Energy Density / Energy Flux / Total Energy in 3D


 Sabrina McKinney
 3 years ago
 Views:
Transcription
1 Lecture 5 Phys 75 Energy Density / Energy Fux / Tota Energy in D Overview and Motivation: In this ecture we extend the discussion of the energy associated with wave otion to waves described by the D wave euation. In fact the first part of the discussion is exacty the sae as the 1D case just extended to D. In the exapes we oo at the energy associated with sphericay syetric waves. Key Matheatics: oe D cacuus especiay the divergence theore and the sphericacoordinates version of the gradient. I. Density Fux and the Continuity Euation As in the 1D case et's assue that we are interested in soe uantity Q () t that has an associated density ρ ( x. ince we are deaing with a density that ives in a D space ρ = Q. the units of density wi be the units of Q divided by. That is [ ] [ ] Let's consider a voue encosed by a surface as iustrated in the foowing figure. At each point on the surface we define a perpendicuar outwardpointing unit vector nˆ ( r ) associated with each point r on the surface. r nˆ nˆ ( r ) nˆ ( r ) encosed voue The aount of Q contained in can be written as () t ρ ( r d (1) Q = r As in the 1D case if Q is a conserved uantity then the change in Q inside D M Riffe 1 /1/1
2 Lecture 5 Phys 75 () t ρ( r dq = d r () dt ust be eua to the net fow of Q into dq dt () t ( r nˆ ( r ) d. () = j The (vector) uantity j is again nown as the Q current density or the Q fux. The diensions of j are the diensions of ρ ties a veocity so [ j ] = [ ρ] s. Thus aso [] j = [ Q]( s). Note that the rhs of E. () can be interpreted as the tota Q current fowing out through the surface. Euating the rhs's of Es. () and () gives us ρ ( r d r = j ( r nˆ ( r) d. (4) We can now use the divergence theore (which is one of severa D extensions of the fundaenta theore of cacuus) A ( r ) d r = A( r) nˆ ( r ) d (5) to rewrite E. (4) as ρ ( r + j ( r d r = (6) Now because the voue is arbitrary the integrand ust vanish. Thus ρ ( r + j ( r =. (7) Euation (7) is the D version of the continuity euation which again is a oca stateent of the conservation of Q. D M Riffe  /1/1
3 Lecture 5 Phys 75 II. Energy Density and Fux for D Waves We now appy this discussion to the energy associated with D waves. In this case Q represents the energy associated with a wave (within soe voue). For waves described by the D wave euation the energy density can be written as µ ρ( r = + c ( ) (8) where ( r is the variabe that is governed by the wave euation. The first ter on the rhs of E. (8) is the inetic energy density ρ T and the second is the potentia energy density ρ V. Now E. (8) is fairy genera as ong as µ is suitaby interpreted. If is a true dispaceent then µ wi be a paraeter with the units of ass density. If represents soething ese say an eectric fied then it wi have soe other units. Fro E. (8) it is fairy easy to see that the units of µ are generay given by [ µ ]= (Joue s )/( [] ). It is not hard to show that the energy fux which can be written as j ( r = µ c (9) together with the energy density in E. (8) satisfy E. (7) the continuity euation. III. evera Exapes Let's oo at soe exapes that invove sphericay syetric waves. A. pherica tanding Wave Let's oo at a standingwave exape. You ay reca that a sphericacoordinates separabe soution that is finite everywhere is of the for iφ iφ ict ict ( r θ φ = C j ( ) P ( cos( θ )) ( C e + D e ) ( A e + B e ) (1) where j is a spherica Besse function (of the first ind) and P is an associated Legendre function (of the first ind). The paraeter is an integer whose absoute vaue can be no arger than the nonnegative integer. If we want a soution with spherica syetry then there can be no θ or φ dependence. This eans that both and ust be zero because the ony associated Legendre function independent of θ is P ( cos( θ )) = 1. Thus the spherica Besse function in E. (1) ust be j ( ) = sin( ) ( ) and so E. (1) sipifies to D M Riffe  /1/1
4 Lecture 5 Phys 75 sin ( ) ( ict ict r θ t = C A e + B e ). (11) φ If we sipify this further by etting A be a rea nuber and et soution expicity rea) then we have B = A (aing the sin ( ) r θ φ t A C cos( c. (1) = Because a parts of the syste osciate with the sae phase this is a sphericay syetric version of a standing wave. Using Es. (8) and (9) we can cacuate the inetic and potentia energy densities and the energy fux associated with the wave in E. (1). To do this in a fairy sipe anner we can use the sphericacoordinates version of the gradient f 1 f 1 f f ( r θ φ) = rˆ + θˆ + φˆ (1) r r θ r sin( θ ) φ where rˆ θˆ and φˆ are unit vectors in the r θ and φ directions respectivey. The nice thing about sphericay syetric soutions is that ony the first ter on the rhs of E. (1) contributes to the gradient. A video of ρ T ρ V and j for the wave in E. (1) Energy in D tanding Wave.avi is avaiabe on the cass web site. As the video shows the dispaceent is indeed a standing wave. Unfortunatey the energy densities and fux fa off with the radia distance r so fast that it is hard to reay see their behavior. Given this we have ade another video Energy in D tanding Wave.avi which pots the surface integrated density and fux 1 and D I () r ρ ( r ) d (14) = () r = j() r nˆ ( r ) d (15) 1 The video separatey shows the inetic and potentia contributions to D ( r). D M Riffe 4 /1/1
