Srednicki Chapter 51
|
|
- Victoria Chambers
- 5 years ago
- Views:
Transcription
1 Srednici Chapter 51 QFT Probems & Soutions A. George September 7, 13 Srednici Derive the fermion-oop correction to the scaar proagator by woring through equation 5., and show that it has an extra minus sign reative to the case of a scaar oop. Note: I don t ie this probem very much because it is poory expained. Given what we have done up to this point, it woud be reasonabe to expect the diagram in question to be the foowing: 7 7 In fact, however, we are not going to consider the ines eading to the sources as propagators; we are rather going to consider the ines themseves as externa sources. This maes sense when phrased in this way, but this seems to contradict the instructions to foow equation 5.. In fact we must change equation 5. to account for our new vertex. Our diagram is: This corresponds to a vertex ining two propagators with one externa φ-fied. We must then modify equation 5. to account for this: [ ( )( ) [ Z(η,η,J) = exp ig d δ 1 δ x φ(x) i exp i d xd yη(x)s(x y)η(y) δη(x) i δη(x) [ i exp d xd yj(x) (x y)j(y) Now we need to write these exponents as a Tayor Series, and eep ony those terms corresponding to our diagram above. In particuar, we can eep ony the owest-order term (1) 1
2 from the fina exponentia, since we have no scaar propagators (the externa scaar fieds do not count as propagators, as discussed above). We have two vertices and two Dirac propagators, so we eep ony the second-order terms from the expansion of the first two exponentias. This gives: Z(η,η,J) = (ig) ( d x φ(x) i δ δη(x) )( 1 i δ δη(x) ) ( d y φ(y) δ i δη(y) (i) d a d b η(a)s(a b)η(b) (i) d c d d η(c)s(c d)η(d) )( 1 i δ δη(y) Now we have to be a itte bit carefu. Before we coud just start mady differentiating stuff, but this time we have to worry about commutation. Let s organize, and pu the scaars to the front. Z(η,η,J) = g d x d y d a d b d c d d φ(x)φ(y) η(a)s(a b)η(b)η(c)s(c d)η(d) ( δ )( δ )( δ )( δ ) δη(x) δη(x) δη(y) δη(y) To start the differentiation process, et s move the ast two functiona derivatives past the η(a)s(a b). We get a negative sign from the anticommutation with the functiona derivative. S(a b) is made up of scaars and therefore commutes, see equation 5.. Then: Z(η,η,J) = g d x d y d a d b d c d d φ(x)φ(y) ( δ )( δ ) δη(x) δη(x) ( )( ) δ δ η(a)s(a b) η(b)η(c)s(c d)η(d) δη(y) δη(y) Now we can tae these two functiona derivatives: reca that functiona derivatives give deta functions, which we integrate over. Further, we coud have chosen to use these functiona derivatives on the other propagators; the resut is the same up to dummy indices, so we mutipy by four (two for η and two for η): Z(η,η,J) = g d x d y d a d d φ(x)φ(y) η(a)s(a y)s(y d)η(d) ( δ )( δ ) δη(x) δη(x) Now we can tae the remaining functiona derivatives: Z(η,η,J) = g d x d y φ(x)φ(y)s(x y)s(y x) Sure enough, this has a negative sign. In the case of a scaar oop, there woud be no anticommutation, and so there woud be no negative sign. Now notice that since we started with equation 5., we ve ept ony one term, but we ve never formay associated that term with the diagram above. Now we use equation 5.6 to do so. The extra minus sign means that we have to buid our minus sign into the vaue of )
3 the diagram, ie into the Feynman Rues. But from this point, the ony way forward (that we ve covered) is to use our Feynman rues to assess the vaues of the diagrams, and then to use the Lehmann-Käen form to turn the 1PI diagrams into a propagator. This is what Srednici did in the chapter, so we are done. The resut is equations 51.5 and Srednici 51.. Finish the computation of V Y (p,p), imposing the condition V Y (,) = igγ 5. Start with 51.7: iv Y (p,p) = g3 8π [ (1 ε 1 1 ( D og µ )) γ Ñ Z g gγ 5 D Now we need to determine D and Ñ given p = p = : D = (x 1 +x )m +x 3 M = (1 x 3 )m +x 3 M = 1 Ñ = m γ 5 dx 1 dx dx 3 δ(x 1 +x +x 3 1) Since there is no dependence on x 1 or x, we can simpify: 1 1 x3 (...) = dx 3 dx (...) Doing the inner integra gives: Putting a this together gives: [{ iv Y (,) = g3 1 8π ε 1 1 Reordering: [{ iv Y (,) = g3 1 8π ε 1 1 Soving these integras, we have: [{ iv Y (,) = g3 1 8π ε 1 ( M og µ (...) = 1 1 dx 3 (1 x 3 )(...) ( )} (1 x3 )m +x 3 M dx 3 (1 x 3 )og γ 5 + µ m γ 5 dx 3 (1 x 3 ) Z (1 x 3 )m +x 3 M g gγ 5 1 ( )} m +x 3 (M m ) dx 3 (1 x 3 )og γ 5 + µ m γ 5 dx 3 (1 x 3 ) Z m +x 3 (M m g gγ 5 ) )+ 3 + m (M m ) m M m (M m ) og 3 ( )} M γ 5 + m
4 m γ 5 M og(m/m) m γ 5 Z (M m ) (M m g gγ 5 ) Two of these terms cance, another two combine: [{ iv Y (,) = g3 1 8π ε + 1 ( M ) m og M m µ (M m ) og m γ 5 M og(m/m) Z (M m ) g gγ 5 Canceing another two terms: which is: iv Y (,) = g3 8π γ 5 Now we use 51.5: which gives: [ 1 ε + 1 og [ V Y (,) = iz g gγ 5 i g3 1 8π γ 5 ε + 1 og igγ 5 = iz g gγ 5 i g3 8π γ 5 ( M ) m M m µ (M m ) og [ 1 ε + 1 og [ Z g = 1+ g 1 8π ε + 1 og ( ) M µ ( ) M µ ( ) M µ m M m og m M m og m M m og This concurs with 51.8, a good sign. Next we have: [ (1 iv Y (p,p) = g3 8π ε 1 1 ( )) D og γ µ Simpifying: [ (1+ g 1 8π ε + 1 og iv Y (p,p) = gγ 5 + g3 8π ( ) M µ [ ( 3 1 m M m og ( D og µ ( )} M γ 5 + m ( ) M Z g gγ 5 m ( ) M m ( ) M m ( ) M m Ñ D ( )) M gγ 5 m )) γ Ñ D [ ( ) ( ) + g3 M m M 8π γ 5 og + µ M m og m Now notice that in the first integra, we can write this as: 1 df3 ogd + 1 df3 ogµ. This second term is then ogµ 1 (1 x)dx = ogµ. This then cances with the denominator of the first term on the second row. Thus: [ ( iv Y (p,p) = gγ 5 + g3 3 8π 1 ( )) m M ogd+og(m)+ M m og γ Ñ m D
