Explicit, analytical solution of scaling quantum graphs. Abstract

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Explicit, analytical solution of scaling quantum graphs. Abstract"

Transcription

1 Explicit, analytical solution of scaling quantum gaphs Yu. Dabaghian and R. Blümel Depatment of Physics, Wesleyan Univesity, Middletown, CT , USA (Januay 6, 2003) Abstact Based on ealie wok on egula quantum gaphs we show that a lage class of scaling quantum gaphs with abitay topology ae explicitly analytically solvable. This is supising since quantum gaphs ae excellent models of quantum chaos and quantum chaotic systems ae not usually explicitly analytically solvable Mt,03.65.Sq 1 Typeset using REVTEX

2 Integable systems ae ae. Many have become textbook classics, such as the hamonic oscillato and the hydogen atom. It is theefoe always a notewothy event when anothe integable system is discoveed. The pupose of this Lette is to add an impotant class of quantum systems to this set: quantum gaphs. Quantum gaphs have a long histoy in mathematics, chemisty and physics (see, e.g. the excellent eview by P. Kuchment [1]). They have ecently come to the attention of many eseaches as models of quantum chaos [2 5]. It may theefoe be supising that a lage and impotant subset of quantum gaphs, scaling quantum gaphs, is explicitly integable in the fom E n = f(n), whee n is an intege, E n ae the enegy levels of the gaph and f is a function that can be constucted explicitly. Because of the quantum chaos connection it aleady caused quite a sti when about a yea ago explicit solutions of a sub-class of quantum gaphs, egula quantum gaphs, wee discoveed [6,7]. This is so, because fo any given n the enegy level E n is individually computable, without knowledge of the othe quantum levels. Thus the counting index n is a quantum numbe which is geneally believed not to exist in a quantum system such as quantum gaphs that show so many featues of quantum chaos [8,9]. While in the case of egula quantum gaphs one might shug off the existence of explicit solutions by pointing to the special, non-geneic natue of egula quantum gaphs (only simply connected, linea and cicula gaphs have so fa been identified as membes of the set of egula quantum gaphs), this is now no longe possible. Thus the esults epoted in this Lette ae a substantial, qualitative step fowad with pofound implications fo quantum gaph theoy and quantum chaos. Focussing on scaling quantum gaphs [6,7] excluding gaphs with tunneling and bound states fo convenience (tunneling and bound states ae discussed in [10]), we will pove below that these quantum gaphs, no matte how complex thei topology, ae explicitly solvable analytically. We emphasize that scaling quantum gaphs [6,7] epesent a huge class of gaphs, fa lage than and including the standad, undessed gaphs pimaily studied in the liteatue [1 5]. The class of quantum gaphs fo which we pove the existence of explicit solutions compise quantum gaphs with abitay topology, abitay scaling potentials on thei bonds and abitay scaling δ-functions on thei vetices. 2

3 It has been shown befoe [6,7] that the spectal function of scaling quantum gaphs is given by N g (0) (k) =cos(s 0 k + ϕ 0 ) a j cos(s j k + ϕ j ), (1) j=1 whee S 0 is the action length of the gaph, S j <S 0 ae cetain combinations of the gaph actions, N is the numbe of diffeent gaph action combinations in (1), ϕ 0, ϕ j ae constant phases and a j ae constant amplitudes. The oots k (0) n = E n of the spectal equation g (0) (k n (0) )=0 (2) ae the wave numbes of the eigenenegies E n of the gaph. It was shown in [7] and put on a solid mathematical foundation in [11] that (2) is explicitly solvable in the fom k (0) n =... fo egula quantum gaphs [6,7,11] which satisfy N a j < 1. (3) j=1 The vast majoity of quantum gaphs does not satisfy the egulaity condition (3). Up to now this was believed to be the hedge that makes geneal quantum gaphs esilient against explicit solution and thus peseves one of the cental tenets of quantum chaos theoy, i.e. the loss of quantum numbes [8,9]. Howeve, thee is a way to solve all scaling quantum gaphs, even if they don t fulfill (3), by constucting a chain of spectal functions teminating with a function that does satisfy (3), and then making full use of (3) and the associated mathematical machiney developed fo egula quantum gaphs [11]. Thus the theoy of egula quantum gaphs povides us with a poweful point of depatue fo solving the moe geneal poblem of all scaling quantum gaphs. Taking the m th deivative of (1) and dividing by S m 0 we obtain whee b (m) j g (m) (k) =cos(s 0 k + ϕ 0 + mπ/2) N j=1 b (m) j cos(s j k + ϕ j + mπ/2), (4) = a j (S j /S 0 ) m. Since S j <S 0, thee always exists an M such that b (M) j < 1, i.e. fo m = M (4) satisfies the egulaity condition (3) and the oots k (M) q 3 of g (M) (k) =0ae