5 Lecture 5 Phys 75 where the surface is of radius r centered at the origin. Now because a sphericay syetric soution is independent of the two anges θ and φ this aounts to utipying the density ρ and fux j by the factor 4π r which is the surface area of the sphere. The uantity D () r (which has units of Joue/) can be though of as a inear energy density (i.e. the energy per unit ength aong the radia direction) whie the uantity I () r (which has units of Joue/s) is the tota (energy) current fowing through. Notice that this new video is very siiar to the 1D standing wave video that we ooed at in the ast ecture. B. pherica Traveing Wave Let's aso oo at a sphericay syetric traveing wave. If we are thining about sound waves this is the sort of wave that woud resut fro a pusating sphere centered at the origin. We can construct a traveing wave soution fro a inear cobination of ineary independent standing waves. We thus need to use both inds of spherica Besse functions. The inear cobination that produces a sphericay syetric outgoing traveing wave is ( r θ φ = C [ j ( ) cos( c y ( ) sin( c ]. (16) which can be written in ters of sine and cosine functions as sin ( ) cos ( ) r θ t = C cos ct sin( c. (17) φ The video Energy in D Traveing Wave.avi shows D ( r) and I ( r) for this wave. Indeed away fro the origin the wave appears to be an outgoing traveing wave. However at the origin soething rather different sees to be happening soething with soe standingwave character perhaps? We as it turns out the current density j has ters with two types of behavior. The first type has a 1 r dependence. These ters describe the radiative part of the wave which carries energy off to infinity. Because the radiative part of j varies as 1 r the radiative part of I () r does not vanish as r. However there are aso nonradiative ters which vary as 1 r. These ters act ore ie a standing wave: the energy associated with these ters just osciates bac and forth and never reay goes anywhere. Because of the 1 r behavior to the nonradiative part of j the current I () r associated with these ters vanishes as 1 r as r. These ters are Note that this soution is ony vaid in the region of space outside the source. In the video you ay thin of the source as being infinitesiay sa so that the soution is vaid infinitesiay cose to the origin. D M Riffe 5 /1/1
6 Lecture 5 Phys 75 thus soeties caed the oca fieds associated with the source. In the video Energy in D Traveing Wave.avi we separatey show the current I ( r) associated with each type of ter. Notice that the radiative piece oos essentiay ie a 1D traveing wave whie the nonradiative piece is reay ony iportant in the vicinity of the origin. Exercises *5.1 how that the expressions for the density ρ and j in Es. (8) and (9) respectivey satisfy E. (7) the continuity euation. **5. pherica Traveing Wave (a) Write the wave in E. (17) ( r ( ) ( ) sin cos θ = C cos( c sin( c φ as an expicit function of ( r c thus showing that it is a traveing wave oving outward fro the origin. (b) Using your resut fro part (a) show that the radiative and nonradiative coponents of the current density j can be written respectivey as j R c ( r = µ cos ( crˆ and j ( r cos( c sin( crˆ r NR c = µ. (c) Cacuate the tie average of each of these currentdensity coponents (defined as 1 T j ( r dt where T is one period of osciation) and show that the average of the T radiative part points in the positive rˆ direction whie the tie average of the nonradiative part is zero. (Note: neither answer shoud have any dependence on T.) *5. Pane Wave Energy Density. Consider the panewave soution to the D wave euation ( x y z = exp{ i( xx + y y + z z c }. (a) Cacuate the inetic potentia and tota energy densities ρ T ( x y z ρ V ( x y z and ρ ( x y z respectivey and the energy current density j ( x y z. (b) how that the D continuity euation is satisfied by your expressions for ρ and j. D M Riffe 6 /1/1
7 Lecture 5 Phys 75 **5.4 pherica tanding Wave Energy Density. Consider the sphericay syetric standing wave soution to the D wave euation sin ( ) r θ φ = cos( c. (a) Cacuate the inetic potentia and tota energy densities ρ T ( r θ φ ρ V ( r θ φ and ρ ( r θ φ respectivey. (b) how for arge distances fro the origin ( >> 1) that the tota energy density for µ this wave is approxiatey sin cos ρ r θ φ t = c sin ct + cos( c. D M Riffe 7 /1/1
11  KINETIC THEORY OF GASES Page 1
 KIETIC THEORY OF GASES Page Introduction The constituent partices of the atter ike atos, oecues or ions are in continuous otion. In soids, the partices are very cose and osciate about their ean positions.
