TwoWay and MultipleAccess Energy Harvesting Systems with Energy Cooperation

 Alisha Scott
 9 months ago
 Views:
Transcription
1 TwoWay and MultpleAccess Energy Harvestng Systems wth Energy Cooperaton Berk Gurakan, Omur Ozel, Jng Yang 2, and Sennur Ulukus Department of Electrcal and Computer Engneerng, Unversty of Maryland, College Park, MD Department of Electrcal Engneerng, Unversty of Arkansas, Fayettevlle, AR 7270 Abstract We study the capacty regons of twoway and multpleaccess harvestng communcaton systems wth oneway wreless transfer. In these systems, requred for data transmsson s harvested by the users from nature throughout the communcaton duraton, and there s a separate unt that enables transfer from the frst user to the second user wth an effcency of. Energy harvests are known by the transmtters a pror. We frst nvestgate the capacty regon of the harvestng Gaussan twoway channel (TWC wth oneway transfer. We show that the boundary of the capacty regon s acheved by a generalzed twodmensonal drectonal waterfllng algorthm. Then, we study the capacty regon of the harvestng Gaussan multple access channel (MAC wth oneway transfer. We show that f the prorty of the frst user s hgher, then transfer s not needed. In addton, f the prorty of the second user s suffcently hgh, then the frst user must transfer all of ts to the second user. I. INTRODUCTION We study the capacty regons of the Gaussan twoway channel (TWC and the Gaussan twouser multple access channel (MAC wth oneway transfer. In both scenaros, there are two users powered by harvestng devces communcatng messages to each other or to an access pont. We model these scenaros as two users havng exogenous arrval processes that recharge ther batteres throughout the communcaton duraton. Addtonally, oneway transfer s possble: The frst user can transmt a porton of ts to the second user through a separate wreless transfer unt subject to an neffcency (.e., loss durng the transfer. Wreless transfer enables a new form of cooperaton whch we call cooperaton; see also []. In contrast to the usual noton of cooperaton, whch s at the sgnal level [2], cooperaton s at the battery level. In ths paper, we study optmal management polces for the users n systems wth cooperaton. Assumng that the users know the realzatons of the arrval processes n advance, as n the exstng lterature [], [3] [4], we characterze the correspondng capacty regons. We frst consder the Gaussan TWC wth transfer. We show that the boundary of the capacty regon s obtaned by a generalzed twodmensonal drectonal waterfllng Ths work was supported by NSF Grants CNS , CCF , CCF and CNS Fg.. E User δ Ē User 2 TWC wth oneway transfer. algorthm. Ths algorthm optmzes the levels n two dmensons, namely the tme and user dmensons, subject to causalty constrants. We then study the Gaussan MAC wth transfer. We show that f the frst user has a hgher prorty, then transfer s not needed and the boundary s acheved by the generalzed backward drectonal waterfllng algorthm gven n [9]. Moreover, we show that f the second user has a suffcently hgh prorty, then transferrng all of the of the frst user to the second user s optmal. In between these two extremes, some nonzero porton of the frst user s s transferred to the second user. II. TWC WITH ONEWAY ENERGY TRANSFER We consder a Gaussan TWC as shown n Fg.. The two s at the nodes are the data and s. The energes that arrve at the nodes are saved n the correspondng s. The s of both users always carry some data packets. The physcal layer s a memoryless Gaussan TWC [5] where the channel nputs and outputs are x, x 2 and y, y 2, respectvely. The nputoutput relatons are y = x + x 2 + n and y 2 = x + x 2 + n 2 where n and n 2 are ndependent Gaussan noses wth zeromean and untvarance. We assume that the tme s slotted and there are a total of T equal length slots. In slot t, the frst and second users harvest n amounts E t and Ēt, respectvely. There s a separate oneway wreless transfer unt from the frst user to the second user wth effcency 0 : When the frst user transfers δ amount of to the second user, δ amount of exts the frst user s and δ amount of enters the second user s n the same slot. The power polcy of user
