Modified Moyal-Weyl Star product in a Curved Non Commutative space-time
|
|
|
- Φιλομήνα Αναγνωστάκης
- 6 χρόνια πριν
- Προβολές:
Transcript
1 EJTP 3, No. 12 ( Electronic Journal of Theoretical Physics Modified Moyal-Weyl Star product in a Curved Non Commutative space-time N.Mebarki,F.Khallili, M.Boussahel, and M.Haouchine Laboratoire de Physique Mathématique et Subatomique. Mentouri University,Constantine,ALGERIA Received 6 July 2006, Accepted 16 August 2006, Published 20 September 2006 Abstract: To generate gravitational terms in a curved noncommutative space-time, ne Moyal- Weyl star product as ell as Weyl ordering are defined. As an example, a complex scalar mass term action is considered. c Electronic Journal of Theoretical Physics. All rights reserved. Keyords: Quantum Groups, Non Commutative Geometry, Moyal-Weyl Star Product PACS (2006: U, Nx, ?q, Gh 1. Introduction During the last to decades many efforts have been made to solve or at least to understand the reaming unsolved outstanding problems of theoretical physics by using ne ideas like quantum groups, deformation theory, noncommutative geometry etc.. 1] 9]. This may shed a light on the real microscopic geometry and structure of our universe. One approach, is to consider a noncommutative space-time here the dynamical variables become operators and therefore, the formalism of the quantum fields theories constructions must be modified. It turns out that in a flat space-time geometry, this amounts basically to replace the ordinary products by a Moyal-Weyl star products and taking into account the Weyl ordering 1] 10]. The goal of this paper is to consider a curved space-time (presence of a gravitational background and define the corresponding ne Moyal-Weyl star product and Weyl ordering. In section 2, e present our mathematical formalism and consider an example a mass term of a complex scalar field. Finally, in section 3, e dra our conclusions.
2 38 Electronic Journal of Theoretical Physics 3, No. 12 ( Formalism 2.1 T he ordinary Moyal W eyl product The Moyal-Weyl -product of any to smooth functions f and g such as 12] f (x = (2π 3 2 d 4 k e ikx f (k (1 g (x = (2π 3 2 d 4 k e ikx g (k (2 can be defined as follos: First e associate to f and g the Weyl operators W (f and W (g defined by W (f = (2π 3 2 W (g = (2π 3 2 d 4 k e ik x f (k (3 d 4 k e ik x g (k here x µ are non commuting operators satisfying x µ, x ν ] = iθ µν (4 Next e define the product W (f W (g as W (f W (g = (2π 3 2 (2π 3 2 d 4 k d 4 p e ik x e ip x f (k g (p (5 Using the C-B-H formula, the Weyl product W (f W (g reads : W (f W (g = (2π 3 2 (2π 3 2 d 4 k d 4 p e ik x+ip x i 2 k µp ν θ µν f (k g (p = W (f g (6 Where f g is a ne classical function defined by: (f g (x = e i 2 θµν x µ y ν f (x g (y (7 This is the ordinary Moyal -product. To the second ordre in θ reads: The Moyal -product (f g (x = f (x g (x + i 2 θµν x µ f (x x + i 2 θµν i 2 θαβ x µ g (x + (8 ν x f (x g (x +... α x ν xβ
3 Electronic Journal of Theoretical Physics 3, No. 12 ( Notice here, that the operators x µ are only defined modulo terms hich vanish at the classical limit, for example x µ and x µ + Σ µ αβ xα x β Σ µ αβ xβ x α are equal but the corresponding non commutating operators are not except if Σ µ αβ is symmetric x µ x µ + Σ µ αβ xα x β Σ µ αβ xβ x α = x µ + iθ αβ Σ µ αβ (9 2.2 T he deformed Moyal W eyl product Here, by deforming the ordinary Moyal -product, e propose a ne Moyal-Weyl -product hich take in consideration the missing terms cited above and hich generate gravitational terms to the order θ 2. To any smooth function f e associate the Weyl operator W (f f (x = (2π 3 2 d 4 x e ikx f (k W (f = (2π 3 2 d 4 k e ik X f (k (10 here X µ are non commuting operators associated to the folloing classical variables X µ = x µ + Γ µ αβ xα x β 1 2 Γµ ρλ Γρ αβ xα x β x λ (11 here Γ µ αβ (x = Γµ βα (x is the symmetric affine connection. The non commuting operators X µ are defined by a symmetrization procedure: X µ = x µ + ( Γµ αβ xα x β 1 ( Γµ 2 ρλ Γ ρ αβ xα x β x λ Where the Weyl ordering is defined by: ith (12 αβ xα x β = ( Σ µ αβ xα x β 2 x α Σµ αβ xβ + x α x β Σµ αβ / g (13 and direct simplifications (see Appendix give: Where e have used the fact that Σ µ αβ g = det g µν (14 αβ xα x β = iθ βλ iθ ασ σ λ Σµ αβ / g (15 is symmetric, and: x µ, x ν ] = iθ µν and ] x µ, f (x = iθ µν ν f (x (16 Thus the second term in eq.( 15 reads:
4 40 Electronic Journal of Theoretical Physics 3, No. 12 ( ( Γµ αβ xα x β = θ βλ θ ασ σ λ Γµ αβ / g (17 It is orth to mention that e can get the same result if e define the Weyl ordering as: Where ( Γµ αβ xα x β = ( Γ µ αβ xα x β x α Γ µ αβ xβ + Γ µ αβ x α x β / g (18 x α x β x α, x β] (19 and ] x µ f (x x µ, f (x (20 No using these relations and the fact that Γ µ αβ eq.(18 can be reritten as x α x β = Γ µ αβ x α, x β] = iθ αβ Γµ αβ (21 ( Γµ αβ xα x β = θ βλ θ ασ σ λ Γµ αβ / g Which is the same result as above. Similarly, one can define the Weyl ordering αβλ xα x β x λ as: αβλ xα x β x λ = ( Σ µ αβλ xα β x λ + Σ µ αβλ xβ α x λ + Σ µ αβλ xλ α x β / g (22 and straightforard simplifications by using the fact that Σ µ αβλ is symmetric ith respect to αβ give: αβλ xα x β x λ = 2iθ βλ iθ ασ σ Σµ αβλ / g (23 and one can deduce that: ( Γµ ρλ Γ ρ αβ xα x β x λ = 2iθ βλ iθ ασ σ ( Γµ ρλ Γ ρ αβ / g (24 and the noncommuting variable X µ can be reritten as: X µ = x µ + ( Γµ αβ xα x β 1 ( Γµ 2 ρλ Γ ρ αβ xα x β x λ here after the corresponding expressions substitutions reads: (25 X µ = x µ iθβλ iθ ασ σ Rµαβλ (x / g (26 Here R µ αβλ stands for the Riemann curvature tensor defined as:
5 Electronic Journal of Theoretical Physics 3, No. 12 ( Using the C-B-H formula, one can rite: R µ αβλ (x = β Γ µ αλ λ Γ µ αβ + Γ µ ρβ Γ ρ αλ Γ µ ρλ Γ ρ αβ (27 e ik Xe ip X = e ik X+ip X+ 1 2ik X, ip X]+... = e ik x+ip x+ik µ x µ +ip ν x ν i 2 θµν k µ p ν... (28 ith Thus, The Moyal-Weyl -product reads: x µ = 1 2 iθβλ iθ ασ σ Rµαβλ / g (29 W (f W (g = (2π 3 2 (2π 3 2 here d 4 kd 4 pe ik x+ip x+ikµ xµ +ip ν x ν i 2 θµν k µp ν f (k g (p = W (f g (30 (f g (x = e xµ and to the second order in θ 2, one obtains: x µ + x ν y ν + i 2 θµν x µ y ν f (x g (y] x=y (31 (f g (x = f (x g (x + x µ µ (f (x g (x + i 2 θµν x µ f (x hich can be reritten as + i 2 θµν i 2 θαβ x µ x α f (x x ν g (x (32 xν g (x +... xβ here (f g (x = f (x g (x + i 2 θµν x µ f (x x i 2 θµν i 2 θαβ x µ g (x + (33 ν x f (x g (x +.. α x ν xβ x µ = x µ + x µ = x µ iθβλ iθ ασ σ R µ αβλ / g. (34 Notice that one can add to the expression of Xµ the term Γ µ αβ Γ ρ ρλ xα x β x λ. Hoever, if e require that the Weyl ordering of a product of non coupled terms like Σ µ αβλ...ϱ xα x β x λ... x ϱ and Λ ν πστ...κ x π x σ x τ... x κ factorizes i.e.: W αβλ...ϱ xα x β x λ... x ϱ Λν πστ...κ x π x σ x τ... x κ = W αβλ...ϱ xα x β x λ... x ϱ W ( Λν πστ...κ x π x σ x τ... x κ Then, this term does not contribute since (35
