Salmon: Lectures on partial differential equations. Consider the general linear, secondorder PDE in the form. ,x 2


 Shana Day
 4 years ago
 Views:
Transcription
1 Salmon: Lectures on partal dfferental equatons 5. Classfcaton of secondorder equatons There are general methods for classfyng hgherorder partal dfferental equatons. One s very general (applyng even to some nonlnear equatons), and seems to have been motvated by the success of the theory of frstorder PDEs. In ths method one rewrtes the hgher order PDE as a system of frstorder PDEs and attempts to generalze the method of characterstcs to that system. Ths turns out to be possble only for a restrcted (but mportant) class of PDEs called hyperbolc. The other classfcaton method apples only to lnear, secondorder equatons. Ths lecture covers ths second method; we postpone the frst method untl the last lecture. Consder the general lnear, secondorder PDE n the form (1) A θ θ + B x x + Cθ F( x) x where θ s a functon of the n varables x ( x 1,x,K, x n ). We assume that all the coeffcents A, B and C are constants. Wth no loss n generalty, we assume that A s symmetrc. We wll show that, because A s symmetrc, t s possble to transform to new coordnates n whch (1) takes the dagonal form () θ A x + θ B + Cθ F( x) x and each A takes the value +1, 1 or 0. For example, the general dmensonal equaton (3) A 11 + A 1 θ xy + A θ yy +l.o.d. F (where l.o.d. means lowerorder dervatves ) can always be transformed nto one of the followng forms: (4) +θ yy + l.o.d. F θ yy + l.o.d. F ±θ y + l.o.d. F plus varous permutatons of these, such as θ yy ± θ x + l.o.d. F. In (4) we wrte only the hghestdervatve terms n each varable, and we do not consder the case θ x +θ y +L because t s a frstorder equaton. It turns out that some of the most mportant propertes of such equatons (such as the approprate form of boundary condtons) depend only on the hghestdervatve terms. Thus there s some pont n consderng the cases (4) n ther purest forms : +θ yy F Posson s equaton 51
2 Salmon: Lectures on partal dfferental equatons (5) θ tt F wave equaton θ t F heat equaton (where the notaton hnts at the typcal physcal meanng). To prove our clam we must frst prove a theorem about symmetrc A. For future purposes, we shall prove more than s actually needed for ths lecture. Theorem. [The spectral decomposton theorem.] If the matrx A s Hermtan (meanng that A A * where * denotes the complex conugate), then all the egenvalues of A are real, and all the egenvectors are, or can be made, orthogonal. Note that real symmetrc A, lke that n (1), are a class of Hermtan matrces. Before provng ths theorem, we note that the problem of dagonalzng (1) s closely related to the problem of dagonalzng the quadratc form (6) A x + B x + Cθ. In both problems the strategy s the same: choose the new coordnates x to be the coeffcents n the expanson (7) x x e of x n the n egenvectors e of A. Then the frst term n (6) becomes (8) x T Ax e x x e T Ae, T A x e ( ) x ( ) x λ e e e T λ e, λ x, where λ s the egenvalue correspondng to egenvector e. Note that we use the supposed orthogonalty of the egenvectors n ways: frst, n the assumpton (7) that any x can be expanded n these egenvectors, and, second, n the fnal step of (8). By an addtonal rescalng of x, we may reduce (8) to smplest form, x. The transformaton of (1) to the dagonal form () proceeds n the same manner. But frst we prove the theorem. Sketch of the proof. Recall that the column vector e s an egenvector of the n n matrx A f Aeλe where λ s the correspondng egenvalue, a generally complex number. Thus the egenvalues of A correspond to the roots of det (A λi) 0. Let λ be the egenvalue of A correspondng to egenvector e : (9) Ae λ e. Let A + A T * (transpose conugate). Thus A A + f A s Hermtan. 5