More information11  KINETIC THEORY OF GASES Page 1. The constituent particles of the matter like atoms, molecules or ions are in continuous motion.
 KIETIC THEORY OF GASES Page Introduction The constituent partices of the atter ike atos, oecues or ions are in continuous otion. In soids, the partices are very cose and osciate about their ean positions.
More informationAngles formed by 2 Lines being cut by a Transversal
Chapter 4 Anges fored by 2 Lines being cut by a Transversa Now we are going to nae anges that are fored by two ines being intersected by another ine caed a transversa. 1 2 3 4 t 5 6 7 8 If I asked you
More informationCHAPTER 30 GAUSS S LAW
CHPTER GUSS S LW. Given : E /5 E î /5 E ĵ E. N/C The pane is parae to yzpane. Hence ony /5 E î passes perpendicuar to the pane whereas /5 E ĵ goes parae. rea.m (given Fux E /5.. Nm /c Nm /c. Given ength
More informationLecture L263D Rigid Body Dynamics: The Inertia Tensor
J. Peraire, S. Widnall 16.07 Dynaics Fall 008 Lecture L63D Rigid Body Dynaics: The Inertia Tensor Version.1 In this lecture, we will derive an expression for the angular oentu of a 3D rigid body. We shall
More informationAnswer, Key Homework 7 David McIntyre 45123 Mar 25, 2004 1
Answer, Key Hoework 7 David McIntyre 453 Mar 5, 004 This printout should have 4 questions. Multiplechoice questions ay continue on the next colun or page find all choices before aking your selection.
More informationUNITI DRIVE CHARACTERISTICS
Eectrica Drives: UNII DRIVE CHARACERISICS Motion contro is required in arge nuber of industria and doestic appications ike transportation systes, roing is, paper achines, textie is, achine toos, fans,
More information3.5 Pendulum period. 20090210 19:40:05 UTC / rev 4d4a39156f1e. g = 4π2 l T 2. g = 4π2 x1 m 4 s 2 = π 2 m s 2. 3.5 Pendulum period 68
68 68 3.5 Penduum period 68 3.5 Penduum period Is it coincidence that g, in units of meters per second squared, is 9.8, very cose to 2 9.87? Their proximity suggests a connection. Indeed, they are connected
More informationAssigning Tasks in a 24Hour Software Development Model
Assigning Tasks in a Hour Software Deveopent Mode Pankaj Jaote, Gourav Jain Departent of Coputer Science & Engineering Indian Institute of Technoogy, Kanpur, INDIA 006 Eai:{jaote, gauravj}@iitk.ac.in
More informationPhysics 100A Homework 11 Chapter 11 (part 1) The force passes through the point A, so there is no arm and the torque is zero.
Physics A Homework  Chapter (part ) Finding Torque A orce F o magnitude F making an ange with the x axis is appied to a partice ocated aong axis o rotation A, at Cartesian coordinates (,) in the igure.
More informationThe Simple Pendulum. by Dr. James E. Parks
by Dr. James E. Parks Department of Physics and Astronomy 401 Niesen Physics Buidin The University of Tennessee Knoxvie, Tennessee 37996100 Copyriht June, 000 by James Edar Parks* *A rihts are reserved.
More informationAn Idiot s guide to Support vector machines (SVMs)
An Idiot s guide to Support vector machines (SVMs) R. Berwick, Viage Idiot SVMs: A New Generation of Learning Agorithms Pre 1980: Amost a earning methods earned inear decision surfaces. Linear earning
More informationSecure Network Coding with a Cost Criterion
Secure Network Coding with a Cost Criterion Jianong Tan, Murie Médard Laboratory for Information and Decision Systems Massachusetts Institute of Technoogy Cambridge, MA 0239, USA Emai: {jianong, medard}@mit.edu
More informationToday in Physics 217: the method of images
Today in Physics 17: the method of images Solving the Laplace and Poisson euations by sleight of hand Introduction to the method of images Caveats Example: a point charge and a grounded conducting sphere
More informationName: Period: 9/28 10/7
Nae: Period: 9/ 0/ LINES & TRANSVERSALS ) I can define, identify and iustrate te foowing ters Transversa Corresponding anges Aternate exterior anges. Aternate interior anges Sae side interior anges Dates,
More information5. Introduction to Robot Geometry and Kinematics
V. Kumar 5. Introduction to Robot Geometry and Kinematics The goa of this chapter is to introduce the basic terminoogy and notation used in robot geometry and kinematics, and to discuss the methods used
More informationMagnetic circuits. Chapter 7. 7.1 Introduction to magnetism and magnetic circuits. At the end of this chapter you should be able to:
Chapter 7 Magnetic circuits At the end of this chapter you shoud be abe to: appreciate some appications of magnets describe the magnetic fied around a permanent magnet state the aws of magnetic attraction
More informationPhysics 211: Lab Oscillations. Simple Harmonic Motion.
