Cosmology with non-minimal derivative coupling
|
|
- Δευκαλίων Βικελίδης
- 6 χρόνια πριν
- Προβολές:
Transcript
1 Kazan Federal University, Kazan, Russia 8th Spontaneous Workshop on Cosmology Institut d Etude Scientifique de Cargèse, Corsica May 13, 2014
2 Plan Plan Scalar fields: minimal and nonminimal coupling to gravity Horndeski theory Scalar fields with nonminimal derivative coupling Cosmological models with nonminimal derivative coupling Perturbations Summary Based on Sushkov, PRD 80, (2009) Saridakis, Sushkov, PRD 81, (2010) Sushkov, PRD 85, (2012) Skugoreva, Sushkov, Toporensky, PRD 88, (2013)
3 Scalar fields minimally coupled to gravity S = d 4 x g [ L GR + L S ] L GR gravitational Lagrangian general relativity; L GR = R square gravity; L GR = R + cr 2 f(r)-theories; L GR = f(r) etc... L S scalar field Lagrangian; ordinary STT; L S = ɛ( φ) 2 2V (φ) ɛ = +1 canonical scalar field ɛ = 1 phantom or ghost scalar field with negative kinetic energy V (φ) potential of self-action K-essence; L s = K(X) [X = ( φ) 2 ] etc...
4 Scalar fields nonminimally coupled to gravity Bergmann-Wagoner-Nordtvedt scalar-tensor theories S = d 4 x g [ f(φ)r h(φ)( φ) 2 2V (φ) ] f(φ)r = nonminimal coupling between φ and R
5 Scalar fields nonminimally coupled to gravity Bergmann-Wagoner-Nordtvedt scalar-tensor theories S = d 4 x g [ f(φ)r h(φ)( φ) 2 2V (φ) ] f(φ)r = nonminimal coupling between φ and R Conformal transformation to the Einstein frame (Wagoner, 1970): g µν = f(φ)g µν ; dφ dψ = f fh ( df dφ ) 2 1/2 S = d 4 x [ ] g R ɛ( ψ) 2 2U(ψ) ψ = new scalar field U(ψ) = new effective potential [ ( ) ] 2 ɛ = sign fh + 3 df 2 dφ
6 Scalar fields nonminimally coupled to gravity General scalar-tensor theories S = d 4 x g [ F (φ, R) ( φ) 2 2V (φ) ] F (φ, R) = generalized nonminimal coupling between φ and R
7 Scalar fields nonminimally coupled to gravity General scalar-tensor theories S = d 4 x g [ F (φ, R) ( φ) 2 2V (φ) ] F (φ, R) = generalized nonminimal coupling between φ and R Conformal transformation to the Einstein frame (Maeda, 1989): g µν = Ω 2 Ω 2 g µν ; 16π F (φ, R) R S = d 4 x g [ ] R 16π h(φ)ψ 1 ( φ) π ψ 2 ( ψ) 2 + U(φ, ψ) ψ Ω 2 = new (second!) scalar field U(φ, ψ) = new effective potential
8 Scalar fields nonminimally coupled to gravity Some remarks: A nonminimal scalar field is conformally equivalent to the minimal one possessing some effective potential V (φ) A behavior of the scalar field is encoded in the potential V (φ) The potential V (φ) is a very important ingredient of various cosmological models: a slowly varying potential behaves like an effective cosmological constat providing one or more than one inflationary phases. An appropriate choice of V (φ) is known as a problem of fine tuning of the cosmological constant.
9 Scalar fields nonminimally coupled to gravity Some remarks: A nonminimal scalar field is conformally equivalent to the minimal one possessing some effective potential V (φ) A behavior of the scalar field is encoded in the potential V (φ) The potential V (φ) is a very important ingredient of various cosmological models: a slowly varying potential behaves like an effective cosmological constat providing one or more than one inflationary phases. An appropriate choice of V (φ) is known as a problem of fine tuning of the cosmological constant.
10 Scalar fields nonminimally coupled to gravity Some remarks: A nonminimal scalar field is conformally equivalent to the minimal one possessing some effective potential V (φ) A behavior of the scalar field is encoded in the potential V (φ) The potential V (φ) is a very important ingredient of various cosmological models: a slowly varying potential behaves like an effective cosmological constat providing one or more than one inflationary phases. An appropriate choice of V (φ) is known as a problem of fine tuning of the cosmological constant.
11 Nonminimal derivative coupling generalization S = d 4 x g [ R g µν φ,µ φ,ν 2V (φ) ] F (φ, R, R µν,... ) nonminimal coupling generalization! K(φ, φ,µ, φ ;µν,..., R, R µν,... ) nonminimal derivative coupling generalization!
12 Nonminimal derivative coupling generalization S = d 4 x g [ R g µν φ,µ φ,ν 2V (φ) ] F (φ, R, R µν,... ) nonminimal coupling generalization! K(φ, φ,µ, φ ;µν,..., R, R µν,... ) nonminimal derivative coupling generalization!
13 Nonminimal derivative coupling generalization S = d 4 x g [ R g µν φ,µ φ,ν 2V (φ) ] F (φ, R, R µν,... ) nonminimal coupling generalization! K(φ, φ,µ, φ ;µν,..., R, R µν,... ) nonminimal derivative coupling generalization!
14 Horndeski (Galileon) theory In 1973, Horndeski derived the action of the most general scalar-tensor theories with second-order equations of motion G.Horndeski, IJTP 10, 363 (1974) Horndeski Lagrangian: L = i L i L 2 =K(φ, X), X = 1 2 ( φ)2 L 3 =G 3 (φ, X) φ, L 4 =G 4 (φ, X)R + G 4,X (φ, X) [ ( φ) 2 ( µ ν φ) 2], L 5 =G 5 (φ, X)G µν µ ν φ G 5,X(φ, X) [ ( φ) 3 3( φ)( µ ν φ) 2 + ( µ ν φ) 3], Notice: Generally, there are no conformal transformations which transform the Horndeski theory to the Einstein frame!