3 Salmon: Lectures on partal dfferental equatons To prove that λ s real, take the transpose conugate of (9) to get (10) e + A + λ * e +. (Note that e + s a row vector.) Now multply (9) on the left by e +, multply (10) on the rght by e, and subtract, usng the fact that A A +. The result s * (11) ( λ λ )e + e 0 whch proves that λ s real. To prove that the e are orthogonal, consder any egenvalues λ and λ. Multply (9) by e +, multply (1) e + A + λ e + (obtaned by takng the Hermtan conugate of (9)) by e on the rght and subtract, agan usng the fact that A s Hermtan, to obtan (13) ( λ λ )e + e 0. Thus f λ λ, then (14) e + e e e 0 and the two egenvectors are orthogonal. Ths proof does not apply to the case λ λ of degenerate egenvalues, whch we postpone untl later n ths lecture. Assume for the moment that the n egenvalues are dstnct. Then we have n orthogonal egenvectors. These can easly be normalzed; we henceforth assume that they are. The normalzed egenvectors can be used to dagonalze A. Consder the n n matrx (15) U ( e 1,e,K,e n ) whose column vectors are the orthonormal egenvectors of A. Then (16) U + U 1, that s, (17) U + U I whch smply restates the fact that the e are orthonormal. Matrces U wth the property (17) are called untary. [Note that snce A and λ are real, t s always possble to choose 53
4 Salmon: Lectures on partal dfferental equatons real egenvectors e. Then U s real and U + U T ]. The matrx U defned by (15) dagonalzes A n the sense that (18) U 1 AU U T AU U T dag( λ 1,λ,K, λ n ) ( ) λ 1 e 1,λ e,k,λ n e n Now we return to the general equaton (1) wth secondorder term (19) A θ x Defnng U as above, we ntroduce the new varables (0) x U T. Thus x s the proecton of x onto e ; (0) s equvalent to (7). Snce (1) x x T U we have () θ A A x x U T T k U m km θ θ A x k x km m km x k x m where, n matrx notaton, (3) A U T AU. But, by (18), A s the dagonal matrx wth the egenvalues of A on the dagonal. Thus (4) A θ x θ λ x. Ths s almost the smplest form. To obtan the smplest form, t s only necessary to redefne (5) x x / λ (provded that λ 0.) In overall summary, to transform the second dervatves n (1) to the canoncal form (), we use the transformaton (6) x U T x 54
5 Salmon: Lectures on partal dfferental equatons where U s the untary matrx defned by (15). The components of x are ust the ampltude of each egenvector s contrbuton to x. Snce untary transformatons have the property that they preserve x x (prove ths!), the transformaton (6) may be nterpretted as a rotaton of the coordnates n ndmensonal space. How does ths work n the case n? We let a A 11, b A 1, and c A, so (1) takes the form (7) a + bθ xy + cθ yy +L F The egenvalues of (8) A a b b c satsfy (9) ( a λ) ( c λ) b. Thus (30) λ a + c ± 1 ( a c) + 4b a + c ± 1 from whch we see that λ s ndeed real for any a, b, c. If (31) b ac 0 (the parabolc case) ( a + c) + 4( b ac) t follows from (30) that at least one of the egenvalues vanshes. Ths s the parabolc case and leads to +θ y +L or θ yy + θ x +L Smlarly, the egenvalues have the same sgn f (3) b ac < 0 (the ellptc case) Ths s the ellptc case and leads to +θ yy +L Fnally, the egenvalues have opposte sgns f (33) b ac > 0 (the hyperbolc case) Ths s the hyperbolc case and leads to θ yy +L We have already mentoned the connecton wth quadratc forms. For the form ax + bxy + cy, the same transformaton produces a dagonalzaton, and the resultng equaton descrbes a parabola, ellpse or hyperbola. Quadratc forms le at the heart of Remannan geometry. There one seeks coordnates n whch the dfferental arc length ds, whch has the general form 55
6 Salmon: Lectures on partal dfferental equatons (34) ds smplfes to A dx d (35) ds ( dx 1 ) ± ( dx ) L ± ( dx n ) The new coordnates are called Cartesan coordnates. Examples nclude Eucldean space ds ( dx) + ( dy) + ( dz) Lorentzan spacetme ds c ( dt) ( dx) ( dy) ( dz) If you have studed general relatvty (where A s always wrtten as g ) then you know that global Cartesan coordnates exst only f the curvature tensor assocated wth g vanshes. Ths bears upon the generalzaton of (1) to the case of nonconstant coeffcents A ( x). Whereas (1) can always be transformed locally to the form (), there s n general no globally vald transformaton. For the case n however, one can show that a global transformaton to the form () s possble provded that the A do not