Physics 11: Lab Oscillations. Siple Haronic Motion. Reading Assignent: Chapter 15 Introduction: As we learned in class, physical systes will undergo an oscillatory otion, when displaced fro a stable equilibriu.
More informationMath 447/547 Partial Differential Equations Prof. Carlson Homework 7 Text section 4.2 1. Solve the diffusion problem. u(t,0) = 0 = u x (t,l).
Math 447/547 Partia Differentia Equations Prof. Carson Homework 7 Text section 4.2 1. Sove the diffusion probem with the mixed boundary conditions u t = ku xx, < x
More informationLecture L9  Linear Impulse and Momentum. Collisions
J. Peraire, S. Widnall 16.07 Dynaics Fall 009 Version.0 Lecture L9  Linear Ipulse and Moentu. Collisions In this lecture, we will consider the equations that result fro integrating Newton s second law,
More information1 Basic concepts in geometry
1 asic concepts in geometry 1.1 Introduction We start geometry with the simpest idea a point. It is shown using a dot, which is abeed with a capita etter. The exampe above is the point. straight ine is
More informationReliability Constrained Packetsizing for Linear Multihop Wireless Networks
Reliability Constrained acketsizing for inear Multihop Wireless Networks Ning Wen, and Randall A. Berry Departent of Electrical Engineering and Coputer Science Northwestern University, Evanston, Illinois
More informationLesson 44: Acceleration, Velocity, and Period in SHM
Lesson 44: Acceleration, Velocity, and Period in SHM Since there is a restoring force acting on objects in SHM it akes sense that the object will accelerate. In Physics 20 you are only required to explain
More informationSymplectic structures and Hamiltonians of a mechanical system
INVESTIGACIÓN REVISTA MEXICANA DE FÍSICA 49 (5) 445 449 OCTUBRE 2003 Syplectic structures and Hailtonians of a echanical syste G.F. Torres del Castillo Departaento de Física Mateática, Instituto de Ciencias,
More informationAA Fixed Rate ISA Savings
AA Fixed Rate ISA Savings For the road ahead The Financia Services Authority is the independent financia services reguator. It requires us to give you this important information to hep you to decide whether
More informationDistances in random graphs with finite mean and infinite variance degrees
E e c t r o n i c J o u r n a o f P r o b a b i i t y Vo. 12 2007, Paper no. 25, pages 70 766. Journa URL http://www.ath.washington.edu/~epecp/ Distances in rando graphs with finite ean and infinite variance
More informationDiscounted Cash Flow Analysis (aka Engineering Economy)
Discounted Cash Fow Anaysis (aka Engineering Economy) Objective: To provide economic comparison of benefits and costs that occur over time Assumptions: Future benefits and costs can be predicted A Benefits,
More informationWork, Energy, Conservation of Energy
This test covers Work, echanical energy, kinetic energy, potential energy (gravitational and elastic), Hooke s Law, Conservation of Energy, heat energy, conservative and nonconservative forces, with soe
More information7. Dry Lab III: Molecular Symmetry
0 7. Dry Lab III: Moecuar Symmetry Topics: 1. Motivation. Symmetry Eements and Operations. Symmetry Groups 4. Physica Impications of Symmetry 1. Motivation Finite symmetries are usefu in the study of moecues.
More informationThe Use of CoolingFactor Curves for Coordinating Fuses and Reclosers
he Use of ooingfactor urves for oordinating Fuses and Recosers arey J. ook Senior Member, IEEE S& Eectric ompany hicago, Iinois bstract his paper describes how to precisey coordinate distribution feeder
More informationPHYSICS 151 Notes for Online Lecture 2.2
PHYSICS 151 otes for Online Lecture. A freebod diagra is a wa to represent all of the forces that act on a bod. A freebod diagra akes solving ewton s second law for a given situation easier, because
More informationCosmology of Einstein s NOW
Aerican Journa of Modern hysics 016; 5(41): 15 ubished onine January 1, 016 (http://www.sciencepubishinggroup.co/j/ajp) doi: 10.11648/j.ajp.s.016050401.11 ISSN: 68867 (rint); ISSN: 68891 (Onine) Cosoogy
More informationFinance 360 Problem Set #6 Solutions
Finance 360 Probem Set #6 Soutions 1) Suppose that you are the manager of an opera house. You have a constant margina cost of production equa to $50 (i.e. each additiona person in the theatre raises your
More informationCONDENSATION. Prabal Talukdar. Associate Professor Department of Mechanical Engineering IIT Delhi Email: prabal@mech.iitd.ac.in
CONDENSATION Praba Taukdar Associate Professor Department of Mechanica Engineering IIT Dehi Emai: praba@mech.iitd.ac.in Condensation When a vapor is exposed to a surface at a temperature beow T sat, condensation