6 42 Electronic Journal of Theoretical Physics 3, No. 12 ( ( Γρ = Γ ρ ρλ xλ / g = iθ λσ σ Γρ ρλ / g (36 No, using the fact that one deduce that: Γ ρ ρλ = λ log (g (37 ( Γρ ρλ xλ = iθ λσ σ Γρ ρλ / g = iθ λσ σ λ log (g / g = 0 ( T he N oncommutative Action Let us calculate the mass term Φ + Φ here Φ is a complex scalar field in this ne noncommuting space-time: Eq.(39 can be reritten as: ( Φ + Φ (x = Φ + (x Φ (x + total derivative (39 ( Φ + Φ (x = Φ + (x + x Φ (x + x = Φ + (x Φ (x+ x µ µ Φ + (x Φ (x ] +total derivative (40 hich after direct simplifications, the action reads: I = d 4 x ( Φ + Φ (x = d 4 xφ + (x Φ (x+ 1 2 d 4 xθ βλ θ ασ R µ αβλ σ µ Φ + (x Φ (x ] g ( d Gravity coupled to a scalar field Let us calculate this action in to dimensions, choosing θ 01 = +η, ith η 1, The gravitational term reads θ βλ θ ασ R µ αβλ σ µ Φ + (x Φ (x ] / g = θ βλ θ ασ R µαβλ µ σ Φ + (x Φ (x ] / g (42 hich can be simplified as: θ βλ θ ασ R µ αβλ σ µ Φ + (x Φ (x ] / g = 2η 2 R 0101 g µν µ ν Φ + (x Φ (x ] / g (43 In to dimensions, the scalar curvature:
7 Electronic Journal of Theoretical Physics 3, No. 12 ( is related to the component R 0101 by the relation : R = g µν R µν (44 Straightforard simplifications lead to: R 0101 = 1 gr (45 2 R 00 = g 11 R 0101 (46 and R 11 = g 00 R 0101 (47 R 01 = R 10 = g 01 R 0101 and consequently, the scalar curvature R reads R = 2g 1 R 0101 (48 Finally the action gets the form: I = d 2 xφ + (x Φ (x 1 2 η2 d 2 x grg µν µ ν Φ + (x Φ (x ] (49 3. Conclusion Thought this ork e have considered a noncommutative curved space-time in a gravitational background and define ne Moyal-Weyl star product and Weyl ordering at the order of θ 2 (θ is the order parameter of the noncommutative of the space timehere the geometric structure is included. As an example, e have considered the mass term of a complex field and sho explicitly the gravitation effect on the noncommutative-space time. (More studies are under investigations. Appendix e have αβ xα x β = ( αβ xα x α Σµ αβ x β x α αβ xβ x β Σµ αβ / g (A-1 hich can be reritten as: αβ xα x β == ( x x α β, Σ ] µ αβ x α, Σ ] µ αβ x β / g (A-2
8 44 Electronic Journal of Theoretical Physics 3, No. 12 ( using the relations eq.(16 e obtain: ( αβ xα x β = iθ βλ x α λ Σµ αβ λ Σ µ βα xα / ( g = iθ βλ x α λ Σµ αβ λ Σ µ αβ xα / g (A-3 thus, ] αβ xα x β = iθ βλ x α, λ Σµ αβ / g = iθ βλ iθ ασ σ λ Σµ αβ / g Acknoledgement (A-4 One of us (N. Mebarki is very grateful to Profs. P. Aurenche and F. Boudjemaa for the hospitality during the visit to LAPTH d Annecy.This ork is supported by the research grant N D2005/13/04
9 Electronic Journal of Theoretical Physics 3, No. 12 ( References 1] J.Madore, S.Schraml,P.Schupp and J.Wess, Eur.Phys.Jour.C16( ] M.R.Douglas and N.A.Nekrasov,Rev.Mod.Phys.73,( ] J.K.Glikman and S.Noak, Int.Journ.Mod.Phys.A12,( ] X.Calmet,B.Jurco,J.Wess and M.Wohlgenannt, Eur.Phys.Jour.C23,( ] X.Calmet,and M.Wohlgenannt,Phys.Rev.D68( ] J.Madore, An introduction to Noncommutetive geometry and its physical applications, Cambridge University Press ] F.Ardalan, H.Arafi and M.M.Sheikh-Jabbari JHEP( ] N.Seiberg and E.Witten, JHEP( ] M.Chaichan, A.Demitchev, P.Presnajder, M.M.Sheick-jabbari and A.Tureanu, Nucl.Phys.B611( ] M.M.Sheick-Jabbari,JHEP( ] J.E.Moyal, Proc.Cambridge Phil. Soc.45,(
Higher 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)
derivation 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
6.1. 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
Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3
Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 1 State vector space and the dual space Space of wavefunctions The space of wavefunctions is the set of all
2 Composition. Invertible Mappings
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,
C.S. 430 Assignment 6, Sample Solutions
C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order
Phys460.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
Congruence Classes of Invertible Matrices of Order 3 over F 2
International Journal of Algebra, Vol. 8, 24, no. 5, 239-246 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.2988/ija.24.422 Congruence Classes of Invertible Matrices of Order 3 over F 2 Ligong An and
Partial Trace and Partial Transpose
Partial Trace and Partial Transpose by José Luis Gómez-Muñoz http://homepage.cem.itesm.mx/lgomez/quantum/ jose.luis.gomez@itesm.mx This document is based on suggestions by Anirban Das Introduction This
Srednicki 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
= {{D α, D α }, D α }. = [D α, 4iσ µ α α D α µ ] = 4iσ µ α α [Dα, D α ] µ.
![= {{D α, D α }, D α }. = [D α, 4iσ µ α α D α µ ] = 4iσ µ α α [Dα, D α ] µ.](/thumbs/93/113370452.jpg "= {{D α, D α }, D α }. = [D α, 4iσ µ α α D α µ ] = 4iσ µ α α [Dα, D α ] µ.")
PHY 396 T: SUSY Solutions for problem set #1. Problem 2(a): First of all, [D α, D 2 D α D α ] = {D α, D α }D α D α {D α, D α } = {D α, D α }D α + D α {D α, D α } (S.1) = {{D α, D α }, D α }. Second, {D
Areas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
forms This gives Remark 1. How to remember the above formulas: Substituting these into the equation we obtain with
Week 03: C lassification of S econd- Order L inear Equations In last week s lectures we have illustrated how to obtain the general solutions of first order PDEs using the method of characteristics. We
HOMEWORK 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
Example 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
Space-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
Partial 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
Jesse 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
Finite 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
Every 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
Areas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
CHAPTER 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) =
MATH423 String Theory Solutions 4. = 0 τ = f(s). (1) dτ ds = dxµ dτ f (s) (2) dτ 2 [f (s)] 2 + dxµ. dτ f (s) (3)
![MATH423 String Theory Solutions 4. = 0 τ = f(s). (1) dτ ds = dxµ dτ f (s) (2) dτ 2 [f (s)] 2 + dxµ. dτ f (s) (3)](/thumbs/81/84712331.jpg "MATH423 String Theory Solutions 4. = 0 τ = f(s). (1) dτ ds = dxµ dτ f (s) (2) dτ 2 [f (s)] 2 + dxµ. dτ f (s) (3)")
1. MATH43 String Theory Solutions 4 x = 0 τ = fs). 1) = = f s) ) x = x [f s)] + f s) 3) equation of motion is x = 0 if an only if f s) = 0 i.e. fs) = As + B with A, B constants. i.e. allowe reparametrisations
A Note on Intuitionistic Fuzzy. Equivalence Relation
International Mathematical Forum, 5, 2010, no. 67, 3301-3307 A Note on Intuitionistic Fuzzy Equivalence Relation D. K. Basnet Dept. of Mathematics, Assam University Silchar-788011, Assam, India dkbasnet@rediffmail.com
SCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018
Journal of rogressive Research in Mathematics(JRM) ISSN: 2395-028 SCITECH Volume 3, Issue 2 RESEARCH ORGANISATION ublished online: March 29, 208 Journal of rogressive Research in Mathematics www.scitecresearch.com/journals
Statistical 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
4.6 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
ST5224: 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
Matrices 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
Section 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.
SCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions
SCHOOL OF MATHEMATICAL SCIENCES GLMA Linear Mathematics 00- Examination Solutions. (a) i. ( + 5i)( i) = (6 + 5) + (5 )i = + i. Real part is, imaginary part is. (b) ii. + 5i i ( + 5i)( + i) = ( i)( + i)
( ) 2 and compare to M.