4 explicitly computable via explicit analytical, convegent peiodic-obit expansions as shown in [6,7] and poved igoously in [11]. Having bootstapped the oots of (1) at the level M by taking the M th deivative of (1), we now have to go backwads to the levels M 1, M 2,..., until we each level 0 and possess k (0) n explicitly. Retacing ou steps is indeed possible by making use of so-called oot-sepaatos ˆk n, which play an impotant ole in the theoy of egula quantum gaphs [11]. A sepaato ˆk n sepaates oots numbe n 1andnumben fom each othe. Because the theoy of egula quantum gaphs applies to g (M) (k), we know the sepaatos go one level up to M 1, we need to know the sepaatos ˆk (M 1) (M) ˆk q.to which sepaate oots numbe 1andnumbe fom each othe. It tuns out [10] that the extema of g (M 1) (k) ae oot sepaatos on the level M 1. But the location of the extema of g (M 1) (k) ae the zeos of g (M) (k), which we know. With the knowledge of the oot sepaatos compute the zeos k (M 1) of g (M 1) (k): ˆk (M 1) we now ˆk(M 1) k (M 1) = ˆk (M 1) 1 k g (M) (k) δ(g (M 1) (k)) dk. (5) It is impotant to ealize that (5) is not just a fomal solution, but yields k (M 1) explicitly, analytically, even numeically, if we wish, to any desied accuacy. This is so since both g (M 1) (k) and the oot sepaatos ˆk (M 1) ae known explicitly. It is futhemoe impotant to ealize that the tick (5) does not geneally wok fo othe quantum systems, even if thei spectal equation g is known. This is so because the oot sepaatos ˆk n,knownand explicitly constuctable in the case of quantum gaphs, ae not in geneal known fo othe quantum systems. So fo moe geneal quantum (chaotic) systems the missing sepaatos is indeed a hedge that, so fa, potects them against explicit solution. Ou etacing pocess fom levels M to M 1 can now be continued until we each level 0. This completes ou poof that all scaling quantum gaphs can be solved analytically in closed fom. 4

5 I. ACKNOWLEDGMENTS The authos acknowledge financial suppot by the National Science Foundation unde Gant No

6 REFERENCES [1] P. Kuchment, Waves in Rand. Media 12(4), R1 (2002). [2] T. Kottos and U. Smilansky, Phys. Rev. Lett. 79, 4794 (1997); Ann. Phys. (N.Y.) 274, 76 (1999). [3] H. Schanz and U. Smilansky, Phys. Rev. Lett. 84, 1427 (2000). [4] T. Kottos and H. Schanz, Physica E 9, 523 (2001). [5] U. Smilansky, in Mesoscopic Quantum Physics, Les Houches, Session LXI, 1994, edited by E. Akkemans, G. Montambaux, J.-L. Pichad and J. Zinn-Justin (Elsevie Science Publishes, Amstedam, 1995), pp [6] Yu. Dabaghian, R. V. Jensen, and R. Blümel, JETP Lett. 74, 235 (2001); JETP 94, 1201 (2002). [7] R. Blümel, Yu. Dabaghian, and R. V. Jensen, Phys. Rev. Lett. 88, (2002); Phys. Rev. E 65, (2002). [8] M. Gutzwille, Chaos in Classical and Quantum Mechanics (Spinge, New Yok, 1990). [9] H.-J. Stöckmann, Quantum Chaos (Cambidge Univesity Pess, Cambidge, 1999). [10] Yu. Dabaghian and R. Blümel, in pepaation fo submittal to Phys. Rev. E (2003). [11] R. Blümel, Yu. Dabaghian, and R. V. Jensen, Mathematical foundations of egula quantum gaphs, to be published (2003). 6

2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,

2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses, 3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects

More information

The Grating Spectrometer and Atomic Spectra

The Grating Spectrometer and Atomic Spectra PHY 19 Gating Spectomete 1 The Gating Spectomete and Atomic Specta Intoduction In the pevious expeiment diffaction and intefeence wee discussed and at the end a diffaction gating was intoduced. In this

More information

est using the formula I = Prt, where I is the interest earned, P is the principal, r is the interest rate, and t is the time in years.

est using the formula I = Prt, where I is the interest earned, P is the principal, r is the interest rate, and t is the time in years. 9.2 Inteest Objectives 1. Undestand the simple inteest fomula. 2. Use the compound inteest fomula to find futue value. 3. Solve the compound inteest fomula fo diffeent unknowns, such as the pesent value,

More information

On Factoring Arbitrary Integers with Known Bits

On Factoring Arbitrary Integers with Known Bits On Factoing Abitay Integes with Known Bits Mathias Hemann, Alexande May Faculty of Compute Science, TU Damstadt, 689 Damstadt, Gemany hemann@bg.infomatik.tu-damstadt.de, may@infomatik.tu-damstadt.de Abstact:

More information

UNIT CIRCLE TRIGONOMETRY

UNIT CIRCLE TRIGONOMETRY UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + = - - -

More information

Figure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360!

Figure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360! 1. What ae angles? Last time, we looked at how the Geeks intepeted measument of lengths. Howeve, as fascinated as they wee with geomety, thee was a shape that was much moe enticing than any othe : the

More information

FXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it.

FXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it. Candidates should be able to : Descibe how a mass ceates a gavitational field in the space aound it. Define gavitational field stength as foce pe unit mass. Define and use the peiod of an object descibing

More information

Some text, some maths and going loopy. This is a fun chapter as we get to start real programming!

Some text, some maths and going loopy. This is a fun chapter as we get to start real programming! Chapte Two Some text, some maths and going loopy In this Chapte you ae going to: Lean how to do some moe with text. Get Python to do some maths fo you. Lean about how loops wok. Lean lots of useful opeatos.

More information

Section 5-3 Angles and Their Measure

Section 5-3 Angles and Their Measure 5 5 TRIGONOMETRIC FUNCTIONS Section 5- Angles and Thei Measue Angles Degees and Radian Measue Fom Degees to Radians and Vice Vesa In this section, we intoduce the idea of angle and two measues of angles,

More information

mv2. Equating the two gives 4! 2. The angular velocity is the angle swept per GM (2! )2 4! 2 " 2 = GM . Combining the results we get !

mv2. Equating the two gives 4! 2. The angular velocity is the angle swept per GM (2! )2 4! 2  2 = GM . Combining the results we get ! Chapte. he net foce on the satellite is F = G Mm and this plays the ole of the centipetal foce on the satellite i.e. mv mv. Equating the two gives = G Mm i.e. v = G M. Fo cicula motion we have that v =!

