d a central role in various aspects of quantum information processing because quantum teleportation, [1] quan-

Size: px
Start display at page:

Download "d a central role in various aspects of quantum information processing because quantum teleportation, [1] quan-"

Transcription

1 Commun. Theor. Phys. Beijing, China pp c Chinese Physical Society Vol. 49, No. 3, March 5, 008 Calculation of Entanglement Entropy for Continuous-Variable Entangled State Based on General Two-Mode Boson Exponential Quadratic Operator in Fock Space DAI Fang-Wen and MA Lei Department of Physics, East China Normal University, Shanghai 0006, China Received February 5, 007 Abstract We obtain an explicit formula to calculate the entanglement entropy of bipartite entangled state of general two-mode boson exponential quadratic operator with continuous variables in Fock space. The simplicity and generality of our formula are shown by some examples. PACS numbers: Ud, p, Ta Key words: entanglement entropy, continuous-variable entangled states, linear quantum transformation Introduction 0 Σ B =, Continuous quantum variables have emerged as an alternative to discrete-level systems for performing quan- 0 tum information processing tasks. Entanglement plays M M M c M Σ B = c d a central role in various aspects of quantum information processing because quantum teleportation, [] quan- Thus, from Eq. one can conveniently obtain the c b c. 3 tum cryptography, [] and quantum dense coding, [3] etc., entanglement entropy of the states in Eq. only by calculating the eigenvalues of negative Hermitian matrix N. are all based on it. Thus, it is interesting to measure the entanglement of continuous variables systems. The von But in general cases, the form of a bipartite continuousvariable entangled state is not written as Eq. explicitly, Neumann entropy [4] of either partial trace of the density operator for any bipartite pure entangled state is considered as a good measurement of quantum entangle- complicated to use Eq.. and then N is more implicitly. In certain cases, it is too ment. Based on the Schmidt decompositions, Parker et In this paper, we shall derive an explicit formula al. [5] have developed an elegant method to calculate the to calculate the entanglement entropy of any bipartite entanglement of the bipartite pure entangled states with continuous-variable entangled states in Fock space. This continuous variables by means of the integral eigenvalue paper is arranged as follows. In Sec., we shall obtain equations in the coordinate-momentum space. But in certain cases, it is somewhat harder to find such a Schmidt the general form of the bipartite continuous-variable entangled states, and then derive the explicit formula to calculate the entanglement entropy of the states. In Sec. 3, basis for the continuous variable systems. By virtue of the linear quantum transformation theory LQTT, [6] Lu et al. [3,4] have provided an alter- by virtue of the explicit formula, we shall calculate the degree of entanglement of some bipartite continuous-variable entangled states. A brief conclusion will close the paper native method to calculate the entanglement entropy for in the last section. any bipartite pure entangled Gaussian state with continuous variables in Fock space. If the density operator of Explicit Formula for Calculating Entanglement Entropy of Bipartite Continuous- the bipartite continuous-variable entangled state can be written as follows: { a ρ = A 0 : exp [a, a a Variable Entangled States M + a a, a M. General Form of Entangled State of Two- a Mode Boson Exponential Quadratic Operator a ]} + a, a M : a where A 0 is a normalization factor, M i i =, are Hermitian matrices, and M = e f f e with e and f being two arbitrary complex numbers. Then the entanglement entropy of any bipartite pure entangled Gaussian state with continuous variables in Fock space can be calculated by the following formula, E = { ln det e N + tr [ N e N ]}, ln where a d N = lnm, M =, ac bd =, b c Let us consider the entangled state of a two-mode boson operator with continuous variables as follows: ψ = U 00, 4 where U is a two-mode boson exponential quadratic operator. If we denote Λ = a, ã a = a, a ã = a, a, 5 a i and a i i =, are two-mode boson creation and annihilation operators, respectively. Thus, without any loss of generality, the ordinary form of U can be written as U = exp ΛNΣ Λ B, 6 where Σ B, N C 4 4, Σ B = 0 I I 0, I = 0 0, and N satisfies NΣ B = NΣ B. The project supported by the National Fundamental Research Program under Grant No. 006CB904 and National Natural Science Foundation of China under Grant No

2 No. 3 Calculation of Entanglement Entropy for Continuous-Variable Entangled State Based on General Two-Mode 59 Denoting M = e N = A D B C, A, B, C, and D are four complex matrices, U and M thus satisfy [6] By using Eq. 8, we immediately have UΛU = ΛM, 7 MΣ B M = Σ B. 8 Ã B BA = 0, Ã C BD =, 9 If C exists, by using Eq. 9, we can rewrite matrix M as I C M = e N D C 0 I 0 = 0 I 0 C BC. 0 I Therefore, from LQTT, the action of U is equivalent to the product of three LQT operators, i.e., U = U + U 0 U, where U +, U 0, and U satisfy the following expressions, I C U + ΛU+ D = ΛM + = Λ, 0 I C U 0 ΛU0 0 = ΛM 0 = Λ, 0 C I 0 U ΛU = ΛM = Λ BC, I By virtue of LQTT, we can easily obtain 0 C D N + = lnm + =, 0 0 lnc 0 N 0 = lnm 0 =, N = lnm = 0 ln C, BC 0 U + = exp ΛN +Σ B Λ = exp a C Dã, U 0 = exp ΛN 0Σ B Λ = detc / : exp [ a C a ] :, U = exp ΛN Σ B Λ = exp ãbc a. 4 U +, U 0, and U only contain the terms of a i a j, a i a j, and a i a j, respectively, where i, j =,. By noticing expa i a j 00 = exp a i a j 00 = 00, 5 the bipartite continuous-variable entangled state 4 can be rewritten as ψ = U 00 = U + U 0 U 00 = det C / exp a C Dã From the above discussions, we obtain the general form of the entangled state of any two-mode boson exponential quadratic operator 4 as the following form, ψ = det C / expαa + βa + γa a 00, 7 where α, β, and γ are determined by C D.. Explicit Formula of Calculating Entanglement Entropy It is easy to see that the entanglement entropy, i.e., the degree of entanglement of the entangled state 7 between a and a, is determined by α, β, and γ, but which parameter is more predominant? Are α and β really symmetrical? If γ = 0, obviously, the state 7 is a separable state, and its entanglement entropy should be zero. When γ 0, the density operator of Eq. 7 can be written as ρ =detc expαa + βa + γa a exp α a + β a + γ a a. 8 By noticing the normal ordering form of the two-mode vacuum state projector = : e a a a a : 9 after substituting Eq. 9 into Eq. 8, we have, { α ρ =det C : exp a [, a α so we get α M = α a a β + a, a β a a γ 0 + a, a 0 γ a ]} : 0 β γ 0, M = β, M = 0 γ. By using Eq. 3, we find the explicit form of M as follows: M = γ 4 4 α 4 β 4 αβγ + α β γ α 4α β + β γ γ. α 4α β + βγ 4 β Then, substituting it into Eq., we obtain the explicit formula for calculating the entanglement entropy for the continuous variables entangled states 7 as follows: L 4 L + L E = log 4 L L 4L log 4 L + L log 4 L L 4 + log 4 + log L, 3 where the logarithm is defined in the complex domain and L = γ + γ 4 α 4 β 4 αβγ γ + α β γ. 4 From Eq. 4, we can see that α and β are symmetrical and exchangeable for calculating the entanglement entropy. Obviously, it is true because the modes a and a are symmetrical in Fock space. a