Problems and Solutions for Section 4.2 4.9 through 4.33) 4.9 Calculate the square root of the matrix 3!0 M!0 8 Hint: Let M / 2 a!b ; calculate M / 2!b c ) 2 and compare to M. Solution: Given: 3!0 M!0 8
Approximation 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
EE512: 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
Math221: 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
Homework 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
Concrete 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)
The 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
3.4 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
A summation formula ramified with hypergeometric function and involving recurrence relation
South Asian Journal of Mathematics 017, Vol. 7 ( 1): 1 4 www.sajm-online.com ISSN 51-151 RESEARCH ARTICLE A summation formula ramified with hypergeometric function and involving recurrence relation Salahuddin
Parametrized Surfaces
Parametrized Surfaces Recall from our unit on vector-valued functions at the beginning of the semester that an R 3 -valued function c(t) in one parameter is a mapping of the form c : I R 3 where I is some
Problem Set 9 Solutions. θ + 1. θ 2 + cotθ ( ) sinθ e iφ is an eigenfunction of the ˆ L 2 operator. / θ 2. φ 2. sin 2 θ φ 2. ( ) = e iφ. = e iφ cosθ.
Chemistry 362 Dr Jean M Standard Problem Set 9 Solutions The ˆ L 2 operator is defined as Verify that the angular wavefunction Y θ,φ) Also verify that the eigenvalue is given by 2! 2 & L ˆ 2! 2 2 θ 2 +
Απόκριση 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
Second 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
Orbital angular momentum and the spherical harmonics
Orbital angular momentum and the spherical harmonics March 8, 03 Orbital angular momentum We compare our result on representations of rotations with our previous experience of angular momentum, defined
Exercises 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
Homework 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
ω 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 +
General 2 2 PT -Symmetric Matrices and Jordan Blocks 1
General 2 2 PT -Symmetric Matrices and Jordan Blocks 1 Qing-hai Wang National University of Singapore Quantum Physics with Non-Hermitian Operators Max-Planck-Institut für Physik komplexer Systeme Dresden,
Solutions to Exercise Sheet 5
Solutions to Eercise Sheet 5 jacques@ucsd.edu. Let X and Y be random variables with joint pdf f(, y) = 3y( + y) where and y. Determine each of the following probabilities. Solutions. a. P (X ). b. P (X
1 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
On a four-dimensional hyperbolic manifold with finite volume
BULETINUL ACADEMIEI DE ŞTIINŢE A REPUBLICII MOLDOVA. MATEMATICA Numbers 2(72) 3(73), 2013, Pages 80 89 ISSN 1024 7696 On a four-dimensional hyperbolic manifold with finite volume I.S.Gutsul Abstract. In
Assalamu `alaikum wr. wb.
LUMP SUM Assalamu `alaikum wr. wb. LUMP SUM Wassalamu alaikum wr. wb. Assalamu `alaikum wr. wb. LUMP SUM Wassalamu alaikum wr. wb. LUMP SUM Lump sum lump sum lump sum. lump sum fixed price lump sum lump
Tridiagonal matrices. Gérard MEURANT. October, 2008
Tridiagonal matrices Gérard MEURANT October, 2008 1 Similarity 2 Cholesy factorizations 3 Eigenvalues 4 Inverse Similarity Let α 1 ω 1 β 1 α 2 ω 2 T =......... β 2 α 1 ω 1 β 1 α and β i ω i, i = 1,...,
w o = R 1 p. (1) R = p =. = 1
Πανεπιστήµιο Κρήτης - Τµήµα Επιστήµης Υπολογιστών ΗΥ-570: Στατιστική Επεξεργασία Σήµατος 205 ιδάσκων : Α. Μουχτάρης Τριτη Σειρά Ασκήσεων Λύσεις Ασκηση 3. 5.2 (a) From the Wiener-Hopf equation we have:
6.3 Forecasting ARMA processes
122 CHAPTER 6. ARMA MODELS 6.3 Forecasting ARMA processes The purpose of forecasting is to predict future values of a TS based on the data collected to the present. In this section we will discuss a linear
From the finite to the transfinite: Λµ-terms and streams
From the finite to the transfinite: Λµ-terms and streams WIR 2014 Fanny He f.he@bath.ac.uk Alexis Saurin alexis.saurin@pps.univ-paris-diderot.fr 12 July 2014 The Λµ-calculus Syntax of Λµ t ::= x λx.t (t)u
Practice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1
Conceptual Questions. State a Basic identity and then verify it. a) Identity: Solution: One identity is cscθ) = sinθ) Practice Exam b) Verification: Solution: Given the point of intersection x, y) of the
Strain 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
Homomorphism in Intuitionistic Fuzzy Automata
International Journal of Fuzzy Mathematics Systems. ISSN 2248-9940 Volume 3, Number 1 (2013), pp. 39-45 Research India Publications http://www.ripublication.com/ijfms.htm Homomorphism in Intuitionistic
Inverse 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
Chapter 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
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ting Zhang Stanford May 11, 2001 Stanford, 5/11/2001 1 Outline Ordinal Classification Ordinal Addition Ordinal Multiplication Ordinal
Symmetric 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
Εργαστήριο Ανάπτυξης Εφαρμογών Βάσεων Δεδομένων. Εξάμηνο 7 ο
Εργαστήριο Ανάπτυξης Εφαρμογών Βάσεων Δεδομένων Εξάμηνο 7 ο Procedures and Functions Stored procedures and functions are named blocks of code that enable you to group and organize a series of SQL and PL/SQL
b. Use the parametrization from (a) to compute the area of S a as S a ds. Be sure to substitute for ds!