15 Theory with nonminimal kinetic coupling Action: S = d 4 x { } R g 8π [ɛgµν + κg µν ]φ,µ φ,ν 2V (φ) Field equations: G µν = 8π [ T µν + κθ µν ], [ɛg µν + κg µν ] µ ν φ = V φ T µν =ɛ µ φ ν φ 1 2 ɛg µν( φ) 2 g µν V (φ), Θ µν = 1 2 µφ ν φ R + 2 α φ (µ φr α ν) 1 2 ( φ)2 G µν + α φ β φ R µανβ + µ α φ ν α φ µ ν φ φ + g µν [ 1 2 α β φ α β φ ( φ)2 α φ β φ R αβ] Notice: The field equations are of second order!
16 Cosmological models: General formulas Ansatz: Sushkov, 2009; Saridakis, Sushkov, 2010 ds 2 = dt 2 + a 2 (t)dx 2, φ = φ(t) Field equations: a(t) cosmological factor, H = ȧ/a Hubble parameter 3H 2 = 4π φ 2 ( ɛ 9κH 2) + 8πV (φ), 2Ḣ + 3H2 = 4π φ [ ( 2 ɛ + κ 2Ḣ + 3H2 + 4H φ φ 1)] + 8πV (φ), [ (ɛ 3κH 2 )a 3 φ] = a 3 dv (φ) dφ
17 Cosmological models: General formulas Ansatz: Sushkov, 2009; Saridakis, Sushkov, 2010 ds 2 = dt 2 + a 2 (t)dx 2, φ = φ(t) Field equations: a(t) cosmological factor, H = ȧ/a Hubble parameter 3H 2 = 4π φ 2 ( ɛ 9κH 2) + 8πV (φ), 2Ḣ + 3H2 = 4π φ [ ( 2 ɛ + κ 2Ḣ + 3H2 + 4H φ φ 1)] + 8πV (φ), [ (ɛ 3κH 2 )a 3 φ] = a 3 dv (φ) dφ V (φ) const = φ = C a 3 (ɛ 3κH 2 )
18 Cosmological models: I. No kinetic coupling Trivial model without kinetic coupling (κ = 0) S = d 4 x [ ] R g 8π ( φ)2
19 Cosmological models: I. No kinetic coupling Trivial model without kinetic coupling (κ = 0) S = d 4 x [ ] R g 8π ( φ)2 Solution: a 0 (t) = t 1/3 ; φ 0 (t) = 1 2 3π ln t ds 2 0 = dt 2 + t 2/3 dx 2 t = 0 is an initial singularity
20 Cosmological models: II. No free kinetic term Model without free kinetic term (ɛ = 0) S = d 4 x [ ] R g 8π κgµν φ,µ φ,ν
21 Cosmological models: II. No free kinetic term Solution: Model without free kinetic term (ɛ = 0) S = d 4 x [ ] R g 8π κgµν φ,µ φ,ν a(t) = t 2/3 ; φ(t) = ds 2 0 = dt 2 + t 4/3 dx 2 t 2 3π κ, κ < 0 t = 0 is an initial singularity
22 Cosmological models: III. Ordinary scalar field without potential Model for an ordinary scalar field (ɛ = 1) without potential (V = 0) S = d 4 x { } R g 8π (gµν + κg µν )φ,µ φ,ν
23 Cosmological models: III. Ordinary scalar field without potential Model for an ordinary scalar field (ɛ = 1) without potential (V = 0) S = d 4 x { } R g 8π (gµν + κg µν )φ,µ φ,ν Asymptotic for t : a(t) a 0 (t) = t 1/3 ; φ(t) φ 0 (t) = 1 2 3π ln t Notice: At large times the model with κ 0 has the same behavior like that with κ = 0
24 Cosmological models: III. Ordinary scalar field without potential Asymptotics for early times The case κ < 0: a t 0 t 2/3 ; t φ t 0 2 3π κ ds 2 t 0 = dt 2 + t 4/3 dx 2 t = 0 is an initial singularity The case κ > 0: a t e Hκt ; φ t Ce t/ κ ds 2 t = dt 2 + e 2Hκt dx 2 de Sitter asymptotic with H κ = 1/ 9κ
25 Cosmological models: III. Ordinary scalar field without potential Plots of α = ln a in case κ 0, ɛ = 1, V = 0. (a) κ < 0; κ = 0; 1; 10; 100 De Sitter asymptotics: α(t) = t/3 κ (b) κ > 0; κ = 0; 1; 10; 100 Notice: In the model with nonmnimal kinetic coupling one get de Sitter phase without any potential!
26 Cosmological models: IV. Phantom scalar field without potential Plots of α(t) in case κ < 0, ɛ = 1, V = 0 (a) (9k 2 ) 1 < α 2 < (3k 2 ) 1 (b) α 2 > (3k 2 ) 1 De Sitter asymptotics: α 1(t) = t/ 9 κ (dash-dotted), α 2(t) = t/ 3 κ (dotted).
27 Cosmological models: V. Constant potential Models with the constant potential V (φ) = Λ/8π = const S = d 4 x [ ] R 2Λ g [ɛg µν + κg µν ]φ,µ φ,ν 8π
28 Cosmological models: V. Constant potential Models with the constant potential V (φ) = Λ/8π = const S = d 4 x [ ] R 2Λ g [ɛg µν + κg µν ]φ,µ φ,ν 8π There are two exact de Sitter solutions: I. α(t) = H Λ t, φ(t) = φ 0 = const, II. α(t) = t, φ(t) = 3κHΛ κ 8πκ H Λ = Λ/3 1/2 t,
29 Cosmological models: V. Constant potential Plots of α(t) in case κ > 0, ɛ = 1, V = Λ (a) H 2 Λ < α2 < 1/9κ (b) 1/9κ < α 2 < 1/3κ < H 2 Λ De Sitter asymptotics: α 1(t) = H Λt (dashed), α 2(t) = t/ 9κ (dash-dotted), α 3(t) = t/ 3κ (dotted).
30 Cosmological models: V. Constant potential Plots of α(t) in cases κ > 0, ɛ = 1 and κ < 0, ɛ = 1 (a) κ < 0, ɛ = 1 (b) κ > 0, ɛ = 1 De Sitter asymptotic: α 1(t) = H Λt (dashed).
31 Cosmological models: V. Constant potential Plots of α(t) in case κ < 0, ɛ = 1 (a) H 2 Λ < 1/9 κ < α 2 < 1/3 κ (b) H 2 Λ < 1/9 κ < 1/3 κ < α 2 (c) α 2 < H 2 Λ < 1/9 κ (d) α 2 < 1/9 κ < H 2 Λ De Sitter asymptotics: α 1(t) = H Λt (dashed), α 2(t) = t/ 9κ (dash-dotted), α 3(t) = t/ 3κ (dotted).