vary n such a way that the equaton changes type (e.g. from ellptc to hyperbolc). For a careful dscusson of ths, see Garabedan, Chapter. We must stll complete the proof of the spectral decomposton theorem by showng that orthonormal egenvectors may be found even n the case where some of the egenvalues are equal. Completon of the proof. Even f all the egenvalues are equal, we can fnd at least one egenvector e 1. Then we defne ( ) (36) U 1 e 1, e ˆ,K, ˆ e n where e ˆ,K, ˆ e n are n 1 other vectors (not necessarly egenvectors of A) whch are orthonormal to e 1 and to each other. (It s always possble to fnd such a set!) By the orthonormalty of ts columns, U 1 s untary. We use U 1 to transform A nto the form λ 1 0 L 0 0 (37) U 1 1 AU 1 M A 0 where A s an ( n 1) ( n 1) matrx. Then we do the same thng to A. That s, we fnd the (untary) U such that 56
7 Salmon: Lectures on partal dfferental equatons λ 1 0 L 0 0 λ 0 L 0 (38) U 1 ( U 1 1 AU 1 )U 0 M M A and keep gong. Each step corresponds to a rotaton n the remanng coordnates. At the end we have (39) U 1 AU dag( λ 1,λ,K, λ n ) where U U 1 U LU n s untary because the product of untary matrces s untary (prove ths!). Eqn (39) mples that the columns of U are the soughtfor orthonormal egenvectors. (Each U has rank n, hence so does U.) References. Mathews and Walker Chap 6, Zauderer Chap
Inner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fntedmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More information1 Matrix representations of canonical matrices
1 Matrx representatons of canoncal matrces 2d rotaton around the orgn: ( ) cos θ sn θ R 0 = sn θ cos θ 3d rotaton around the xaxs: R x = 1 0 0 0 cos θ sn θ 0 sn θ cos θ 3d rotaton around the yaxs:
More informationQuantum Mechanics for Scientists and Engineers. David Miller
Quantum Mechancs for Scentsts and Engneers Davd Mller Types of lnear operators Types of lnear operators Blnear expanson of operators Blnear expanson of lnear operators We know that we can expand functons
More informationQuantum Mechanics I  Session 4
Quantum Mechancs I  Sesson 4 Aprl 3, 05 Contents Operators Change of Bass 4 3 Egenvectors and Egenvalues 5 3. Denton....................................... 5 3. Rotaton n D....................................
More information763622S ADVANCED QUANTUM MECHANICS Solution Set 1 Spring c n a n. c n 2 = 1.
7636S ADVANCED QUANTUM MECHANICS Soluton Set 1 Sprng 013 1 Warmup Show that the egenvalues of a Hermtan operator Â are real and that the egenkets correspondng to dfferent egenvalues are orthogonal (b)
More information2.3 Nilpotent endomorphisms
s a block dagonal matrx, wth A Mat dm U (C) In fact, we can assume that B = B 1 B k, wth B an ordered bass of U, and that A = [f U ] B, where f U : U U s the restrcton of f to U 40 23 Nlpotent endomorphsms
More informationLectures  Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures  Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More informationC/CS/Phy191 Problem Set 3 Solutions Out: Oct 1, 2008., where ( 00. ), so the overall state of the system is ) ( ( ( ( 00 ± 11 ), Φ ± = 1
C/CS/Phy9 Problem Set 3 Solutons Out: Oct, 8 Suppose you have two qubts n some arbtrary entangled state ψ You apply the teleportaton protocol to each of the qubts separately What s the resultng state obtaned
More informationLecture 6/7 (February 10/12, 2014) DIRAC EQUATION. The nonrelativistic Schrödinger equation was obtained by noting that the Hamiltonian 2
P470 Lecture 6/7 (February 10/1, 014) DIRAC EQUATION The nonrelatvstc Schrödnger equaton was obtaned by notng that the Hamltonan H = P (1) m can be transformed nto an operator form wth the substtutons
More informationComposite Hypotheses testing
Composte ypotheses testng In many hypothess testng problems there are many possble dstrbutons that can occur under each of the hypotheses. The output of the source s a set of parameters (ponts n a parameter
More information= = = (a) Use the MATLAB command rref to solve the system. (b) Let A be the coefficient matrix and B be the righthand side of the system.
