Consider a 1-D stationary state diffusion-type equation, which we will call the generalized diffusion equation from now on:
|
|
- Evangeline Richard
- 7 years ago
- Views:
Transcription
1 Chapter 1 Boundary value problems Numercal lnear algebra technques can be used for many physcal problems. In ths chapter we wll gve some examples of how these technques can be used to solve certan boundary value problems that occur n physcs. 1.1 A 1-D generalzed dffuson equaton Consder a 1-D statonary state dffuson-type equaton, whch we wll call the generalzed dffuson equaton from now on: d ( D(x dy(x + K(x y(x = q(x (1.1 where D(x > 0 s the dffuson coeffcent, K(x > 0 s some gven functon (ts meanng wll be elucdated later and q(x > 0 s a source functon. To solve ths problem, we have to specfy a fnte doman x left x x rght and gve the boundary condtons. For problems of ths knd (dffuson-lke there are two classcal types of boundary condtons that are usually mposed: Drchlet boundary condton: Here you smply specfy the value of the functon y(x at the boundary/boundares. Neumann boundary condton: Here you specfy the value of the dervatve of y(x at the boundary/boundares. but one can also specfy a mxture of the two. A second order ordnary dfferental equaton (ODE requres two boundary condtons. They could n prncple be specfed both at one sde (ether x = x left or x = x rght or one at each sde. The latter wll turn out to be the best (and usually physcally most reasonable choce. But n the lecture course we have so far not encountered methods of soluton that can deal wth boundary condtons mposed at both ends of the doman. We have so far only dealt wth problems that can be ntegrated n tme or space from one poston (e.g. x = x left to the other (e.g. x = x rght, whch requres us to mpose all boundary condtons at the startng pont. In ths chapter we deal wth methods to handle problems whch requre us to mpose boundary condtons at opposte sdes of the doman. Such problem are called boundary value problems. 1
2 1.1.1 The dscretzed verson of the equaton To solve ths equaton numercally, we have to defne a grd n x,.e. a dscrete set of samplng ponts x wth = 1 N. We often regard these ponts as the centers of cells. These cells have cell walls located between the x samplng ponts. Usually ther locaton are denoted wth x +1/2, where the + 1/2 s just meant to say between and + 1 and equvalently x 1/2, where the 1/2 s just meant to say between 1 and. We can then defne the cell walls to be located at: x +1/2 = 1 2 (x + x +1 x 1/2 = 1 2 (x 1 + x (1.2 In a computer we cannot use half ntegers as array ndces, so we wll have to defne for nstance the varables: xc[1] xc[n]: The cell center postons x. x[1] x[n+1]: The cell wall postons x 1/2. Note that for N cells we have N +1 walls! Now let s make a regularly spaced grd: x = x left + ( 1/2 (1.3 x 1/2 = x left + ( 1 (1.4 where = 1 N. We place our varables on the x postons. Frst dervatves of some functon f(x at the x ±1/2 postons: ( df = f +1 f = f +1 f (1.5 x +1 x +1/2 Ths expresson s accurate to second order n (you can verfy ths by usng a Taylor expanson of the true functon f(x around x = x. Lkewse we can express the frst dervatve of f(x at the x poston s: ( df = f +1 f 1 = f +1 f 1 x +1 x 1 2 (1.6 whch s also second order accurate. The second dervatve of f(x at the x = x poston can be found by takng the dervatve of Eq. (1.5: ( d 2 f 2 = 1 [ (df +1/2 ( ] df = f +1 2f + f 1 (1.7 1/2 2 In Eq. (1.1, however, we have to not just take the second dervatve, but the dervatve of some functon D(x tmes a dervatve: ( [ d D(x df ] [ = 1 ( ( ] df df D +1/2 D 1/2 +1/2 1/2 = 1 [ ( ( ] (1.8 f+1 f f f 1 D +1/2 D 1/2 2
3 Usng these defntons we can dscretze Eq. (1.1: [ ( ( ] 1 y y 1 y+1 y D 1/2 D +1/2 + K y = q (1.9 We now must fnd a way to solve ths equaton numercally. We wll frst look at the shootng method, but soon after we wll dscuss a better (more stable method based on a trdagonal matrx equaton. Note: Eq. (1.9 s already the fully defned dscretzed verson of Eq. (1.1, whch s second order accurate n. If you would nstead wsh to ntegrate Eq. (1.1 usng e.g. Runge-Kutta, then that Runge-Kutta procedure tself wll defne a dscretzaton. Here, however, we wll smply stck to the second order dscretzaton of Eq. ( Dscrete versons of Drchlet or Neumann boundary condtons Whle Eq. (1.9 defnes the dscrete verson of the ODE, the dscrete versons of the boundary condtons have to be specfed separately. For a Drchlet boundary condton (let s take the left boundary as an example we smply replace Eq. (1.9 for = 1 wth y 1 = y left (1.10 A Neumann boundary condton would replace Eq. (1.9 for = 1 wth and lkewse for the rght boundary. y 2 y 1 = y left ( The shootng method One way to solve Eq. (1.9 s to use the shootng method. Consder a case when we wsh to mpose Drchlet boundary condtons at both ends: y(x left = y 0 and y(x rght = y 1. We could n prncple make an ntal guess for the dervatve y left at x = x left, and then, together wth y(x left = y 0, ntegrate from x left to x rght. Usually we wll then obtan a value of y(x rght that s not equal to y 1. But we can now adjust our ntal guess of y left and redo the