3 59 DAI Fang-Wen and MA Lei Vol. 49 When 0 L <, equation 3 can be simplified as E = + L L + i 4 L L arctan log L. 5 When L >, equation 3 can be simplified as E = L + L log L + L 4 log L. 6 With Eqs. 5 and 6, one can easily have E L and E L +, so we have E L=. 7 Equation 7 shows that if, the entangled state 7 will be maximally entangled. The entanglement entropy is plotted as a function of L in Fig., and figure shows the entanglement entropy as a function of α and γ β is fixed to be 0.5. Fig. Entanglement entropy as a function of L E when L. Fig. Entanglement entropy as a function of α and γ β is fixed to be Some Applications 3. Calculating Entanglement Entropy of Common Two-Mode Squeezed Vacuum State Now, let us consider two-mode squeezed vacuum state ξ = exp ξ a a ξa a 00, where ξ = r e iθ is the squeezing parameter, by virtue of LQTT, [6] we can obtain ξ = sech rexp a a eiθ tanhr 00, where α = 0, β = 0, γ = e iθ tanhr, according to Eq. 4, we have By using Eq. 6, we can easily obtain L = tanh r + coth r >. E = cosh r log cosh r sinh r log sinh r. This result has been obtained by other authors through different methods, [5,68] but our formula is more explicit. figure 3a shows that the amount of entanglement entropy of two-mode squeezed vacuum state is approximately linear against the amount of squeezing parameter r. 3. Calculating Entanglement Entropy of Two-Mode One-Sided Squeezed Vacuum State Reference [5] has constructed two-mode one-sided squeezed vacuum state { λ S = exp a + a 4[ a + ]} [ a 00 = sech / λ exp where α = β = tanhλ/4, γ = tanhλ/, so tanhλ 4 a + ] a 00, L = 4 coth λ > and then by Eq. 6 we obtain E = coshλ log + cosh λ cosh λ log sinh λ log cosh λ.

4 No. 3 Calculation of Entanglement Entropy for Continuous-Variable Entangled State Based on General Two-Mode 593 Figure 3b shows that the amount of entanglement entropy of two-mode one-sided squeezed vacuum state is also approximately linear against the amount of squeezing parameter λ, but is about only half of amount of entanglement entropy of the two-mode squeezed vacuum state with the squeezing parameter λ = r. Fig. 3 a Entanglement entropy of two-mode squeezed vacuum state as a function of squeezing parameter r. b The entanglement entropy of two-mode one-sided squeezed vacuum state as a function of squeezing parameter λ. 3.3 Calculating the Entanglement Entropy of Entangled States Produced by a Beam Splitter with Squeezed States Inputs When the two input fields are squeezed, the output state from a beam splitter is [6] B θ, φs ζ S ζ 00, 8 where B θ, φ is the beam splitter operator, [ θ B θ, φ = exp a a e iφ a a e iφ]. t = cosθ/ and r = sinθ/ are the amplitude reflection and transmission coefficients. The beam splitter gives the phase difference between the reflected and transmitted fields. S ζ is the single-mode squeezed operator S ζ = exp ζ a ζa, ζ = s e iϕ is the squeezing parameter. By virtue of LQTT, [6] we can obtain B θ, φs ζ S ζ 00 = coshs coshs exp αa + βa + γa a 00, where α = [ e iϕ cos θ tanhs + e iϕ+φ sin θ ] tanhs, β = [ e iϕ φ sin θ tanhs + e iϕ cos θ ] tanhs, γ = sinθ e iϕ φ tanhs e iϕ+φ tanhs. 9 Thus, substituting Eq. 9 into Eq. 4, the entanglement entropy of the state 8 can be calculated explicitly. The entanglement entropy of the state 8 is plotted in Fig. 4 against the squeezing parameter s and reflection coefficient for s = 0.5. The relative phase φ = 0 in Fig. 4a and φ = π/ in Fig. 4b. These two figures have been obtained in Ref. [6], but it did not give the explicit form of the entanglement entropy. It is necessary to point out that in Ref. [6], natural logarithm is used to calculate the entanglement entropy, but in this paper, the logarithm with base is used. 3.4 Calculate the Entanglement Entropy of the Thermal Vacuum State of a Free Boson Now, we investigate the entanglement entropy of the thermal vacuum state of a free boson, [8], 0 β = e βω / exp e βω/ a ã 00. Substituting α = β = 0, γ = e βω/ into Eq. 4, we have and then, by using Eq. 6, we can easily get L = e βω + e βω >, E = log e βω e βω e βω log e βω.