5 which is: V Y (p,p) = igγ 5 + ig3 8π [ (3 + 1 ogd og(m) m M m og ( )) M γ 5 1 m Ñ D Note that const = const. Thus, we can combine two of these terms: [ (3 V Y (p,p) = igγ 5 + ig3 8π + 1 ( ) D og M m M m og ( )) M γ 5 1 m Ñ D which is the fina answer. Note: In Srednici s soutions, the first + sign on the right hand side is a negative sign. Someone is off by a sign, I suspect it is him. He aso has the γ 5 matrix distributed to the Ñ term, which is definitey incorrect. Srednici Consider maing φ a scaar rather than a pseudoscaar, so that the Yuawa interaction is L Y u = gφψψ. In this case, renormaizabiity requires us to add a term L φ 3 = 1 6 Z κκφ 3, as we as a term inear in φ to cance tadpoes. Find the one-oop contributions to the renormaizing Z factors for this theory in the MS scheme. Let s rewrite equation 51.5 (the other equations sti hod, though we must add, as mentioned, a Y φ term to the counterterm Lagrangian). L 1 = Z g gφψψ+ 1 6 Z κκφ Z λλφ These three terms represent the ony possibe interaction terms with mass dimensions. Note that we added a renormaization factor to the Yuawa interaction, as in the text. Now we have seven renormaization factors (three above, and four in equation Y is a rea number, as discussed on page 66; we coud sove for it, but the resut woud not be very interesting, and has nothing to do with the renormaizing Z factors we are ased to find). The rest of the probem is to figure out these seven Z-factors. Reca that we are using the MS renormaization scheme, so we need ony to cance a the divergent terms. The finite terms do not affect the Z factors. We start by trying to correct our scaar propagator at the one-oop eve. The diagrams are: 5
6 + + X Now we need to assess the vaue of each diagram. Diagram 1 We have: Then: (-1) because there is a Fermion oop ( 1 i) S(/ + /) S(/) from the oop, aong with an integra over. (ig) from the two vertices, since Z g = 1+O(g ). Π = g d (π) S(/ +/) S(/) We can tae a trace over these propagators, just be writing in index notation and reordering. Thus: Π = g d [ S(/ (π) Tr + /) S(/) Now we use equation Let s oo at the numerator: numer = Tr [ ( / / +m)( / +m) Dropping those terms with an odd number of gamma matrices we have: which is: Simpifying: which we define to be which we define in anaogy to equation numer = Tr [ // +// +m numer = [ ( ) ( )+m numer = [ m (+) numer = N As for the denominator, we use equation Putting a this together, we have: 1 iπ = g d N dx π (q +D) 6
7 where q = +x and D = x(1 x) +m. Now et s change the integration variabe q. iπ = g π 1 Now et s put in our N, in terms of q rather than : iπ = g π 1 The terms inear in q integrate to zero, so: iπ = g π N dx d q (51.3.1) (q +D) dx d q m q +xq x q +x (q +D) 1 dx d q m q x +x (q +D) Next et s mae g g µ ε/, shifting the mass dimensionaity off of g. Π = g π iπ = g π µε 1 Now we use and 51.19, and simpify: 1 { dx (m x +x ) dx d q m q x +x (q +D) ( ε og ( )) D µ ( +D ε + 1 og ( ))} D Now a we reay care about in the MS scheme is the divergent part, so we can write this as: 1 { } Π = g (m x +x )+D dx +(finite) π ε Now et s put the D in, and simpify: 1 { Π = g 6m +6x 6 x dx π ε Doing the integra: Diagram { } Π = g 6m + +(finite) π ε } +(finite) Nothing has changed from the text, so we can quote the answer (equation 51.1): Π = λ [ 1 (π) ε ( ) M og M µ µ Dropping the finite part: Π = λ [ 1 (π) ε +(finite) M 7
8 Diagram 3 This is the same diagram as in φ 3 theory. However, we cannot quote the answer from section 1, because that was done in six dimensions, but now we are woring in four dimensions. We can start from scratch, but everything we did up to equation (51.3.1) sti hods with the foowing modifications: There is no negative sign, since there is no fermion oop The vertex factor is iκ There is a symmetry factor of g g µ ε/ This gives: Using equation 51.18: This is: Diagram iπ = κ µ ε Π = κ 1 1 dx d q (π) 1 (q +D) dx 1 ( 16π ε og Π = κ 16π ε +(finite) ( )) D µ This is just a vertex, so we can tae the vaue straight from the chapter 5 Feynman Rues (or more recenty, equation 51.). Π = (Z φ 1) (Z M 1)M Now we sum a these sef-energies, and choose the Zs to cance the infinite parts. which impies: Simiary: (Z φ 1) g π ε = Z φ = 1 g π ε (Z M 1)M + κ 16π ε + λ 16π ε M g 6m π ε = Dividing through by M : (Z M 1)+ κ 16π εm + λ 16π ε g 6m π M ε = 8
9 Soving this: Z M = 1+ 1 [ κ 16π ε M +λ g m M Note: this is different from Srednici s soution, but it agrees with an independent soution from Andre Schneider at the University of Indiana. Moreover, Srenici s soution is necessariy incorrect because the mass φ 3 vertex forces κ to have a mass dimension of 1; Srednici s soution therefore adds a term with mass dimension = to terms of mass dimension =, which is obviousy wrong. Now for the fermion propagator. The diagrams are: + X Diagram Using the counterterm Lagrangian, we change the / to i/ and rub out the fieds, we get the vertex factor for the counterterm vertex (don t forget to mutipy by i, as with a vertex factors) as: iπ = i(z Ψ 1)/ i(z m 1)m Then: Diagram 1 Π = (Z Ψ 1)/ +(Z m 1)m Diagram one has two vertices, two propagators, and an integra over the oop; the resut is: ( ) 1 iπ = (ig) d i (π) S(/ +/) ( ) which is: iπ = g d / / +m (π) [( +) +m [ +m Using to combine these denominators: iπ = g d 1 dx / / +m (π) (q +D) where q = +x and D = x(1 x) +m. Now we change the integration variabe q. Thus: iπ = g d q 1 dx / /q +x/ +m (π) (q +D) 9
10 We can drop the terms in the numerator that are odd in q, since those integrate to zero. Further, we tae g gµ ε/, shifting the mass dimensionaity onto µ. Then: iπ = g µ ε d q (π) We use equation to do the q-integra: iπ = ig 16π 1 1 dx[m+/(x 1) dx m+/(x 1) (q +D) [ ε og ( ) D µ Doing the x integra: This is: [ iπ = ig m / [ 16π ε og ( ) D µ ( Π = g m / ) +(finite) 8π ε Combining these two diagrams, and requiring the m terms to cance, we have: Thus: and simiary with the / terms: 1 ε g 8π m (Z m 1)m = Z m = 1+ g 8π ε Thus: (Z Ψ 1)/ / 1 ε g Z Ψ = 1 g 16π ε 8π = Now we consider the correction to the φψψ vertex. The diagrams are: p p p + p+ p p p + p+ p p Assessing the vaues of these diagrams, we have: 1