ntegraton, agan comparng the resultng y(x rght wth what we want t to be (y 1. By usng a clever scheme for adjustng the guess of y left to obtan values of y(x rght that are closer and closer to the desred value y 1 we eventually obtan the soluton we seek. Ths s the shootng method. It allows us to use any of the numercal ntegraton schemes we learned before (Euler, Numerov, Runge-Kutta etc to ntegrate the equaton from x left to x rght. All we need to do s develop a clever method for adjustng our guess for y left. One way to do ths s to create a functon f(ξ, wth ξ beng our guess for y left, where f(ξ = y(x rght y 1 for the gven value of ξ. We can then use any off-the-shelf root-fndng routne (such as the zbrent( routne of Numercal Recpes to fnd the value for ξ for whch f(ξ = 0. The functon y(x obtaned for that value of ξ s then the soluton. Ths method works well for the classc dffuson equaton: d ( D(x dy(x 3 = q(x (1.12
4 whch s the generalzed dffuson equaton wth K(x = 0. But f K(x > 0 then ths shootng method dverges hopelessly! You can verfy ths, for nstance, wth the smple example of x left = 10, x rght = 10, D(x = 1, q(x = 1 and K(x = 1 wth boundary condtons at x = x left of y(x left = 1 and y left = 1. Ths yelds y(x rght The shootng method may stll work, but t s far less relable when the equaton tends to dverge so quckly. Moreover, the precson of the choce of the value of y left requred to meet the boundary condton at x = x rght may be hgher than the float or double precson arthmatc can deal wth Why does the shootng method often fal? Consder the followng verson of the above generalzed dffuson equaton: 1 d 2 y(x = y(x q(x ( Ths s an equaton that often appears n the theory of radatve transfer: the dffuse movement of radaton through opaque meda, e.g. clouds n the Earth s atmosphere. To see why the shootng method tends to create dvergences, we can rewrte ths equaton nto the form of two coupled frst order dfferental equatons: In matrx form: d dy(x dz(x ( y = z = 3z(x (1.14 = q(x y(x (1.15 ( ( y z + ( 0 q (1.16 The matrx has egenvalues λ = ± 3. Evdently t has one negatve and one postve egenvalue. We already know that an equaton of the type dy/ = λy dverges f you ntegrate from left to rght f λ > 0. Snce we have at least one such dvergng egenvalue, ths must exponentally dverge! Ths s the cause of the problem. In fact, ntegratng n the opposte drecton won t help: there, also, there s a dvergng component. By the way (as a sde-note: If you try to do the same analyss (.e. splttng the second order equaton nto a matrx equaton of frst order to the classc dffuson equaton (Eq you wll encounter a defectve matrx: A matrx for whch no complete set of Egenvectors exsts. Exercse: verfy ths. Lesson: The above analyss of a system s dvergng or convergng egencomponents does not always apply; but often t does. In ths partcular case of a classc dffuson equaton the shoothng method works fne Castng the problem nto a matrx equaton Clearly, for problems for whch the shootng method fals, we must fnd an alternatve method: one n whch we can mpose one boundary condton on one sde and one on the other sde from the start. Ths can be done by constructng the matrx equaton correspondng to Eq. (1.9 and ts boundary condtons. We defne the N-dmensonal vectors y and q as y = (y 1, y 2,, y N 1, y N T q = (q 1, q 2,, q N 1, q N T (1.17 4
5 We now want to construct an N N matrx A such that the equaton A y = q (1.18 (where s the matrx-vector product represents Eq. (1.9 wth the correspondng boundary condtons. You can see that Eq. (1.9 for some contans y 1, y and y +1, but no other y-values. And the Drchlet or Neumann boundary condtons only nvolve y 1 and y 2 (for the left boundary and y N 1 and y N (for the rght boundary. In other words: the matrx A s a trdagonal matrx: b 1 c a 2 b 2 c a 3 b 3 c A = ( a N 2 b N 2 c N a N 1 b N 1 c N a N b N The matrx elements and rght-hand-sde elements can be found from Eq. (1.9. For 2 N 1 we have a b = D 1/2 2 (1.20 = D 1/2 + D +1/2 2 + K (1.21 c = D +1/2 2 (1.22 r = q (1.23 Now suppose we put a Neumann boundary condton on the left sde and a Drchlet boundary condton on the rght sde. We then get: b 1 = 1/ (1.24 c 1 = 1/ (1.25 r 1 = y left (1.26 a N = 0 (1.27 b N = 1 (1.28 r N = y rght (1.29 We can now use a trdagonal matrx equaton solver (see exercses to solve ths system of equatons. Ths mmedately gves us our soluton, because we have a lnear problem. 1.2 Heat conducton The generalzed dffuson equaton has many applcatons. One of them s heat transport. It s an every-day experence that heat tends to move from hot regons to cooler regons. One can express ths quanttatvely as F( x = D( x T( x (1.30 5