5 594 DAI Fang-Wen and MA Lei Vol. 49 Reference 8] obtained the same result by virtue of the technique of IWOP, [5] but we get the result more simply. Fig. 4 Entanglement entropy of the beam-splitter output field. The squeezing parameter for one squeezed input is fixed to S = 0.5 while the squeezing parameter for other squeezed state is varied from S = 0 to. The transitivity is R R r. The beam splitter gives phase difference φ = 0 a and φ = π/ b between reflected and transmitted fields. 4 Conclusions In Fock space, by virtue of LQTT, we show that the entangled state of any two-mode boson exponential quadratic operator ψ = U 00 can be rewritten as ψ = detc / expαa + βa + γa a 00, and we obtain the explicit formula to calculate the entanglement entropy of the state. The simplicity and generality of our formula are shown by several entangled states of continuous variables. These states include the common two-mode squeezed vacuum state, the two-mode one-sided squeezed vacuum state, the output states produced by a beam splitter with squeezed states inputs and the thermal vacuum state of a free single boson. It is shown that LQTT and the explicit formula we obtained are power tools for investigating the degree of entanglement for entangled states of two-mode boson system with exponential quadratic operator. Acknowledgments We thank Dr. Jin-Ming Liu for beneficial discussions. References [] C.H. Bennett, G. Brassard, et al., Phys. Rev. Lett [] K. Ekert, Phys. Rev. Lett [3] C.H. Bennett and S.J. Wiesner, Phys. Rev. Lett [4] C.H. Bennett, H.J. Herbert, S. Popescu, and B. Schumacher, Phys. Rev. A [5] S. Parker, S. Bose, and M.B. Plenio, Phys. Rev. A [6] Y.D. Zhang and Z. Tang, J. Math. Phys [7] Y.D. Zhang and Z. Tang, Nuovo Cimento B [8] L. Ma and Y.D. Zhang, Nuovo Cimento B [9] Y.D. Zhang and Z. Tang, Commun. Theor. Phys. Beijing, China [0] Y.D. Zhang, L. Ma, X.B. Wang, et al., Commun. Theor. Phys. Beijing, China [] S.X. Yu and Y.D. Zhang, Commun. Theor. Phys. Beijing, China [] L. Ma, Doctoral Dissertation, University of Science and Technology of China 995. [3] H.X. Lu, Z.B. Chen, J.W. Pan, and Y.D. Zhang, LANL eprint quant-ph/ [4] H.X. Lu, Doctoral Dissertation, University of Science and Technology of China 003. [5] H.Y. Fan, Int. J. Mod. Phys. B [6] M.S. Kim, W. Son, V. Buzek, and P.L. Knight, Phys. Rev. A [7] S.J. van Enk, Phys. Rev. A [8] X.T. Liang, Commun. Theor. Phys. Beijing, China

Phillips Policy Rules 3.1 A simple textbook macrodynamic model

Phillips Policy Rules 3.1 A simple textbook macrodynamic model 1 2 3 ( ) ( ) = ( ) + ( ) + ( ) ( ) ( ) ( ) [ ] &( ) = α ( ) ( ) α > 0 4 ( ) ( ) = ( ) 0 < < 1 ( ) = [ ] &( ) = α ( 1) ( ) + + ( ) 0 < < 1 + = 1 5 ( ) ( ) &( ) = β ( ( ) ( )) β > 0 ( ) ( ) ( ) β ( ) =

More information

Quantum Discord in Two-Qubit System Constructed from the Yang Baxter Equation

Quantum Discord in Two-Qubit System Constructed from the Yang Baxter Equation Commun. Theor. Phys. 61 (014) 349 353 Vol. 61, No. 3, March 1, 014 Quantum Discord in Two-Qubit System Constructed from the Yang Baxter Equation GOU Li-Dan ( ), 1 WANG Xiao-Qian ( ¼), 1, XU Yu-Mei (Å Ö),

More information

Mixed states and pure states

Mixed states and pure states Mixed states and pure states (Dated: April 9, 2009) These are brief notes on the abstract formalism of quantum mechanics. They will introduce the concepts of pure and mixed quantum states. Some statements

More information

Many-body entanglement in decoherence processes

Many-body entanglement in decoherence processes PHYSICAL REVIEW A 68, 63814 3 Many-body entanglement in decoherence processes Helen McAneney, Jinhyoung Lee, and M. S. Kim School of Mathematics and Physics, Queen s University, Belfast BT7 1NN, United

More information

classical vs. quantum statistics, quasi-probability distributions

classical vs. quantum statistics, quasi-probability distributions Lecture 6: Quantum states in phase space classical vs. quantum statistics, quasi-probability distributions operator expansion in phase space Classical vs. quantum statistics, quasi-probability distributions:

More information

Exact Jacobian Elliptic Function Solutions to sine-gordon Equation

Exact Jacobian Elliptic Function Solutions to sine-gordon Equation Commun. Theor. Phys. (Beijing, China 44 (005 pp. 3 30 c International Academic Publishers Vol. 44, No., July 5, 005 Exact Jacobian Elliptic Function Solutions to sine-gordon Equation FU Zun-Tao,,, YAO

More information

Physics 505 Fall 2007 Homework Assignment #2 Solutions. Textbook problems: Ch. 2: 2.2, 2.8, 2.10, 2.11

Physics 505 Fall 2007 Homework Assignment #2 Solutions. Textbook problems: Ch. 2: 2.2, 2.8, 2.10, 2.11 Physics 55 Fall 27 Homework Assignment #2 Solutions Textbook problems: Ch. 2: 2.2, 2.8, 2., 2. 2.2 Using the method of images, discuss the problem of a point charge q inside a hollow, grounded, conducting

More information

Teleportation of a two-particle entangled state via W class states

Teleportation of a two-particle entangled state via W class states Teleportation of a two-particle entangled state via W class states Zhuo-Liang Cao Wei Song Department of Physics nhui University Hefei 339 P. R. of China bstract scheme for teleporting an unknown two-particle

More information

Quantum key distribution via quantum encryption

Quantum key distribution via quantum encryption Quantum key distribution via quantum encryption Yong-Sheng Zhang, Chuan-Feng Li, Guang-Can Guo Laboratory of Quantum Communication and Quantum Computation and Department of Physics, University of Science

More information

Quantification of entanglement via uncertainties

Quantification of entanglement via uncertainties Quantification of entanglement via uncertainties Barış Öztop Bilkent University Department of Physics September 2007 In blessed memory of Alexander Stanislaw Shumovsky Outline Introduction what can we

More information

Ph 219a/CS 219a. Exercises Due: Friday 2 December 2005

Ph 219a/CS 219a. Exercises Due: Friday 2 December 2005 1 Ph 219a/CS 219a Exercises Due: Friday 2 December 2005 3.1 Two inequivalent types of tripartitite entangled pure states Alice, Bob, and Charlie share a GHZ state of three qubits: GHZ = 1 2 ( 000 + 111

More information

arxiv:quant-ph/9607009v1 11 Jul 1996

arxiv:quant-ph/9607009v1 11 Jul 1996 Distillability of Inseparable Quantum Systems Micha l Horodecki Department of Mathematics and Physics University of Gdańsk, 80 952 Gdańsk, Poland arxiv:quant-ph/9607009v1 11 Jul 1996 Pawe l Horodecki Faculty