11 Diagram 1 The ony contribution comes from the vertex, so: iπ = iz g g so: Diagram 3 Π = Z g g Note that there is ony one fermion propagator, so the numerator wi at most have terms of ony O(). There are three tota propagators, so the denominator wi be of O( 6 ). Thus, this integra goes as d 1 5. Even after four integras, this wi remain convergent assuming reasonabe boundary conditions. This therefore does not contribute to the Z-factors in this renormaiation scheme. Diagram Assessing the vaue of this diagram, we have: iπ = (ig) 3 ( 1 i ) 3 d (π) S(/p +/) S(/p+/) ( ) Writing these propagators: iπ = g 3 d (π) ( /p / +m)( /p / +m) ((p+) +m ))((p+) +m )( M ) Comparing this to equation 51.1, ony the definition of N has changed. Thus: iπ = g 3 d N df (π) 3 (q +D) 3 We coud cacuate N, but this is a ot of messy agebra. Reca that we ony care about the divergent part. By equation 1.7, the ony divergent part has a q in the numerator. We have: N / /q = q Aso, we shift the mass dimensionaity onto g: g gµ ε/. This gives: iπ = g 3 µ ε/ d q df (π) 3 (q +D) +(finite) 3 Now we use 1.7 to sove the integra (reca that the dimensionaity is ε).. We aso perform a Wic Rotation, which adds a factor of i. This gives: ( iπ = ig3 ε df (π) 3 Γ +(finite) ) 11
12 which is: Using equation 1.11, we have: Π = g3 8π ε +(finite) Π = g3 8π ε +(finite) Putting these together, we have: iv Y = iz g g g3 8π ε +(finite) Choosing Z g to absorb this divergence, we achieve, up to order g : Z g = 1+ g 8π ε Next we turn to the φ 3 vertex. The diagrams are: p p p p 1 p 3 +p +p p 1 p 1 p 3 p 3 p 1 p 1 p p 1 p 3 p 1 Diagram 1 The ony contribution comes from the vertex factor. Dropping the i, we have: Diagram Π = Z κ κ This is the same diagram as in φ 3 theory. Since we are now woring in four dimensions, we cannot simpy quote the resut from section 16. We can, however, use equation 16.6: Π = g d d q 1 (π) d (q +D) 3 1
13 Equation 16.7 gives: Now tae d = ε: Taing ε : Now we use 16.5: Π = g (π) Π = g Π = g Π = g Γ(3 d ) (π) d/d (3 d/) Γ ( 1+ ε D 1+ε/ (π) ) 1 (π) D 1 [ x3 x 1 1 +x 3 x +x 1 x 3 +m 1 Now we can set the externa momenta equa to zero, since they don t contribute to the divergent part (actuay we can set a of this equa to zero, since it is aready manifesty convergent, but et s stic with our usua set of trics): Dropping the convergent part, we have: Diagram 3 Π = g 3π m Π = (finite) Here we have to start from scratch. We have: ( ) 3 1 iπ = ( 1)(ig) 3 d i (π) S() S(p +) S( p 1 ) where the negative sign is necessary because we have a fermion oop. Expanding this: iπ = g 3 d ( / +m)( /p / +m)( / +/p 1 +m) (π) [( p 1 ) +m [(p +) +m [ +m Consider the numerator. We set the externa momenta to zero. The terms odd in integrate to zero, so we have: numer = 3m// +m 3 Using our usua tric (Feynman s formua), we can combine the denominator. Then: iπ = g 3 d (π) 1 1m +m 3 (q +D) 3 Switching our integration variabe to q: iπ = g 3 d q 1 1mq +(finite term of O(q )) df (π) 3 (q +D) 3 13
14 The constant term is of O(q 6 ), which wi ceary be convergent. Then: iπ = 1mg 3 d q 1 q df (π) 3 (q +D) 3 Now we tae a Wic Rotation and use equation 1.7: iπ = 1mg 3 i 1 Keeping the divergent terms ony, we have: which is: Doing the integra: Diagram iπ = 1mg 3 i 1 Π = mg3 16π ε Γ ( ε ) Γ(3) (π) Γ(3)Γ() D ε/ 1 (π) 1 [ ε +(finite) +(finite) Π = 3mg3 π ε +(finite) There are actuay three such diagrams: due to the two different vertices, the choice of which vertex goes on the eft is a substantia difference. Due to the oop, each diagram has a symmetry factor of two. Swapping the two externa propagators on the right does not yied a substantivey different diagram. Thus, we assess the vaues of these diagrams at: iπ = 3! (iκ)( iλ) ( ) 1 i Our usua manipuations give: iπ = 3κλ Using equation 51.18, we have: iπ = 3iκλ (π) d 1 (π) ( +M )((+p 1 ) +M ) d 1 1 (π) (q +D) [ ε og ( D µ ) which is: Π = 3κλ [ 1 16π ε +(finite) Now we combine these: Z κ κ+ 3mg3 π ε 3κλ 16π ε 1
15 We want Z κ to absorb these divergences ony. These give the O(ε 1 ) contributions to Z κ : Z κ = 1+ 1 ε Finay, we have the φ vertex. The diagrams are: ( ) 3λ 16π 3mg3 π κ p p 1 p p 3 p p p 1 +p +p 3 p p +p p 1 p 3 p +p +p 3 p +p p 3 In addition, there are the diagrams of figure Diagram 1 Ony the vertex factor contributes. Diagram 3 Π = Z λ λ Fortunatey, this is exacty the same diagram that was considered in the chapter. The factors of γ 5 in the numerator are different, but since we ony care about determining the divergent part which Srednici tes us is, none of this matters. We coud introduce a negative sign because we do not need to anticommute γ 5 across S, but since this happens twice, any effect wi be ost. We can therefore quote Srednici s resut, equation 51.5: Diagrams of figure 31.5 Π = 3g π ( 1 ε +(finite) Fortunatey for us, φ theory naturay uses four dimensions, and so we can quote equation 51.51: ( ) Π = 3λ 1 16π ε +(finite) Diagram Here we wi have terms of O( ) in the numerator since these are a scaars, but of O( 8 ) in the denominator. Even after four integras, this wi be convergent. Then: Π = (finite) ) 15
16 We combine these in the usua way: Π = Z λ λ 3g π ε + 3λ 16π ε Choosing the contributions to Z λ at O(ε 1 ) to cance these divergences, we have: Z λ = 1+ 1 ε ( ) 3λ 16π 3g λπ 16
Separation of Variables and a Spherical Shell with Surface Charge