32 Cosmological models: VI. Power-law potential S = d 4 x { R g 8π [ g µν + κg µν] } φ,µ φ,ν m 2 φ 2 Asymptotics: H t 1/ 9κ( κm2 ) primary inflation H <t< 1/ ( ) 1 3κ 1 ± 6 κm2 secondary inflation H t 2 [ ] sin 2mt 1 oscillatory asymptotic or graceful exit 3t 2mt from inflation
33 Cosmological scenarios with nonminimal kinetic coupling and matter Sushkov, PRD 85 (2012) S = d 4 x { } R 2Λ g [g µν + κg µν ]φ,µ φ,ν + S matter 8π
34 Cosmological scenarios with nonminimal kinetic coupling and matter Sushkov, PRD 85 (2012) S = d 4 x { } R 2Λ g [g µν + κg µν ]φ,µ φ,ν + S matter 8π Stress-energy tensor: T µν = diag(ρ, p, p, p) Field equations: 3H 2 = 4π φ 2 ( 1 9κH 2) + Λ + 8πρ, 2Ḣ + 3H2 = 4π φ [ ( κ 2Ḣ + 3H2 + 4H φ φ 1)] + Λ 8πp [ (1 3κH 2 )a 3 φ] = 0
35 Cosmological scenarios with nonminimal kinetic coupling and matter Sushkov, PRD 85 (2012) S = d 4 x { } R 2Λ g [g µν + κg µν ]φ,µ φ,ν + S matter 8π Stress-energy tensor: T µν = diag(ρ, p, p, p) Field equations: 3H 2 = 4π φ 2 ( 1 9κH 2) + Λ + 8πρ, 2Ḣ + 3H2 = 4π φ [ ( κ 2Ḣ + 3H2 + 4H φ φ 1)] + Λ 8πp [ (1 3κH 2 )a 3 φ] = 0 = φ = C a 3 (1 3κH 2 )
36 Realistic cosmological scenario: Master equation Perfect fluid: ρ i = ρ i0 a, p 3(1+wi) i = w i ρ i Density parameters: Ω mi0 = ρ i0, Ω φ0 = C2 /2, Ω Λ0 = Λ ρ cr ρ cr 8πρ cr ρ cr = 3H0 2 /8π Modified Friedmann equation: H 2 = H 2 0 [ Ω Λ0 + i Constraint for parameters: ] Ω mi0 a + Ω φ0(1 9κH 2 ) 3(1+wi) a 6 (1 3κH 2 ) 2 Ω Λ0 + i Ω mi0 + Ω φ0(1 9κH 2 0 ) (1 3κH 2 0 )2 = 1
37 Realistic cosmological scenario: Simple model No coupling: κ = 0; No cosmological constant: Λ = 0 Pressureless matter: p = 0; ρ = ρ 0 /a 3 H 2 = H 2 0 [ Ωm0 a 3 + Ω ] φ0 a 6 ; Ω m0 + Ω φ0 = 1 Scale factor a(t) a(t) (t t ) δ Hubble parameter H(a) H = ȧ/a H(a) at a 0 H(a) 0 at a Acceleration parameter q(a) q = äa/ȧ 2 q is negative Ω φ0 = 0; 0.25; 0.5; 0.75; 1, and Ω m0 = 1 Ω φ0
38 Realistic cosmological scenario: Simple model with cosmological constant No coupling: κ = 0 H 2 = H 2 0 [ Ω Λ0 + Ω m0 a 3 + Ω ] φ0 a 6 ; Ω Λ0 + Ω m0 + Ω φ0 = 1 Scale factor a(t) a(t) (t t ) 1/3 at t t a(t) e H Λt at t Hubble parameter H(a) H = ȧ/a H(a) at a 0 H(a) H Λ at a Acceleration parameter q(a) q = äa/ȧ 2 q changes its sign Ω φ0 = 0.23, Ω Λ0 = 0; 0.07; 0.27; 0.47; 0.73, and Ω m0 = 1 Ω φ0 Ω Λ0
39 Realistic cosmological scenario with kinetic coupling κ > 0 H 2 = H 2 0 [ Ω Λ0 + Ω m0 a 3 + Ω φ0(1 9κH 2 ] ) a 6 (1 3κH 2 ) 2 Universal asymptotic: H H κ = 1/ 9κ at a 0 Notice: The asymptotic H H κ at early cosmological times is only determined by the coupling parameter κ and does not depend on other parameters!
40 Realistic cosmological scenario: Numerical solutions Scale factor a(t) Hubble parameter H(a) Acceleration parameter q
41 Realistic cosmological scenario: Estimations H κ t f 60 e-folds t f sec the end of initial inflationary stage H κ = 1/ 9κ sec 1 κ sec 2 or l κ = κ 1/ cm
42 Realistic cosmological scenario: Estimations H κ t f 60 e-folds t f sec the end of initial inflationary stage H κ = 1/ 9κ sec 1 κ sec 2 or l κ = κ 1/ cm H 0 70 (km/sec)/mpc sec 1 Present Hubble parameter γ = 3κH Extremely small! [ H 2 = H0 2 Ω Λ0 + Ω m0 a 3 + Ω φ0(1 9κH 2 ] ) a 6 (1 3κH 2 ) 2 Ω Λ0 + Ω m0 + Ω φ0 1 Ω Λ0 = 0.73, Ω φ0 = 0.23, Ω m0 = 0.04 q 0 = 0.25
43 Perturbations Perturbed FRW metric: g µν = ḡ µν + h µν background metric: ḡ 00 = 1, ḡ i0 = ḡ 0i = 0, ḡ ij = a 2 (t)δ ij metric perturbations (Newtonian gauge): h 00 = Φ, h 0i = h i0 = 0, h ij =a 2 (Ψδ ij + i C j + j C i + Θ ij ) Perturbed SET: δt 00 =ρφ + δρ, δt i0 = (ρ + p)δu i, δt ij =ph ij + a 2 δ ij δp Perturbed scalar field: φ (x, t) = φ 0 (t) + δφ (x, t)
44 Perturbations: Perturbed equations Perturbed Einstein equations P 1 Φ + P 2 Ψ + P3 2 Ψ + P 4 δ φ + P 5 2 δφ + P 6 δρ = 0, Q 1 Φ + Q 2 Φ + Q3 2 Φ + Q 4 Ψ + Q5 Ψ + Q6 2 Ψ +Q 7 δ φ + Q 8 δ φ + Q 9 2 δφ + Q 10 δp = 0, R 1 i Φ + R 2 i Ψ + R3 i δφ + R 4 i δ φ + R 5 2 Ċ i + R 6 δu i = 0, S 1 i j Φ + S 2 i j Ψ + S 3 i j δφ ( ) ( ) +S 4 i Ċ j + j Ċ i + S 5 i Cj + j Ci +S 6 θij + S 7 θij + S 8 2 θ ij = 0 P i, Q j, R k, S l coefficients depending on unperturbed values P 1 = 8π (ρ + 92 κh2 φ ) 2, and so on...
