormalized functions is 0 Pm (cos )Pn (cos ) sin d = 2 2m+1 m,n With this condition we can derive the rule to obtain the radial expansion coefcients, i (r), for the electrostatic potential, (r, ) in the standard way, i.e. multiplication of the basic expansion with Pj (cos ) followed by integration over which yields, j (r) = 2j+1 0 2 (r, )Pj (cos ) sin d A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.29/30 Numerical Integration Uniform Charge Distribution With the just presented expression we can get a more general method to calculate the electrostatic potential. We just need to determine the expansion coefcients i (r) and we have xp = rp cos (p ) and yp = rp sin (p ). y (rp, p) x A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.30/30

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:

UVA - ASTR - 130
v 4 # 5 4 # 4 5 # 5 5 !$#&quot;q$C!Df!$`&quot;%k$9Dg&quot;!&quot;u 4 5 DGxC4gfC&quot;gg Ih4kk4~$k4&quot;5!Do&quot;kgp$k#D!D&quot;45!Dl$#&quot;!&quot;$!&quot; | 4 #55 4 5 4 5 9Cu&quot;!D4B$d!&quot;!Df!&quot;l&quot;!9D9x&quot;4Dr&quot;9w&quot;%gj&quot;D s 45 5 # 4 @
North-West Uni. - MMSS - 532
The Effect of Congestion on Flight Delays Experienced by Departing Aircraft at Chicago OHare International Airport and Illustrative Congestion Fees That Could Alleviate the Problemby Tracy Johnson June 6, 2005 Senior Thesis for the Mathematical Met
Mich Tech - GENENG - 1102
ENG 1102 Graphics in Engineering (Technical Communication) Design Project g j Sketching Sectional V Views1G:\common\eng1102\1102_200508\01graphics\en2.g08a.sectional_des.sxiSectiona Views* alcutting planecutting plane line2*Bertoline,
North-West Uni. - APA - 522
Monopoly with resaleGiacomo Calzolari* Alessandro Pavan* Abstract This paper examines the intricacies associated with the design of revenue-maximizing mechanisms for a monopolist who expects her buyers to resell. We consider two cases: resale to a t
Mich Tech - JH - 1101
ENG1101 Presentations Chemical Engineering I t to Engineering Design Intro tAs a Team Team. Get out a piece of paper For each of the following slides, answer: Wh t was the objective? What th bj ti ? What were the constraints? What do you se
Arizona - CS - 620
c m#Es c m#Es Ga d ycGysyEd9pGpyp G9pE Ypf 9e9vyeUp a See eyepye jp CChpEjfC #p yyedph ye 9eC%py yyyyCdCpypy
Mich Tech - CE - 5403
Lecture 16PERFORMANCE MODELSInstructional Objectives Understand use of performance models Identify common modeling approaches Understand methods for evaluating reliability Describe requirements for updating modelsOverview Serviceability-pe
Mich Tech - CE - 5403
Lecture1L 12 LecturePAVEMENT CONDITION INDICESInstructional ObjectivessHistoric development of pavement condition indices The basic functions of condition indices in PMS Different types of condition indices Development of a pavement condition
Mich Tech - CE - 4501
CE4501 Environmental Engineering Chemical Processes Problem Set 2 Fall 2008 Due: Friday, 10/3 by 5 p.m. Solutions will be posted on the Web. Problem sets will be graded for completeness, and one problem (selected at random) will be graded in detail.