Chapter Matlab Exercses Chapter Matlab Exercses. Consder the lnear system of Example n Secton.. x x x y z y y z (a) Use the MATLAB command rref to solve the system. (b) Let A be the coeffcent matrx and
More informationMathematical Preparations
1 Introducton Mathematcal Preparatons The theory of relatvty was developed to explan experments whch studed the propagaton of electromagnetc radaton n movng coordnate systems. Wthn expermental error the
More informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
More informationTHEOREMS OF QUANTUM MECHANICS
THEOREMS OF QUANTUM MECHANICS In order to develop methods to treat manyelectron systems (atoms & molecules), many of the theorems of quantum mechancs are useful. Useful Notaton The matrx element A mn
More informationTensor Analysis. For orthogonal curvilinear coordinates, ˆ ˆ (98) Expanding the derivative, we have, ˆ. h q. . h q h q
For orthogonal curvlnear coordnates, eˆ grad a a= ( aˆ ˆ e). h q (98) Expandng the dervatve, we have, eˆ aˆ ˆ e a= ˆ ˆ a h e + q q 1 aˆ ˆ ˆ a e = ee ˆˆ ˆ + e. h q h q Now expandng eˆ / q (some of the detals
More information9 Characteristic classes
THEODORE VORONOV DIFFERENTIAL GEOMETRY. Sprng 2009 [under constructon] 9 Characterstc classes 9.1 The frst Chern class of a lne bundle Consder a complex vector bundle E B of rank p. We shall construct
More informationCanonical transformations
Canoncal transformatons November 23, 2014 Recall that we have defned a symplectc transformaton to be any lnear transformaton M A B leavng the symplectc form nvarant, Ω AB M A CM B DΩ CD Coordnate transformatons,
More informationThe Feynman path integral
The Feynman path ntegral Aprl 3, 205 Hesenberg and Schrödnger pctures The Schrödnger wave functon places the tme dependence of a physcal system n the state, ψ, t, where the state s a vector n Hlbert space
More information8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 493 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces you have studed thus far n the text are real vector spaces because the scalars
More information1 Vectors over the complex numbers
Vectors for quantum mechancs 1 D. E. Soper 2 Unversty of Oregon 5 October 2011 I offer here some background for Chapter 1 of J. J. Sakura, Modern Quantum Mechancs. 1 Vectors over the complex numbers What
More informationThe Order Relation and Trace Inequalities for. Hermitian Operators
Internatonal Mathematcal Forum, Vol 3, 08, no, 50757 HIKARI Ltd, wwwmhkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence
More informationTransfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system
Transfer Functons Convenent representaton of a lnear, dynamc model. A transfer functon (TF) relates one nput and one output: x t X s y t system Y s The followng termnology s used: x y nput output forcng
More informationELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM
ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look
More informationBézier curves. Michael S. Floater. September 10, These notes provide an introduction to Bézier curves. i=0
Bézer curves Mchael S. Floater September 1, 215 These notes provde an ntroducton to Bézer curves. 1 Bernsten polynomals Recall that a real polynomal of a real varable x R, wth degree n, s a functon of
More informationPHYS 705: Classical Mechanics. Calculus of Variations II
1 PHYS 705: Classcal Mechancs Calculus of Varatons II 2 Calculus of Varatons: Generalzaton (no constrant yet) Suppose now that F depends on several dependent varables : We need to fnd such that has a statonary
More informationLECTURE 9 CANONICAL CORRELATION ANALYSIS
LECURE 9 CANONICAL CORRELAION ANALYSIS Introducton he concept of canoncal correlaton arses when we want to quantfy the assocatons between two sets of varables. For example, suppose that the frst set of
More informationWorkshop: Approximating energies and wave functions Quantum aspects of physical chemistry
Workshop: Approxmatng energes and wave functons Quantum aspects of physcal chemstry http://quantum.bu.edu/pltl/6/6.pdf Last updated Thursday, November 7, 25 7:9:55: Copyrght 25 Dan Dll (dan@bu.edu) Department
More informationThe Prncpal Component Transform The Prncpal Component Transform s also called KarhunenLoeve Transform (KLT, Hotellng Transform, oregenvector Transfor
Prncpal Component Transform Multvarate Random Sgnals A real tme sgnal x(t can be consdered as a random process and ts samples x m (m =0; ;N, 1 a random vector: The mean vector of X s X =[x0; ;x N,1] T
More informationAffine transformations and convexity
Affne transformatons and convexty The purpose of ths document s to prove some basc propertes of affne transformatons nvolvng convex sets. Here are a few onlne references for background nformaton: http://math.ucr.edu/
More information332600_08_1.qxp 4/17/08 11:29 AM Page 481
336_8_.qxp 4/7/8 :9 AM Page 48 8 Complex Vector Spaces 8. Complex Numbers 8. Conjugates and Dvson of Complex Numbers 8.3 Polar Form and DeMovre s Theorem 8.4 Complex Vector Spaces and Inner Products 8.5
More informationLinear Approximation with Regularization and Moving Least Squares
Lnear Approxmaton wth Regularzaton and Movng Least Squares Igor Grešovn May 007 Revson 4.6 (Revson : March 004). 5 4 3 0.5 3 3.5 4 Contents: Lnear Fttng...4. Weghted Least Squares n Functon Approxmaton...