45 Tensor perturbations Two polarizations: θ ij θ +, θ Equation for tensor modes (1 + 4πκ φ ( 2 ) θ + 3H + 4πκ(2 φ φ + 3H φ ) 2 k ) θ 2 + a 2 (1 4πκ φ 2 )θ = 0 The case 4πκ φ 2 1: The case 4πκ φ 2 1: θ + θ + 3H θ + k2 a 2 θ = 0 ( 2 φ ) φ + 3H θ k2 a 2 θ = 0
46 Conclusions The non-minimal kinetic coupling provides an essentially new inflationary mechanism which does not need any fine-tuned potential. At early cosmological times the coupling κ-terms in the field equations are dominating and provide the quasi-de Sitter behavior of the scale factor: a(t) e Hκt with H κ = 1/ 9κ and κ sec 2 (or l κ κ 1/ cm) The model provides a natural mechanism of epoch change without any fine-tuned potential.
47 THANKS FOR YOUR ATTENTION!
Nonminimal derivative coupling scalar-tensor theories: odd-parity perturbations and black hole stability
Nonminimal derivative coupling scalar-tensor theories: odd-parity perturbations and black hole stability A. Cisterna 1 M. Cruz 2 T. Delsate 3 J. Saavedra 4 1 Universidad Austral de Chile 2 Facultad de
Διαβάστε περισσότεραDark matter from Dark Energy-Baryonic Matter Couplings
Dark matter from Dark Energy-Baryonic Matter Coulings Alejandro Avilés 1,2 1 Instituto de Ciencias Nucleares, UNAM, México 2 Instituto Nacional de Investigaciones Nucleares (ININ) México January 10, 2010
Διαβάστε περισσότεραHigher Derivative Gravity Theories
Higher Derivative Gravity Theories Black Holes in AdS space-times James Mashiyane Supervisor: Prof Kevin Goldstein University of the Witwatersrand Second Mandelstam, 20 January 2018 James Mashiyane WITS)
Διαβάστε περισσότεραPhys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)
Phys460.nb 81 ψ n (t) is still the (same) eigenstate of H But for tdependent H. The answer is NO. 5.5.5. Solution for the tdependent Schrodinger s equation If we assume that at time t 0, the electron starts
Διαβάστε περισσότεραSpace-Time Symmetries
Chapter Space-Time Symmetries In classical fiel theory any continuous symmetry of the action generates a conserve current by Noether's proceure. If the Lagrangian is not invariant but only shifts by a
Διαβάστε περισσότεραLifting Entry (continued)
ifting Entry (continued) Basic planar dynamics of motion, again Yet another equilibrium glide Hypersonic phugoid motion Planar state equations MARYAN 1 01 avid. Akin - All rights reserved http://spacecraft.ssl.umd.edu
Διαβάστε περισσότερα6.1. Dirac Equation. Hamiltonian. Dirac Eq.
6.1. Dirac Equation Ref: M.Kaku, Quantum Field Theory, Oxford Univ Press (1993) η μν = η μν = diag(1, -1, -1, -1) p 0 = p 0 p = p i = -p i p μ p μ = p 0 p 0 + p i p i = E c 2 - p 2 = (m c) 2 H = c p 2
Διαβάστε περισσότεραCosmological dynamics and perturbations in Light mass Galileon models
Cosmological dynamics and perturbations in Light mass Galileon models Md. Wali Hossain APCTP Pohang, Korea Collaborators: A. Ali, R. Gannouji and M. Sami 13 th December, 2017, CosPA 2017, YITP, Kyoto University,
Διαβάστε περισσότεραECE Spring Prof. David R. Jackson ECE Dept. Notes 2
ECE 634 Spring 6 Prof. David R. Jackson ECE Dept. Notes Fields in a Source-Free Region Example: Radiation from an aperture y PEC E t x Aperture Assume the following choice of vector potentials: A F = =
Διαβάστε περισσότεραThe Simply Typed Lambda Calculus
Type Inference Instead of writing type annotations, can we use an algorithm to infer what the type annotations should be? That depends on the type system. For simple type systems the answer is yes, and
Διαβάστε περισσότεραHOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:
HOMEWORK 4 Problem a For the fast loading case, we want to derive the relationship between P zz and λ z. We know that the nominal stress is expressed as: P zz = ψ λ z where λ z = λ λ z. Therefore, applying
Διαβάστε περισσότεραForced Pendulum Numerical approach
Numerical approach UiO April 8, 2014 Physical problem and equation We have a pendulum of length l, with mass m. The pendulum is subject to gravitation as well as both a forcing and linear resistance force.
Διαβάστε περισσότεραGeodesic Equations for the Wormhole Metric
Geodesic Equations for the Wormhole Metric Dr R Herman Physics & Physical Oceanography, UNCW February 14, 2018 The Wormhole Metric Morris and Thorne wormhole metric: [M S Morris, K S Thorne, Wormholes
Διαβάστε περισσότεραExercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1.