2 s composed of the sequences P,δ, and the power polcy of user 2 s the sequence P. For both users, the that has not arrved yet cannot be used for data transmsson or transfer. In addton, transfer amounts cannot be larger than the harvested. These constrants yeld the followng set F of feasble power control and transfer polces: F = (δ, P, P : P P δ (E δ, k (Ē +δ, k } E, k For the Gaussan TWC wth ndvdual power constrants P and P 2, rate pars (R,R 2 wth R 2 log(+p,r 2 2 log(+p 2 are achevable [5]. For a fxed transfer vector δ, and feasble power control polces P and P, the set of achevable rates s: C δ (P, P = (R,R 2 : R R 2 2 log(+p 2 log(+ P } The notaton shows the dependence of the regon on the transfer vector δ. Ths regon s shown n Fg. 2 for dfferent values of δ. Each of these regons are rectangles of the form R C where C s the mum throughput acheved for user found by mzng (2 constraned to the feasblty constrants F. As δ s ncreased, s transferred from user to user 2 therefore C decreases whle C 2 ncreases. By takng the unon of the regons over all possble transfer vectors and power polces for the users, we obtan the capacty regon of the Gaussan TWC as: C(E, Ē = C δ (P, P (3 (δ,p, P F III. CAPACITY REGION OF THE GAUSSIAN TWC In ths secton, we characterze the capacty regon as well as the optmal power allocaton and transfer polces. We start by notng that the capacty regon s convex. Lemma C(E, Ē s a convex regon. Snce C(E, Ē s convex, each boundary pont can be found by solvng the followng weghted rate mzaton problem: P,P,δ θ 2 log(+p +θ 2 2 log(+ P ( (2 s.t. (δ, P, P F (4 The problem n (4 s a convex optmzaton problem as the objectve functon s concave and the feasble set s a convex R Fg. 2. θr 2 3 R 2 Capacty regon of the Gaussan TWC. set [6]. We wrte the Lagrangan functon for (4 as L = θ log(+p +θ 2 log(+ P + k( P (E δ k=µ + k( P (Ē +δ k=η + k( δ E ρ k δ k (5 k=γ k= The Lagrange multplerρ k s due to the constrantδ k 0. We exclude the nonnegatvty constrants for P and P as P and P are always nonzero n the optmal polcy for θ,θ 2 > 0. Smlarly, we elmnate the constrants k δ k E and the multplers γ k n the followng analyss snce these constrants can never be satsfed wth equalty n the optmal polcy for the Gaussan TWC for θ,θ 2 > 0, as that would requre P = 0 for some. However, we note that these constrants and the multplers γ k play an mportant role for the analyss of the capacty regon of the MAC and hence we renstate these constrants when necessary. The KKT condtons for the case of TWC are: θ + µ k = 0, (6 +P θ 2 + P + η k = 0, (7 µ k η k ρ = 0, (8 wth the addtonal complementary slackness condtons as: µ k ( P (E δ η k ( P (Ē +δ = 0, k (9 = 0, k (0 ρ k δ k = 0, k (
3 From (6, (7 and (8 we get: P = θ µ, (2 k θ 2 P = η, k (3 ρ = µ k η k, (4 We wll gve the soluton for general θ,θ 2 > 0 n the sequel. Before that, we note that n the extreme case when θ 2 = 0, the problem reduces to mzng the frst user s throughput only and hence any transfer s strctly suboptmal,.e., δ = 0 s optmal. Ths corresponds to pont n Fg. 2. Smlarly, when θ = 0, the problem reduces to mzng the second user s throughput only and the frst user must transfer all of ts to the second user,.e., δ = E s optmal. Ths corresponds to pont 3 n Fg. 2. When θ,θ 2 > 0, we obtan the ponts between ponts and 3 n Fg. 2. In ths case, for a gven transfer profle δ,...,δ T, the optmzaton problem can be separated nto two optmzaton problems, each only n terms of the power control polcy of the correspondng user. Lemma 2 The optmal power sequences P and P are monotoncally ncreasng sequences:p+ P, P + P. Next, we provde the necessary optmalty condton for a nonzero transfer. Lemma 3 For the optmal power sequences P, P and transfer sequence δ, f δ 0 for a slot, then, +P + P = θ θ 2 Proof: From (2(4 we have +P + P = θ η k θ 2 ( η k +ρ (5 (6 If there s a nonzero transfer, δ 0, we have from (, ρ = 0. Therefore, (5 must be satsfed f δ 0. In order to devse an algorthmc soluton, we apply a change of varable P = P and rewrte the optmzaton problem n terms of P, P,δ as follows: P,P,δ s.t. θ 2 log(+p +θ 2 2 log(+ P P P δ (E δ, k (Ē +δ, k E, k (7 The optmal power allocaton for ths problem s: where ν and ν n slot are defned by ν = P = θ ν, (8 P = θ 2 ν, (9 µ k and ν = η k (20 The power level expressons n (8(9 lead to a drectonal waterfllng nterpretaton [5]. In partcular, we note that has to be jontly allocated n tme and user dmensons together. Ths calls for a twodmensonal drectonal waterfllng algorthm where s allowed to flow n two dmensons, from left to rght (n tme and from up to down (among users. We utlze rght permeable taps to account for whch wll be used n the future and down permeable taps to account for that wll be transferred from user to user 2. We see from the KKT optmalty condtons that ν = ν n slots where there s nonzero transfer. We note that n the orgnal problem, ths mples that f some s transferred, then the power levels n that slot need to satsfy (5. The base levels for users and 2 are and, respectvely. Moreover, to facltate the water flow nterpretaton, we scale the arrvals of user 2 by as seen n (7. If the resultng water levels are hgher for user or not monotoncally ncreasng n tme for both users, then water has to flow untl the levels are balanced. Whle fndng the balanced water levels, the two dmensons of the water flow (.e., n tme and among users are coupled and therefore t s not easy to determne beforehand whch taps wll be open or closed n the optmal soluton. In partcular, the water flow of user 2 from tme slot to tme slot + j, j > 0, may become redundant f some s transferred from user. To crcumvent ths dffculty, we let each tap (rght/down permeable have a meter measurng the water that has already passed through t and we allow that tap to let the water flow back f an update n the allocaton necesstates t. Ths way, we keep track of the source of the and whether t s transferred to future tme slots or to the other user. Frst, we fll nto the slots wth all taps closed. Then, we open only the rght permeable taps and perform drectonal waterfllng for both users ndvdually [5]. Then, we open the down taps one by one n a backward fashon. If water flows down through a tap, the amount s measured by the meter. Water levels n the slots connected by the bdrectonal horzontal taps need to be equal. Whenever water flows down through a down permeable tap, the water levels must satsfy the proportonalty relatonshp n (5. When the water levels are properly balanced, the optmal soluton s obtaned. IV. MAC WITH ONEWAY ENERGY TRANSFER In ths secton, we consder the MAC scenaro as shown n Fg. 3. In MAC, the receved sgnal s y = x +x 2 +n where x and x 2 are sgnals of user and user 2, respectvely, and n s a Gaussan nose wth zeromean and untvarance. For the
4 E δ Ē R 2 User User 2 3 Recever < = 4 4 R 2 Fg. 3. MAC wth oneway transfer. Fg. 4. Capacty regon of the Gaussan MAC. Gaussan twouser MAC wth ndvdual power constrants P and P 2, rate pars (R,R 2 wth R 2 log(+p,r 2 2 log(+p 2, R +R 2 2 log(+p +P 2 are achevable [7]. For a fxed transfer vector δ, and feasble power control polces P and P, the set of achevable rates s a pentagon defned as [9]: C δ (P, P = (R,R 2 :R R 2 R +R 2 2 log(+p 2 log(+ P 2 log(+ P } +P (2 For each feasble (P, P, δ the regon s a pentagon. The capacty regon of the Gaussan MAC wth transfer s the unon over all feasble power allocatons and transfer profles: C(E, Ē = C δ (P, P (22 (δ,p, P F where F s gven n (. Ths regon s shown n Fg. 4. V. CAPACITY REGION OF THE GAUSSIAN MAC In ths secton, we characterze the capacty regon of the Gaussan MAC wth oneway transfer. Frst, we note that the capacty regon s convex. Lemma 4 C(E, Ē s a convex regon. Snce the regon s convex, each boundary pont s a soluton to R C M θr [8] for some θ = [θ,θ 2 ]. We examne two cases separately, θ θ 2 and θ < θ 2. A. θ θ 2 We show that when θ θ 2, no transfer from user to user 2 s needed. Note that as θ θ 2, the boundary ponts between, 2 and 3 n Fg. 4 are found by solvng the followng problem: P,P, δ (θ θ 2 2 log(+p +θ 2 2 log(+ P +P s.t. (δ, P, P F (23 The problem n (23 s a convex optmzaton problem and the correspondng KKT condtons are: θ θ 2 θ 2 +P +P + P + µ k = 0, (24 θ 2 +P + P + η k = 0, (25 µ k η k + γ k ρ = 0, (26 Snce θ θ 2, from (24(25, we have µ k η k, whch s satsfed wth equalty ff θ = θ 2. Ths together wth (26 mples that ρ γ k 0, whch s satsfed wth equalty ff θ = θ 2 and =. Therefore, unless we have exactlyθ = θ 2 and =, then we must haveρ > 0 for all. Ths together wth the complementary slackness condtons n ( mples that we must have δ = 0 for all,.e., no transfer s needed. However, when θ = θ 2 and addtonally f =, then there may exst multple dfferent optmal transfer profles, ncludng the one wth no transfer. Snce transfer s not needed, optmal power control polces for the two users are the same as those n the harvestng MAC wth no transfer and can be found by the generalzed backward drectonal waterfllng algorthm descrbed n [9]. That s, the capacty regon boundary from pont to pont 3 n Fg. 4 s found by the algorthm n [9]. Specfcally, for θ = θ 2, we have η k = µ k for all k and the sumrate optmal power polces are obtaned by applyng sngleuser drectonal waterfllng algorthm to the sum of the profles of the users [9]. B. θ < θ 2 Here, we consder the remanng parts of the boundary, namely the ponts from pont 3 to pont 4 n Fg. 4. In ths
5 case, we need to solve the followng optmzaton problem: P,P,δ (θ 2 θ log(+ P +θ log(+ P +P s.t. (δ, P, P F (27 whch s a convex optmzaton problem and the correspondng KKT condtons are: θ +P + P + µ k = 0, (28 θ 2 θ θ + P +P + P + µ k η k + η k = 0, (29 γ k ρ = 0, (30 We do not have an analytcal closed form soluton for (28 (30. Snce (27 s a convex optmzaton problem, standard numercal methods for convex optmzaton may be employed. We fnd that the soluton of (27 has a smple form n some specal cases, whch we nvestgate next. When =, we fnd that the optmal soluton of (27 requres all the of user transferred to user 2. To verfy ths fact, we note from (28(29 that η T > µ T, snce θ 2 > θ. Combnng ths wth (30, we obtan γ T ρ T > 0. Note that f δ < E, then γ T = 0 and hence ρ T < 0, whch s not possble. Thus, n the optmal soluton, we must have δ = E. Therefore, user should not transmt any data, and nstead should transfer all of ts to user 2 by the end of T slots. Ths polcy corresponds to pont4n Fg. 4. On the other hand, sumrate optmal pont, pont 3, acheves the same throughput as pont 4. Ths mples that when =, ponts 2, 3 and 4 n Fg. 4 le on the 45 o lne. In partcular, the optmal throughput of user 2, whch s obtaned by sngleuser throughput mzaton subject to harvested energes of user 2 plus the harvested energes of user, concdes wth the optmal sumthroughput. When <, ponts 2, 3 and 4 n Fg. 4 are not on the same lne. However, we observe that when θ2 θ s suffcently large, user transfers all of ts to user 2. In order to verfy ths clam, we note that, f user transfers some but not all of ts at the end of T slots, then γ T = 0. In ths case, from (28(30 and as ρ T 0, we have + P T + P T +P T (θ 2 θ (θ (3 + Snce P T + P T+P T <, we conclude that f (θ2θ (θ, then (3 cannot be satsfed whch forces all of the of user to be transferred to user 2 so thatγ T > 0. Note that (θ2θ (θ s equvalent to θ2 θ θ2. Hence, f θ, n the optmal soluton, user transfers all of ts to user 2. When θ2 θ, some nonzero porton of the frst user s may need to be transferred to the second user n the optmal soluton. VI. CONCLUSIONS In ths paper, we consdered the Gaussan TWC and the Gaussan twouser MAC under harvestng and oneway wreless transfer condtons. For the Gaussan TWC, we showed that a generalzed twodmensonal drectonal waterfllng algorthm, whch dstrbutes the overall harvested optmally over the tme and user dmensons subject to causalty constrants acheves the boundary of the capacty regon. For the Gaussan twouser MAC, wth transfer from the frst user to the second user, we showed that, f the frst user has hgher prorty over the second user, then transfer s not needed. In addton, when the second user s prorty s suffcently hgh, the frst user must transfer all of ts to the second user. REFERENCES [] B. Gurakan, O. Ozel, J. Yang, and S. Ulukus, Energy cooperaton n harvestng wreless communcatons, n IEEE ISIT, July 202. [2] A. Sendonars, E. Erkp, and B. Aazhang, User cooperaton dversty. Part I. System descrpton, IEEE Trans. Comm., vol. 5, pp , November [3] J. Yang and S. Ulukus, Optmal packet schedulng n an harvestng communcaton system, IEEE Trans. Comm., vol. 60, pp , January 