More information

Math Spring 2014 Solutions to Assignment # 4 Completion Date: Friday May 16, f(z) = 3x + y + i (3y x)

Math Spring 2014 Solutions to Assignment # 4 Completion Date: Friday May 16, f(z) = 3x + y + i (3y x) Math 311 - Spring 2014 Solutions to Assignment # 4 Completion Date: Friday May 16, 2014 Question 1. [p 77, #1 (a)] Apply the theorem in Sec. 22 to verify that the function is entire. f(z) = 3x + y + i

More information

Teleportation improvement by inconclusive photon subtraction

Teleportation improvement by inconclusive photon subtraction PHYSICAL REVIEW A 67, 03314 003 Teleportation improvement by inconclusive photon subtraction Stefano Olivares, 1 Matteo G. A. Paris, and Rodolfo Bonifacio 1 1 Dipartimento di Fisica and Unità INFM, Università

More information

Small oscillations with many degrees of freedom

Small oscillations with many degrees of freedom Analytical Dynamics - Graduate Center CUNY - Fall 007 Professor Dmitry Garanin Small oscillations with many degrees of freedom General formalism Consider a dynamical system with N degrees of freedom near

More information

Quantum Squeezing of Dark Solitons in Optical Fibers

Quantum Squeezing of Dark Solitons in Optical Fibers Commun. Theor. Phys. 56 (11) 3 36 Vol. 56, No., August 15, 11 Quantum Squeezing of Dark Solitons in Optical Fibers G.R. Honarasa, 1,, M. Hatami, 1 and M.K. Tavassoly 1 1 Atomic and Molecular Group, Faculty

More information

5 Indefinite integral

5 Indefinite integral 5 Indefinite integral The most of the mathematical operations have inverse operations: the inverse operation of addition is subtraction, the inverse operation of multiplication is division, the inverse

More information

CONSTRUCTING FAMILIES TRAVELING WAVE SOLUTIONS IN TERMS OF SPECIAL FUNCTION FOR THE ASYMMETRIC NIZHNIK NOVIKOV VESSELOV EQUATION

CONSTRUCTING FAMILIES TRAVELING WAVE SOLUTIONS IN TERMS OF SPECIAL FUNCTION FOR THE ASYMMETRIC NIZHNIK NOVIKOV VESSELOV EQUATION International Journal of Modern Physics C Vol. 15, No. 4 (004 595 606 c World Scientific Publishing Company CONSTRUCTING FAMILIES TRAVELING WAVE SOLUTIONS IN TERMS OF SPECIAL FUNCTION FOR THE ASYMMETRIC

More information

ON THE FIBONACCI NUMBERS

ON THE FIBONACCI NUMBERS ON THE FIBONACCI NUMBERS Prepared by Kei Nakamura The Fibonacci numbers are terms of the sequence defined in a quite simple recursive fashion. However, despite its simplicity, they have some curious properties

More information

3.7 Non-autonomous linear systems of ODE. General theory

3.7 Non-autonomous linear systems of ODE. General theory 3.7 Non-autonomous linear systems of ODE. General theory Now I will study the ODE in the form ẋ = A(t)x + g(t), x(t) R k, A, g C(I), (3.1) where now the matrix A is time dependent and continuous on some

More information

Is bound entanglement for continuous variables a rare phenomenon?

Is bound entanglement for continuous variables a rare phenomenon? Is bound entanglement for continuous variables a rare phenomenon? Pawe l Horodecki 1 and Maciej Lewenstein 2 1 Faculty of Applied Physics and Mathematics Technical University of Gdańsk, 80 952 Gdańsk,

More information

(x) = lim. x 0 x. (2.1)

(x) = lim. x 0 x. (2.1) Differentiation. Derivative of function Let us fi an arbitrarily chosen point in the domain of the function y = f(). Increasing this fied value by we obtain the value of independent variable +. The value

More information

MATH36001 Background Material 2015

MATH36001 Background Material 2015 MATH3600 Background Material 205 Matrix Algebra Matrices and Vectors An ordered array of mn elements a ij (i =,, m; j =,, n) written in the form a a 2 a n A = a 2 a 22 a 2n a m a m2 a mn is said to be

More information

A note on the symmetry properties of Löwdin s orthogonalization schemes. Abstract

A note on the symmetry properties of Löwdin s orthogonalization schemes. Abstract A note on the symmetry properties of Löwdin s orthogonalization schemes T. A. Rokob Chemical Research Center, ungarian Academy of Sciences, -155 Budapest, POB 17, ungary Á. Szabados and P. R. Surján Eötvös

More information

Paper I ( ALGEBRA AND TRIGNOMETRY )

Paper I ( ALGEBRA AND TRIGNOMETRY ) Paper I ( ALGEBRA AND TRIGNOMETRY ) Dr. J. N. Chaudhari Prof. P. N. Tayade Prof. Miss. R. N. Mahajan Prof. P. N. Bhirud Prof. J. D. Patil M. J. College, Jalgaon Dr. A. G. D. Bendale Mahila Mahavidyalaya,

More information

Module 3F2: Systems and Control EXAMPLES PAPER 1 - STATE-SPACE MODELS

Module 3F2: Systems and Control EXAMPLES PAPER 1 - STATE-SPACE MODELS Cambridge University Engineering Dept. Third year Module 3F2: Systems and Control EXAMPLES PAPER - STATE-SPACE MODELS. A feedback arrangement for control of the angular position of an inertial load is

More information

Rutgers - Physics Graduate Qualifying Exam Quantum Mechanics: September 1, 2006

Rutgers - Physics Graduate Qualifying Exam Quantum Mechanics: September 1, 2006 Rutgers - Physics Graduate Qualifying Exam Quantum Mechanics: September 1, 2006 QA J is an angular momentum vector with components J x, J y, J z. A quantum mechanical state is an eigenfunction of J 2 J

More information

Linear Algebra In Dirac Notation

Linear Algebra In Dirac Notation Chapter 3 Linear Algebra In Dirac Notation 3.1 Hilbert Space and Inner Product In Ch. 2 it was noted that quantum wave functions form a linear space in the sense that multiplying a function by a complex

More information

1 Ordinary Differential Equations Separation of Variables

1 Ordinary Differential Equations Separation of Variables Ordinary Differential Equations Separation of Variables. Introduction Calculus is fundamentally important for the simple reason that almost everything we stu is subject to change. In many if not most such