Separation of Variabes and a Spherica She with Surface Charge In cass we worked out the eectrostatic potentia due to a spherica she of radius R with a surface charge density σθ = σ cos θ. This cacuation
More informationCS229 Lecture notes. Andrew Ng
CS229 Lecture notes Andrew Ng Part IX The EM agorithm In the previous set of notes, we taked about the EM agorithm as appied to fitting a mixture of Gaussians. In this set of notes, we give a broader view
More informationMATH 172: MOTIVATION FOR FOURIER SERIES: SEPARATION OF VARIABLES
MATH 172: MOTIVATION FOR FOURIER SERIES: SEPARATION OF VARIABLES Separation of variabes is a method to sove certain PDEs which have a warped product structure. First, on R n, a inear PDE of order m is
More informationMath 124B January 17, 2012
Math 124B January 17, 212 Viktor Grigoryan 3 Fu Fourier series We saw in previous ectures how the Dirichet and Neumann boundary conditions ead to respectivey sine and cosine Fourier series of the initia
More informationMath 124B January 31, 2012
Math 124B January 31, 212 Viktor Grigoryan 7 Inhomogeneous boundary vaue probems Having studied the theory of Fourier series, with which we successfuy soved boundary vaue probems for the homogeneous heat
More information$, (2.1) n="# #. (2.2)
Chapter. Eectrostatic II Notes: Most of the materia presented in this chapter is taken from Jackson, Chap.,, and 4, and Di Bartoo, Chap... Mathematica Considerations.. The Fourier series and the Fourier
More information4 1-D Boundary Value Problems Heat Equation
4 -D Boundary Vaue Probems Heat Equation The main purpose of this chapter is to study boundary vaue probems for the heat equation on a finite rod a x b. u t (x, t = ku xx (x, t, a < x < b, t > u(x, = ϕ(x
More informationSEMINAR 2. PENDULUMS. V = mgl cos θ. (2) L = T V = 1 2 ml2 θ2 + mgl cos θ, (3) d dt ml2 θ2 + mgl sin θ = 0, (4) θ + g l
Probem 7. Simpe Penduum SEMINAR. PENDULUMS A simpe penduum means a mass m suspended by a string weightess rigid rod of ength so that it can swing in a pane. The y-axis is directed down, x-axis is directed
More informationb n n=1 a n cos nx (3) n=1
Fourier Anaysis The Fourier series First some terminoogy: a function f(x) is periodic if f(x ) = f(x) for a x for some, if is the smaest such number, it is caed the period of f(x). It is even if f( x)
More informationJackson 4.10 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell
Jackson 4.10 Homework Probem Soution Dr. Christopher S. Baird University of Massachusetts Lowe PROBLEM: Two concentric conducting spheres of inner and outer radii a and b, respectivey, carry charges ±.
More informationDavid Eigen. MA112 Final Paper. May 10, 2002
David Eigen MA112 Fina Paper May 1, 22 The Schrodinger equation describes the position of an eectron as a wave. The wave function Ψ(t, x is interpreted as a probabiity density for the position of the eectron.
More informationProblem set 6 The Perron Frobenius theorem.
Probem set 6 The Perron Frobenius theorem. Math 22a4 Oct 2 204, Due Oct.28 In a future probem set I want to discuss some criteria which aow us to concude that that the ground state of a sef-adjoint operator
More informationMore Scattering: the Partial Wave Expansion
More Scattering: the Partia Wave Expansion Michae Fower /7/8 Pane Waves and Partia Waves We are considering the soution to Schrödinger s equation for scattering of an incoming pane wave in the z-direction
More informationLecture Notes 4: Fourier Series and PDE s
Lecture Notes 4: Fourier Series and PDE s 1. Periodic Functions A function fx defined on R is caed a periodic function if there exists a number T > such that fx + T = fx, x R. 1.1 The smaest number T for
More informationLecture Notes for Math 251: ODE and PDE. Lecture 32: 10.2 Fourier Series
Lecture Notes for Math 251: ODE and PDE. Lecture 32: 1.2 Fourier Series Shawn D. Ryan Spring 212 Last Time: We studied the heat equation and the method of Separation of Variabes. We then used Separation
More informationAndre Schneider P622
Andre Schneder P6 Probem Set #0 March, 00 Srednc 7. Suppose that we have a theory wth Negectng the hgher order terms, show that Souton Knowng β(α and γ m (α we can wrte β(α =b α O(α 3 (. γ m (α =c α O(α
More information( ) is just a function of x, with
II. MULTIVARIATE CALCULUS The first ecture covered functions where a singe input goes in, and a singe output comes out. Most economic appications aren t so simpe. In most cases, a number of variabes infuence
More informationLECTURE NOTES 8 THE TRACELESS SYMMETRIC TENSOR EXPANSION AND STANDARD SPHERICAL HARMONICS
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.07: Eectromagnetism II October, 202 Prof. Aan Guth LECTURE NOTES 8 THE TRACELESS SYMMETRIC TENSOR EXPANSION AND STANDARD SPHERICAL HARMONICS
More informationFourier Series. 10 (D3.9) Find the Cesàro sum of the series. 11 (D3.9) Let a and b be real numbers. Under what conditions does a series of the form
Exercises Fourier Anaysis MMG70, Autumn 007 The exercises are taken from: Version: Monday October, 007 DXY Section XY of H F Davis, Fourier Series and orthogona functions EÖ Some exercises from earier
More informationFOURIER SERIES ON ANY INTERVAL
FOURIER SERIES ON ANY INTERVAL Overview We have spent considerabe time earning how to compute Fourier series for functions that have a period of 2p on the interva (-p,p). We have aso seen how Fourier series
More information14 Separation of Variables Method
14 Separation of Variabes Method Consider, for exampe, the Dirichet probem u t = Du xx < x u(x, ) = f(x) < x < u(, t) = = u(, t) t > Let u(x, t) = T (t)φ(x); now substitute into the equation: dt
More informationStrauss PDEs 2e: Section Exercise 2 Page 1 of 12. For problem (1), complete the calculation of the series in case j(t) = 0 and h(t) = e t.