MTH U341 urface Integrals, tokes theorem, the divergence theorem To be turned in Wed., Dec. 1. 1. Let be the sphere of radius a, x 2 + y 2 + z 2 a 2. a. Use spherical coordinates (with ρ a) to parametrize.
The 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
SOLVING CUBICS AND QUARTICS BY RADICALS
SOLVING CUBICS AND QUARTICS BY RADICALS The purpose of this handout is to record the classical formulas expressing the roots of degree three and degree four polynomials in terms of radicals. We begin with
Other Test Constructions: Likelihood Ratio & Bayes Tests
Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :
Notes on the Open Economy
Notes on the Open Econom Ben J. Heijdra Universit of Groningen April 24 Introduction In this note we stud the two-countr model of Table.4 in more detail. restated here for convenience. The model is Table.4.
Coefficient Inequalities for a New Subclass of K-uniformly Convex Functions
International Journal of Computational Science and Mathematics. ISSN 0974-89 Volume, Number (00), pp. 67--75 International Research Publication House http://www.irphouse.com Coefficient Inequalities for
Reminders: 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
Lecture 13 - Root Space Decomposition II
Lecture 13 - Root Space Decomposition II October 18, 2012 1 Review First let us recall the situation. Let g be a simple algebra, with maximal toral subalgebra h (which we are calling a CSA, or Cartan Subalgebra).
Phys624 Quantization of Scalar Fields II Homework 3. Homework 3 Solutions. 3.1: U(1) symmetry for complex scalar
Homework 3 Solutions 3.1: U(1) symmetry for complex scalar 1 3.: Two complex scalars The Lagrangian for two complex scalar fields is given by, L µ φ 1 µ φ 1 m φ 1φ 1 + µ φ µ φ m φ φ (1) This can be written
Commutative Monoids in Intuitionistic Fuzzy Sets
Commutative Monoids in Intuitionistic Fuzzy Sets S K Mala #1, Dr. MM Shanmugapriya *2 1 PhD Scholar in Mathematics, Karpagam University, Coimbatore, Tamilnadu- 641021 Assistant Professor of Mathematics,
DiracDelta. Notations. Primary definition. Specific values. General characteristics. Traditional name. Traditional notation
DiracDelta Notations Traditional name Dirac delta function Traditional notation x Mathematica StandardForm notation DiracDeltax Primary definition 4.03.02.000.0 x Π lim ε ; x ε0 x 2 2 ε Specific values
Fractional Colorings and Zykov Products of graphs
Fractional Colorings and Zykov Products of graphs Who? Nichole Schimanski When? July 27, 2011 Graphs A graph, G, consists of a vertex set, V (G), and an edge set, E(G). V (G) is any finite set E(G) is
( 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
CRASH 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
Numerical 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) =
Bounding Nonsplitting Enumeration Degrees
Bounding Nonsplitting Enumeration Degrees Thomas F. Kent Andrea Sorbi Università degli Studi di Siena Italia July 18, 2007 Goal: Introduce a form of Σ 0 2-permitting for the enumeration degrees. Till now,
Econ 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
DESIGN 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
The challenges of non-stable predicates
The challenges of non-stable predicates Consider a non-stable predicate Φ encoding, say, a safety property. We want to determine whether Φ holds for our program. The challenges of non-stable predicates
SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM
SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM Solutions to Question 1 a) The cumulative distribution function of T conditional on N n is Pr T t N n) Pr max X 1,..., X N ) t N n) Pr max
Second 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
k A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +
[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +](/thumbs/73/69566903.jpg "k A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +")
Chapter 3. Fuzzy Arithmetic 3- Fuzzy arithmetic: ~Addition(+) and subtraction (-): Let A = [a and B = [b, b in R If x [a and y [b, b than x+y [a +b +b Symbolically,we write A(+)B = [a (+)[b, b = [a +b
Written Examination. Antennas and Propagation (AA ) April 26, 2017.