Mich Tech - CE - 4501
CE4501 Environmental Engineering Chemical Processes Problem Set 3 - SOLUTION Fall 2008 Due: Monday, 10/13 by 5 p.m. DO 10 OF THE 12 PROBLEMS. Solutions will be posted on the Web. Problem sets will be graded for completeness, and one problem (selected
Mich Tech - CE - 4501
Fall 2008 CE4501. Environmental Chemical Processes LaboratoryThe laboratory section of this course has specific objectives that are distinct from the lecture/recitation part of the course. Nonetheless, the two are closely synchronized, and activitie
Mich Tech - CE - 3502
CE3502, EMMA Project Guidelines and List of Potential Projects Objectives 1. To plan the gathering of data suitable for evaluation of a hypothesis; 2. To use statistical analyses to evaluate a hypothesis; 3. To learn about environmental conditions; 4
Mich Tech - CE - 3502
NAME _ CE3502 Environmental Measurements, Monitoring &amp; Data Analysis Spring 2009 Midterm Section 1. True or False Questions (2 pts each) 1. For any list of numbers, half of them will be below the mean. True _ False _X__ 2. The sample mean is always t
Mich Tech - CE - 4620
Dynamic Wave EquationsUnsteady Flow Modeling with HEC-RASCE 5666 Fall 2006 ContinuityA V y y + VB + B = q x x twhere A = cross sectional area V = average velocity x = distance along channel y = depth t = time q = lateral inflow S f = friction slo
Mich Tech - FW - 5115
Peatland RestorationDifferent types of peatlandsBogs-Poor Fens acid peat deposits with no to little significant inflow and outflow of surface or ground waterFens Open peatland systems significant drainage from surrounding mineral soils and grou
Mich Tech - FW - 4260
FW4260 Population Ecology Problem Set #2 Logistic Growth1.Comparing exponential and logistic growth models. Construct two 50-year data sets each with a starting population size of 5 individuals and a population growth rate r = 0.1. One data set sho
Mich Tech - CHEMISTRY - 2421
IDENTIFICATION OF AN ALCOHOL OR CARBONYL COMPOUND BY MEANS OF THREE REAGENTS (12/12/2004)The reactions of alcohols, aldehydes, and ketones with the reagents, chromic acid in acetone, 2,4-dinitrophenylhydrazine, and sodium hypoiodite, are useful for
Mich Tech - CHEMISTRY - 2411
COLUMN CHROMATOGRAPHY: SEPARATION OF A MIXTURE OF FLUORENE AND FLUORENONE (7/20/04)We have already seen how mixtures of compounds may be separated by thin-layer chromatography (TLC). However, TLC works only on a very small (milligram) scale. When l
Mich Tech - CHEMISTRY - 2421
GRIGNARD REACTION: PREPARATION OF TRIPHENYLMETHANOL (12/12/2004)Grignard reagents are among the most versatile organometallic reagents, and they are the easiest organometallic reagent to prepare. Grignard reagents may be readily prepared from alkyl
Mich Tech - CE - 5403
CE 5403Assignment 2Develop an inventory form that you can use when you collect the data for the condition assessment and evaluation required as part of a typical PMS. The basic data should fit on one page but may include supplement data to make t
Arizona - AZ - 1331
Particle Film Technologies: Pest Management and Yield Enhancement Qualities in Lemons1David L. Kerns and Glenn C. WrightAbstractSurround WP and Snow were evaluated for their ability to manage citrus thrips populations in lemons on the Yuma Mesa,
Mich Tech - CS - 6461
An Architecture for Privacy-Sensitive Ubiquitous ComputingJason I. HongGroup for User Interface Research Computer Science Division University of California at Berkeley Berkeley, CA, 94720-1776 USAJames A. LandayDUB Group Computer Science and Eng
Mich Tech - CS - 6461
IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 22, NO. 1, JANUARY 20041Guest Editorial Recent Advances in Service Overlay NetworksI. SERVICE OVERLAY NETWORKSTHE best effort Internet was designed when host connectivity was the primary
Mich Tech - FW - 5411
FW5411 Applied Regression Analysis03/18/2009Assignment #4 Adequacy and TransformationsIntroduction: Tent caterpillars are a native insect to North America. Populations of tent caterpillar fluctuate, with outbreaks every few years. During outbr
Mich Tech - FW - 3200
Mich Tech - FW - 3200
FW3200 Inventory, Monitoring and Data Analysis03-16-2009Homework No. 7 Stand InventoryStand Level Structure SummariesTimber and non-timber resource values are related to stand structure. Clearly, the value of a stand for timber products is re
Mich Tech - FW - 3200
FW3200 Inventory, Monitoring and Data Analysis01-26-2009Homework No. 2 Basic Statistics by SimulationObjectives:In this assignment we will work with a population that follows the continuous uniform distribution on the interval {0,1}. The obje
Mich Tech - FW - 3200
Mich Tech - EE - 3306
users_guide_33250a.book Page 1 Monday, April 10, 2000 9:09 AMUsers GuidePublication Number 33250-90001 April 2000 Copyright Agilent Technologies 2000 All Rights Reserved.Agilent 33250A 80 MHz Function / Arbitrary Waveform Generatorusers_gui
UVA - ANTH - 589
Spatial Data Analysis in ArchaeologyAnthropology 589b Fraser D. Neiman 2.05.07 University of Virginia Spring 2007Problem Set 2: More Correlograms and VariogramsThe Problem Archaeologists have long used temporal and spatial variation in Maya termi
Mich Tech - ECE - 3140