More informationChapter 14 Oscillations
Chapter 4 Oscillations Conceptual Probles 3 n object attached to a spring exhibits siple haronic otion with an aplitude o 4. c. When the object is. c ro the equilibriu position, what percentage o its total
More informationThe Mathematics of Pumping Water
The Matheatics of Puping Water AECOM Design Build Civil, Mechanical Engineering INTRODUCTION Please observe the conversion of units in calculations throughout this exeplar. In any puping syste, the role
More informationConstruction Economics & Finance. Module 3 Lecture1
Depreciation: Construction Econoics & Finance Module 3 Lecture It represents the reduction in arket value of an asset due to age, wear and tear and obsolescence. The physical deterioration of the asset
More informationVectors & Newton's Laws I
Physics 6 Vectors & Newton's Laws I Introduction In this laboratory you will eplore a few aspects of Newton s Laws ug a force table in Part I and in Part II, force sensors and DataStudio. By establishing
More informationSimultaneous Routing and Power Allocation in CDMA Wireless Data Networks
Simutaneous Routing and Power Aocation in CDMA Wireess Data Networks Mikae Johansson *,LinXiao and Stephen Boyd * Department of Signas, Sensors and Systems Roya Institute of Technoogy, SE 00 Stockhom,
More informationSAT Math Facts & Formulas
Numbers, Sequences, Factors SAT Mat Facts & Formuas Integers:..., 3, 2, 1, 0, 1, 2, 3,... Reas: integers pus fractions, decimas, and irrationas ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences:
More informationCell Coverage Optimization for the Multicell Massive MIMO Uplink
Ce Coverage Optiization for the Mutice Massive MIMO Upink Jin, S., Wang, J., Sun, Q., Matthaiou, M., & Gao, X. 04. Ce Coverage Optiization for the Mutice Massive MIMO Upink. IEEE Transactions on Vehicuar
More informationSTRONGLY CONSISTENT ESTIMATES FOR FINITES MIX'IURES OF DISTRIBUTION FUNCTIONS ABSTRACT. An estimator for the mixing measure
STRONGLY CONSISTENT ESTIMATES FOR FINITES MIX'IURES OF DISTRIBUTION FUNCTIONS Keewhan Choi Cornell University ABSTRACT The probles with which we are concerned in this note are those of identifiability
More informationKey Features of Life Insurance
Key Features of Life Insurance Life Insurance Key Features The Financia Conduct Authority is a financia services reguator. It requires us, Aviva, to give you this important information to hep you to decide
More informationTeaching fractions in elementary school: A manual for teachers
Teaching fractions in eementary schoo: A manua for teachers H. Wu Apri 0, 998 [Added December, 200] I have decided to resurrect this fie of 998 because, as a reativey short summary of the basic eements
More informationHW 2. Q v. kt Step 1: Calculate N using one of two equivalent methods. Problem 4.2. a. To Find:
HW 2 Proble 4.2 a. To Find: Nuber of vacancies per cubic eter at a given teperature. b. Given: T 850 degrees C 1123 K Q v 1.08 ev/ato Density of Fe ( ρ ) 7.65 g/cc Fe toic weight of iron ( c. ssuptions:
More informationFace Hallucination and Recognition
Face Haucination and Recognition Xiaogang Wang and Xiaoou Tang Department of Information Engineering, The Chinese University of Hong Kong {xgwang1, xtang}@ie.cuhk.edu.hk http://mmab.ie.cuhk.edu.hk Abstract.
More informationThermal properties. Heat capacity atomic vibrations, phonons temperature dependence contribution of electrons
Therma properties Heat capacity atomic vibrations, phonons temperature dependence contribution of eectrons Therma expansion connection to anharmonicity of interatomic potentia inear and voume coefficients
More informationThe width of single glazing. The warmth of double glazing.
Therma Insuation CI/SfB (31) Ro5 (M5) September 2012 The width of singe gazing. The warmth of doube gazing. Pikington Spacia Revoutionary vacuum gazing. Image courtesy of Lumen Roofight Ltd. Pikington
More informationPayondelivery investing
Payondeivery investing EVOLVE INVESTment range 1 EVOLVE INVESTMENT RANGE EVOLVE INVESTMENT RANGE 2 Picture a word where you ony pay a company once they have deivered Imagine striking oi first, before
More informationDivergence and Curl of the Magnetic Field
Divergence and Curl of the Magnetic Field The static electric field E(x,y,z such as the field of static charges obeys equations E = 1 ǫ ρ, (1 E =. (2 The static magnetic field B(x,y,z such as the field
More informationReview B: Coordinate Systems
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of hysics 8.02 Review B: Coordinate Systems B.1 Cartesian Coordinates... B2 B.1.1 Infinitesimal Line Element... B4 B.1.2 Infinitesimal Area Element...