6 where x s the spatal poston vector and F( x s the heat flux and D( x the heat conductvty coeffcent (whch we allow to vary n space. Consder now a source functon q( x. The statonary-state conservaton of energy now reads: In 1-D ths equaton becomes F( x = [D( x T( x] = q( x (1.31 d ( D(x dt(x = q(x (1.32 We can now solve ths usng the above matrx equaton. In the exercse class we wll work out an explct example D Dffuson problem So what about a 2-D heat conducton problem? The equaton s d ( dt(x, y D(x, y d ( dt(x, y D(x, y = q(x, y (1.33 dy dy Let us, for smplcty, take D(x, y = D =constant, so that the equaton becomes ( d 2 T(x, y D + d2 T(x, y = q(x, y ( dy 2 How do we cast ths nto matrx form? Let us frst defne a grd n x and y: We have x wth 1 N and y j wth 1 j M. We construct both the x and y grds n the same way as we dd for the 1-D case (Secton We thus have an N M grd. We wsh to solve for T,j. In order to be able to make a matrx equaton for ths, we must defne a large vector z: z = (T 1,1,, T N,1, T 1,2,, T N,2,,, T 1,M 1,, T N,M 1, T 1,M,, T N,M T (1.35.e. a vector wth NM components. So for a grd of (.e. N = 20, M = 30 we have a vector of 600 components: z 1, z 600. We also create a vector r for the rght-hand-sde of the equaton: r = (q 1,1,, q N,1, q 1,2,, q N,2,,, q 1,M 1,, q N,M 1, q 1,M,, q N,M T (1.36 We then want to construct a matrx A such that A z = r (1.37 s the matrx-equaton correspondng to Eq. (1.34. It turns out that, lke the case of a trdagonal matrx, also here most of the elements of the matrx A are zero, except for some specal locatons. However, n ths case the matrx s not trdagonal anymore. It s trdagonal wth sdebands : A k,k = 4D/ 2 (1.38 A k+1,k = D/ 2 (1.39 A k 1,k = D/ 2 (1.40 A k+n,k = D/ 2 (1.41 A k N,k = D/ 2 (1.42 6
7 for k = + (j 1N wth 2 N 1 and 2 j M 1. The boundary condtons have to be specfed all around the doman,.e. at all y j for x = x 1, all y j for x = x N, all x for y j = y 1 and all x for y j = y M. In spte of the fact that also ths matrx A has mostly 0-elements (t s a so-called sparse matrx, the presence of the sde-bands now makes t less easy to solve the matrx equaton than for a trdagonal matrx. So we are forced back to the LU-decomposton method. But for very large N and M ths may become extremely computatonally expensve. Fortunately there s a whole class of methods that can solve such sparse matrx equatons relatvely effcently even for large NM. Most of these are based on teraton. Some famous methods are GMRES, BCG, BCGSTAB. We wll not go nto these methods here. It s just mportant to remember that for large mult-dmensonal problems, the correspondng matrx equatons usually are very large sparse matrx equatons that requre such specal methods such as GMRES, BCG or BCGSTAB to be solved wthn reasonable tme. Note that n a smlar manner one can do to 3-D. Ths adds two more sde bands, even further away from the dagonal. 1.4 The Cable Equaton The cable equaton s a smple model of sgnal propagaton through neuronal fbres (dendrtes/axons. The model s related to the telegraph equaton for sgnal propagaton through electrc wres. The model descrbes a neuronal fbre as a cylnder of radus a contanng ntracellular flud (cytosol. The surface of the cylnder conssts of two membranes, the phospholpd blayer, that act as a capactance that can store charge. We wrte the capactance per unt surface of the par of membranes as c m (unt: Farad/m 2. The blayer s a reasonably good nsulator, but not perfect. It has a certan resstance tmes unt surface r m (unt: Ohm m 2. Ths means that some of the current that flows through the fbre can leak out sdeways through the double membrane of the fbre. Let us call ths current-per-untsurface m (unt: Ampère/m 2, and defne t postve for current flowng out of the fbre. Let us defne the voltage at the nsde of the blayer as V whle the voltage outsde the blayer as 0. Ohm s law then states m = V r m (1.43 If V changes wth tme, then part of ths current s consumed n readjustng the charge n the capactor. Ths s called the dsplacement current, whch we wll wrte as d : d = c m V t (1.44 The remander of m s actual current from the cytosol to the outsde of the fbre,.e. ths s the leakng current l : l = m d (1.45 Now let us look at how a current moves along the cylnder. Defne the current n the cylnder as I(x, t, where x s the coordnate along the cylnder and t s tme, and I > 0 for current flowng toward postve x. Defne the resstance along the cable as R. If we frst assume that there s no loss through the membranes r m = and that the membranes 7
8 have zero capacty c m = 0, then we can look at a steady-state stuaton. The voltage V (x and current I must obey V (x = RI (1.46 x where I(x = I =constant. However, wth the membrane capactance and fnte resstance the equaton becomes more complex. Let us ntroduce the current through the membrane per unt length of cable I m, as well as I l and I d as I m = 2πa m I l = 2πa l I d = 2πa d (1.47 the capactance per unt length of cable C m as and the membrane resstance tmes unt length of cable R m as C m = 2πa c m (1.48 R m = r m 2πa (1.49 We can then wrte the contnuty equaton for I(x (gnorng the / t term as I(x x = I l = I d I m = C m V t V R m (1.50 and nsertng Eq. (1.46 we obtan 1 2 V R x = C V 2 m t + V (1.51 R m where we wrte V nstead of V (x, t just for clarty of the equaton. Ths equaton s called the cable equaton. We can make ths equaton dmensonless by ntroducng a length scale λ = R m /R and a tme scale τ = C m R m. Then the equaton becomes Statonary state solutons For a steady-state stuaton the cable equaton becomes λ 2 2 V x 2 = τ V t + V (1.52 λ 2 2 V x 2 = V (1.53 whch s, for the choce λ = 1/ 3 dentcal to the dffuson equaton for radatve transfer, Eq. (1.13 wth q(x = 0. So also for the cable equaton we expect that a shootng method approach wll fal, so we need to solve a matrx equaton nstead. Ths goes perfectly analogous to what we dd n Secton