Exercises 0 More exercises are available in Elementary Differential Equations. If you have a problem to solve any of them, feel free to come to office hour. Problem Find a fundamental matrix of the given
Διαβάστε περισσότεραCHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS
CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =
Διαβάστε περισσότεραEcon 2110: Fall 2008 Suggested Solutions to Problem Set 8 questions or comments to Dan Fetter 1
Eon : Fall 8 Suggested Solutions to Problem Set 8 Email questions or omments to Dan Fetter Problem. Let X be a salar with density f(x, θ) (θx + θ) [ x ] with θ. (a) Find the most powerful level α test
Διαβάστε περισσότεραStatistical Inference I Locally most powerful tests
Statistical Inference I Locally most powerful tests Shirsendu Mukherjee Department of Statistics, Asutosh College, Kolkata, India. shirsendu st@yahoo.co.in So far we have treated the testing of one-sided
Διαβάστε περισσότερα3+1 Splitting of the Generalized Harmonic Equations
3+1 Splitting of the Generalized Harmonic Equations David Brown North Carolina State University EGM June 2011 Numerical Relativity Interpret general relativity as an initial value problem: Split spacetime
Διαβάστε περισσότεραSecond Order RLC Filters
ECEN 60 Circuits/Electronics Spring 007-0-07 P. Mathys Second Order RLC Filters RLC Lowpass Filter A passive RLC lowpass filter (LPF) circuit is shown in the following schematic. R L C v O (t) Using phasor
Διαβάστε περισσότεραMatrices and Determinants
Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z
Διαβάστε περισσότεραSymmetric Stress-Energy Tensor
Chapter 3 Symmetric Stress-Energy ensor We noticed that Noether s conserved currents are arbitrary up to the addition of a divergence-less field. Exploiting this freedom the canonical stress-energy tensor
Διαβάστε περισσότερα4.6 Autoregressive Moving Average Model ARMA(1,1)
84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this
Διαβάστε περισσότεραEE512: Error Control Coding
EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3
Διαβάστε περισσότερα3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β
3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α NOTE: cos(αβ cos α cos β cos(α-β cos α -cos β Proof of cos(α-β cos α cos β sin α Let s use a unit circle
Διαβάστε περισσότεραHomework 3 Solutions
Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For
Διαβάστε περισσότεραHomework 8 Model Solution Section
MATH 004 Homework Solution Homework 8 Model Solution Section 14.5 14.6. 14.5. Use the Chain Rule to find dz where z cosx + 4y), x 5t 4, y 1 t. dz dx + dy y sinx + 4y)0t + 4) sinx + 4y) 1t ) 0t + 4t ) sinx
Διαβάστε περισσότεραDESIGN OF MACHINERY SOLUTION MANUAL h in h 4 0.
DESIGN OF MACHINERY SOLUTION MANUAL -7-1! PROBLEM -7 Statement: Design a double-dwell cam to move a follower from to 25 6, dwell for 12, fall 25 and dwell for the remader The total cycle must take 4 sec
Διαβάστε περισσότεραΑπόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο. Απόστολος Σ.
Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο The time integral of a force is referred to as impulse, is determined by and is obtained from: Newton s 2 nd Law of motion states that the action
Διαβάστε περισσότεραApproximation of distance between locations on earth given by latitude and longitude
Approximation of distance between locations on earth given by latitude and longitude Jan Behrens 2012-12-31 In this paper we shall provide a method to approximate distances between two points on earth
Διαβάστε περισσότεραNon-Gaussianity from Lifshitz Scalar
COSMO/CosPA 200 September 27, 200 Non-Gaussianity from Lifshitz Scalar Takeshi Kobayashi (Tokyo U.) based on: arxiv:008.406 with Keisuke Izumi, Shinji Mukohyama Lifshitz scalars with z=3 obtain super-horizon,
Διαβάστε περισσότεραDifferential equations
Differential equations Differential equations: An equation inoling one dependent ariable and its deriaties w. r. t one or more independent ariables is called a differential equation. Order of differential
Διαβάστε περισσότεραIf we restrict the domain of y = sin x to [ π, π ], the restrict function. y = sin x, π 2 x π 2
Chapter 3. Analytic Trigonometry 3.1 The inverse sine, cosine, and tangent functions 1. Review: Inverse function (1) f 1 (f(x)) = x for every x in the domain of f and f(f 1 (x)) = x for every x in the
Διαβάστε περισσότεραDr. D. Dinev, Department of Structural Mechanics, UACEG
Lecture 4 Material behavior: Constitutive equations Field of the game Print version Lecture on Theory of lasticity and Plasticity of Dr. D. Dinev, Department of Structural Mechanics, UACG 4.1 Contents
Διαβάστε περισσότεραChapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
Διαβάστε περισσότεραThe kinetic and potential energies as T = 1 2. (m i η2 i k(η i+1 η i ) 2 ). (3) The Hooke s law F = Y ξ, (6) with a discrete analog
Lecture 12: Introduction to Analytical Mechanics of Continuous Systems Lagrangian Density for Continuous Systems The kinetic and potential energies as T = 1 2 i η2 i (1 and V = 1 2 i+1 η i 2, i (2 where
Διαβάστε περισσότεραIf we restrict the domain of y = sin x to [ π 2, π 2
Chapter 3. Analytic Trigonometry 3.1 The inverse sine, cosine, and tangent functions 1. Review: Inverse function (1) f 1 (f(x)) = x for every x in the domain of f and f(f 1 (x)) = x for every x in the
Διαβάστε περισσότεραHigher spin gauge theories and their CFT duals
Higher spin gauge theories and their CFT duals E-mail: hikida@phys-h.keio.ac.jp 2 AdS Vasiliev AdS/CFT 4 Vasiliev 3 O(N) 3 Vasiliev 2 W N 1 AdS/CFT g µν Vasiliev AdS [1] AdS/CFT anti-de Sitter (AdS) (CFT)