More informationLecture 09 Nuclear Physics Part 1
Lecture 09 Nuclear Physics Part 1 Structure and Size of the Nucleus Νuclear Masses Binding Energy The Strong Nuclear Force Structure of the Nucleus Discovered by Rutherford, Geiger and Marsden in 1909
More informationOur Goals for our Students
Hoe Courses Registration Logon Contact Introducing King s Onine Bibe Schoo Our Goas for our Students Options for Study Bibe Schoo Dipoa What we ask of our students Downoad this page as a PDF Our Goas
More information5.7 Chebyshev Multisection Matching Transformer
/9/ 5_7 Chebyshev Multisection Matching Transforers / 5.7 Chebyshev Multisection Matching Transforer Reading Assignent: pp. 555 We can also build a ultisection atching network such that Γ f is a Chebyshev
More informationRisk Margin for a NonLife Insurance RunOff
Risk Margin for a NonLife Insurance RunOff Mario V. Wüthrich, Pau Embrechts, Andreas Tsanakas February 2, 2011 Abstract For sovency purposes insurance companies need to cacuate socaed bestestimate
More informationA quantum model for the stock market
A quantum mode for the stock market Authors: Chao Zhang a,, Lu Huang b Affiiations: a Schoo of Physics and Engineering, Sun Yatsen University, Guangzhou 5175, China b Schoo of Economics and Business Administration,
More informationDEGREES OF ORDERS ON TORSIONFREE ABELIAN GROUPS
DEGREES OF ORDERS ON TORSIONFREE ABELIAN GROUPS ASHER M. KACH, KAREN LANGE, AND REED SOLOMON Abstract. We show that if H is an effectivey competey decomposabe computabe torsionfree abeian group, then
More informationTERM INSURANCE CALCULATION ILLUSTRATED. This is the U.S. Social Security Life Table, based on year 2007.
This is the U.S. Socia Security Life Tabe, based on year 2007. This is avaiabe at http://www.ssa.gov/oact/stats/tabe4c6.htm. The ife eperiences of maes and femaes are different, and we usuay do separate
More informationPhys101 Lectures 14, 15, 16 Momentum and Collisions
Phs0 Lectures 4, 5, 6 Moentu and ollisions Ke points: Moentu and ipulse ondition for conservation of oentu and wh How to solve collision probles entre of ass Ref: 9,,3,4,5,6,7,8,9. Page Moentu is a vector:
More information2. The acceleration of a simple harmonic oscillator is zero whenever the oscillating object is at the equilibrium position.
CHAPTER : Vibrations and Waes Answers to Questions The acceleration o a siple haronic oscillator is zero wheneer the oscillating object is at the equilibriu position 5 The iu speed is gien by = A k Various
More informationand that of the outgoing water is mv f
Week 6 hoework IMPORTANT NOTE ABOUT WEBASSIGN: In the WebAssign ersions of these probles, arious details hae been changed, so that the answers will coe out differently. The ethod to find the solution is
More informationThe Lagrangian Method
Chapter 6 The Lagrangian Method Copyright 2007 by David Morin, orin@physics.harvard.edu (draft version In this chapter, we re going to learn about a whole new way of looking at things. Consider the syste
More informationThe Concept of the Effective Mass Tensor in GR. The Equation of Motion
The Concept of the Effective Mass Tensor in GR The Equation of Motion Mirosław J. Kubiak Zespół Szkół Technicznych, Gruziąz, Polan Abstract: In the papers [, ] we presente the concept of the effective
More information( C) CLASS 10. TEMPERATURE AND ATOMS
CLASS 10. EMPERAURE AND AOMS 10.1. INRODUCION Boyle s understanding of the pressurevolue relationship for gases occurred in the late 1600 s. he relationships between volue and teperature, and between
More informationVector surface area Differentials in an OCS
Calculus and Coordinate systems EE 311  Lecture 17 1. Calculus and coordinate systems 2. Cartesian system 3. Cylindrical system 4. Spherical system In electromagnetics, we will often need to perform integrals
More informationFlow Pattern Map for In Tube Evaporation and Condensation
Fow Pattern Map for In Tube Evaporation and Condensation Lászó Garbai Dr. *, Róbert Sánta ** * Budapest University of Technoogy and Economics, Hungary ** Coege of Appied Sciences Subotica Tech, Serbia
More informationCONTRIBUTION OF INTERNAL AUDITING IN THE VALUE OF A NURSING UNIT WITHIN THREE YEARS
Dehi Business Review X Vo. 4, No. 2, Juy  December 2003 CONTRIBUTION OF INTERNAL AUDITING IN THE VALUE OF A NURSING UNIT WITHIN THREE YEARS John N.. Var arvatsouakis atsouakis DURING the present time,
More informationDOING BUSINESS WITH THE REGION OF PEEL A GUIDE FOR NEW AND CURRENT VENDORS
DOING BUSINESS WITH THE REGION OF PEEL A GUIDE FOR NEW AND CURRENT VENDORS TABLE OF CONTENTS INTRODUCTION... 1 GOVERNANCE... 1 COMMONLY PURCHASED GOODS AND SERVICES... 1 HOW TO REGISTER YOUR COMPANY...