More information

Linear systems of ordinary differential equations

Linear systems of ordinary differential equations Linear systems of ordinary differential equations (This is a draft and preliminary version of the lectures given by Prof. Colin Atkinson FRS on 2st, 22nd and 25th April 2008 at Tecnun Introduction. This

More information

9 Matrices, determinants, inverse matrix, Cramer s Rule

9 Matrices, determinants, inverse matrix, Cramer s Rule AAC - Business Mathematics I Lecture #9, December 15, 2007 Katarína Kálovcová 9 Matrices, determinants, inverse matrix, Cramer s Rule Basic properties of matrices: Example: Addition properties: Associative:

More information

Quantum implementation of elementary arithmetic operations

Quantum implementation of elementary arithmetic operations Quantum implementation of elementary arithmetic operations G. Florio and D. Picca INFN (Sezione di Bari and Dipartimento di Fisica, Università di Bari, Via Orabona 4, 706, Bari, Italy Abstract Quantum

More information

Definition of entanglement for pure and mixed states

Definition of entanglement for pure and mixed states Definition of entanglement for pure and mixed states seminar talk given by Marius Krumm in the master studies seminar course Selected Topics in Mathematical Physics: Quantum Information Theory at the University

More information

Lecture Quantitative Finance Spring Term 2015

Lecture Quantitative Finance Spring Term 2015 (1,1) Lecture Quantitative Finance Spring Term 2015 Prof. Dr. Erich Walter Farkas : May 21, 2015 1 / 58 Outline (1,1) 1 (1,1) 2 / 58 Outline of the Presentation (1,1) 1 (1,1) 3 / 58 (1,1) The goal of this

More information

Harmonic Oscillator Physics

Harmonic Oscillator Physics Physics 34 Lecture 9 Harmonic Oscillator Physics Lecture 9 Physics 34 Quantum Mechanics I Friday, February th, 00 For the harmonic oscillator potential in the time-independent Schrödinger equation: d ψx

More information

THE LAWS OF COSINES FOR NON-EUCLIDEAN TETRAHEDRA

THE LAWS OF COSINES FOR NON-EUCLIDEAN TETRAHEDRA THE LAWS OF COSINES FOR NON-EUCLIDEAN TETRAHEDRA B.D.S. MCCONNELL Darko Veljan s article The 500-Year-Old Pythagorean Theorem 1 discusses the history and lore of probably the only nontrivial theorem in

More information

A matrix over a field F is a rectangular array of elements from F. The symbol

A matrix over a field F is a rectangular array of elements from F. The symbol Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted

More information

t-h ey are.. r.. v-e.. an t.. bo th for th e.. s-t ru c.. tur alan d.. t he.. quait.. ta ive un de \ centerline {. Inrodu \quad c t i o na r s

t-h ey are.. r.. v-e.. an t.. bo th for th e.. s-t ru c.. tur alan d.. t he.. quait.. ta ive un de \ centerline {. Inrodu \quad c t i o na r s - - - I - ˆ - - - q I q I ˆ I q R R q I q q I R R R R - - - ˆ @ & q k 7 q k O q k 8 & q & k P S q k q k ˆ q k 3 q k ˆ A & [ 7 O [8 P & S & [ [ 3 ˆ A @ q ˆ U q - : U [ U φ : U D φ φ Dφ A ψ A : I SS N :

More information

Bit error rate in multipath wireless channels with several specular paths

Bit error rate in multipath wireless channels with several specular paths Bit error rate in multipath wireless channels with several specular paths C. Chen and A. Adi In this letter a recursive and computationally efficient new formula for it error rate (BER) in multipath channels

More information

MITES 2010: Physics III Survey of Modern Physics Final Exam Solutions

MITES 2010: Physics III Survey of Modern Physics Final Exam Solutions MITES 2010: Physics III Survey of Modern Physics Final Exam Solutions Exercises 1. Problem 1. Consider a particle with mass m that moves in one-dimension. Its position at time t is x(t. As a function of

More information

Fourier Transforms The Fourier Transform Properties of the Fourier Transform Some Special Fourier Transform Pairs 27

Fourier Transforms The Fourier Transform Properties of the Fourier Transform Some Special Fourier Transform Pairs 27 24 Contents Fourier Transforms 24.1 The Fourier Transform 2 24.2 Properties of the Fourier Transform 14 24.3 Some Special Fourier Transform Pairs 27 Learning outcomes In this Workbook you will learn about

More information

Mathematical Background

Mathematical Background Appendix A Mathematical Background A.1 Joint, Marginal and Conditional Probability Let the n (discrete or continuous) random variables y 1,..., y n have a joint joint probability probability p(y 1,...,

More information

Lecture Quantitative Finance

Lecture Quantitative Finance Lecture Quantitative Finance Spring 2011 Prof. Dr. Erich Walter Farkas Lecture 12: May 19, 2011 Chapter 8: Estimating volatility and correlations Prof. Dr. Erich Walter Farkas Quantitative Finance 11:

More information

From the first principles, we define the complex exponential function as a complex function f(z) that satisfies the following defining properties:

From the first principles, we define the complex exponential function as a complex function f(z) that satisfies the following defining properties: 3. Exponential and trigonometric functions From the first principles, we define the complex exponential function as a complex function f(z) that satisfies the following defining properties: 1. f(z) is

More information

1 Review of complex numbers

1 Review of complex numbers 1 Review of complex numbers 1.1 Complex numbers: algebra The set C of complex numbers is formed by adding a square root i of 1 to the set of real numbers: i = 1. Every complex number can be written uniquely

More information

Module 3: Second-Order Partial Differential Equations

Module 3: Second-Order Partial Differential Equations Module 3: Second-Order Partial Differential Equations In Module 3, we shall discuss some general concepts associated with second-order linear PDEs. These types of PDEs arise in connection with various

More information

3.12 SL(2, C) and Its Subgroups: SL(2, R), SU(2), SU(1, 1) and SO(1, 2)

3.12 SL(2, C) and Its Subgroups: SL(2, R), SU(2), SU(1, 1) and SO(1, 2) The Ricci curvature tensor of a bi-invariant metric for semi-simple group is proportional to the metric and, therefore, the scalar curvature is Ric = 1 4 g, (3.211) R = n 4. (3.212) 3.12 SL(2, ) and Its