More information

Chapter 3 Savings, Present Value and Ricardian Equivalence

Chapter 3 Savings, Present Value and Ricardian Equivalence Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,

More information

ON THE (Q, R) POLICY IN PRODUCTION-INVENTORY SYSTEMS

ON THE (Q, R) POLICY IN PRODUCTION-INVENTORY SYSTEMS ON THE R POLICY IN PRODUCTION-INVENTORY SYSTEMS Saifallah Benjaafa and Joon-Seok Kim Depatment of Mechanical Engineeing Univesity of Minnesota Minneapolis MN 55455 Abstact We conside a poduction-inventoy

More information

1240 ev nm 2.5 ev. (4) r 2 or mv 2 = ke2

1240 ev nm 2.5 ev. (4) r 2 or mv 2 = ke2 Chapte 5 Example The helium atom has 2 electonic enegy levels: E 3p = 23.1 ev and E 2s = 20.6 ev whee the gound state is E = 0. If an electon makes a tansition fom 3p to 2s, what is the wavelength of the

More information

MULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION

MULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION MULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION K.C. CHANG AND TAN ZHANG In memoy of Pofesso S.S. Chen Abstact. We combine heat flow method with Mose theoy, supe- and subsolution method with

More information

Butterfly Network Analysis and The Beneˇ s Network

Butterfly Network Analysis and The Beneˇ s Network 6.895 Theoy of Paallel Systems Lectue 17 Buttefly Netwok Analysis and The Beneˇ s Netwok Lectue: Chales Leiseson Lectue Summay 1. Netwok with N Nodes This section poves pat of the lowe bound on expected

More information

Financing Terms in the EOQ Model

Financing Terms in the EOQ Model Financing Tems in the EOQ Model Habone W. Stuat, J. Columbia Business School New Yok, NY 1007 hws7@columbia.edu August 6, 004 1 Intoduction This note discusses two tems that ae often omitted fom the standad

More information

New proofs for the perimeter and area of a circle

New proofs for the perimeter and area of a circle New poofs fo the peimete and aea of a cicle K. Raghul Kuma Reseach Schola, Depatment of Physics, Nallamuthu Gounde Mahalingam College, Pollachi, Tamil Nadu 64001, India 1 aghul_physics@yahoo.com aghulkumak5@gmail.com

More information

2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES

2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES . TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES In ode to etend the definitions of the si tigonometic functions to geneal angles, we shall make use of the following ideas: In a Catesian coodinate sstem, an

More information

Concept and Experiences on using a Wiki-based System for Software-related Seminar Papers

Concept and Experiences on using a Wiki-based System for Software-related Seminar Papers Concept and Expeiences on using a Wiki-based System fo Softwae-elated Semina Papes Dominik Fanke and Stefan Kowalewski RWTH Aachen Univesity, 52074 Aachen, Gemany, {fanke, kowalewski}@embedded.wth-aachen.de,

More information

Trigonometric Functions of Any Angle

Trigonometric Functions of Any Angle Tigonomet Module T2 Tigonometic Functions of An Angle Copight This publication The Nothen Albeta Institute of Technolog 2002. All Rights Reseved. LAST REVISED Decembe, 2008 Tigonometic Functions of An

More information

TALLINN UNIVERSITY OF TECHNOLOGY, INSTITUTE OF PHYSICS 14. NEWTON'S RINGS

TALLINN UNIVERSITY OF TECHNOLOGY, INSTITUTE OF PHYSICS 14. NEWTON'S RINGS 4. NEWTON'S RINGS. Obective Detemining adius of cuvatue of a long focal length plano-convex lens (lage adius of cuvatue).. Equipment needed Measuing micoscope, plano-convex long focal length lens, monochomatic

More information

PAN STABILITY TESTING OF DC CIRCUITS USING VARIATIONAL METHODS XVIII - SPETO - 1995. pod patronatem. Summary

PAN STABILITY TESTING OF DC CIRCUITS USING VARIATIONAL METHODS XVIII - SPETO - 1995. pod patronatem. Summary PCE SEMINIUM Z PODSTW ELEKTOTECHNIKI I TEOII OBWODÓW 8 - TH SEMIN ON FUNDMENTLS OF ELECTOTECHNICS ND CICUIT THEOY ZDENĚK BIOLEK SPŠE OŽNO P.., CZECH EPUBLIC DLIBO BIOLEK MILITY CDEMY, BNO, CZECH EPUBLIC

More information

Trajectory Following Method on Output Regulation of Affine Nonlinear Control Systems with Relative Degree not Well Defined

Trajectory Following Method on Output Regulation of Affine Nonlinear Control Systems with Relative Degree not Well Defined ITB J. Sci., Vol. 43 A, No., 011, 73-86 73 Taectoy Following Method on Output Regulation of Affine Nonlinea Contol Systems with Relative Degee not Well Defined Janson Naibohu Industial Financial Mathematics

More information

Converting knowledge Into Practice

Converting knowledge Into Practice Conveting knowledge Into Pactice Boke Nightmae srs Tend Ride By Vladimi Ribakov Ceato of Pips Caie 20 of June 2010 2 0 1 0 C o p y i g h t s V l a d i m i R i b a k o v 1 Disclaime and Risk Wanings Tading

More information

Week 3-4: Permutations and Combinations

Week 3-4: Permutations and Combinations Week 3-4: Pemutations and Combinations Febuay 24, 2016 1 Two Counting Pinciples Addition Pinciple Let S 1, S 2,, S m be disjoint subsets of a finite set S If S S 1 S 2 S m, then S S 1 + S 2 + + S m Multiplication