More informationCapacity of Multiservice Cellular Networks with TransmissionRate Control: A Queueing Analysis
Capacity of Mutiservice Ceuar Networs with TransmissionRate Contro: A Queueing Anaysis Eitan Atman INRIA, BP93, 2004 Route des Lucioes, 06902 SophiaAntipois, France aso CESIMO, Facutad de Ingeniería,
More informationSAT Math MustKnow Facts & Formulas
SAT Mat MustKnow Facts & Formuas Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Rationas: fractions, tat is, anyting expressabe as a ratio of integers Reas: integers pus rationas
More informationDistributed Strategic Interleaving with Load Balancing
Distributed Strategic Intereaving with Load Baancing J.A. Bergstra 1,2 and C.A. Middeburg 1,3 1 Programming Research Group, University of Amsterdam, P.O. Box 41882, 1009 DB Amsterdam, the Netherands 2
More informationDEGREES OF ORDERS ON TORSIONFREE ABELIAN GROUPS
1 DEGREES OF ORDERS ON TORSIONFREE ABELIAN GROUPS 2 ASHER M. KACH, KAREN LANGE, AND REED SOLOMON Abstract. We show that if H is an effectivey competey decomposabe computabe torsionfree abeian group,
More informationFederal Reserve Bank of New York Staff Reports
Federa Reserve Bank of New York Staff Reports ESOP Fabes: The Ipact of Epoyee Stock Ownership Pans on Labor Disputes Peter Craton Haid Mehran Joseph Tracy Staff Report no. 347 Septeber 2008 This paper
More informationExample: Suppose that we deposit $1000 in a bank account offering 3% interest, compounded monthly. How will our money grow?
Finance 111 Finance We have to work with oney every day. While balancing your checkbook or calculating your onthly expenditures on espresso requires only arithetic, when we start saving, planning for retireent,
More informationRisk Margin for a NonLife Insurance RunOff
Risk Margin for a NonLife Insurance RunOff Mario V. Wüthrich, Pau Embrechts, Andreas Tsanakas August 15, 2011 Abstract For sovency purposes insurance companies need to cacuate socaed bestestimate reserves
More informationProblem Set 2: Solutions ECON 301: Intermediate Microeconomics Prof. Marek Weretka. Problem 1 (Marginal Rate of Substitution)
Proble Set 2: Solutions ECON 30: Interediate Microeconoics Prof. Marek Weretka Proble (Marginal Rate of Substitution) (a) For the third colun, recall that by definition MRS(x, x 2 ) = ( ) U x ( U ). x
More informationFigure 1. A Simple Centrifugal Speed Governor.
ENGINE SPEED CONTROL Peter Westead and Mark Readman, contro systems principes.co.uk ABSTRACT: This is one of a series of white papers on systems modeing, anaysis and contro, prepared by Contro Systems
More informationManifold Technology.  made in Germany
Manifod Technoogy.  made in Germany I EVERYTHING UNDER CONTROL. Manifod Technoogy BEULCO heating and cooing manifods made of highquaity brass ensure
More informationCARBON FOOTPRINT REPORT
CARBON FOOTPRINT REPORT 07.03.2012 ATEA ASA REPORT: CARBON FOOTPRINT ANALYSIS 2011 PROVIDED BY: CO2FOCUS Content Introduction... 2 Method... 2 Resuts... 4 Atea Group... 4 Atea Norway... 5 Atea Denmark...
More informationASYMPTOTIC DIRECTION FOR RANDOM WALKS IN RANDOM ENVIRONMENTS arxiv:math/0512388v2 [math.pr] 11 Dec 2007
ASYMPTOTIC DIRECTION FOR RANDOM WALKS IN RANDOM ENVIRONMENTS arxiv:math/0512388v2 [math.pr] 11 Dec 2007 FRANÇOIS SIMENHAUS Université Paris 7, Mathématiques, case 7012, 2, pace Jussieu, 75251 Paris, France
More information2.2 Magic with complex exponentials
2.2. MAGIC WITH COMPLEX EXPONENTIALS 97 2.2 Magic with complex exponentials We don t really know what aspects of complex variables you learned about in high school, so the goal here is to start more or
More informationChapter 22: The Electric Field. Read Chapter 22 Do Ch. 22 Questions 3, 5, 7, 9 Do Ch. 22 Problems 5, 19, 24
Chapter : The Electric Field Read Chapter Do Ch. Questions 3, 5, 7, 9 Do Ch. Problems 5, 19, 4 The Electric Field Replaces actionatadistance Instead of Q 1 exerting a force directly on Q at a distance,
More informationMarch 14, 1967 _ A. GABOR ETAL I 3,309,597 MOTOR ACCELERATION CONTROL SYSTEM. l Filed April 20, 1964. T1.l _. ,57m/Az.
March 14, 1967 _ A. GABOR ETAL I 3,309,597 MOTOR ACCELERATION CONTROL SYSTEM Fied Apri 20, 1964 T1. _,57m/Az. SVGA/,QL ATTORNEY United States Patent O 1 3,309,597 MOTOR ACCELERATION CONTRGL SYSTEM Andrew
More informationKinetic Molecular Theory of Ideal Gases
ecture /. Kinetic olecular Theory of Ideal Gases ast ecture. IG is a purely epirical law  solely the consequence of eperiental obserations Eplains the behaior of gases oer a liited range of conditions.