More information

Introduction II. The Fieller Method

Introduction II. The Fieller Method ρ ψ = φ ρ = β φ = β = β + ε ε σ 2 β = σ = 㭗匷 㭗匷 ( ( ) ) β β σ ψ ψ ρ = φ ρ = β φ = β % ψ ψ + ψ + = = β σ = σ β β = β σ ± > ( ψ ψ ) = β = β = > > < é ( )% { } ( ) ( ) = β β ψ {( β ) ( β ) ψ } ± { ( ) ( σ

More information

On a Problem of Large-Amplitude Oscillation of a Non-linear Conservative System with Inertia and Static Non-linearity

On a Problem of Large-Amplitude Oscillation of a Non-linear Conservative System with Inertia and Static Non-linearity Applied Mathematics & Information Sciences 1(2)(2007), 173-183 An International Journal c 2007 Dixie W Publishing Corporation, U. S. A. On a Problem of Large-Amplitude Oscillation of a Non-linear Conservative

More information

Analysis of the Probability Model of Wind Load on the Offshore Wind Turbine

Analysis of the Probability Model of Wind Load on the Offshore Wind Turbine Open Journal of Civil Engineering, 015, 5, 61-67 Published Online June 015 in SciRes. http://www.scirp.org/journal/ojce http://dx.doi.org/10.436/ojce.015.506 Analysis of the Probability Model of Wind Load

More information

AREAS OF SPHERICAL AND HYPERBOLIC TRIANGLES IN TERMS OF THEIR MIDPOINTS. G.M. Tuynman

AREAS OF SPHERICAL AND HYPERBOLIC TRIANGLES IN TERMS OF THEIR MIDPOINTS. G.M. Tuynman ARAS OF SPHRICA AND HYPRBOIC TRIANGS IN TRMS OF THIR MIDPOINTS GM Tuynman Abstract et M be either the 2-sphere S 2 R 3 or the hyperbolic plane H 2 R 3 If abc is a geodesic triangle on M with corners at

More information

5. Möbius Transformations

5. Möbius Transformations 5. Möbius Transformations 5.1. The linear transformation and the inversion. In this section we investigate the Möbius transformation which provides very convenient methods of finding a one-to-one mapping

More information

Matrix Algebra LECTURE 1. Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 = a 11 x 1 + a 12 x 2 + +a 1n x n,

Matrix Algebra LECTURE 1. Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 = a 11 x 1 + a 12 x 2 + +a 1n x n, LECTURE 1 Matrix Algebra Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 a 11 x 1 + a 12 x 2 + +a 1n x n, (1) y 2 a 21 x 1 + a 22 x 2 + +a 2n x n, y m a m1 x 1 +a m2 x

More information

arxiv:cond-mat/ v3 11 Feb 1998

arxiv:cond-mat/ v3 11 Feb 1998 Maxwell-Schrödinger Equation for Polarized Light and Evolution of the Stokes Parameters Hiroshi Kuratsuji and Shouhei Kakigi Department of Physics, Ritsumeikan University-BKC, Kusatsu City 55-8577, Japan

More information

MATH2001 Development of Mathematical Ideas History of Solving Polynomial Equations

MATH2001 Development of Mathematical Ideas History of Solving Polynomial Equations MATH2001 Development of Mathematical Ideas History of Solving Polynomial Equations 19/24 April 2012 Lagrange s work on general solution formulae for polynomial equations The formulae for the cubic and

More information

Introduction to Quantum Information Theory. Carlos Palazuelos Instituto de Ciencias Matemáticas (ICMAT)

Introduction to Quantum Information Theory. Carlos Palazuelos Instituto de Ciencias Matemáticas (ICMAT) Introduction to Quantum Information Theory Carlos Palazuelos Instituto de Ciencias Matemáticas (ICMAT) carlospalazuelos@icmat.es Madrid, Spain March 013 Contents Chapter 1. A comment on these notes 3

More information

VI. Transcendental Functions. x = ln y. In general, two functions f, g are said to be inverse to each other when the

VI. Transcendental Functions. x = ln y. In general, two functions f, g are said to be inverse to each other when the VI Transcendental Functions 6 Inverse Functions The functions e x and ln x are inverses to each other in the sense that the two statements y = e x, x = ln y are equivalent statements In general, two functions

More information

Project description a) What is quantum information theory with continuous variables?

Project description a) What is quantum information theory with continuous variables? Project description a) What is quantum information theory with continuous variables? According to pre-quantum physics, any theory should display the property of locality defined in three almost equivalent

More information

Pre-Calculus Review Lesson 1 Polynomials and Rational Functions

Pre-Calculus Review Lesson 1 Polynomials and Rational Functions If a and b are real numbers and a < b, then Pre-Calculus Review Lesson 1 Polynomials and Rational Functions For any real number c, a + c < b + c. For any real numbers c and d, if c < d, then a + c < b

More information

Quantum Mechanics I: Basic Principles

Quantum Mechanics I: Basic Principles Quantum Mechanics I: Basic Principles Michael A. Nielsen University of Queensland I ain t no physicist but I know what matters - Popeye the Sailor Goal of this and the next lecture: to introduce all the

More information

UNIT 1: ANALYTICAL METHODS FOR ENGINEERS

UNIT 1: ANALYTICAL METHODS FOR ENGINEERS UNIT : ANALYTIAL METHODS FOR ENGINEERS Unit code: A/6/ QF Level: redit value: 5 OUTOME - ALULUS TUTORIAL INTEGRATION The calculus: the concept of the limit and continuity; definition of the derivative;

More information

Methods - Solutions to Sheet B

Methods - Solutions to Sheet B Methods - Solutions to Sheet B 1. Equation is u t + u = 0 with, t > 0 and u(0, t) = sin t and u(, 0) = sin. First parametrise PDE: ds = 1...(1); d ds = 1...(2); du ds = ds u t + d ds u = 0...(3) Net, parametrise

More information

MISO Capacity with Per-Antenna Power Constraint

MISO Capacity with Per-Antenna Power Constraint 1 MISO Capacity with Per-Antenna Power Constraint Mai Vu Department of Electrical and Computer Engineering, McGill University, Montreal, HAA7 Email: mai.h.vu@mcgill.ca arxiv:100.178v [cs.it] 18 Jan 011

More information

ENTANGLEMENT OF BOSONIC AND FERMIONIC FIELDS IN AN EXPANDING UNIVERSE. Ivette Fuentes University of Nottingham

ENTANGLEMENT OF BOSONIC AND FERMIONIC FIELDS IN AN EXPANDING UNIVERSE. Ivette Fuentes University of Nottingham ENTANGLEMENT OF BOSONIC AND FERMIONIC FIELDS IN AN EXPANDING UNIVERSE Ivette Fuentes University of Nottingham with: Robert B. Mann (U Waterloo) Shahpoor Moradi (U Razi) Eduardo Martin Martinez (CISC Madrid)

More information

MATHEMATICS SPECIALIST ATAR COURSE FORMULA SHEET

MATHEMATICS SPECIALIST ATAR COURSE FORMULA SHEET MATHEMATICS SPECIALIST ATAR COURSE FORMULA SHEET 06 Copyright School Curriculum and Standards Authority, 06 This document apart from any third party copyright material contained in it may be freely copied,

More information

Modern Geometry Homework.