More information

Chapter 23: Gauss s Law

Chapter 23: Gauss s Law Chapte 3: Gauss s Law Homewok: Read Chapte 3 Questions, 5, 1 Poblems 1, 5, 3 Gauss s Law Gauss s Law is the fist of the fou Maxwell Equations which summaize all of electomagnetic theoy. Gauss s Law gives

More information

Macroeconomics I. Antonio Zabalza. University of Valencia 1. Class 5. The IS-LM model and Aggregate Demand

Macroeconomics I. Antonio Zabalza. University of Valencia 1. Class 5. The IS-LM model and Aggregate Demand Macoeconomics I. Antonio Zabalza. Univesity of Valencia 1 Class 5. The IS-LM model and Aggegate Demand 1. Use the Keynesian coss to pedict the impact of: a) An incease in govenment puchases. b) An incease

More information

Universal Cycles. Yu She. Wirral Grammar School for Girls. Department of Mathematical Sciences. University of Liverpool

Universal Cycles. Yu She. Wirral Grammar School for Girls. Department of Mathematical Sciences. University of Liverpool Univesal Cycles 2011 Yu She Wial Gamma School fo Gils Depatment of Mathematical Sciences Univesity of Livepool Supeviso: Pofesso P. J. Giblin Contents 1 Intoduction 2 2 De Buijn sequences and Euleian Gaphs

More information

Gravitation. AP Physics C

Gravitation. AP Physics C Gavitation AP Physics C Newton s Law of Gavitation What causes YOU to be pulled down? THE EARTH.o moe specifically the EARTH S MASS. Anything that has MASS has a gavitational pull towads it. F α Mm g What

More information

Seshadri constants and surfaces of minimal degree

Seshadri constants and surfaces of minimal degree Seshadi constants and sufaces of minimal degee Wioletta Syzdek and Tomasz Szembeg Septembe 29, 2007 Abstact In [] we showed that if the multiple point Seshadi constants of an ample line bundle on a smooth

More information

Channel selection in e-commerce age: A strategic analysis of co-op advertising models

Channel selection in e-commerce age: A strategic analysis of co-op advertising models Jounal of Industial Engineeing and Management JIEM, 013 6(1):89-103 Online ISSN: 013-0953 Pint ISSN: 013-843 http://dx.doi.og/10.396/jiem.664 Channel selection in e-commece age: A stategic analysis of

More information

The transport performance evaluation system building of logistics enterprises

The transport performance evaluation system building of logistics enterprises Jounal of Industial Engineeing and Management JIEM, 213 6(4): 194-114 Online ISSN: 213-953 Pint ISSN: 213-8423 http://dx.doi.og/1.3926/jiem.784 The tanspot pefomance evaluation system building of logistics

More information

Ilona V. Tregub, ScD., Professor

Ilona V. Tregub, ScD., Professor Investment Potfolio Fomation fo the Pension Fund of Russia Ilona V. egub, ScD., Pofesso Mathematical Modeling of Economic Pocesses Depatment he Financial Univesity unde the Govenment of the Russian Fedeation

More information

Lecture 18: Multicut

Lecture 18: Multicut Advanced Appoximation Algoithms (CMU 15-854B, Sping 008) Lectue 18: Multicut Mach 0, 008 Lectue: Anupam Gupta Scibe: Kanat Tangwongsan 1 The Multicut Poblem In this lectue, we will study a diffeent type

More information

Ch. 8 Universal Gravitation. Part 1: Kepler s Laws. Johannes Kepler. Tycho Brahe. Brahe. Objectives: Section 8.1 Motion in the Heavens and on Earth

Ch. 8 Universal Gravitation. Part 1: Kepler s Laws. Johannes Kepler. Tycho Brahe. Brahe. Objectives: Section 8.1 Motion in the Heavens and on Earth Ch. 8 Univesal Gavitation Pat 1: Keple s Laws Objectives: Section 8.1 Motion in the Heavens and on Eath Objectives Relate Keple s laws of planetay motion to Newton s law of univesal gavitation. Calculate

More information

Th Po er of th Cir l. Lesson3. Unit UNIT 6 GEOMETRIC FORM AND ITS FUNCTION

Th Po er of th Cir l. Lesson3. Unit UNIT 6 GEOMETRIC FORM AND ITS FUNCTION Lesson3 Th Po e of th Ci l Quadilateals and tiangles ae used to make eveyday things wok. Right tiangles ae the basis fo tigonometic atios elating angle measues to atios of lengths of sides. Anothe family

More information

CHAT Pre-Calculus Section 10.7. Polar Coordinates

CHAT Pre-Calculus Section 10.7. Polar Coordinates CHAT Pe-Calculus Pola Coodinates Familia: Repesenting gaphs of equations as collections of points (, ) on the ectangula coodinate sstem, whee and epesent the diected distances fom the coodinate aes to

More information

Saturated and weakly saturated hypergraphs

Saturated and weakly saturated hypergraphs Satuated and weakly satuated hypegaphs Algebaic Methods in Combinatoics, Lectues 6-7 Satuated hypegaphs Recall the following Definition. A family A P([n]) is said to be an antichain if we neve have A B

More information

A study modeling of 15 days cumulative rainfall at Purajaya Region, Bandar Lampung, Indonesia

A study modeling of 15 days cumulative rainfall at Purajaya Region, Bandar Lampung, Indonesia A study modeling of 15 days cumulative ainfall at Puajaya Region, Banda Lampung, Indonesia Ahmad Zakaia* Abstact Aim of this eseach is to study peiodic modeling of 15 days cumulative ainfall time seies.

More information

Classical Lifetime of a Bohr Atom

Classical Lifetime of a Bohr Atom 1 Poblem Classical Lifetime of a Boh Atom James D. Olsen and Kik T. McDonald Joseph Heny Laboatoies, Pinceton Univesity, Pinceton, NJ 85 (Mach 7, 5) In the Boh model of the hydogen atom s gound state,

More information

USB True Emulation for KVM Switches. Transparent and reliable USB KVM switching technology.