More informationIn some states, however, the couple must live apart for a period of months or years before they can obtain a no fault divorce.
What is a "no faut" divorce? "No faut" divorce describes any divorce where the spouse asking for a divorce does not have to prove that the other spouse did something wrong. A states aow no faut divorces.
More informationChapter 1 Structural Mechanics
Chapter Structura echanics Introduction There are many different types of structures a around us. Each structure has a specific purpose or function. Some structures are simpe, whie others are compex; however
More informationData Set Generation for Rectangular Placement Problems
Data Set Generation for Rectangular Placeent Probles Christine L. Valenzuela (Muford) Pearl Y. Wang School of Coputer Science & Inforatics Departent of Coputer Science MS 4A5 Cardiff University George
More informationBudgeting Loans from the Social Fund
Budgeting Loans from the Socia Fund tes sheet Pease read these notes carefuy. They expain the circumstances when a budgeting oan can be paid. Budgeting Loans You may be abe to get a Budgeting Loan if:
More informationMODELING OF WHEELSOIL CONTACT FOR THE EDRES MOBILE ROBOT SIMULATOR
MODELING OF WHEELSOIL CONTACT FOR THE EDRES MOBILE ROBOT SIMULATOR D. LhommeDesages 1, C. Grand 1, and M. Maurette 2 1 ISIR, 18 Route du Panorama BP61 92265 FontenayAuxRoses, France, {homme,grand}@isir.fr
More informationLeakage detection in water pipe networks using a Bayesian probabilistic framework
Probabiistic Engineering Mechanics 18 (2003) 315 327 www.esevier.com/ocate/probengmech Leakage detection in water pipe networks using a Bayesian probabiistic framework Z. Pouakis, D. Vaougeorgis, C. Papadimitriou*
More informationA Gas Law And Absolute Zero
A Gas Law And Absolute Zero Equipent safety goggles, DataStudio, gas bulb with pressure gauge, 10 C to +110 C theroeter, 100 C to +50 C theroeter. Caution This experient deals with aterials that are very
More informationPROPAGATION OF SURGE WAVES ON NONHOMOGENEOUS TRANSMISSION LINES INDUCED BY LIGHTNING STROKE
Advances in Eectrica and Eectronic Engineering 98 PROPAGATION OF SRGE WAVES ON NONHOMOGENEOS TRANSMISSION LINES INDCED BY LIGHTNING STROKE Z. Benešová, V. Kotan WB Pisen, Facuty of Eectrica Engineering,
More informationWeek 3: Consumer and Firm Behaviour: The WorkLeisure Decision and Profit Maximization
AROEOOIS 2006 Week 3: onsumer and Firm Behaviour: The WorkLeisure Decision and Profit aximization Questions for Review 1. How are a consumer s preferences over goods represented? By utiity functions:
More informationThe definition of insanity is doing the same thing over and over again and expecting different results
insurance services Sma Business Insurance a market opportunity being missed Einstein may not have known much about insurance, but if you appy his definition to the way existing brands are deveoping their
More informationGREEN: An Active Queue Management Algorithm for a Self Managed Internet
: An Active Queue Management Agorithm for a Sef Managed Internet Bartek Wydrowski and Moshe Zukerman ARC Specia Research Centre for UtraBroadband Information Networks, EEE Department, The University of
More informationNordic Ecolabelling of Copy and printing paper  supplementary module
rdic Ecoabeing of Copy and printing paper  suppementary modue Version 4.1 22 June 2011 30 June 2016 rdic Ecoabeing Content What is rdic Ecoabeed copy and printing paper? 3 Why choose the rdic Ecoabe?
More informationOptimum Design of Drip Irrigation System using Microtubes as Emitters
Optiu Design of Drip Irrigation Syste using Microtubes as Eitters A thesis subitte in fufient of the requireents for the egree of Master of Engineering Air Keshtgar B. Eng (Irrigation an Drainage Engineering)
More informationFundamental Theorems of Vector Calculus
Fundamental Theorems of Vector Calculus We have studied the techniques for evaluating integrals over curves and surfaces. In the case of integrating over an interval on the real line, we were able to use
More informationZZIP250z. Innowacje w technice. Innovations in Technology
MODULE DESCRIPTION Modue code ZZIP250z Modue name Innowacje w technice Modue name in Engish Innovations in Technoogy Vaid from academic year 2016/2017 A. MODULE PLACEMENT IN THE SYLLABUS Subject Leve
More informationComparison of Traditional and OpenAccess Appointment Scheduling for Exponentially Distributed Service Time
Journa of Heathcare Engineering Vo. 6 No. 3 Page 34 376 34 Comparison of Traditiona and OpenAccess Appointment Scheduing for Exponentiay Distributed Service Chongjun Yan, PhD; Jiafu Tang *, PhD; Bowen
More information