Notes on the FCCs 700 MHz Auction for EE 3140 Professor MeyerBackgroundWith digital TV gaining ubiquity, broadcasters are slowly moving away from the old analog channels of 5090 or so, which occupy about 700900 MHz of the wireless spectrum. When t
Arizona - SCHEDULE - 091
Crs. #(units) Call # Sec #Course/Section Title Time DaysInstructor Bldg. Room #Crs. #(units) Call # Sec #Course/Section Title Time DaysInstructor Bldg. Room # BOESENPHARMACY PRACTICE (PHPR)Dr. Marie Chisholm, Head, Drachman B2011D, 6
Mich Tech - EE - 5223
FEATUREARTICLEA Review of Faults Detectable by Gas-in-Oil Analysis in TransformersKey Words: Transformer, fault, dissolved gas analysisEC Publication 60599 [1] provides a coded list of faults detectable by dissolved gas
Mich Tech - CE - 4505
CE4505 Surface Water Quality EngineeringLecture 15. Numerical MethodsNumerical Methods:Techniques by which mathematical problems are formulated so that they can be solved with arithmetic operations Typically involve multiple, repetitive (tedious)
Mich Tech - FW - 5560
Digital Image Processing: A Remote Sensing Perspective FW5560 Lecture Evaluation of Training Sets (Training Signatures)Once training sets (training signatures) are created, they can be evaluated, deleted, renamed, and merged with signatures from ot
Mich Tech - CS - 6461
COVER FEATUREPrioritized Overlay Multicast in Mobile Ad Hoc EnvironmentsThe authors propose a model to improve the efciency and robustness of overlay multicast in manets by building multiple role-based prioritized trees, possibly with the help of
Harvard - ES - 151
THE PARAXIAL WAVE EQUATIONGAUSSIAN BEAMS IN UNIFORM MEDIA :In point-to-point communication, we may think of the electromagnetic field as propagating in a kind of &quot;searchlight&quot; mode - i.e. a beam of finite width that propagates in some particular di
UVA - CHEM - 181
CHEM 181L: SyllabusThe course has two components, a weekly one-hour lecture period and a three-and-half hour laboratory period. The syllabus for both components, shown below, is organized by weeks. The &quot;course week&quot; is defined to begin with Friday l
Arizona - ECE - 533
OPTICAL IMAGE FORMATIONGEOMETRICAL IMAGING First-order image is perfectobject (input) scaled (by magnification) version of objectmagnification = image distance/object distanceoptical systemimage (output)object distance image distance no bl
Mich Tech - FW - 4220
Scientific NameAbies balsamea Abutilon theophrasti Acalypha rhomboidea Acalypha virginica Acer negundo Acer pensylvanicum Acer rubrum Acer rubrum Acer rubrum Acer saccharinum Acer saccharum Acer spicatum Achillea millefolium Aconitum columbianum Aco
Harvard - ASTRO - 200
The black hole in the Milky WayAstronomy 200 16 Feb 2005 Alexandre TchekhovskoiWhat can we say about the black hole in the Milky Way? Mmm. this one is probably black, too Location, velocity Mass, Spin, Charge (&quot;no hair&quot; theorem) How do we meas
UVA - CS - 462
SDB - Simple Relational Database System A User Guideby David Betz 114 Davenport Ave. Manchester, NH 03103 revised: Fall 1990 Robert C. Beckinger John L. Pfaltz Dept. of Computer Science University of VirginiaSDB - a Simple Database System11. I
UVA - PHYS - 304
Physics of the Human Body Chapter 545 BioenergeticsBioenergeticsIn Chapter 8 (Thermodynamics), we find the maximum theoretical efficiency of a heat engine, in turning heat to work1, to be Carnot = T&gt; T&lt; . T&gt; molecule gives up one of its phospha
UVA - CS - 451
AgendaLast timeChpt 12: Coordination and Agreement (election)SunBefore we start: ScheduleTues Thurs FriThis timeHW#3 back Chpt 12: Coordination and Agreement (group communication, consensus, Byzantine Generals) HW #5 out (due last day of cla
UVA - CS - 451
AgendaLast time (today)HW#3 back Chpt 12: Coordination and Agreement (group communication, consensus, Byzantine Generals) HW #5 out (due last day of class, May 1) NO LATE SUBMISSIONS ACCEPTED! SunBefore we start: ScheduleTues Thurs FriToday (
UVA - CS - 615
Cooperative Bug IsolationBen Liblit et al.#1Whats This? I decided that that sigma calculus for objects was too heavy for our final lecture. OO slides are available on the webpage. Instead, well talk about the work that won the 2005 ACM Doctor
UVA - CS - 150
Programming with State &amp; Golden Ages#1One-Slide Summary The substitution model for evaluating Scheme does not allow us to reason about mutation. In the environment model: A name is a place for storing a value. define, cons and function applicati
Harvard - CHS - 116
AN APOBATIC MOMENT FOR ACHILLES AS ATHLETE AT THE FESTIVAL OF THE PANATHENAIAProf. Nagy GregoryHarvard UniversityThis presentation focuses on two Black Figure paintings, both dated around 510 BCE, that depict the athletic event of the apobaton a
Harvard - CHS - 119
MODEL RESPONSE PAPER II The juxtaposition of the terms spiritual and primal in Moses and Monotheism is troubling. According to Freud, the succession of early events in civilization is as follows: first, the primal horde; second, the killing of the fa
Harvard - CHS - 119
HOW TO WRITE A RESPONSE PAPER IN OLYMPIAIn each weekly course you will be required to write a response paper. The length of the response paper is one to one-and-a-half pages, and it is due at the end, or shortly after the end, of the weekly module.