Written Examination Antennas and Propagation (AA. 6-7) April 6, 7. Problem ( points) Let us consider a wire antenna as in Fig. characterized by a z-oriented linear filamentary current I(z) = I cos(kz)ẑ
SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM
SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM Solutions to Question 1 a) The cumulative distribution function of T conditional on N n is Pr (T t N n) Pr (max (X 1,..., X N ) t N n) Pr (max
THE SECOND ISOMORPHISM THEOREM ON ORDERED SET UNDER ANTIORDERS. Daniel A. Romano
235 Kragujevac J. Math. 30 (2007) 235 242. THE SECOND ISOMORPHISM THEOREM ON ORDERED SET UNDER ANTIORDERS Daniel A. Romano Department of Mathematics and Informatics, Banja Luka University, Mladena Stojanovića
Sequent Calculi for the Modal µ-calculus over S5. Luca Alberucci, University of Berne. Logic Colloquium Berne, July 4th 2008
Sequent Calculi for the Modal µ-calculus over S5 Luca Alberucci, University of Berne Logic Colloquium Berne, July 4th 2008 Introduction Koz: Axiomatisation for the modal µ-calculus over K Axioms: All classical
Section 9.2 Polar Equations and Graphs
180 Section 9. Polar Equations and Graphs In this section, we will be graphing polar equations on a polar grid. In the first few examples, we will write the polar equation in rectangular form to help identify
Uniform Convergence of Fourier Series Michael Taylor
Uniform Convergence of Fourier Series Michael Taylor Given f L 1 T 1 ), we consider the partial sums of the Fourier series of f: N 1) S N fθ) = ˆfk)e ikθ. k= N A calculation gives the Dirichlet formula
Empirical best prediction under area-level Poisson mixed models
Noname manuscript No. (will be inserted by the editor Empirical best prediction under area-level Poisson mixed models Miguel Boubeta María José Lombardía Domingo Morales eceived: date / Accepted: date
Dynamic types, Lambda calculus machines Section and Practice Problems Apr 21 22, 2016
Harvard School of Engineering and Applied Sciences CS 152: Programming Languages Dynamic types, Lambda calculus machines Apr 21 22, 2016 1 Dynamic types and contracts (a) To make sure you understand the
ECE 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 = =
Main source: "Discrete-time systems and computer control" by Α. ΣΚΟΔΡΑΣ ΨΗΦΙΑΚΟΣ ΕΛΕΓΧΟΣ ΔΙΑΛΕΞΗ 4 ΔΙΑΦΑΝΕΙΑ 1
Main source: "Discrete-time systems and computer control" by Α. ΣΚΟΔΡΑΣ ΨΗΦΙΑΚΟΣ ΕΛΕΓΧΟΣ ΔΙΑΛΕΞΗ 4 ΔΙΑΦΑΝΕΙΑ 1 A Brief History of Sampling Research 1915 - Edmund Taylor Whittaker (1873-1956) devised a
Section 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)
(As on April 16, 2002 no changes since Dec 24.)
~rprice/area51/documents/roswell.tex ROSWELL COORDINATES FOR TWO CENTERS As on April 16, 00 no changes since Dec 4. I. Definitions of coordinates We define the Roswell coordinates χ, Θ. A better name will
Arithmetical applications of lagrangian interpolation. Tanguy Rivoal. Institut Fourier CNRS and Université de Grenoble 1
Arithmetical applications of lagrangian interpolation Tanguy Rivoal Institut Fourier CNRS and Université de Grenoble Conference Diophantine and Analytic Problems in Number Theory, The 00th anniversary