USB True Emulation for KVM Switches. Transparent and reliable USB KVM switching technology. Tanspaent and eliable KVM switching technology. 0118 724-746-5500 965 6000 www.blackbox.co.uk blackbox.com Table of Contents Intoduction... 3 Enumeated Switching... 4 Emulated Switching... 5 Tue Emulation...

More information

Skills Needed for Success in Calculus 1

Skills Needed for Success in Calculus 1 Skills Needed fo Success in Calculus Thee is much appehension fom students taking Calculus. It seems that fo man people, "Calculus" is snonmous with "difficult." Howeve, an teache of Calculus will tell

More information

Chapter 13 Gravitation. Problems: 1, 4, 5, 7, 18, 19, 25, 29, 31, 33, 43

Chapter 13 Gravitation. Problems: 1, 4, 5, 7, 18, 19, 25, 29, 31, 33, 43 Chapte 13 Gavitation Poblems: 1, 4, 5, 7, 18, 19, 5, 9, 31, 33, 43 Evey object in the univese attacts evey othe object. This is called gavitation. We e use to dealing with falling bodies nea the Eath.

More information

CHAPTER 10 Aggregate Demand I

CHAPTER 10 Aggregate Demand I CHAPTR 10 Aggegate Demand I Questions fo Review 1. The Keynesian coss tells us that fiscal policy has a multiplied effect on income. The eason is that accoding to the consumption function, highe income

More information

The Role of Gravity in Orbital Motion

The Role of Gravity in Orbital Motion ! The Role of Gavity in Obital Motion Pat of: Inquiy Science with Datmouth Developed by: Chistophe Caoll, Depatment of Physics & Astonomy, Datmouth College Adapted fom: How Gavity Affects Obits (Ohio State

More information

SUPPORT VECTOR MACHINE FOR BANDWIDTH ANALYSIS OF SLOTTED MICROSTRIP ANTENNA

SUPPORT VECTOR MACHINE FOR BANDWIDTH ANALYSIS OF SLOTTED MICROSTRIP ANTENNA Intenational Jounal of Compute Science, Systems Engineeing and Infomation Technology, 4(), 20, pp. 67-7 SUPPORT VECTOR MACHIE FOR BADWIDTH AALYSIS OF SLOTTED MICROSTRIP ATEA Venmathi A.R. & Vanitha L.

More information

YARN PROPERTIES MEASUREMENT: AN OPTICAL APPROACH

YARN PROPERTIES MEASUREMENT: AN OPTICAL APPROACH nd INTERNATIONAL TEXTILE, CLOTHING & ESIGN CONFERENCE Magic Wold of Textiles Octobe 03 d to 06 th 004, UBROVNIK, CROATIA YARN PROPERTIES MEASUREMENT: AN OPTICAL APPROACH Jana VOBOROVA; Ashish GARG; Bohuslav

More information

Evidence for the exponential distribution of income in the USA

Evidence for the exponential distribution of income in the USA Eu. Phys. J. B 2, 585 589 (21) THE EUROPEAN PHYSICAL JOURNAL B c EDP Sciences Società Italiana di Fisica Spinge-Velag 21 Evidence fo the exponential distibution of income in the USA A. Dăgulescu and V.M.

More information

Learning Objectives. Decreasing size. ~10 3 m. ~10 6 m. ~10 10 m 1/22/2013. Describe ionic, covalent, and metallic, hydrogen, and van der Waals bonds.

Learning Objectives. Decreasing size. ~10 3 m. ~10 6 m. ~10 10 m 1/22/2013. Describe ionic, covalent, and metallic, hydrogen, and van der Waals bonds. Lectue #0 Chapte Atomic Bonding Leaning Objectives Descibe ionic, covalent, and metallic, hydogen, and van de Waals bonds. Which mateials exhibit each of these bonding types? What is coulombic foce of

More information

Infinite-dimensional Bäcklund transformations between isotropic and anisotropic plasma equilibria.

Infinite-dimensional Bäcklund transformations between isotropic and anisotropic plasma equilibria. Infinite-dimensional äcklund tansfomations between isotopic and anisotopic plasma equilibia. Infinite symmeties of anisotopic plasma equilibia. Alexei F. Cheviakov Queen s Univesity at Kingston, 00. Reseach

More information

1.4 Phase Line and Bifurcation Diag

1.4 Phase Line and Bifurcation Diag Dynamical Systems: Pat 2 2 Bifucation Theoy In pactical applications that involve diffeential equations it vey often happens that the diffeential equation contains paametes and the value of these paametes

More information

On Some Functions Involving the lcm and gcd of Integer Tuples

On Some Functions Involving the lcm and gcd of Integer Tuples SCIENTIFIC PUBLICATIONS OF THE STATE UNIVERSITY OF NOVI PAZAR SER. A: APPL. MATH. INFORM. AND MECH. vol. 6, 2 (2014), 91-100. On Some Functions Involving the lcm and gcd of Intege Tuples O. Bagdasa Abstact:

More information

Simple Harmonic Motion

Simple Harmonic Motion Simple Hamonic Motion Intoduction Simple hamonic motion occus when the net foce acting on an object is popotional to the object s displacement fom an equilibium position. When the object is at an equilibium

More information

Research on Risk Assessment of the Transformer Based on Life Cycle Cost

Research on Risk Assessment of the Transformer Based on Life Cycle Cost ntenational Jounal of Smat Gid and lean Enegy eseach on isk Assessment of the Tansfome Based on Life ycle ost Hui Zhou a, Guowei Wu a, Weiwei Pan a, Yunhe Hou b, hong Wang b * a Zhejiang Electic Powe opoation,