9 1.4.2 Tme-dependent solutons A straghtforward dscretzaton of the cable equaton for tme-dependent Euler ntegraton reads τ V n+1 V n = λ 2V +1 n 2V n + V 1 n V n t 2 (1.54 where V n s the potental at poston and tme step n. Fndng V n+1 n ths manner can work, but only f t τ 2 /λ 2. For t τ 2 /λ 2 ths drect forward Euler ntegraton becomes numercally volently unstable! It means that one may need to do very many tme steps, meanng a computatonally expensve ntegraton. An often-used trck to speed up the calculaton s to rewrte the equaton n the followng way: τ V n+1 V n n+1 n+1 = λ 2V+1 2V + V 1 n+1 V n+1 t 2 (1.55.e. take as the rght-hand-sde the future varables nstead of the current ones. Puttng all future varables to the left and all current ones to the rght, we obtan τ V n+1 t n+1 λ 2V+1 n+1 2V + V n V n+1 2 = τ V n t (1.56 whch can be cast nto a matrx equaton and solved for V n+1. In other words, the matrx equaton gves us the values of V at the next tme step. Ths s called mplct ntegraton (or mplct dfferencng of the partal dfferental equaton. It turns out that wth ths method the ntegraton s numercally stable, even for relatvely large t. 1.5 Afterword In ths chapter we have dealt wth lnear boundary value problems. However, n practce, many boundary value problems are non-lnear. Nevertheless one can use the same methods as above, but now n an teratve scheme. For each teraton we wrte the vector y = y prev + δy, where y prev s the vector y from the prevous teraton. We then lnearze the equatons n δy and solve for δy. We wll not go nto detal on ths, because the prncples are dentcal to those of Newton-Raphson teraton for root-fndng. 9
8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by
6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng
More informationRecurrence. 1 Definitions and main statements
Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.
More informationv a 1 b 1 i, a 2 b 2 i,..., a n b n i.
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are
More informationwhere the coordinates are related to those in the old frame as follows.
Chapter 2 - Cartesan Vectors and Tensors: Ther Algebra Defnton of a vector Examples of vectors Scalar multplcaton Addton of vectors coplanar vectors Unt vectors A bass of non-coplanar vectors Scalar product
More informationWhat is Candidate Sampling
What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble
More information1 Example 1: Axis-aligned rectangles
COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture # 6 Scrbe: Aaron Schld February 21, 2013 Last class, we dscussed an analogue for Occam s Razor for nfnte hypothess spaces that, n conjuncton
More informationWe are now ready to answer the question: What are the possible cardinalities for finite fields?
Chapter 3 Fnte felds We have seen, n the prevous chapters, some examples of fnte felds. For example, the resdue class rng Z/pZ (when p s a prme) forms a feld wth p elements whch may be dentfed wth the
More information+ + + - - This circuit than can be reduced to a planar circuit
MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to
More informationSupport Vector Machines
Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.
More informationPERRON FROBENIUS THEOREM
PERRON FROBENIUS THEOREM R. CLARK ROBINSON Defnton. A n n matrx M wth real entres m, s called a stochastc matrx provded () all the entres m satsfy 0 m, () each of the columns sum to one, m = for all, ()
More informationLuby s Alg. for Maximal Independent Sets using Pairwise Independence
Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent
More informationbenefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).
REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or
More informationLoop Parallelization
- - Loop Parallelzaton C-52 Complaton steps: nested loops operatng on arrays, sequentell executon of teraton space DECLARE B[..,..+] FOR I :=.. FOR J :=.. I B[I,J] := B[I-,J]+B[I-,J-] ED FOR ED FOR analyze
More informationFaraday's Law of Induction
Introducton Faraday's Law o Inducton In ths lab, you wll study Faraday's Law o nducton usng a wand wth col whch swngs through a magnetc eld. You wll also examne converson o mechanc energy nto electrc energy
More informationLinear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits
Lnear Crcuts Analyss. Superposton, Theenn /Norton Equalent crcuts So far we hae explored tmendependent (resste) elements that are also lnear. A tmendependent elements s one for whch we can plot an / cure.
More informationBERNSTEIN POLYNOMIALS
On-Lne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful
More informationNMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING. Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582
NMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582 7. Root Dynamcs 7.2 Intro to Root Dynamcs We now look at the forces requred to cause moton of the root.e. dynamcs!!