Strauss PDEs e: Section 5.6 - Exercise Page 1 of 1 Exercise For probem (1, compete the cacuation of the series in case j(t = and h(t = e t. Soution With j(t = and h(t = e t, probem (1 on page 147 becomes
More information2M2. Fourier Series Prof Bill Lionheart
M. Fourier Series Prof Bi Lionheart 1. The Fourier series of the periodic function f(x) with period has the form f(x) = a 0 + ( a n cos πnx + b n sin πnx ). Here the rea numbers a n, b n are caed the Fourier
More informationNotes: Most of the material presented in this chapter is taken from Jackson, Chap. 2, 3, and 4, and Di Bartolo, Chap. 2. 2π nx i a. ( ) = G n.
Chapter. Eectrostatic II Notes: Most of the materia presented in this chapter is taken from Jackson, Chap.,, and 4, and Di Bartoo, Chap... Mathematica Considerations.. The Fourier series and the Fourier
More information1D Heat Propagation Problems
Chapter 1 1D Heat Propagation Probems If the ambient space of the heat conduction has ony one dimension, the Fourier equation reduces to the foowing for an homogeneous body cρ T t = T λ 2 + Q, 1.1) x2
More informationPhysics 235 Chapter 8. Chapter 8 Central-Force Motion
Physics 35 Chapter 8 Chapter 8 Centra-Force Motion In this Chapter we wi use the theory we have discussed in Chapter 6 and 7 and appy it to very important probems in physics, in which we study the motion
More informationSection 6: Magnetostatics
agnetic fieds in matter Section 6: agnetostatics In the previous sections we assumed that the current density J is a known function of coordinates. In the presence of matter this is not aways true. The
More informationhole h vs. e configurations: l l for N > 2 l + 1 J = H as example of localization, delocalization, tunneling ikx k
Infinite 1-D Lattice CTDL, pages 1156-1168 37-1 LAST TIME: ( ) ( ) + N + 1 N hoe h vs. e configurations: for N > + 1 e rij unchanged ζ( NLS) ζ( NLS) [ ζn unchanged ] Hund s 3rd Rue (Lowest L - S term of
More informationFFTs in Graphics and Vision. Spherical Convolution and Axial Symmetry Detection
FFTs in Graphics and Vision Spherica Convoution and Axia Symmetry Detection Outine Math Review Symmetry Genera Convoution Spherica Convoution Axia Symmetry Detection Math Review Symmetry: Given a unitary
More informationHomework #04 Answers and Hints (MATH4052 Partial Differential Equations)
Homework #4 Answers and Hints (MATH452 Partia Differentia Equations) Probem 1 (Page 89, Q2) Consider a meta rod ( < x < ), insuated aong its sides but not at its ends, which is initiay at temperature =
More information(Refer Slide Time: 2:34) L C V
Microwave Integrated Circuits Professor Jayanta Mukherjee Department of Eectrica Engineering Indian Intitute of Technoogy Bombay Modue 1 Lecture No 2 Refection Coefficient, SWR, Smith Chart. Heo wecome
More informationAssignment 7 Due Tuessday, March 29, 2016
Math 45 / AMCS 55 Dr. DeTurck Assignment 7 Due Tuessday, March 9, 6 Topics for this week Convergence of Fourier series; Lapace s equation and harmonic functions: basic properties, compuations on rectanges
More information6 Wave Equation on an Interval: Separation of Variables
6 Wave Equation on an Interva: Separation of Variabes 6.1 Dirichet Boundary Conditions Ref: Strauss, Chapter 4 We now use the separation of variabes technique to study the wave equation on a finite interva.
More informationVI.G Exact free energy of the Square Lattice Ising model
VI.G Exact free energy of the Square Lattice Ising mode As indicated in eq.(vi.35), the Ising partition function is reated to a sum S, over coections of paths on the attice. The aowed graphs for a square
More informationC. Fourier Sine Series Overview
12 PHILIP D. LOEWEN C. Fourier Sine Series Overview Let some constant > be given. The symboic form of the FSS Eigenvaue probem combines an ordinary differentia equation (ODE) on the interva (, ) with a
More informationChapter 7 PRODUCTION FUNCTIONS. Copyright 2005 by South-Western, a division of Thomson Learning. All rights reserved.