More informationPHYS 705: Classical Mechanics. Canonical Transformation II
1 PHYS 705: Classcal Mechancs Canoncal Transformaton II Example: Harmonc Oscllator f ( x) x m 0 x U( x) x mx x LT U m Defne or L p p mx x x m mx x H px L px p m p x m m H p 1 x m p m 1 m H x p m x m m
More informationThe exponential map of GL(N)
The exponental map of GLN arxv:hepth/9604049v 9 Apr 996 Alexander Laufer Department of physcs Unversty of Konstanz P.O. 5560 M 678 78434 KONSTANZ Aprl 9, 996 Abstract A fnte expanson of the exponental
More informationLecture 3. Ax x i a i. i i
18.409 The Behavor of Algorthms n Practce 2/14/2 Lecturer: Dan Spelman Lecture 3 Scrbe: Arvnd Sankar 1 Largest sngular value In order to bound the condton number, we need an upper bound on the largest
More informationFrom BiotSavart Law to Divergence of B (1)
From BotSavart Law to Dvergence of B (1) Let s prove that BotSavart gves us B (r ) = 0 for an arbtrary current densty. Frst take the dvergence of both sdes of BotSavart. The dervatve s wth respect to
More informationAffine and Riemannian Connections
Affne and Remannan Connectons Semnar Remannan Geometry Summer Term 2015 Prof Dr Anna Wenhard and Dr GyeSeon Lee Jakob Ullmann Notaton: X(M) space of smooth vector felds on M D(M) space of smooth functons
More informationMEM 255 Introduction to Control Systems Review: Basics of Linear Algebra
MEM 255 Introducton to Control Systems Revew: Bascs of Lnear Algebra Harry G. Kwatny Department of Mechancal Engneerng & Mechancs Drexel Unversty Outlne Vectors Matrces MATLAB Advanced Topcs Vectors A
More information5 The Rational Canonical Form
5 The Ratonal Canoncal Form Here p s a monc rreducble factor of the mnmum polynomal m T and s not necessarly of degree one Let F p denote the feld constructed earler n the course, consstng of all matrces
More informationBOUNDEDNESS OF THE RIESZ TRANSFORM WITH MATRIX A 2 WEIGHTS
BOUNDEDNESS OF THE IESZ TANSFOM WITH MATIX A WEIGHTS Introducton Let L = L ( n, be the functon space wth norm (ˆ f L = f(x C dx d < For a d d matrx valued functon W : wth W (x postve semdefnte for all
More informationU.C. Berkeley CS294: Beyond WorstCase Analysis Luca Trevisan September 5, 2017
U.C. Berkeley CS94: Beyond WorstCase Analyss Handout 4s Luca Trevsan September 5, 07 Summary of Lecture 4 In whch we ntroduce semdefnte programmng and apply t to Max Cut. Semdefnte Programmng Recall that
More informationDifference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More informationThis model contains two bonds per unit cell (one along the xdirection and the other along y). So we can rewrite the Hamiltonian as:
1 Problem set #1 1.1. A oneband model on a square lattce Fg. 1 Consder a square lattce wth only nearestneghbor hoppngs (as shown n the fgure above): H t, j a a j (1.1) where,j stands for nearest neghbors
More informationBezier curves. Michael S. Floater. August 25, These notes provide an introduction to Bezier curves. i=0
Bezer curves Mchael S. Floater August 25, 211 These notes provde an ntroducton to Bezer curves. 1 Bernsten polynomals Recall that a real polynomal of a real varable x R, wth degree n, s a functon of the
More informationAsymptotics of the Solution of a Boundary Value. Problem for OneCharacteristic Differential. Equation Degenerating into a Parabolic Equation
Nonl. Analyss and Dfferental Equatons, ol., 4, no., 5  HIKARI Ltd, www.mhar.com http://dx.do.org/.988/nade.4.456 Asymptotcs of the Soluton of a Boundary alue Problem for OneCharacterstc Dfferental Equaton
More informationNOTES ON SIMPLIFICATION OF MATRICES
NOTES ON SIMPLIFICATION OF MATRICES JONATHAN LUK These notes dscuss how to smplfy an (n n) matrx In partcular, we expand on some of the materal from the textbook (wth some repetton) Part of the exposton
More informationWeek3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity
Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1s tme nterval. The velocty of the partcle
More informationRepresentation theory and quantum mechanics tutorial Representation theory and quantum conservation laws
Representaton theory and quantum mechancs tutoral Representaton theory and quantum conservaton laws Justn Campbell August 1, 2017 1 Generaltes on representaton theory 1.1 Let G GL m (R) be a real algebrac
More information(A and B must have the same dmensons to be able to add them together.) Addton s commutatve and assocatve, just lke regular addton. A matrx A multpled
CNS 185: A Bref Revew of Lnear Algebra An understandng of lnear algebra s crtcal as a steppngo pont for understandng neural networks. Ths handout ncludes basc dentons, then quckly progresses to elementary
More informationAdvanced Quantum Mechanics
Advanced Quantum Mechancs Rajdeep Sensarma! sensarma@theory.tfr.res.n ecture #9 QM of Relatvstc Partcles Recap of ast Class Scalar Felds and orentz nvarant actons Complex Scalar Feld and Charge conjugaton
More informationAPPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More informationGeorgia Tech PHYS 6124 Mathematical Methods of Physics I
Georga Tech PHYS 624 Mathematcal Methods of Physcs I Instructor: Predrag Cvtanovć Fall semester 202 Homework Set #7 due October 30 202 == show all your work for maxmum credt == put labels ttle legends
More informationCausal Diamonds. M. Aghili, L. Bombelli, B. Pilgrim
Causal Damonds M. Aghl, L. Bombell, B. Plgrm Introducton The correcton to volume of a causal nterval due to curvature of spacetme has been done by Myrhem [] and recently by Gbbons & Solodukhn [] and later
More informationFINITELYGENERATED MODULES OVER A PRINCIPAL IDEAL DOMAIN
FINITELYGENERTED MODULES OVER PRINCIPL IDEL DOMIN EMMNUEL KOWLSKI Throughout ths note, s a prncpal deal doman. We recall the classfcaton theorem: Theorem 1. Let M be a fntelygenerated module. (1) There
More informationn α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0
MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector
More informationA New Refinement of Jacobi Method for Solution of Linear System Equations AX=b
Int J Contemp Math Scences, Vol 3, 28, no 17, 819827 A New Refnement of Jacob Method for Soluton of Lnear System Equatons AX=b F Naem Dafchah Department of Mathematcs, Faculty of Scences Unversty of Gulan,
More informationMath 217 Fall 2013 Homework 2 Solutions
Math 17 Fall 013 Homework Solutons Due Thursday Sept. 6, 013 5pm Ths homework conssts of 6 problems of 5 ponts each. The total s 30. You need to fully justfy your answer prove that your functon ndeed has
More informationSolutions to Problem Set 6
Solutons to Problem Set 6 Problem 6. (Resdue theory) a) Problem 4.7.7 Boas. n ths problem we wll solve ths ntegral: x sn x x + 4x + 5 dx: To solve ths usng the resdue theorem, we study ths complex ntegral:
More informationHomework Notes Week 7
Homework Notes Week 7 Math 4 Sprng 4 #4 (a Complete the proof n example 5 that s an nner product (the Frobenus nner product on M n n (F In the example propertes (a and (d have already been verfed so we
More informationLecture Note 3. Eshelby s Inclusion II
ME340B Elastcty of Mcroscopc Structures Stanford Unversty Wnter 004 Lecture Note 3. Eshelby s Incluson II Chrs Wenberger and We Ca c All rghts reserved January 6, 004 Contents 1 Incluson energy n an nfnte
More informationPhysics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian1
P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the
More informationSTAT 309: MATHEMATICAL COMPUTATIONS I FALL 2018 LECTURE 16
STAT 39: MATHEMATICAL COMPUTATIONS I FALL 218 LECTURE 16 1 why teratve methods f we have a lnear system Ax = b where A s very, very large but s ether sparse or structured (eg, banded, Toepltz, banded plus
More informationSL n (F ) Equals its Own Derived Group
Internatonal Journal of Algebra, Vol. 2, 2008, no. 12, 585594 SL n (F ) Equals ts Own Derved Group Jorge Macel BMCCThe Cty Unversty of New York, CUNY 199 Chambers street, New York, NY 10007, USA macel@cms.nyu.edu
More informationSome Comments on Accelerating Convergence of Iterative Sequences Using Direct Inversion of the Iterative Subspace (DIIS)
Some Comments on Acceleratng Convergence of Iteratve Sequences Usng Drect Inverson of the Iteratve Subspace (DIIS) C. Davd Sherrll School of Chemstry and Bochemstry Georga Insttute of Technology May 1998
More informationLinear, affine, and convex sets and hulls In the sequel, unless otherwise specified, X will denote a real vector space.
Lnear, affne, and convex sets and hulls In the sequel, unless otherwse specfed, X wll denote a real vector space. Lnes and segments. Gven two ponts x, y X, we defne xy = {x + t(y x) : t R} = {(1 t)x +
More information6.854J / J Advanced Algorithms Fall 2008
MIT OpenCourseWare http://ocw.mt.edu 6.854J / 18.415J Advanced Algorthms Fall 2008 For nformaton about ctng these materals or our Terms of Use, vst: http://ocw.mt.edu/terms. 18.415/6.854 Advanced Algorthms
More informationThe Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction
ECONOMICS 5*  NOTE (Summary) ECON 5*  NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also
More informationLINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity
LINEAR REGRESSION ANALYSIS MODULE IX Lecture  30 Multcollnearty Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur 2 Remedes for multcollnearty Varous technques have
More informationAn Introduction to Morita Theory
An Introducton to Morta Theory Matt Booth October 2015 Nov. 2017: made a few revsons. Thanks to Nng Shan for catchng a typo. My man reference for these notes was Chapter II of Bass s book Algebrac KTheory
More informationAssortment Optimization under MNL
Assortment Optmzaton under MNL Haotan Song Aprl 30, 2017 1 Introducton The assortment optmzaton problem ams to fnd the revenuemaxmzng assortment of products to offer when the prces of products are fxed.