More informationLaws of Electromagnetism
There are four laws of electromagnetsm: Laws of Electromagnetsm The law of Bot-Savart Ampere's law Force law Faraday's law magnetc feld generated by currents n wres the effect of a current on a loop of
More information1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP)
6.3 / -- Communcaton Networks II (Görg) SS20 -- www.comnets.un-bremen.de Communcaton Networks II Contents. Fundamentals of probablty theory 2. Emergence of communcaton traffc 3. Stochastc & Markovan Processes
More informationConversion between the vector and raster data structures using Fuzzy Geographical Entities
Converson between the vector and raster data structures usng Fuzzy Geographcal Enttes Cdála Fonte Department of Mathematcs Faculty of Scences and Technology Unversty of Combra, Apartado 38, 3 454 Combra,
More informationRotation Kinematics, Moment of Inertia, and Torque
Rotaton Knematcs, Moment of Inerta, and Torque Mathematcally, rotaton of a rgd body about a fxed axs s analogous to a lnear moton n one dmenson. Although the physcal quanttes nvolved n rotaton are qute
More informationLeast Squares Fitting of Data
Least Squares Fttng of Data Davd Eberly Geoetrc Tools, LLC http://www.geoetrctools.co/ Copyrght c 1998-2016. All Rghts Reserved. Created: July 15, 1999 Last Modfed: January 5, 2015 Contents 1 Lnear Fttng
More information1. Measuring association using correlation and regression
How to measure assocaton I: Correlaton. 1. Measurng assocaton usng correlaton and regresson We often would lke to know how one varable, such as a mother's weght, s related to another varable, such as a
More informationComputational Fluid Dynamics II
Computatonal Flud Dynamcs II Eercse 2 1. Gven s the PDE: u tt a 2 ou Formulate the CFL-condton for two possble eplct schemes. 2. The Euler equatons for 1-dmensonal, unsteady flows s dscretsed n the followng
More informationCalculation of Sampling Weights
Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a two-stage stratfed cluster desgn. 1 The frst stage conssted of a sample
More informationDescription of the Force Method Procedure. Indeterminate Analysis Force Method 1. Force Method con t. Force Method con t
Indeternate Analyss Force Method The force (flexblty) ethod expresses the relatonshps between dsplaceents and forces that exst n a structure. Prary objectve of the force ethod s to deterne the chosen set
More informationFisher Markets and Convex Programs
Fsher Markets and Convex Programs Nkhl R. Devanur 1 Introducton Convex programmng dualty s usually stated n ts most general form, wth convex objectve functons and convex constrants. (The book by Boyd and
More informationLecture 2: Single Layer Perceptrons Kevin Swingler
Lecture 2: Sngle Layer Perceptrons Kevn Sngler kms@cs.str.ac.uk Recap: McCulloch-Ptts Neuron Ths vastly smplfed model of real neurons s also knon as a Threshold Logc Unt: W 2 A Y 3 n W n. A set of synapses
More informationTime Domain simulation of PD Propagation in XLPE Cables Considering Frequency Dependent Parameters
Internatonal Journal of Smart Grd and Clean Energy Tme Doman smulaton of PD Propagaton n XLPE Cables Consderng Frequency Dependent Parameters We Zhang a, Jan He b, Ln Tan b, Xuejun Lv b, Hong-Je L a *
More informationLecture 3: Force of Interest, Real Interest Rate, Annuity
Lecture 3: Force of Interest, Real Interest Rate, Annuty Goals: Study contnuous compoundng and force of nterest Dscuss real nterest rate Learn annuty-mmedate, and ts present value Study annuty-due, and
More informationInstitute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic
Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange
More informationHedging Interest-Rate Risk with Duration
FIXED-INCOME SECURITIES Chapter 5 Hedgng Interest-Rate Rsk wth Duraton Outlne Prcng and Hedgng Prcng certan cash-flows Interest rate rsk Hedgng prncples Duraton-Based Hedgng Technques Defnton of duraton
More informationNumerical Methods 數 值 方 法 概 說. Daniel Lee. Nov. 1, 2006
Numercal Methods 數 值 方 法 概 說 Danel Lee Nov. 1, 2006 Outlnes Lnear system : drect, teratve Nonlnear system : Newton-lke Interpolatons : polys, splnes, trg polys Approxmatons (I) : orthogonal polys Approxmatons
More informationAn Alternative Way to Measure Private Equity Performance
An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate
More informationFinite difference method
grd ponts x = mesh sze = X NÜÆ Fnte dfference method Prncple: dervatves n the partal dfferental eqaton are approxmated by lnear combnatons of fncton vales at the grd ponts 1D: Ω = (0, X), (x ), = 0,1,...,
More informationDEFINING %COMPLETE IN MICROSOFT PROJECT
CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMI-SP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,
More informationHÜCKEL MOLECULAR ORBITAL THEORY
1 HÜCKEL MOLECULAR ORBITAL THEORY In general, the vast maorty polyatomc molecules can be thought of as consstng of a collecton of two electron bonds between pars of atoms. So the qualtatve pcture of σ
More informationLogistic Regression. Lecture 4: More classifiers and classes. Logistic regression. Adaboost. Optimization. Multiple class classification
Lecture 4: More classfers and classes C4B Machne Learnng Hlary 20 A. Zsserman Logstc regresson Loss functons revsted Adaboost Loss functons revsted Optmzaton Multple class classfcaton Logstc Regresson
More informationRing structure of splines on triangulations
www.oeaw.ac.at Rng structure of splnes on trangulatons N. Vllamzar RICAM-Report 2014-48 www.rcam.oeaw.ac.at RING STRUCTURE OF SPLINES ON TRIANGULATIONS NELLY VILLAMIZAR Introducton For a trangulated regon
More informationThe OC Curve of Attribute Acceptance Plans
The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4
More informationPoint cloud to point cloud rigid transformations. Minimizing Rigid Registration Errors
Pont cloud to pont cloud rgd transformatons Russell Taylor 600.445 1 600.445 Fall 000-014 Copyrght R. H. Taylor Mnmzng Rgd Regstraton Errors Typcally, gven a set of ponts {a } n one coordnate system and
More informationSIMPLE LINEAR CORRELATION
SIMPLE LINEAR CORRELATION Smple lnear correlaton s a measure of the degree to whch two varables vary together, or a measure of the ntensty of the assocaton between two varables. Correlaton often s abused.