More information

An Introduction to Omega

An Introduction to Omega An Intoduction to Omega Con Keating and William F. Shadwick These distibutions have the same mean and vaiance. Ae you indiffeent to thei isk-ewad chaacteistics? The Finance Development Cente 2002 1 Fom

More information

Theory and practise of the g-index

Theory and practise of the g-index Theoy and pactise of the g-index by L. Egghe (*), Univesiteit Hasselt (UHasselt), Campus Diepenbeek, Agoalaan, B-3590 Diepenbeek, Belgium Univesiteit Antwepen (UA), Campus Die Eiken, Univesiteitsplein,

More information

Continuous Compounding and Annualization

Continuous Compounding and Annualization Continuous Compounding and Annualization Philip A. Viton Januay 11, 2006 Contents 1 Intoduction 1 2 Continuous Compounding 2 3 Pesent Value with Continuous Compounding 4 4 Annualization 5 5 A Special Poblem

More information

The Supply of Loanable Funds: A Comment on the Misconception and Its Implications

The Supply of Loanable Funds: A Comment on the Misconception and Its Implications JOURNL OF ECONOMICS ND FINNCE EDUCTION Volume 7 Numbe 2 Winte 2008 39 The Supply of Loanable Funds: Comment on the Misconception and Its Implications. Wahhab Khandke and mena Khandke* STRCT Recently Fields-Hat

More information

Gravitational Mechanics of the Mars-Phobos System: Comparing Methods of Orbital Dynamics Modeling for Exploratory Mission Planning

Gravitational Mechanics of the Mars-Phobos System: Comparing Methods of Orbital Dynamics Modeling for Exploratory Mission Planning Gavitational Mechanics of the Mas-Phobos System: Compaing Methods of Obital Dynamics Modeling fo Exploatoy Mission Planning Alfedo C. Itualde The Pennsylvania State Univesity, Univesity Pak, PA, 6802 This

More information

Applicable Analysis and Discrete Mathematics available online at

Applicable Analysis and Discrete Mathematics available online at Applicable Analysis and Discete Mathematics available online at http://pefmath.etf.s Appl. Anal. Discete Math. 5 (2011, 212 229. doi:10.2298/aadm110615015d POLYNOMIALS RELATED TO HARMONIC NUMBERS AND EVALUATION

More information

AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM

AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM Main Golub Faculty of Electical Engineeing and Computing, Univesity of Zageb Depatment of Electonics, Micoelectonics,

More information

Chapte 3 Is Gavitation A Results Of Asymmetic Coulomb Chage Inteactions? Jounal of Undegaduate Reseach èjurè Univesity of Utah è1992è, Vol. 3, No. 1, pp. 56í61. Jeæey F. Gold Depatment of Physics, Depatment

More information

The Clustering Coefficient of Multiple Parallel Airlines AANET

The Clustering Coefficient of Multiple Parallel Airlines AANET Intenational Jounal of Futue Geneation Communication and Netwoking Vol. 9, No. 7 (016), pp. 15-144 http://dx.doi.og/10.1457/ijfgcn.016.9.7.1 The Clusteing Coefficient of Multiple Pael Ailines AANET Xue

More information

An Efficient Group Key Agreement Protocol for Ad hoc Networks

An Efficient Group Key Agreement Protocol for Ad hoc Networks An Efficient Goup Key Ageement Potocol fo Ad hoc Netwoks Daniel Augot, Raghav haska, Valéie Issany and Daniele Sacchetti INRIA Rocquencout 78153 Le Chesnay Fance {Daniel.Augot, Raghav.haska, Valéie.Issany,

More information

STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION

STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION Page 1 STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION C. Alan Blaylock, Hendeson State Univesity ABSTRACT This pape pesents an intuitive appoach to deiving annuity fomulas fo classoom use and attempts

More information

SELF-INDUCTANCE AND INDUCTORS

SELF-INDUCTANCE AND INDUCTORS MISN-0-144 SELF-INDUCTANCE AND INDUCTORS SELF-INDUCTANCE AND INDUCTORS by Pete Signell Michigan State Univesity 1. Intoduction.............................................. 1 A 2. Self-Inductance L.........................................

More information

The Binomial Distribution

The Binomial Distribution The Binomial Distibution A. It would be vey tedious if, evey time we had a slightly diffeent poblem, we had to detemine the pobability distibutions fom scatch. Luckily, thee ae enough similaities between

More information

Mechanics 1: Motion in a Central Force Field

Mechanics 1: Motion in a Central Force Field Mechanics : Motion in a Cental Foce Field We now stud the popeties of a paticle of (constant) ass oving in a paticula tpe of foce field, a cental foce field. Cental foces ae ve ipotant in phsics and engineeing.

More information

Automatic Testing of Neighbor Discovery Protocol Based on FSM and TTCN*

Automatic Testing of Neighbor Discovery Protocol Based on FSM and TTCN* Automatic Testing of Neighbo Discovey Potocol Based on FSM and TTCN* Zhiliang Wang, Xia Yin, Haibin Wang, and Jianping Wu Depatment of Compute Science, Tsinghua Univesity Beijing, P. R. China, 100084 Email:

More information

Software Engineering and Development

Software Engineering and Development I T H E A 67 Softwae Engineeing and Development SOFTWARE DEVELOPMENT PROCESS DYNAMICS MODELING AS STATE MACHINE Leonid Lyubchyk, Vasyl Soloshchuk Abstact: Softwae development pocess modeling is gaining

More information

Instructions to help you complete your enrollment form for HPHC's Medicare Supplemental Plan