Harvard - CHS - 119
MODEL RESPONSE PAPER I I am surprised by how much discourses such as Balkanism or Orientalism lump vastly different people together. Orientalist treatment of the Middle East has been a particularly vicious affair. The political divides of the Middle
Harvard - CHS - 116
Performances and Texts Oxford Handbook of Hellenic Studies ed. George Boys-Stones, Barbara Graziosi, and Phiroze Vasunia Title: Performance and text in ancient Greece Section: Performances and Texts [Permission has been granted by Oxford University P
Harvard - CHS - 116
1 The Sign of the Hero: A Prologue to the Heroikos of Philostratus by Gregory Nagy (Originally published in J. K. Berenson Maclean and E. B. Aitken, eds., Flavius Philostratus, Heroikos (Atlanta 2001) xv-xxxv.) The traditional practice of worshipping
Harvard - CHS - 116
Did Sappho and Alcaeus ever meet?Symmetries of myth and ritual in performing the songs of ancient Lesbos[This is an electronic version of an article that appeared in Literatur und Religion I. Wege zu einer mythischrituellen Poetik bei den Griechen
Harvard - CHS - 116
The fragmentary Muse and the poetics of refraction in Sappho, Sophocles, OffenbachGregory Nagy[text highlighted in gray indicates technical parts that the reader may want to skip]The idea of a fragmentary Muse comes from a fragmentary opera, The
Harvard - CHS - 116
AN INTRODUCTION TO CLOSE READING DAVID SCHUR 1998 David Schur Harvard University Second version (2/99) PLEASE DO NOT DUPLICATECONTENTS Introduction Preparation Assumptions and Goals Straightforward Reading Descriptive Analysis Interpretation Cohe
Mich Tech - EET - 2411
6-3 Addition in the 2's Complement System Perform normal binary addition of magnitudes. g The sign bits are added with the magnitude bits. If addition results in a carry of the sign bit, the carry bit is ignored. If the result is positive it is i
Mich Tech - EET - 2411
5-2 NOR Gate Latch The NOR latch is similar to the NAND latch except that the Q and Q outputs are reversed reversed. The set and clear inputs are active high, that is, the output will change when the input is pulsed high. In order to ensure that a
Mich Tech - EET - 4141
EET4141 Fall2008 (Week 13 &amp; 14), Experiment 10 - Project Assignment 4 Name: _ Partner: _ Keypad Interface and Digital Thermometer 10.1 Upgraded Version of the Digital Thermometer In this part of the lab, you will integrate the Digital Thermometer (Pr
Mich Tech - EET - 2411
Chapter 5 Flip Flops Sequential circuits - Introduction Logic circuits studied so far have outputs that respond immediately to inputs at some instant in time. We now introduce the concept of memory. The flip-flop, abbreviated FF, is a key memory
Mich Tech - EET - 4141
EET 4141 Fall 2008 (Week7 + Week8), Experiment 6 Name: _ Partner: _ Hardware Timing Before starting the lab exercise, Enter Program 6.1 below and verify that the switch can be read and that the LEDs can be controlled.Figure 6.1 Program 6.1 PORTB EQ