Διαβάστε περισσότεραThree coupled amplitudes for the πη, K K and πη channels without data
Three coupled amplitudes for the πη, K K and πη channels without data Robert Kamiński IFJ PAN, Kraków and Łukasz Bibrzycki Pedagogical University, Kraków HaSpect meeting, Kraków, V/VI 216 Present status
Διαβάστε περισσότερα( y) Partial Differential Equations
Partial Dierential Equations Linear P.D.Es. contains no owers roducts o the deendent variables / an o its derivatives can occasionall be solved. Consider eamle ( ) a (sometimes written as a ) we can integrate
Διαβάστε περισσότεραΗ αλληλεπίδραση ανάμεσα στην καθημερινή γλώσσα και την επιστημονική ορολογία: παράδειγμα από το πεδίο της Κοσμολογίας
Η αλληλεπίδραση ανάμεσα στην καθημερινή γλώσσα και την επιστημονική ορολογία: παράδειγμα από το πεδίο της Κοσμολογίας ΠΕΡΙΛΗΨΗ Αριστείδης Κοσιονίδης Η κατανόηση των εννοιών ενός επιστημονικού πεδίου απαιτεί
Διαβάστε περισσότεραSecond Order Partial Differential Equations
Chapter 7 Second Order Partial Differential Equations 7.1 Introduction A second order linear PDE in two independent variables (x, y Ω can be written as A(x, y u x + B(x, y u xy + C(x, y u u u + D(x, y
Διαβάστε περισσότεραJesse Maassen and Mark Lundstrom Purdue University November 25, 2013
Notes on Average Scattering imes and Hall Factors Jesse Maassen and Mar Lundstrom Purdue University November 5, 13 I. Introduction 1 II. Solution of the BE 1 III. Exercises: Woring out average scattering
Διαβάστε περισσότεραCosmological Space-Times
Cosmological Space-Times Lecture notes compiled by Geoff Bicknell based primarily on: Sean Carroll: An Introduction to General Relativity plus additional material 1 Metric of special relativity ds 2 =
Διαβάστε περισσότεραSrednicki Chapter 55
Srednicki Chapter 55 QFT Problems & Solutions A. George August 3, 03 Srednicki 55.. Use equations 55.3-55.0 and A i, A j ] = Π i, Π j ] = 0 (at equal times) to verify equations 55.-55.3. This is our third
Διαβάστε περισσότεραderivation of the Laplacian from rectangular to spherical coordinates
derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 03-03- :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used
Διαβάστε περισσότεραST5224: Advanced Statistical Theory II
ST5224: Advanced Statistical Theory II 2014/2015: Semester II Tutorial 7 1. Let X be a sample from a population P and consider testing hypotheses H 0 : P = P 0 versus H 1 : P = P 1, where P j is a known
Διαβάστε περισσότεραLifting Entry 2. Basic planar dynamics of motion, again Yet another equilibrium glide Hypersonic phugoid motion MARYLAND U N I V E R S I T Y O F
ifting Entry Basic planar dynamics of motion, again Yet another equilibrium glide Hypersonic phugoid motion MARYAN 1 010 avid. Akin - All rights reserved http://spacecraft.ssl.umd.edu ifting Atmospheric
Διαβάστε περισσότερα6.003: Signals and Systems. Modulation
6.003: Signals and Systems Modulation May 6, 200 Communications Systems Signals are not always well matched to the media through which we wish to transmit them. signal audio video internet applications
Διαβάστε περισσότεραthe total number of electrons passing through the lamp.
1. A 12 V 36 W lamp is lit to normal brightness using a 12 V car battery of negligible internal resistance. The lamp is switched on for one hour (3600 s). For the time of 1 hour, calculate (i) the energy
Διαβάστε περισσότεραwave energy Superposition of linear plane progressive waves Marine Hydrodynamics Lecture Oblique Plane Waves:
3.0 Marine Hydrodynamics, Fall 004 Lecture 0 Copyriht c 004 MIT - Department of Ocean Enineerin, All rihts reserved. 3.0 - Marine Hydrodynamics Lecture 0 Free-surface waves: wave enery linear superposition,
Διαβάστε περισσότεραStrain gauge and rosettes
Strain gauge and rosettes Introduction A strain gauge is a device which is used to measure strain (deformation) on an object subjected to forces. Strain can be measured using various types of devices classified
Διαβάστε περισσότεραVariational Wavefunction for the Helium Atom
Technische Universität Graz Institut für Festkörperphysik Student project Variational Wavefunction for the Helium Atom Molecular and Solid State Physics 53. submitted on: 3. November 9 by: Markus Krammer
Διαβάστε περισσότεραLecture 34 Bootstrap confidence intervals
Lecture 34 Bootstrap confidence intervals Confidence Intervals θ: an unknown parameter of interest We want to find limits θ and θ such that Gt = P nˆθ θ t If G 1 1 α is known, then P θ θ = P θ θ = 1 α
Διαβάστε περισσότερα2. THEORY OF EQUATIONS. PREVIOUS EAMCET Bits.
EAMCET-. THEORY OF EQUATIONS PREVIOUS EAMCET Bits. Each of the roots of the equation x 6x + 6x 5= are increased by k so that the new transformed equation does not contain term. Then k =... - 4. - Sol.
Διαβάστε περισσότεραExample Sheet 3 Solutions
Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note
Διαβάστε περισσότεραChapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
Διαβάστε περισσότεραYou may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
MATHEMATICAL TRIPOS Part III Monday 6 June, 2005 9 to 12 PAPER 60 GENERAL RELATIVITY Attempt THREE questions. There are FOUR questions in total. The questions carry equal weight. The signature is ( + ),
Διαβάστε περισσότεραHartree-Fock Theory. Solving electronic structure problem on computers
Hartree-Foc Theory Solving electronic structure problem on computers Hartree product of non-interacting electrons mean field molecular orbitals expectations values one and two electron operators Pauli