Chapter 7 PRODUCTION FUNCTIONS Copyright 2005 by South-Western, a division of Thomson Learning. A rights reserved. 1 Production Function The firm s production function for a particuar good (q) shows the
More informationMAT 167: Advanced Linear Algebra
< Proem 1 (15 pts) MAT 167: Advanced Linear Agera Fina Exam Soutions (a) (5 pts) State the definition of a unitary matrix and expain the difference etween an orthogona matrix and an unitary matrix. Soution:
More informationFRIEZE GROUPS IN R 2
FRIEZE GROUPS IN R 2 MAXWELL STOLARSKI Abstract. Focusing on the Eucidean pane under the Pythagorean Metric, our goa is to cassify the frieze groups, discrete subgroups of the set of isometries of the
More informationPHYS 110B - HW #1 Fall 2005, Solutions by David Pace Equations referenced as Eq. # are from Griffiths Problem statements are paraphrased
PHYS 110B - HW #1 Fa 2005, Soutions by David Pace Equations referenced as Eq. # are from Griffiths Probem statements are paraphrased [1.] Probem 6.8 from Griffiths A ong cyinder has radius R and a magnetization
More informationExpectation-Maximization for Estimating Parameters for a Mixture of Poissons
Expectation-Maximization for Estimating Parameters for a Mixture of Poissons Brandon Maone Department of Computer Science University of Hesini February 18, 2014 Abstract This document derives, in excrutiating
More informationBohr s atomic model. 1 Ze 2 = mv2. n 2 Z
Bohr s atomic mode Another interesting success of the so-caed od quantum theory is expaining atomic spectra of hydrogen and hydrogen-ike atoms. The eectromagnetic radiation emitted by free atoms is concentrated
More informationPhys 7654 (Basic Training in CMP- Module III)/ Physics 7636 (Solid State II) Homework 1 Solutions
Phys 7654 Basic Training in CMP- Modue III/ Physics 7636 Soid State II Homework 1 Soutions by Hitesh Changani adapted from soutions provided by Shivam Ghosh Apri 19, 011 Ex. 6.4.3 Phase sips in a wire
More informationPhysics 566: Quantum Optics Quantization of the Electromagnetic Field
Physics 566: Quantum Optics Quantization of the Eectromagnetic Fied Maxwe's Equations and Gauge invariance In ecture we earned how to quantize a one dimensiona scaar fied corresponding to vibrations on
More informationc 2007 Society for Industrial and Applied Mathematics
SIAM REVIEW Vo. 49,No. 1,pp. 111 1 c 7 Society for Industria and Appied Mathematics Domino Waves C. J. Efthimiou M. D. Johnson Abstract. Motivated by a proposa of Daykin [Probem 71-19*, SIAM Rev., 13 (1971),
More informationTheory and implementation behind: Universal surface creation - smallest unitcell
Teory and impementation beind: Universa surface creation - smaest unitce Bjare Brin Buus, Jaob Howat & Tomas Bigaard September 15, 218 1 Construction of surface sabs Te aim for tis part of te project is
More informationWave Equation Dirichlet Boundary Conditions
Wave Equation Dirichet Boundary Conditions u tt x, t = c u xx x, t, < x 1 u, t =, u, t = ux, = fx u t x, = gx Look for simpe soutions in the form ux, t = XxT t Substituting into 13 and dividing
More information221B Lecture Notes Notes on Spherical Bessel Functions
Definitions B Lecture Notes Notes on Spherica Besse Functions We woud ike to sove the free Schrödinger equation [ h d r R(r) = h k R(r). () m r dr r m R(r) is the radia wave function ψ( x) = R(r)Y m (θ,
More informationMath 220B - Summer 2003 Homework 1 Solutions
Math 0B - Summer 003 Homework Soutions Consider the eigenvaue probem { X = λx 0 < x < X satisfies symmetric BCs x = 0, Suppose f(x)f (x) x=b x=a 0 for a rea-vaued functions f(x) which satisfy the boundary
More informationStat 155 Game theory, Yuval Peres Fall Lectures 4,5,6
Stat 155 Game theory, Yuva Peres Fa 2004 Lectures 4,5,6 In the ast ecture, we defined N and P positions for a combinatoria game. We wi now show more formay that each starting position in a combinatoria
More informationTHE THREE POINT STEINER PROBLEM ON THE FLAT TORUS: THE MINIMAL LUNE CASE
THE THREE POINT STEINER PROBLEM ON THE FLAT TORUS: THE MINIMAL LUNE CASE KATIE L. MAY AND MELISSA A. MITCHELL Abstract. We show how to identify the minima path network connecting three fixed points on
More informationLECTURE NOTES 9 TRACELESS SYMMETRIC TENSOR APPROACH TO LEGENDRE POLYNOMIALS AND SPHERICAL HARMONICS
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.07: Eectromagnetism II October 7, 202 Prof. Aan Guth LECTURE NOTES 9 TRACELESS SYMMETRIC TENSOR APPROACH TO LEGENDRE POLYNOMIALS AND SPHERICAL
More informationSrednicki Chapter 62
Srednicki Chapter 62 QFT Problems & Solutions A. George September 28, 213 Srednicki 62.1. Show that adding a gauge fixing term 1 2 ξ 1 ( µ A µ ) 2 to L results in equation 62.9 as the photon propagator.
More information4 Separation of Variables
4 Separation of Variabes In this chapter we describe a cassica technique for constructing forma soutions to inear boundary vaue probems. The soution of three cassica (paraboic, hyperboic and eiptic) PDE
More informationMA 201: Partial Differential Equations Lecture - 10
MA 201: Partia Differentia Equations Lecture - 10 Separation of Variabes, One dimensiona Wave Equation Initia Boundary Vaue Probem (IBVP) Reca: A physica probem governed by a PDE may contain both boundary
More informationJoel Broida UCSD Fall 2009 Phys 130B QM II. Homework Set 2 DO ALL WORK BY HAND IN OTHER WORDS, DON T USE MATHEMAT- ICA OR ANY CALCULATORS.