Modern Geometry Homework. Modern Geometry Homework. 1. Rigid motions of the line. Let R be the real numbers. We define the distance between x, y R by where is the usual absolute value. distance between x and y = x y z = { z, z

More information

arxiv: v1 [gr-qc] 21 Jan 2010

arxiv: v1 [gr-qc] 21 Jan 2010 Notes Concerning On the Origin of Gravity and the Laws of Newton by E. Verlinde Jarmo Mäkelä Vaasa University of Applied Sciences, Wolffintie 30, 65200 Vaasa, Finland We point out that certain equations

More information

RADIO PROPAGATION MODELS

RADIO PROPAGATION MODELS RADIO PROPAGATION MODELS 1 Radio Propagation Models 1 Path Loss Free Space Loss Ground Reflections Surface Waves Diffraction Channelization Shadowing 3 Multipath Reception and Scattering Dispersion Time

More information

Mathematical Formulation of the Superposition Principle

Mathematical Formulation of the Superposition Principle Mathematical Formulation of the Superposition Principle Superposition add states together, get new states. Math quantity associated with states must also have this property. Vectors have this property.

More information

Report Assignment 1: A Cancer Model

Report Assignment 1: A Cancer Model Report Assignment 1: A Cancer Model The first report assignment concerns a mathematical model of cancer. It is based on an article by John D. Nagy [Competition and natural selection in a mathematical model

More information

arxiv:hep-ph/ v1 4 Mar 1996

arxiv:hep-ph/ v1 4 Mar 1996 LU TP 96-8 February 996 arxiv:hep-ph/96039v 4 Mar 996 Colour: A Computer Program for QCD Colour Factor Calculations Jari Häkkinen, Hamid Kharraziha Department of Theoretical Physics, Lund University, Sölvegatan

More information

Damped neutrino oscillations

Damped neutrino oscillations Damped neutrino oscillations Erroneous measurements of neutrino oscillation parameters? Based on hep-ph/0502147 Mattias Blennow Division of Mathematical Physics Department of Physics Royal Institute of

More information

M J M j {1,..., J M } i j u ij = X j β αp j + ξ j + ζ ig + (1 σ)ϵ ij, X j ξ j j ϵ ij ζ ig g i ζ + (1 σ)ϵ σ δ j = X j β α i p j + ξ j j D g = ( ) k g δk 1 σ s j = ( ) δj 1 σ Dg σ (1 + Dg 1 σ ), l ij > 0

More information

Full-wave synthetic acoustic logs in porous media

Full-wave synthetic acoustic logs in porous media Global Geology 15 2 151-155 2012 doi 10. 3969 /j. issn. 1673-9736. 2012. 02. 11 Article ID 1673-9736 2012 02-0151-05 Full-wave synthetic acoustic logs in porous media LI Hanqing WANG Zhuwen and ZHANG Xueang

More information

EIGENVALUES AND EIGENVECTORS

EIGENVALUES AND EIGENVECTORS Chapter 6 EIGENVALUES AND EIGENVECTORS 61 Motivation We motivate the chapter on eigenvalues b discussing the equation ax + hx + b = c, where not all of a, h, b are zero The expression ax + hx + b is called

More information

Quantitative Methods for Economics Tutorial 3. Katherine Eyal

Quantitative Methods for Economics Tutorial 3. Katherine Eyal Quantitative Methods for Economics Tutorial 3 Katherine Eyal TUTORIAL 3 9th-13th August 2010 ECO3021S PART 1 1. Let the national income model be: Y = C + I + G C = a + b(y T ) a > 0, 0 < b < 1 T = d +

More information

Tangent and normal lines to conics

Tangent and normal lines to conics 4.B. Tangent and normal lines to conics Apollonius work on conics includes a study of tangent and normal lines to these curves. The purpose of this document is to relate his approaches to the modern viewpoints

More information

EE 321 Analog Electronics, Fall 2013 Homework #5 solution

EE 321 Analog Electronics, Fall 2013 Homework #5 solution EE 321 Analog Electronics, Fall 2013 Homework #5 solution 3.26. For the circuit shown in Fig. P3.26, both diodes are identical, conducting 10mA at 0.7V, and 100mA at 0.8V. Find the value of for which V

More information

3. A LITTLE ABOUT GROUP THEORY

3. A LITTLE ABOUT GROUP THEORY 3. A LITTLE ABOUT GROUP THEORY 3.1 Preliminaries It is an apparent fact that nature exhibits many symmetries, both exact and approximate. A symmetry is an invariance property of a system under a set of

More information

, < x < Using separation of variables, u(x, t) = Φ(x)h(t) (4) we obtain the differential equations. d 2 Φ = λφ (6) Φ(± ) < (7)

, < x < Using separation of variables, u(x, t) = Φ(x)h(t) (4) we obtain the differential equations. d 2 Φ = λφ (6) Φ(± ) < (7) Chapter1: Fourier Transform Solutions of PDEs In this chapter we show how the method of separation of variables may be extended to solve PDEs defined on an infinite or semi-infinite spatial domain. Several

More information

Convex Optimization 3. Convex Functions

Convex Optimization 3. Convex Functions Convex Optimization 3. Convex Functions Goele Pipeleers KU Leuven, Feb. 9-13, 2015 Overview definition properties of convex functions Jensen s inequality restriction to line first-order condition second-order

More information

2.2 Second quantization

2.2 Second quantization 2.2 Second quantization We introduced a compact notation for Slater determinants Ψ S = ψ 1... ψ N. This notation hides the fact that ψ 1... ψ N really stands for a full anti-symmetric state, which can

More information

Entangling rates and the quantum holographic butterfly

Entangling rates and the quantum holographic butterfly Entangling rates and the quantum holographic butterfly David Berenstein DAMTP/UCSB. C. Asplund, D.B. arxiv:1503.04857 Work in progress with C. Asplund, A. Garcia Garcia Questions If a black hole is in

More information

Inverse Problems for Selfadjoint Matrix Polynomials

Inverse Problems for Selfadjoint Matrix Polynomials Peter Lancaster, University of Calgary, Canada Castro Urdiales, SPAIN, 2013. Castro Urdiales, SPAIN, 2013. 1 / Preliminaries Given A 0, A 1,, A l C n n (or possibly in R n n ), L(λ) := l A j λ j, λ C,

More information

Lesson A - Natural Exponential Function and Natural Logarithm Functions

Lesson A - Natural Exponential Function and Natural Logarithm Functions A- Lesson A - Natural Exponential Function and Natural Logarithm Functions Natural Exponential Function In Lesson 2, we explored the world of logarithms in base 0. The natural logarithm has a base of e.