Instructions to help you complete your enrollment form for HPHC's Medicare Supplemental Plan Instuctions to help you complete you enollment fom fo HPHC's Medicae Supplemental Plan Thank you fo applying fo membeship to HPHC s Medicae Supplement plan. Pio to submitting you enollment fom fo pocessing,

More information

Database Management Systems

Database Management Systems Contents Database Management Systems (COP 5725) D. Makus Schneide Depatment of Compute & Infomation Science & Engineeing (CISE) Database Systems Reseach & Development Cente Couse Syllabus 1 Sping 2012

More information

Integer sequences from walks in graphs

Integer sequences from walks in graphs otes on umbe Theoy and Discete Mathematics Vol. 9, 3, o. 3, 78 84 Intege seuences fom walks in gahs Enesto Estada, and José A. de la Peña Deatment of Mathematics and Statistics, Univesity of Stathclyde

More information

Coordinate Systems L. M. Kalnins, March 2009

Coordinate Systems L. M. Kalnins, March 2009 Coodinate Sstems L. M. Kalnins, Mach 2009 Pupose of a Coodinate Sstem The pupose of a coodinate sstem is to uniquel detemine the position of an object o data point in space. B space we ma liteall mean

More information

A framework for the selection of enterprise resource planning (ERP) system based on fuzzy decision making methods

A framework for the selection of enterprise resource planning (ERP) system based on fuzzy decision making methods A famewok fo the selection of entepise esouce planning (ERP) system based on fuzzy decision making methods Omid Golshan Tafti M.s student in Industial Management, Univesity of Yazd Omidgolshan87@yahoo.com

More information

Lab #7: Energy Conservation

Lab #7: Energy Conservation Lab #7: Enegy Consevation Photo by Kallin http://www.bungeezone.com/pics/kallin.shtml Reading Assignment: Chapte 7 Sections 1,, 3, 5, 6 Chapte 8 Sections 1-4 Intoduction: Pehaps one of the most unusual

More information

PROBLEM 12. ANALYTICAL AND FINITE ELEMENT SOLUTION OF A PLATE WITH CENTRAL HOLE

PROBLEM 12. ANALYTICAL AND FINITE ELEMENT SOLUTION OF A PLATE WITH CENTRAL HOLE PROBLEM. ANALYTICAL AND FINITE ELEMENT SOLUTION OF A PLATE WITH CENTRAL HOLE Autho: D. Andás Szekényes D. Andás Szekényes, BME www.tankonyvta.hu Analytical and finite element solution of a plate with cental

More information

Physics 235 Chapter 5. Chapter 5 Gravitation

Physics 235 Chapter 5. Chapter 5 Gravitation Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus

More information

Power and Sample Size Calculations for the 2-Sample Z-Statistic

Power and Sample Size Calculations for the 2-Sample Z-Statistic Powe and Sample Size Calculations fo the -Sample Z-Statistic James H. Steige ovembe 4, 004 Topics fo this Module. Reviewing Results fo the -Sample Z (a) Powe and Sample Size in Tems of a oncentality Paamete.

More information

UNIT 21: ELECTRICAL AND GRAVITATIONAL POTENTIAL Approximate time two 100-minute sessions

UNIT 21: ELECTRICAL AND GRAVITATIONAL POTENTIAL Approximate time two 100-minute sessions Name St.No. - Date(YY/MM/DD) / / Section Goup# UNIT 21: ELECTRICAL AND GRAVITATIONAL POTENTIAL Appoximate time two 100-minute sessions OBJECTIVES I began to think of gavity extending to the ob of the moon,

More information

Lab 5: Circular Motion

Lab 5: Circular Motion Lab 5: Cicula motion Physics 193 Fall 2006 Lab 5: Cicula Motion I. Intoduction The lab today involves the analysis of objects that ae moving in a cicle. Newton s second law as applied to cicula motion

More information

Valuation of Floating Rate Bonds 1

Valuation of Floating Rate Bonds 1 Valuation of Floating Rate onds 1 Joge uz Lopez us 316: Deivative Secuities his note explains how to value plain vanilla floating ate bonds. he pupose of this note is to link the concepts that you leaned

More information

Problem Set 6: Solutions

Problem Set 6: Solutions UNIVESITY OF ALABAMA Depatment of Physics and Astonomy PH 16-4 / LeClai Fall 28 Poblem Set 6: Solutions 1. Seway 29.55 Potons having a kinetic enegy of 5. MeV ae moving in the positive x diection and ente

More information

BA 351 CORPORATE FINANCE LECTURE 4 TAXES AND THE MARGINAL INVESTOR. John R. Graham Adapted from S. Viswanathan FUQUA SCHOOL OF BUSINESS

BA 351 CORPORATE FINANCE LECTURE 4 TAXES AND THE MARGINAL INVESTOR. John R. Graham Adapted from S. Viswanathan FUQUA SCHOOL OF BUSINESS BA 351 CORPORATE FINANCE LECTURE 4 TAXES AND THE MARGINAL INVESTOR John R. Gaham Adapted fom S. Viswanathan FUQUA SCHOOL OF BUSINESS DUKE UNIVERSITY 1 In this lectue we conside the effect of govenment

More information

F G r. Don't confuse G with g: "Big G" and "little g" are totally different things.