Joe Broida UCSD Fa 009 Phys 30B QM II Homework Set DO ALL WORK BY HAND IN OTHER WORDS, DON T USE MATHEMAT- ICA OR ANY CALCULATORS. You may need to use one or more of these: Y 0 0 = 4π Y 0 = 3 4π cos Y
More informationGauss Law. 2. Gauss s Law: connects charge and field 3. Applications of Gauss s Law
Gauss Law 1. Review on 1) Couomb s Law (charge and force) 2) Eectric Fied (fied and force) 2. Gauss s Law: connects charge and fied 3. Appications of Gauss s Law Couomb s Law and Eectric Fied Couomb s
More information<C 2 2. λ 2 l. λ 1 l 1 < C 1
Teecommunication Network Contro and Management (EE E694) Prof. A. A. Lazar Notes for the ecture of 7/Feb/95 by Huayan Wang (this document was ast LaT E X-ed on May 9,995) Queueing Primer for Muticass Optima
More informationPhysicsAndMathsTutor.com
. Two points A and B ie on a smooth horizonta tabe with AB = a. One end of a ight eastic spring, of natura ength a and moduus of easticity mg, is attached to A. The other end of the spring is attached
More informationPhysics 505 Fall 2007 Homework Assignment #5 Solutions. Textbook problems: Ch. 3: 3.13, 3.17, 3.26, 3.27
Physics 55 Fa 7 Homework Assignment #5 Soutions Textook proems: Ch. 3: 3.3, 3.7, 3.6, 3.7 3.3 Sove for the potentia in Proem 3., using the appropriate Green function otained in the text, and verify that
More informationPhysics 505 Fall Homework Assignment #4 Solutions
Physics 505 Fa 2005 Homework Assignment #4 Soutions Textbook probems: Ch. 3: 3.4, 3.6, 3.9, 3.0 3.4 The surface of a hoow conducting sphere of inner radius a is divided into an even number of equa segments
More informationA Brief Introduction to Markov Chains and Hidden Markov Models
A Brief Introduction to Markov Chains and Hidden Markov Modes Aen B MacKenzie Notes for December 1, 3, &8, 2015 Discrete-Time Markov Chains You may reca that when we first introduced random processes,
More informationLSZ reduction for spin-1/2 particles
LSZ reduction for spin-1/2 particles based on S-41 In order to describe scattering experiments we need to construct appropriate initial and final states and calculate scattering amplitude. Summary of free
More informationREVIEW REVIEW. A guess for a suitable initial state: Similarly, let s consider a final state: Summary of free theory:
LSZ reduction for spin-1/2 particles based on S-41 In order to describe scattering experiments we need to construct appropriate initial and final states and calculate scattering amplitude. Summary of free
More informationLegendre Polynomials - Lecture 8
Legendre Poynomias - Lecture 8 Introduction In spherica coordinates the separation of variabes for the function of the poar ange resuts in Legendre s equation when the soution is independent of the azimutha
More informationSUPPLEMENTARY MATERIAL TO INNOVATED SCALABLE EFFICIENT ESTIMATION IN ULTRA-LARGE GAUSSIAN GRAPHICAL MODELS
ISEE 1 SUPPLEMENTARY MATERIAL TO INNOVATED SCALABLE EFFICIENT ESTIMATION IN ULTRA-LARGE GAUSSIAN GRAPHICAL MODELS By Yingying Fan and Jinchi Lv University of Southern Caifornia This Suppementary Materia
More informationStrauss PDEs 2e: Section Exercise 1 Page 1 of 7
Strauss PDEs 2e: Section 4.3 - Exercise 1 Page 1 of 7 Exercise 1 Find the eigenvaues graphicay for the boundary conditions X(0) = 0, X () + ax() = 0. Assume that a 0. Soution The aim here is to determine
More informationXSAT of linear CNF formulas
XSAT of inear CN formuas Bernd R. Schuh Dr. Bernd Schuh, D-50968 Kön, Germany; bernd.schuh@netcoogne.de eywords: compexity, XSAT, exact inear formua, -reguarity, -uniformity, NPcompeteness Abstract. Open
More informationIdentites and properties for associated Legendre functions
Identites and properties for associated Legendre functions DBW This note is a persona note with a persona history; it arose out off y incapacity to find references on the internet that prove reations that
More informationNOISE-INDUCED STABILIZATION OF STOCHASTIC DIFFERENTIAL EQUATIONS
NOISE-INDUCED STABILIZATION OF STOCHASTIC DIFFERENTIAL EQUATIONS TONY ALLEN, EMILY GEBHARDT, AND ADAM KLUBALL 3 ADVISOR: DR. TIFFANY KOLBA 4 Abstract. The phenomenon of noise-induced stabiization occurs
More informationPhysics 506 Winter 2006 Homework Assignment #6 Solutions
Physics 506 Winter 006 Homework Assignment #6 Soutions Textbook probems: Ch. 10: 10., 10.3, 10.7, 10.10 10. Eectromagnetic radiation with eiptic poarization, described (in the notation of Section 7. by
More informationPricing Multiple Products with the Multinomial Logit and Nested Logit Models: Concavity and Implications
Pricing Mutipe Products with the Mutinomia Logit and Nested Logit Modes: Concavity and Impications Hongmin Li Woonghee Tim Huh WP Carey Schoo of Business Arizona State University Tempe Arizona 85287 USA
More informationMP203 Statistical and Thermal Physics. Solutions to Problem Set 3
MP03 Statistica and Therma Physics Soutions to Probem Set 3 1. Consider a cyinder containing 1 mo of pure moecuar nitrogen (N, seaed off withamovabepiston,sothevoumemayvary. Thecyinderiskeptatatmospheric
More information(1 ) = 1 for some 2 (0; 1); (1 + ) = 0 for some > 0:
Answers, na. Economics 4 Fa, 2009. Christiano.. The typica househod can engage in two types of activities producing current output and studying at home. Athough time spent on studying at home sacrices
More informationarxiv: v1 [quant-ph] 18 Nov 2014
Overcoming efficiency constraints on bind quantum computation Caros A Pérez-Degado 1 and oseph F Fitzsimons1, 2, 1 Singapore University of Technoogy and Design, 20 Dover Drive, Singapore 138682 2 Centre
More informationCluster modelling. Collisions. Stellar Dynamics & Structure of Galaxies handout #2. Just as a self-gravitating collection of objects.
Stear Dynamics & Structure of Gaaxies handout # Custer modeing Just as a sef-gravitating coection of objects. Coisions Do we have to worry about coisions? Gobuar custers ook densest, so obtain a rough
More informationSupplementary Appendix (not for publication) for: The Value of Network Information
Suppementary Appendix not for pubication for: The Vaue of Network Information Itay P. Fainmesser and Andrea Gaeotti September 6, 03 This appendix incudes the proof of Proposition from the paper "The Vaue
More information1 Heat Equation Dirichlet Boundary Conditions
Chapter 3 Heat Exampes in Rectanges Heat Equation Dirichet Boundary Conditions u t (x, t) = ku xx (x, t), < x (.) u(, t) =, u(, t) = u(x, ) = f(x). Separate Variabes Look for simpe soutions in the
More information22.615, MHD Theory of Fusion Systems Prof. Freidberg Lecture 2: The Moment Equations
.615, MHD Theory of Fusion ystems Prof. Freidberg Lecture : The Moment Equations Botzmann-Maxwe Equations 1. Reca that the genera couped Botzmann-Maxwe equations can be written as f q + v + E + v B f =
More informationCONGRUENCES. 1. History
CONGRUENCES HAO BILLY LEE Abstract. These are notes I created for a seminar tak, foowing the papers of On the -adic Representations and Congruences for Coefficients of Moduar Forms by Swinnerton-Dyer and
More informationHigher dimensional PDEs and multidimensional eigenvalue problems
Higher dimensiona PEs and mutidimensiona eigenvaue probems 1 Probems with three independent variabes Consider the prototypica equations u t = u (iffusion) u tt = u (W ave) u zz = u (Lapace) where u = u
More informationProblem Set 6: Solutions
University of Aabama Department of Physics and Astronomy PH 102 / LeCair Summer II 2010 Probem Set 6: Soutions 1. A conducting rectanguar oop of mass M, resistance R, and dimensions w by fas from rest
More informationCryptanalysis of PKP: A New Approach
Cryptanaysis of PKP: A New Approach Éiane Jaumes and Antoine Joux DCSSI 18, rue du Dr. Zamenhoff F-92131 Issy-es-Mx Cedex France eiane.jaumes@wanadoo.fr Antoine.Joux@ens.fr Abstract. Quite recenty, in
More informationPattern Frequency Sequences and Internal Zeros
Advances in Appied Mathematics 28, 395 420 (2002 doi:10.1006/aama.2001.0789, avaiabe onine at http://www.ideaibrary.com on Pattern Frequency Sequences and Interna Zeros Mikós Bóna Department of Mathematics,
More informationPhysics 127c: Statistical Mechanics. Fermi Liquid Theory: Collective Modes. Boltzmann Equation. The quasiparticle energy including interactions
Physics 27c: Statistica Mechanics Fermi Liquid Theory: Coective Modes Botzmann Equation The quasipartice energy incuding interactions ε p,σ = ε p + f(p, p ; σ, σ )δn p,σ, () p,σ with ε p ε F + v F (p p
More informationFRST Multivariate Statistics. Multivariate Discriminant Analysis (MDA)
1 FRST 531 -- Mutivariate Statistics Mutivariate Discriminant Anaysis (MDA) Purpose: 1. To predict which group (Y) an observation beongs to based on the characteristics of p predictor (X) variabes, using
More informationBourgain s Theorem. Computational and Metric Geometry. Instructor: Yury Makarychev. d(s 1, s 2 ).