More informationA frequency decomposition time domain model of broadband frequency-dependent absorption: Model II
A frequenc decomposton tme doman model of broadband frequenc-dependent absorpton: Model II W. Chen Smula Research Laborator, P. O. Box. 134, 135 Lsaker, Norwa (1 Aprl ) (Proect collaborators: A. Bounam,
More informationCalibration and Linear Regression Analysis: A Self-Guided Tutorial
Calbraton and Lnear Regresson Analyss: A Self-Guded Tutoral Part The Calbraton Curve, Correlaton Coeffcent and Confdence Lmts CHM314 Instrumental Analyss Department of Chemstry, Unversty of Toronto Dr.
More informationModule 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..
More information4 Cosmological Perturbation Theory
4 Cosmologcal Perturbaton Theory So far, we have treated the unverse as perfectly homogeneous. To understand the formaton and evoluton of large-scale structures, we have to ntroduce nhomogenetes. As long
More informationL10: Linear discriminants analysis
L0: Lnear dscrmnants analyss Lnear dscrmnant analyss, two classes Lnear dscrmnant analyss, C classes LDA vs. PCA Lmtatons of LDA Varants of LDA Other dmensonalty reducton methods CSCE 666 Pattern Analyss
More informationImmersed interface methods for moving interface problems
Numercal Algorthms 14 (1997) 69 93 69 Immersed nterface methods for movng nterface problems Zhln L Department of Mathematcs, Unversty of Calforna at Los Angeles, Los Angeles, CA 90095, USA E-mal: zhln@math.ucla.edu
More informationVision Mouse. Saurabh Sarkar a* University of Cincinnati, Cincinnati, USA ABSTRACT 1. INTRODUCTION
Vson Mouse Saurabh Sarkar a* a Unversty of Cncnnat, Cncnnat, USA ABSTRACT The report dscusses a vson based approach towards trackng of eyes and fngers. The report descrbes the process of locatng the possble
More informationLogistic Regression. Steve Kroon
Logstc Regresson Steve Kroon Course notes sectons: 24.3-24.4 Dsclamer: these notes do not explctly ndcate whether values are vectors or scalars, but expects the reader to dscern ths from the context. Scenaro
More informationSection C2: BJT Structure and Operational Modes
Secton 2: JT Structure and Operatonal Modes Recall that the semconductor dode s smply a pn juncton. Dependng on how the juncton s based, current may easly flow between the dode termnals (forward bas, v
More information21 Vectors: The Cross Product & Torque
21 Vectors: The Cross Product & Torque Do not use our left hand when applng ether the rght-hand rule for the cross product of two vectors dscussed n ths chapter or the rght-hand rule for somethng curl
More informationThe Mathematical Derivation of Least Squares
Pscholog 885 Prof. Federco The Mathematcal Dervaton of Least Squares Back when the powers that e forced ou to learn matr algera and calculus, I et ou all asked ourself the age-old queston: When the hell
More informationHow Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence
1 st Internatonal Symposum on Imprecse Probabltes and Ther Applcatons, Ghent, Belgum, 29 June 2 July 1999 How Sets of Coherent Probabltes May Serve as Models for Degrees of Incoherence Mar J. Schervsh
More informationTHE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES
The goal: to measure (determne) an unknown quantty x (the value of a RV X) Realsaton: n results: y 1, y 2,..., y j,..., y n, (the measured values of Y 1, Y 2,..., Y j,..., Y n ) every result s encumbered
More informationBrigid Mullany, Ph.D University of North Carolina, Charlotte
Evaluaton And Comparson Of The Dfferent Standards Used To Defne The Postonal Accuracy And Repeatablty Of Numercally Controlled Machnng Center Axes Brgd Mullany, Ph.D Unversty of North Carolna, Charlotte
More informationn + d + q = 24 and.05n +.1d +.25q = 2 { n + d + q = 24 (3) n + 2d + 5q = 40 (2)
MATH 16T Exam 1 : Part I (In-Class) Solutons 1. (0 pts) A pggy bank contans 4 cons, all of whch are nckels (5 ), dmes (10 ) or quarters (5 ). The pggy bank also contans a con of each denomnaton. The total
More informationImperial College London
F. Fang 1, C.C. Pan 1, I.M. Navon 2, M.D. Pggott 1, G.J. Gorman 1, P.A. Allson 1 and A.J.H. Goddard 1 1 Appled Modellng and Computaton Group Department of Earth Scence and Engneerng Imperal College London,
More informationPeriod and Deadline Selection for Schedulability in Real-Time Systems
Perod and Deadlne Selecton for Schedulablty n Real-Tme Systems Thdapat Chantem, Xaofeng Wang, M.D. Lemmon, and X. Sharon Hu Department of Computer Scence and Engneerng, Department of Electrcal Engneerng
More information"Research Note" APPLICATION OF CHARGE SIMULATION METHOD TO ELECTRIC FIELD CALCULATION IN THE POWER CABLES *
Iranan Journal of Scence & Technology, Transacton B, Engneerng, ol. 30, No. B6, 789-794 rnted n The Islamc Republc of Iran, 006 Shraz Unversty "Research Note" ALICATION OF CHARGE SIMULATION METHOD TO ELECTRIC