More information

The Characteristic Polynomial

The Characteristic Polynomial Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem

More information

A numerical solution of Nagumo telegraph equation by Adomian decomposition method

A numerical solution of Nagumo telegraph equation by Adomian decomposition method Mathematics Scientific Journal Vol. 6, No. 2, S. N. 13, (2011), 73-81 A numerical solution of Nagumo telegraph equation by Adomian decomposition method H. Rouhparvar a,1 a Department of Mathematics, Islamic

More information

Quantum Fields in Curved Spacetime

Quantum Fields in Curved Spacetime Quantum Fields in Curved Spacetime Lecture 1: Introduction Finn Larsen Michigan Center for Theoretical Physics Yerevan, August 20, 2016. Intro: Fields Setting: many microscopic degrees of freedom interacting

More information

Projectile Motion. Newton s Laws of Motion Click to edit Master title style. Mechanical Phenomena. Physics and Astronomy Department

Projectile Motion. Newton s Laws of Motion Click to edit Master title style. Mechanical Phenomena. Physics and Astronomy Department Projectile Motion PHY 5200 Mechanical Phenomena PHY 5200 Mechanical Phenomena Newton s Laws of Motion Click to edit Master title style Claude A Pruneau Physics and Astronomy Department Wayne State University

More information

Strategy. Theorem Fermat s Last Theorem: If n > 2, then there are no nontrivial integer solutions to x n + y n = z n.

Strategy. Theorem Fermat s Last Theorem: If n > 2, then there are no nontrivial integer solutions to x n + y n = z n. 1. Rewrite equation as α n + β n + γ 3 = 0. 1. Rewrite equation as α n + β n + γ 3 = 0. 2. Exactly one of α, β, γ is divisible by 3. 1. Rewrite equation as α n + β n + γ 3 = 0. 2. Exactly one of α, β,

More information

VON-NEUMANN STABILITY ANALYSIS

VON-NEUMANN STABILITY ANALYSIS university-logo VON-NEUMANN STABILITY ANALYSIS School of Mathematics Semester 1 2008 OUTLINE 1 REVIEW 2 EXPONENTIAL GROWTH Stability under exponential growth 3 VON-NEUMANN STABILITY ANALYSIS 4 SUMMARY

More information

Homework One Solutions. Keith Fratus

Homework One Solutions. Keith Fratus Homework One Solutions Keith Fratus June 8, 011 1 Problem One 1.1 Part a In this problem, we ll assume the fact that the sum of two complex numbers is another complex number, and also that the product

More information

Notes on spherical geometry

Notes on spherical geometry Notes on spherical geometry Math 130 Course web site: www.courses.fas.harvard.edu/5811 This handout covers some spherical trigonometry (following yan s exposition) and the topic of dual triangles. 1. Some

More information

Lecture 10: Characteristic Functions

Lecture 10: Characteristic Functions Lecture 0: Characteristic Functions. Definition of characteristic functions. Complex random variables.2 Definition and basic properties of characteristic functions.3 Examples.4 Inversion formulas 2. Applications

More information

How is a vector rotated?

How is a vector rotated? How is a vector rotated? V. Balakrishnan Department of Physics, Indian Institute of Technology, Madras 600 036 Appeared in Resonance, Vol. 4, No. 10, pp. 61-68 (1999) Introduction In an earlier series

More information

QUADRATIC, EXPONENTIAL AND LOGARITHMIC FUNCTIONS

QUADRATIC, EXPONENTIAL AND LOGARITHMIC FUNCTIONS QUADRATIC, EXPONENTIAL AND LOGARITHMIC FUNCTIONS Content 1. Parabolas... 1 1.1. Top of a parabola... 2 1.2. Orientation of a parabola... 2 1.3. Intercept of a parabola... 3 1.4. Roots (or zeros) of a parabola...

More information

Quadratic Polynomials

Quadratic Polynomials Math 210 Quadratic Polynomials Jerry L. Kazdan Polynomials in One Variable. After studying linear functions y = ax + b, the next step is to study quadratic polynomials, y = ax 2 + bx + c, whose graphs

More information

Module M6.3 Solving second order differential equations

Module M6.3 Solving second order differential equations F L E X I B L E L E A R N I N G A P P R O A C H T O P H Y S I C S Module M6. Solving second order differential equations Opening items. Module introduction.2 Fast track questions. Rea to stu? 2 Methods

More information

RAMANUJAN S HARMONIC NUMBER EXPANSION INTO NEGATIVE POWERS OF A TRIANGULAR NUMBER

RAMANUJAN S HARMONIC NUMBER EXPANSION INTO NEGATIVE POWERS OF A TRIANGULAR NUMBER Volume 9 (008), Issue 3, Article 89, pp. RAMANUJAN S HARMONIC NUMBER EXPANSION INTO NEGATIVE POWERS OF A TRIANGULAR NUMBER MARK B. VILLARINO DEPTO. DE MATEMÁTICA, UNIVERSIDAD DE COSTA RICA, 060 SAN JOSÉ,

More information

Density Evolution. Telecommunications Laboratory. Alex Balatsoukas-Stimming. Technical University of Crete. October 21, 2009

Density Evolution. Telecommunications Laboratory. Alex Balatsoukas-Stimming. Technical University of Crete. October 21, 2009 Density Evolution Telecommunications Laboratory Alex Balatsoukas-Stimming Technical University of Crete October 21, 2009 Telecommunications Laboratory (TUC) Density Evolution October 21, 2009 1 / 27 Outline

More information