Bourgain s Theorem Computationa and Metric Geometry Instructor: Yury Makarychev 1 Notation Given a metric space (X, d) and S X, the distance from x X to S equas d(x, S) = inf d(x, s). s S The distance
More informationLecture 8 February 18, 2010
Sources of Eectromagnetic Fieds Lecture 8 February 18, 2010 We now start to discuss radiation in free space. We wi reorder the materia of Chapter 9, bringing sections 6 7 up front. We wi aso cover some
More informationLecture Note 3: Stationary Iterative Methods
MATH 5330: Computationa Methods of Linear Agebra Lecture Note 3: Stationary Iterative Methods Xianyi Zeng Department of Mathematica Sciences, UTEP Stationary Iterative Methods The Gaussian eimination (or
More informationCourse 2BA1, Section 11: Periodic Functions and Fourier Series
Course BA, 8 9 Section : Periodic Functions and Fourier Series David R. Wikins Copyright c David R. Wikins 9 Contents Periodic Functions and Fourier Series 74. Fourier Series of Even and Odd Functions...........
More informationMidterm 2 Review. Drew Rollins
Midterm 2 Review Drew Roins 1 Centra Potentias and Spherica Coordinates 1.1 separation of variabes Soving centra force probems in physics (physica systems described by two objects with a force between
More information= 1 u 6x 2 4 2x 3 4x + 5. d dv (3v2 + 9v) = 6v v + 9 3v 2 + 9v dx = ln 3v2 + 9v + C. dy dx ay = eax.
Math 220- Mock Eam Soutions. Fin the erivative of n(2 3 4 + 5). To fin the erivative of n(2 3 4+5) we are going to have to use the chain rue. Let u = 2 3 4+5, then u = 62 4. (n(2 3 4 + 5) = (nu) u u (
More information(f) is called a nearly holomorphic modular form of weight k + 2r as in [5].
PRODUCTS OF NEARLY HOLOMORPHIC EIGENFORMS JEFFREY BEYERL, KEVIN JAMES, CATHERINE TRENTACOSTE, AND HUI XUE Abstract. We prove that the product of two neary hoomorphic Hece eigenforms is again a Hece eigenform
More information12.2. Maxima and Minima. Introduction. Prerequisites. Learning Outcomes
Maima and Minima 1. Introduction In this Section we anayse curves in the oca neighbourhood of a stationary point and, from this anaysis, deduce necessary conditions satisfied by oca maima and oca minima.
More informationIntroduction to Simulation - Lecture 14. Multistep Methods II. Jacob White. Thanks to Deepak Ramaswamy, Michal Rewienski, and Karen Veroy
Introduction to Simuation - Lecture 14 Mutistep Methods II Jacob White Thans to Deepa Ramaswamy, Micha Rewiensi, and Karen Veroy Outine Sma Timestep issues for Mutistep Methods Reminder about LTE minimization
More informationFirst-Order Corrections to Gutzwiller s Trace Formula for Systems with Discrete Symmetries
c 26 Noninear Phenomena in Compex Systems First-Order Corrections to Gutzwier s Trace Formua for Systems with Discrete Symmetries Hoger Cartarius, Jörg Main, and Günter Wunner Institut für Theoretische
More informationParallel-Axis Theorem
Parae-Axis Theorem In the previous exampes, the axis of rotation coincided with the axis of symmetry of the object For an arbitrary axis, the paraeaxis theorem often simpifies cacuations The theorem states
More informationSample Problems for Third Midterm March 18, 2013
Mat 30. Treibergs Sampe Probems for Tird Midterm Name: Marc 8, 03 Questions 4 appeared in my Fa 000 and Fa 00 Mat 30 exams (.)Let f : R n R n be differentiabe at a R n. (a.) Let g : R n R be defined by
More informationGeneralized Bell polynomials and the combinatorics of Poisson central moments
Generaized Be poynomias and the combinatorics of Poisson centra moments Nicoas Privaut Division of Mathematica Sciences Schoo of Physica and Mathematica Sciences Nanyang Technoogica University SPMS-MAS-05-43,
More informationLecture 9. Stability of Elastic Structures. Lecture 10. Advanced Topic in Column Buckling
Lecture 9 Stabiity of Eastic Structures Lecture 1 Advanced Topic in Coumn Bucking robem 9-1: A camped-free coumn is oaded at its tip by a oad. The issue here is to find the itica bucking oad. a) Suggest
More informationarxiv: v1 [math.fa] 23 Aug 2018
An Exact Upper Bound on the L p Lebesgue Constant and The -Rényi Entropy Power Inequaity for Integer Vaued Random Variabes arxiv:808.0773v [math.fa] 3 Aug 08 Peng Xu, Mokshay Madiman, James Mebourne Abstract
More information