More informationImplementation of Deutsch's Algorithm Using Mathcad
Implementaton of Deutsch's Algorthm Usng Mathcad Frank Roux The followng s a Mathcad mplementaton of Davd Deutsch's quantum computer prototype as presented on pages - n "Machnes, Logc and Quantum Physcs"
More informationThe Noether Theorems: from Noether to Ševera
14th Internatonal Summer School n Global Analyss and Mathematcal Physcs Satellte Meetng of the XVI Internatonal Congress on Mathematcal Physcs *** Lectures of Yvette Kosmann-Schwarzbach Centre de Mathématques
More informationMean Molecular Weight
Mean Molecular Weght The thermodynamc relatons between P, ρ, and T, as well as the calculaton of stellar opacty requres knowledge of the system s mean molecular weght defned as the mass per unt mole of
More informationSection 5.4 Annuities, Present Value, and Amortization
Secton 5.4 Annutes, Present Value, and Amortzaton Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today
More informationExtending Probabilistic Dynamic Epistemic Logic
Extendng Probablstc Dynamc Epstemc Logc Joshua Sack May 29, 2008 Probablty Space Defnton A probablty space s a tuple (S, A, µ), where 1 S s a set called the sample space. 2 A P(S) s a σ-algebra: a set
More informationChapter 12 Inductors and AC Circuits
hapter Inductors and A rcuts awrence B. ees 6. You may make a sngle copy of ths document for personal use wthout wrtten permsson. Hstory oncepts from prevous physcs and math courses that you wll need for
More informationA hybrid global optimization algorithm based on parallel chaos optimization and outlook algorithm
Avalable onlne www.ocpr.com Journal of Chemcal and Pharmaceutcal Research, 2014, 6(7):1884-1889 Research Artcle ISSN : 0975-7384 CODEN(USA) : JCPRC5 A hybrd global optmzaton algorthm based on parallel
More informationSIMULATION OF THERMAL AND CHEMICAL RELAXATION IN A POST-DISCHARGE AIR CORONA REACTOR
XVIII Internatonal Conference on Gas Dscharges and Ther Applcatons (GD 2010) Grefswald - Germany SIMULATION OF THERMAL AND CHEMICAL RELAXATION IN A POST-DISCHARGE AIR CORONA REACTOR M. Mezane, J.P. Sarrette,
More informationA Penalty Method for American Options with Jump Diffusion Processes
A Penalty Method for Amercan Optons wth Jump Dffuson Processes Y. d Hallun, P.A. Forsyth, and G. Labahn March 9, 23 Abstract The far prce for an Amercan opton where the underlyng asset follows a jump dffuson
More informationApplication of Quasi Monte Carlo methods and Global Sensitivity Analysis in finance
Applcaton of Quas Monte Carlo methods and Global Senstvty Analyss n fnance Serge Kucherenko, Nlay Shah Imperal College London, UK skucherenko@mperalacuk Daro Czraky Barclays Captal DaroCzraky@barclayscaptalcom
More informationCompiling for Parallelism & Locality. Dependence Testing in General. Algorithms for Solving the Dependence Problem. Dependence Testing
Complng for Parallelsm & Localty Dependence Testng n General Assgnments Deadlne for proect 4 extended to Dec 1 Last tme Data dependences and loops Today Fnsh data dependence analyss for loops General code
More informationCHAPTER 14 MORE ABOUT REGRESSION
CHAPTER 14 MORE ABOUT REGRESSION We learned n Chapter 5 that often a straght lne descrbes the pattern of a relatonshp between two quanttatve varables. For nstance, n Example 5.1 we explored the relatonshp
More information5.74 Introductory Quantum Mechanics II
MIT OpenCourseWare http://ocw.mt.edu 5.74 Introductory Quantum Mechancs II Sprng 9 For nformaton about ctng these materals or our Terms of Use, vst: http://ocw.mt.edu/terms. 4-1 4.1. INTERACTION OF LIGHT
More informationChapter 6 Inductance, Capacitance, and Mutual Inductance
Chapter 6 Inductance Capactance and Mutual Inductance 6. The nductor 6. The capactor 6.3 Seres-parallel combnatons of nductance and capactance 6.4 Mutual nductance 6.5 Closer look at mutual nductance Oerew
More informationActuator forces in CFD: RANS and LES modeling in OpenFOAM
Home Search Collectons Journals About Contact us My IOPscence Actuator forces n CFD: RANS and LES modelng n OpenFOAM Ths content has been downloaded from IOPscence. Please scroll down to see the full text.
More informationGoals Rotational quantities as vectors. Math: Cross Product. Angular momentum
Physcs 106 Week 5 Torque and Angular Momentum as Vectors SJ 7thEd.: Chap 11.2 to 3 Rotatonal quanttes as vectors Cross product Torque expressed as a vector Angular momentum defned Angular momentum as a
More informationAnswer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy
4.02 Quz Solutons Fall 2004 Multple-Choce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multple-choce questons. For each queston, only one of the answers s correct.