Διαβάστε περισσότερα[Note] Geodesic equation for scalar, vector and tensor perturbations
[Note] Geodesic equation for scalar, vector and tensor perturbations Toshiya Namikawa 212 1 Curl mode induced by vector and tensor perturbation 1.1 Metric Perturbation and Affine Connection The line element
Διαβάστε περισσότεραConcrete Mathematics Exercises from 30 September 2016
Concrete Mathematics Exercises from 30 September 2016 Silvio Capobianco Exercise 1.7 Let H(n) = J(n + 1) J(n). Equation (1.8) tells us that H(2n) = 2, and H(2n+1) = J(2n+2) J(2n+1) = (2J(n+1) 1) (2J(n)+1)
Διαβάστε περισσότεραCRASH COURSE IN PRECALCULUS
CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 01, Brooks/Cole Chapter
Διαβάστε περισσότεραNumerical Analysis FMN011
Numerical Analysis FMN011 Carmen Arévalo Lund University carmen@maths.lth.se Lecture 12 Periodic data A function g has period P if g(x + P ) = g(x) Model: Trigonometric polynomial of order M T M (x) =
Διαβάστε περισσότεραSPONTANEOUS GENERATION OF GEOMETRY IN 4D
SPONTANEOUS GENERATION OF GEOMETRY IN 4D Dani Puigdomènch Dual year Russia-Spain: Particle Physics, Nuclear Physics and Astroparticle Physics 10/11/11 Based on arxiv: 1004.3664 and wor on progress. by
Διαβάστε περισσότεραω ω ω ω ω ω+2 ω ω+2 + ω ω ω ω+2 + ω ω+1 ω ω+2 2 ω ω ω ω ω ω ω ω+1 ω ω2 ω ω2 + ω ω ω2 + ω ω ω ω2 + ω ω+1 ω ω2 + ω ω+1 + ω ω ω ω2 + ω
0 1 2 3 4 5 6 ω ω + 1 ω + 2 ω + 3 ω + 4 ω2 ω2 + 1 ω2 + 2 ω2 + 3 ω3 ω3 + 1 ω3 + 2 ω4 ω4 + 1 ω5 ω 2 ω 2 + 1 ω 2 + 2 ω 2 + ω ω 2 + ω + 1 ω 2 + ω2 ω 2 2 ω 2 2 + 1 ω 2 2 + ω ω 2 3 ω 3 ω 3 + 1 ω 3 + ω ω 3 +
Διαβάστε περισσότεραExercises to Statistics of Material Fatigue No. 5
Prof. Dr. Christine Müller Dipl.-Math. Christoph Kustosz Eercises to Statistics of Material Fatigue No. 5 E. 9 (5 a Show, that a Fisher information matri for a two dimensional parameter θ (θ,θ 2 R 2, can
Διαβάστε περισσότεραReview Test 3. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Review Test MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the exact value of the expression. 1) sin - 11π 1 1) + - + - - ) sin 11π 1 ) ( -
Διαβάστε περισσότεραMassive gravitons in arbitrary spacetimes
Massive gravitons in arbitrary spacetimes Mikhail S. Volkov LMPT, University of Tours, FRANCE Orsay, Modern aspects of Gravity and Cosmology, 8-th November 2017 C.Mazuet and M.S.V. arxiv: 1708.03554 Mikhail
Διαβάστε περισσότεραSection 8.3 Trigonometric Equations
99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.
Διαβάστε περισσότεραDerivation of Optical-Bloch Equations
Appendix C Derivation of Optical-Bloch Equations In this appendix the optical-bloch equations that give the populations and coherences for an idealized three-level Λ system, Fig. 3. on page 47, will be
Διαβάστε περισσότεραInstruction Execution Times
1 C Execution Times InThisAppendix... Introduction DL330 Execution Times DL330P Execution Times DL340 Execution Times C-2 Execution Times Introduction Data Registers This appendix contains several tables
Διαβάστε περισσότεραPartial Differential Equations in Biology The boundary element method. March 26, 2013
The boundary element method March 26, 203 Introduction and notation The problem: u = f in D R d u = ϕ in Γ D u n = g on Γ N, where D = Γ D Γ N, Γ D Γ N = (possibly, Γ D = [Neumann problem] or Γ N = [Dirichlet
Διαβάστε περισσότερα6.4 Superposition of Linear Plane Progressive Waves
.0 - Marine Hydrodynamics, Spring 005 Lecture.0 - Marine Hydrodynamics Lecture 6.4 Superposition of Linear Plane Progressive Waves. Oblique Plane Waves z v k k k z v k = ( k, k z ) θ (Looking up the y-ais
Διαβάστε περισσότεραAn Inventory of Continuous Distributions
Appendi A An Inventory of Continuous Distributions A.1 Introduction The incomplete gamma function is given by Also, define Γ(α; ) = 1 with = G(α; ) = Z 0 Z 0 Z t α 1 e t dt, α > 0, >0 t α 1 e t dt, α >
Διαβάστε περισσότεραMean bond enthalpy Standard enthalpy of formation Bond N H N N N N H O O O
Q1. (a) Explain the meaning of the terms mean bond enthalpy and standard enthalpy of formation. Mean bond enthalpy... Standard enthalpy of formation... (5) (b) Some mean bond enthalpies are given below.
Διαβάστε περισσότερα[1] P Q. Fig. 3.1
1 (a) Define resistance....... [1] (b) The smallest conductor within a computer processing chip can be represented as a rectangular block that is one atom high, four atoms wide and twenty atoms long. One
Διαβάστε περισσότεραMagnetically Coupled Circuits
DR. GYURCSEK ISTVÁN Magnetically Coupled Circuits Sources and additional materials (recommended) Dr. Gyurcsek Dr. Elmer: Theories in Electric Circuits, GlobeEdit, 2016, ISBN:978-3-330-71341-3 Ch. Alexander,
Διαβάστε περισσότεραReminders: linear functions
Reminders: linear functions Let U and V be vector spaces over the same field F. Definition A function f : U V is linear if for every u 1, u 2 U, f (u 1 + u 2 ) = f (u 1 ) + f (u 2 ), and for every u U
Διαβάστε περισσότεραHigh order interpolation function for surface contact problem
3 016 5 Journal of East China Normal University Natural Science No 3 May 016 : 1000-564101603-0009-1 1 1 1 00444; E- 00030 : Lagrange Lobatto Matlab : ; Lagrange; : O41 : A DOI: 103969/jissn1000-56410160300
Διαβάστε περισσότεραAspects of the BMS/CFT correspondence
DAMTP, Cambridge. February 17, 2010 Aspects of the BMS/CFT correspondence Glenn Barnich Physique théorique et mathématique Université Libre de Bruxelles & International Solvay Institutes Overview Classical
Διαβάστε περισσότεραTorsional Newton-Cartan gravity from a pre-newtonian expansion of GR