More informationON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECONDORDER HYPERBOLIC EQUATION
Advanced Mathematcal Models & Applcatons Vol.3, No.3, 2018, pp.215222 ON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECONDORDER HYPERBOLIC EUATION
More information2. Differentiable Manifolds and Tensors
. Dfferentable Manfolds and Tensors.1. Defnton of a Manfold.. The Sphere as a Manfold.3. Other Examples of Manfolds.4. Global Consderatons.5. Curves.6. Functons on M.7. Vectors and Vector Felds.8. Bass
More informationSolutions Homework 4 March 5, 2018
1 Solutons Homework 4 March 5, 018 Soluton to Exercse 5.1.8: Let a IR be a translaton and c > 0 be a rescalng. ˆb1 (cx + a) cx n + a (cx 1 + a) c x n x 1 cˆb 1 (x), whch shows ˆb 1 s locaton nvarant and
More informationProjective change between two Special (α, β) Finsler Metrics
Internatonal Journal of Trend n Research and Development, Volume 2(6), ISSN 23949333 www.jtrd.com Projectve change between two Specal (, β) Fnsler Metrcs Gayathr.K 1 and Narasmhamurthy.S.K 2 1 Assstant
More informationModule 9. Lecture 6. Duality in Assignment Problems
Module 9 1 Lecture 6 Dualty n Assgnment Problems In ths lecture we attempt to answer few other mportant questons posed n earler lecture for (AP) and see how some of them can be explaned through the concept
More information14 The Postulates of Quantum mechanics
14 The Postulates of Quantum mechancs Postulate 1: The state of a system s descrbed completely n terms of a state vector Ψ(r, t), whch s quadratcally ntegrable. Postulate 2: To every physcally observable
More information12. The HamiltonJacobi Equation Michael Fowler
1. The HamltonJacob Equaton Mchael Fowler Back to Confguraton Space We ve establshed that the acton, regarded as a functon of ts coordnate endponts and tme, satsfes ( ) ( ) S q, t / t+ H qpt,, = 0, and
More informationHowever, since P is a symmetric idempotent matrix, of P are either 0 or 1 [Eigenvalues
Fall 007 Soluton to Mdterm Examnaton STAT 7 Dr. Goel. [0 ponts] For the general lnear model = X + ε, wth uncorrelated errors havng mean zero and varance σ, suppose that the desgn matrx X s not necessarly
More informationOn the symmetric character of the thermal conductivity tensor
On the symmetrc character of the thermal conductvty tensor Al R. Hadjesfandar Department of Mechancal and Aerospace Engneerng Unversty at Buffalo, State Unversty of New York Buffalo, NY 146 USA ah@buffalo.edu
More informationProgramming Project 1: Molecular Geometry and Rotational Constants
Programmng Project 1: Molecular Geometry and Rotatonal Constants Center for Computatonal Chemstry Unversty of Georga Athens, Georga 30602 Summer 2012 1 Introducton Ths programmng project s desgned to provde
More informationDensity matrix. c α (t)φ α (q)
Densty matrx Note: ths s supplementary materal. I strongly recommend that you read t for your own nterest. I beleve t wll help wth understandng the quantum ensembles, but t s not necessary to know t n
More informationLecture 10 Support Vector Machines II
Lecture 10 Support Vector Machnes II 22 February 2016 Taylor B. Arnold Yale Statstcs STAT 365/665 1/28 Notes: Problem 3 s posted and due ths upcomng Frday There was an early bug n the faketest data; fxed
More informationFormulas for the Determinant
page 224 224 CHAPTER 3 Determnants e t te t e 2t 38 A = e t 2te t e 2t e t te t 2e 2t 39 If 123 A = 345, 456 compute the matrx product A adj(a) What can you conclude about det(a)? For Problems 40 43, use
More informationρ some λ THE INVERSE POWER METHOD (or INVERSE ITERATION) , for , or (more usually) to
THE INVERSE POWER METHOD (or INVERSE ITERATION)  applcaton of the Power method to A some fxed constant ρ (whch s called a shft), x λ ρ If the egenpars of A are { ( λ, x ) } ( ), or (more usually) to,
More informationLinear Algebra and its Applications
Lnear Algebra and ts Applcatons 4 (00) 5 56 Contents lsts avalable at ScenceDrect Lnear Algebra and ts Applcatons journal homepage: wwwelsevercom/locate/laa Notes on Hlbert and Cauchy matrces Mroslav Fedler
More informationON MECHANICS WITH VARIABLE NONCOMMUTATIVITY