More informationModelling of Hot Water Flooding
Unversty of Readng Modellng of Hot Water Floodng as an Enhanced Ol Recovery Method by Zenab Zargar August 013 Department of Mathematcs Submtted to the Department of Mathematcs, Unversty of Readng, n Partal
More informationHALL EFFECT SENSORS AND COMMUTATION
OEM770 5 Hall Effect ensors H P T E R 5 Hall Effect ensors The OEM770 works wth three-phase brushless motors equpped wth Hall effect sensors or equvalent feedback sgnals. In ths chapter we wll explan how
More informationA Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy S-curve Regression
Novel Methodology of Workng Captal Management for Large Publc Constructons by Usng Fuzzy S-curve Regresson Cheng-Wu Chen, Morrs H. L. Wang and Tng-Ya Hseh Department of Cvl Engneerng, Natonal Central Unversty,
More informationCausal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting
Causal, Explanatory Forecastng Assumes cause-and-effect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of
More informationChapter 4 ECONOMIC DISPATCH AND UNIT COMMITMENT
Chapter 4 ECOOMIC DISATCH AD UIT COMMITMET ITRODUCTIO A power system has several power plants. Each power plant has several generatng unts. At any pont of tme, the total load n the system s met by the
More informationMultiple stage amplifiers
Multple stage amplfers Ams: Examne a few common 2-transstor amplfers: -- Dfferental amplfers -- Cascode amplfers -- Darlngton pars -- current mrrors Introduce formal methods for exactly analysng multple
More information1. Math 210 Finite Mathematics
1. ath 210 Fnte athematcs Chapter 5.2 and 5.3 Annutes ortgages Amortzaton Professor Rchard Blecksmth Dept. of athematcal Scences Northern Illnos Unversty ath 210 Webste: http://math.nu.edu/courses/math210
More informationAddendum to: Importing Skill-Biased Technology
Addendum to: Importng Skll-Based Technology Arel Bursten UCLA and NBER Javer Cravno UCLA August 202 Jonathan Vogel Columba and NBER Abstract Ths Addendum derves the results dscussed n secton 3.3 of our
More informationThe Full-Wave Rectifier
9/3/2005 The Full Wae ectfer.doc /0 The Full-Wae ectfer Consder the followng juncton dode crcut: s (t) Power Lne s (t) 2 Note that we are usng a transformer n ths crcut. The job of ths transformer s to
More informationSimple Interest Loans (Section 5.1) :
Chapter 5 Fnance The frst part of ths revew wll explan the dfferent nterest and nvestment equatons you learned n secton 5.1 through 5.4 of your textbook and go through several examples. The second part
More informationCHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES
CHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES In ths chapter, we wll learn how to descrbe the relatonshp between two quanttatve varables. Remember (from Chapter 2) that the terms quanttatve varable
More informationFinite Math Chapter 10: Study Guide and Solution to Problems
Fnte Math Chapter 10: Study Gude and Soluton to Problems Basc Formulas and Concepts 10.1 Interest Basc Concepts Interest A fee a bank pays you for money you depost nto a savngs account. Prncpal P The amount
More informationLecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression.
Lecture 3: Annuty Goals: Learn contnuous annuty and perpetuty. Study annutes whose payments form a geometrc progresson or a arthmetc progresson. Dscuss yeld rates. Introduce Amortzaton Suggested Textbook
More information1 What is a conservation law?
MATHEMATICS 7302 (Analytcal Dynamcs) YEAR 2015 2016, TERM 2 HANDOUT #6: MOMENTUM, ANGULAR MOMENTUM, AND ENERGY; CONSERVATION LAWS In ths handout we wll develop the concepts of momentum, angular momentum,
More informationThursday, December 10, 2009 Noon - 1:50 pm Faraday 143
1. ath 210 Fnte athematcs Chapter 5.2 and 4.3 Annutes ortgages Amortzaton Professor Rchard Blecksmth Dept. of athematcal Scences Northern Illnos Unversty ath 210 Webste: http://math.nu.edu/courses/math210
More informationForecasting the Demand of Emergency Supplies: Based on the CBR Theory and BP Neural Network
700 Proceedngs of the 8th Internatonal Conference on Innovaton & Management Forecastng the Demand of Emergency Supples: Based on the CBR Theory and BP Neural Network Fu Deqang, Lu Yun, L Changbng School
More informationQuantization Effects in Digital Filters
Quantzaton Effects n Dgtal Flters Dstrbuton of Truncaton Errors In two's complement representaton an exact number would have nfntely many bts (n general). When we lmt the number of bts to some fnte value
More informationSPEE Recommended Evaluation Practice #6 Definition of Decline Curve Parameters Background:
SPEE Recommended Evaluaton Practce #6 efnton of eclne Curve Parameters Background: The producton hstores of ol and gas wells can be analyzed to estmate reserves and future ol and gas producton rates and
More informationTraffic State Estimation in the Traffic Management Center of Berlin
Traffc State Estmaton n the Traffc Management Center of Berln Authors: Peter Vortsch, PTV AG, Stumpfstrasse, D-763 Karlsruhe, Germany phone ++49/72/965/35, emal peter.vortsch@ptv.de Peter Möhl, PTV AG,
More informationHow To Understand The Results Of The German Meris Cloud And Water Vapour Product
Ttel: Project: Doc. No.: MERIS level 3 cloud and water vapour products MAPP MAPP-ATBD-ClWVL3 Issue: 1 Revson: 0 Date: 9.12.1998 Functon Name Organsaton Sgnature Date Author: Bennartz FUB Preusker FUB Schüller
More informationInterest Rate Fundamentals
Lecture Part II Interest Rate Fundamentals Topcs n Quanttatve Fnance: Inflaton Dervatves Instructor: Iraj Kan Fundamentals of Interest Rates In part II of ths lecture we wll consder fundamental concepts
More informationProduction. 2. Y is closed A set is closed if it contains its boundary. We need this for the solution existence in the profit maximization problem.
Producer Theory Producton ASSUMPTION 2.1 Propertes of the Producton Set The producton set Y satsfes the followng propertes 1. Y s non-empty If Y s empty, we have nothng to talk about 2. Y s closed A set
More informationSUMMARY. Topology optimization, buckling, eigenvalue, derivative, structural optimization 1. INTRODUCTION
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 0000; 00:1 20 Publshed onlne n Wley InterScence (www.nterscence.wley.com). A fast method for solvng bnary programmng
More information