F G r. Don't confuse G with g: Big G and little g are totally different things. G-1 Gavity Newton's Univesal Law of Gavitation (fist stated by Newton): any two masses m 1 and m exet an attactive gavitational foce on each othe accoding to m m G 1 This applies to all masses, not just

More information

EXPERIENCE OF USING A CFD CODE FOR ESTIMATING THE NOISE GENERATED BY GUSTS ALONG THE SUN- ROOF OF A CAR

EXPERIENCE OF USING A CFD CODE FOR ESTIMATING THE NOISE GENERATED BY GUSTS ALONG THE SUN- ROOF OF A CAR EXPERIENCE OF USING A CFD CODE FOR ESTIMATING THE NOISE GENERATED BY GUSTS ALONG THE SUN- ROOF OF A CAR Liang S. Lai* 1, Geogi S. Djambazov 1, Choi -H. Lai 1, Koulis A. Peicleous 1, and Fédéic Magoulès

More information

Top K Nearest Keyword Search on Large Graphs

Top K Nearest Keyword Search on Large Graphs Top K Neaest Keywod Seach on Lage Gaphs Miao Qiao, Lu Qin, Hong Cheng, Jeffey Xu Yu, Wentao Tian The Chinese Univesity of Hong Kong, Hong Kong, China {mqiao,lqin,hcheng,yu,wttian}@se.cuhk.edu.hk ABSTRACT

More information

In the lecture on double integrals over non-rectangular domains we used to demonstrate the basic idea

In the lecture on double integrals over non-rectangular domains we used to demonstrate the basic idea Double Integals in Pola Coodinates In the lectue on double integals ove non-ectangula domains we used to demonstate the basic idea with gaphics and animations the following: Howeve this paticula example

More information

Magnetic Bearing with Radial Magnetized Permanent Magnets

Magnetic Bearing with Radial Magnetized Permanent Magnets Wold Applied Sciences Jounal 23 (4): 495-499, 2013 ISSN 1818-4952 IDOSI Publications, 2013 DOI: 10.5829/idosi.wasj.2013.23.04.23080 Magnetic eaing with Radial Magnetized Pemanent Magnets Vyacheslav Evgenevich

More information

ESCAPE VELOCITY EXAMPLES

ESCAPE VELOCITY EXAMPLES ESCAPE VELOCITY EXAMPLES 1. Escape velocity is the speed that an object needs to be taveling to beak fee of planet o moon's gavity and ente obit. Fo example, a spacecaft leaving the suface of Eath needs

More information

VISCOSITY OF BIO-DIESEL FUELS

VISCOSITY OF BIO-DIESEL FUELS VISCOSITY OF BIO-DIESEL FUELS One of the key assumptions fo ideal gases is that the motion of a given paticle is independent of any othe paticles in the system. With this assumption in place, one can use

More information

Today in Physics 217: multipole expansion

Today in Physics 217: multipole expansion Today in Physics 17: multipole expansion Multipole expansions Electic multipoles and thei moments Monopole and dipole, in detail Quadupole, octupole, Example use of multipole expansion as appoximate solution

More information

Over-encryption: Management of Access Control Evolution on Outsourced Data

Over-encryption: Management of Access Control Evolution on Outsourced Data Ove-encyption: Management of Access Contol Evolution on Outsouced Data Sabina De Capitani di Vimecati DTI - Univesità di Milano 26013 Cema - Italy decapita@dti.unimi.it Stefano Paaboschi DIIMM - Univesità

More information

I- LACUNARY GENERALIZED DIFFERENCE CONVERGENT SEQUENCES IN n-normed SPACES

I- LACUNARY GENERALIZED DIFFERENCE CONVERGENT SEQUENCES IN n-normed SPACES Jounal of Mathematical Analysis ISSN: 17-341, URL: http://www.iliias.com Volume Issue 1(011), Pages 18-4. I- LACUNARY GENERALIZED DIFFERENCE CONVERGENT SEQUENCES IN n-normed SPACES NURTEN FISTIKÇI, MEHMET

More information

Alarm transmission through Radio and GSM networks

Alarm transmission through Radio and GSM networks Alam tansmission though Radio and GSM netwoks 2015 Alam tansmission though Radio netwok RR-IP12 RL10 E10C E10C LAN RL1 0 R11 T10 (T10U) Windows MONAS MS NETWORK MCI > GNH > GND > +E > DATA POWER DATA BUS

More information

Risk Sensitive Portfolio Management With Cox-Ingersoll-Ross Interest Rates: the HJB Equation

Risk Sensitive Portfolio Management With Cox-Ingersoll-Ross Interest Rates: the HJB Equation Risk Sensitive Potfolio Management With Cox-Ingesoll-Ross Inteest Rates: the HJB Equation Tomasz R. Bielecki Depatment of Mathematics, The Notheasten Illinois Univesity 55 Noth St. Louis Avenue, Chicago,

More information

Chapter 13 Gravitation

Chapter 13 Gravitation Chapte 13 Gavitation Newton, who extended the concept of inetia to all bodies, ealized that the moon is acceleating and is theefoe subject to a centipetal foce. He guessed that the foce that keeps the

More information

ON NEW CHALLENGES FOR CFD SIMULATION IN FILTRATION

ON NEW CHALLENGES FOR CFD SIMULATION IN FILTRATION ON NEW CHALLENGES FOR CFD SIMULATION IN FILTRATION Michael Dedeing, Wolfgang Stausbeg, IBS Filtan, Industiestasse 19 D-51597 Mosbach-Lichtenbeg, Gemany. Oleg Iliev(*), Zaha Lakdawala, Faunhofe Institut

More information

Approximation Algorithms for Data Management in Networks

Approximation Algorithms for Data Management in Networks Appoximation Algoithms fo Data Management in Netwoks Chistof Kick Heinz Nixdof Institute and Depatment of Mathematics & Compute Science adebon Univesity Gemany kueke@upb.de Haald Räcke Heinz Nixdof Institute

More information

Problem Set # 9 Solutions

Problem Set # 9 Solutions Poblem Set # 9 Solutions Chapte 12 #2 a. The invention of the new high-speed chip inceases investment demand, which shifts the cuve out. That is, at evey inteest ate, fims want to invest moe. The incease

More information