Torsional Newton-Cartan gravity from a pre-newtonian expansion of GR Dieter Van den Bleeken arxiv.org/submit/1828684 Supported by Bog azic i University Research Fund Grant nr 17B03P1 SCGP 10th March 2017
Διαβάστε περισσότεραCoupling of a Jet-Slot Oscillator With the Flow-Supply Duct: Flow-Acoustic Interaction Modeling
1th AIAA/CEAS Aeroacoustics Conference, May 006 interactions Coupling of a Jet-Slot Oscillator With the Flow-Supply Duct: Interaction M. Glesser 1, A. Billon 1, V. Valeau, and A. Sakout 1 mglesser@univ-lr.fr
Διαβάστε περισσότεραThermodynamics of the motility-induced phase separation
1/9 Thermodynamics of the motility-induced phase separation Alex Solon (MIT), Joakim Stenhammar (Lund U), Mike Cates (Cambridge U), Julien Tailleur (Paris Diderot U) July 21st 2016 STATPHYS Lyon Active
Διαβάστε περισσότεραLecture 26: Circular domains
Introductory lecture notes on Partial Differential Equations - c Anthony Peirce. Not to be copied, used, or revised without eplicit written permission from the copyright owner. 1 Lecture 6: Circular domains
Διαβάστε περισσότεραAquinas College. Edexcel Mathematical formulae and statistics tables DO NOT WRITE ON THIS BOOKLET
Aquinas College Edexcel Mathematical formulae and statistics tables DO NOT WRITE ON THIS BOOKLET Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical
Διαβάστε περισσότεραSpherical Coordinates
Spherical Coordinates MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 2011 Spherical Coordinates Another means of locating points in three-dimensional space is known as the spherical
Διαβάστε περισσότεραExercise 1.1. Verify that if we apply GS to the coordinate basis Gauss form ds 2 = E(u, v)du 2 + 2F (u, v)dudv + G(u, v)dv 2
Math 209 Riemannian Geometry Jeongmin Shon Problem. Let M 2 R 3 be embedded surface. Then the induced metric on M 2 is obtained by taking the standard inner product on R 3 and restricting it to the tangent
Διαβάστε περισσότεραΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΤΜΗΜΑ ΟΔΟΝΤΙΑΤΡΙΚΗΣ ΕΡΓΑΣΤΗΡΙΟ ΟΔΟΝΤΙΚΗΣ ΚΑΙ ΑΝΩΤΕΡΑΣ ΠΡΟΣΘΕΤΙΚΗΣ
ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΤΜΗΜΑ ΟΔΟΝΤΙΑΤΡΙΚΗΣ ΕΡΓΑΣΤΗΡΙΟ ΟΔΟΝΤΙΚΗΣ ΚΑΙ ΑΝΩΤΕΡΑΣ ΠΡΟΣΘΕΤΙΚΗΣ ΣΥΓΚΡΙΤΙΚΗ ΜΕΛΕΤΗ ΤΗΣ ΣΥΓΚΡΑΤΗΤΙΚΗΣ ΙΚΑΝΟΤΗΤΑΣ ΟΡΙΣΜΕΝΩΝ ΠΡΟΚΑΤΑΣΚΕΥΑΣΜΕΝΩΝ ΣΥΝΔΕΣΜΩΝ ΑΚΡΙΒΕΙΑΣ
Διαβάστε περισσότεραInverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------
Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin
Διαβάστε περισσότεραEvery set of first-order formulas is equivalent to an independent set
Every set of first-order formulas is equivalent to an independent set May 6, 2008 Abstract A set of first-order formulas, whatever the cardinality of the set of symbols, is equivalent to an independent
Διαβάστε περισσότεραLecture 21: Scattering and FGR
ECE-656: Fall 009 Lecture : Scattering and FGR Professor Mark Lundstrom Electrical and Computer Engineering Purdue University, West Lafayette, IN USA Review: characteristic times τ ( p), (, ) == S p p
Διαβάστε περισσότεραDERIVATION OF MILES EQUATION FOR AN APPLIED FORCE Revision C
DERIVATION OF MILES EQUATION FOR AN APPLIED FORCE Revision C By Tom Irvine Email: tomirvine@aol.com August 6, 8 Introduction The obective is to derive a Miles equation which gives the overall response
Διαβάστε περισσότεραSection 7.6 Double and Half Angle Formulas
09 Section 7. Double and Half Angle Fmulas To derive the double-angles fmulas, we will use the sum of two angles fmulas that we developed in the last section. We will let α θ and β θ: cos(θ) cos(θ + θ)
Διαβάστε περισσότερα1 String with massive end-points
1 String with massive end-points Πρόβλημα 5.11:Θεωρείστε μια χορδή μήκους, τάσης T, με δύο σημειακά σωματίδια στα άκρα της, το ένα μάζας m, και το άλλο μάζας m. α) Μελετώντας την κίνηση των άκρων βρείτε
Διαβάστε περισσότεραMath221: HW# 1 solutions
Math: HW# solutions Andy Royston October, 5 7.5.7, 3 rd Ed. We have a n = b n = a = fxdx = xdx =, x cos nxdx = x sin nx n sin nxdx n = cos nx n = n n, x sin nxdx = x cos nx n + cos nxdx n cos n = + sin
Διαβάστε περισσότεραTheory of Cosmological Perturbations
1 heory of Cosmological Perturbations Part I gauge-invariant formalism Misao Sasaki PDF files available at http://www2.yukawa.kyoto-u.ac.jp/~misao/pdf/cp/ 1. Formulation Background spacetime Friedmann-Lematre-Robertson-Walker
Διαβάστε περισσότεραLocal Approximation with Kernels
Local Approximation with Kernels Thomas Hangelbroek University of Hawaii at Manoa 5th International Conference Approximation Theory, 26 work supported by: NSF DMS-43726 A cubic spline example Consider
Διαβάστε περισσότεραGalatia SIL Keyboard Information
Galatia SIL Keyboard Information Keyboard ssignments The main purpose of the keyboards is to provide a wide range of keying options, so many characters can be entered in multiple ways. If you are typing
Διαβάστε περισσότεραFinite Field Problems: Solutions
Finite Field Problems: Solutions 1. Let f = x 2 +1 Z 11 [x] and let F = Z 11 [x]/(f), a field. Let Solution: F =11 2 = 121, so F = 121 1 = 120. The possible orders are the divisors of 120. Solution: The
Διαβάστε περισσότεραPARTIAL NOTES for 6.1 Trigonometric Identities
PARTIAL NOTES for 6.1 Trigonometric Identities tanθ = sinθ cosθ cotθ = cosθ sinθ BASIC IDENTITIES cscθ = 1 sinθ secθ = 1 cosθ cotθ = 1 tanθ PYTHAGOREAN IDENTITIES sin θ + cos θ =1 tan θ +1= sec θ 1 + cot
Διαβάστε περισσότεραMath 6 SL Probability Distributions Practice Test Mark Scheme
Math 6 SL Probability Distributions Practice Test Mark Scheme. (a) Note: Award A for vertical line to right of mean, A for shading to right of their vertical line. AA N (b) evidence of recognizing symmetry
Διαβάστε περισσότερα