ON MECHANICS WITH VARIABLE NONCOMMUTATIVITY CIPRIAN ACATRINEI Natonal Insttute of Nuclear Physcs and Engneerng P.O. Box MG6, 07725Bucharest, Romana Emal: acatrne@theory.npne.ro. Receved March 6, 2008
More informationThe Second AntiMathima on Game Theory
The Second AntMathma on Game Theory Ath. Kehagas December 1 2006 1 Introducton In ths note we wll examne the noton of game equlbrum for three types of games 1. 2player 2acton zerosum games 2. 2player
More informationLecture 20: Noether s Theorem
Lecture 20: Noether s Theorem In our revew of Newtonan Mechancs, we were remnded that some quanttes (energy, lnear momentum, and angular momentum) are conserved That s, they are constant f no external
More informationConvexity preserving interpolation by splines of arbitrary degree
Computer Scence Journal of Moldova, vol.18, no.1(52), 2010 Convexty preservng nterpolaton by splnes of arbtrary degree Igor Verlan Abstract In the present paper an algorthm of C 2 nterpolaton of dscrete
More informationPerron Vectors of an Irreducible Nonnegative Interval Matrix
Perron Vectors of an Irreducble Nonnegatve Interval Matrx Jr Rohn August 4 2005 Abstract As s well known an rreducble nonnegatve matrx possesses a unquely determned Perron vector. As the man result of
More informatione  c o m p a n i o n
OPERATIONS RESEARCH http://dxdoorg/0287/opre007ec e  c o m p a n o n ONLY AVAILABLE IN ELECTRONIC FORM 202 INFORMS Electronc Companon Generalzed Quantty Competton for Multple Products and Loss of Effcency
More informationDIFFERENTIAL FORMS BRIAN OSSERMAN
DIFFERENTIAL FORMS BRIAN OSSERMAN Dfferentals are an mportant topc n algebrac geometry, allowng the use of some classcal geometrc arguments n the context of varetes over any feld. We wll use them to defne
More informationCSci 6974 and ECSE 6966 Math. Tech. for Vision, Graphics and Robotics Lecture 21, April 17, 2006 Estimating A Plane Homography
CSc 6974 and ECSE 6966 Math. Tech. for Vson, Graphcs and Robotcs Lecture 21, Aprl 17, 2006 Estmatng A Plane Homography Overvew We contnue wth a dscusson of the major ssues, usng estmaton of plane projectve
More informationMAE140  Linear Circuits  Fall 13 Midterm, October 31
Instructons ME140  Lnear Crcuts  Fall 13 Mdterm, October 31 () Ths exam s open book. You may use whatever wrtten materals you choose, ncludng your class notes and textbook. You may use a hand calculator
More informationψ = i c i u i c i a i b i u i = i b 0 0 b 0 0
Quantum Mechancs, Advanced Course FMFN/FYSN7 Solutons Sheet Soluton. Lets denote the two operators by Â and ˆB, the set of egenstates by { u }, and the egenvalues as Â u = a u and ˆB u = b u. Snce the
More information2 More examples with details
Physcs 129b Lecture 3 Caltech, 01/15/19 2 More examples wth detals 2.3 The permutaton group n = 4 S 4 contans 4! = 24 elements. One s the dentty e. Sx of them are exchange of two objects (, j) ( to j and
More informationPHYS 705: Classical Mechanics. Newtonian Mechanics
1 PHYS 705: Classcal Mechancs Newtonan Mechancs Quck Revew of Newtonan Mechancs Basc Descrpton: An dealzed pont partcle or a system of pont partcles n an nertal reference frame [Rgd bodes (ch. 5 later)]
More informationKernel Methods and SVMs Extension
Kernel Methods and SVMs Extenson The purpose of ths document s to revew materal covered n Machne Learnng 1 Supervsed Learnng regardng support vector machnes (SVMs). Ths document also provdes a general
More informationAERODYNAMICS I LECTURE 6 AERODYNAMICS OF A WING FUNDAMENTALS OF THE LIFTINGLINE THEORY
LECTURE 6 AERODYNAMICS OF A WING FUNDAMENTALS OF THE LIFTINGLINE THEORY The BotSavart Law The velocty nduced by the sngular vortex lne wth the crculaton can be determned by means of the Bot Savart formula
More informationCSCE 790S Background Results
CSCE 790S Background Results Stephen A. Fenner September 8, 011 Abstract These results are background to the course CSCE 790S/CSCE 790B, Quantum Computaton and Informaton (Sprng 007 and Fall 011). Each
More information(δr i ) 2. V i. r i 2,
Cartesan coordnates r, = 1, 2,... D for Eucldean space. Dstance by Pythagoras: (δs 2 = (δr 2. Unt vectors ê, dsplacement r = r ê Felds are functons of poston, or of r or of {r }. Scalar felds Φ( r, Vector
More information