202. [4] K. Tutuncuoglu and A. Yener, Optmum transmsson polces for battery lmted harvestng nodes, IEEE Trans. Wreless Comm., vol., pp , March 202. [5] O. Ozel, K. Tutuncuoglu, J. Yang, S. Ulukus, and A. Yener, Transmsson wth harvestng nodes n fadng wreless channels: Optmal polces, IEEE Jour. on Selected Areas n Commun., vol. 29, pp , September 20. [6] J. Yang, O. Ozel, and S. Ulukus, Broadcastng wth an harvestng rechargeable transmtter, IEEE Trans. Wreless Comm., vol., pp , February 202. [7] M. A. Antepl, E. UysalBykoglu, and H. Erkal, Optmal packet schedulng on an harvestng broadcast lnk, IEEE Jour. on Selected Areas n Commun., vol. 29, pp , September 20. [8] O. Ozel, J. Yang, and S. Ulukus, Optmal broadcast schedulng for an harvestng rechargeable transmtter wth a fnte capacty battery, IEEE Trans. Wreless Comm., vol., pp , June 202. [9] J. Yang and S. Ulukus, Optmal packet schedulng n a multple access channel wth harvestng transmtters, Journal of Communcatons and Networks, vol. 4, pp , Aprl 202. [0] K. Tutuncuoglu and A. Yener, Sumrate optmal power polces for harvestng transmtters n an nterference channel, Journal of Communcatons and Networks, vol. 4, pp. 5 6, Aprl 202. [] C. Huang, R. Zhang, and S. Cu, Throughput mzaton for the Gaussan relay channel wth harvestng constrants, IEEE Jour. on Selected Areas n Commun., submtted, September 20. Avalable at [arxv: ]. [2] D. Gunduz and B. Devllers, Twohop communcaton wth harvestng, n IEEE CAMSAP, December 20. [3] K. Tutuncuoglu and A. Yener, Communcatng usng an harvestng transmtter: Optmum polces under storage losses, IEEE Trans. Wreless Comm., submtted August 202, avalable at [arxv: ]. [4] O. Orhan, D. Gunduz, and E. Erkp, Throughput mzaton for an harvestng communcaton system wth processng cost, n IEEE ITW, September 202. [5] T. S. Han, A general codng scheme for the twoway channel, IEEE Trans. Inform. Theory, vol. 30, pp , January 984. [6] S. Boyd and L. Vandenberghe, Convex Optmzaton. Unted Kngdom: Cambrdge Unversty Press, [7] T. Cover and J. Thomas, Elements of Informaton Theory. Wley Seres n Telecommuncatons and Sgnal Processng, John Wley & Sons, [8] D. Tse and S. Hanly, Multaccess fadng channels  Part I: Polymatrod structure, optmal resource allocaton and throughput capactes, IEEE Trans. Inf. Theory, vol. 44, pp , November 998.
Explicit and Implicit Temperature Constraints in Energy Harvesting Communications
Explct and Implct Temperature Constrants n Energy Harvestng Communcatons Abdulrahman Baknna, Omur Ozel 2, and Sennur Ulukus Department of Electrcal and Computer Engneerng, Unversty of Maryland, College
More informationECE559VV Project Report
ECE559VV Project Report (Supplementary Notes Loc Xuan Bu I. MAX SUMRATE SCHEDULING: THE UPLINK CASE We have seen (n the presentaton that, for downlnk (broadcast channels, the strategy maxmzng the sumrate
More informationIEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 33, NO. 3, MARCH
IEEE JOURAL O SELECTED AREAS I COMMUICATIOS, VOL. 33, O. 3, MARCH 205 467 Optmum Polces for an Energy Harvestng Transmtter Under Energy Storage Losses Kaya Tutuncuoglu, Student Member, IEEE, Ayln Yener,
More informationModule 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:
More informationWhich Separator? Spring 1
Whch Separator? 6.034  Sprng 1 Whch Separator? Mamze the margn to closest ponts 6.034  Sprng Whch Separator? Mamze the margn to closest ponts 6.034  Sprng 3 Margn of a pont " # y (w $ + b) proportonal
More informationJoint Scheduling and Resource Allocation in CDMA Systems
ACCEPTED TO IEEE TRANSACTIONS ON INFORMATION THEORY 1 Jont Schedulng and Resource Allocaton n CDMA Systems Vjay G. Subramanan, Randall A. Berry, and Rajeev Agrawal Abstract We consder schedulng and resource
More informationLossy Compression. Compromise accuracy of reconstruction for increased compression.
Lossy Compresson Compromse accuracy of reconstructon for ncreased compresson. The reconstructon s usually vsbly ndstngushable from the orgnal mage. Typcally, one can get up to 0:1 compresson wth almost
More informationError Probability for M Signals
Chapter 3 rror Probablty for M Sgnals In ths chapter we dscuss the error probablty n decdng whch of M sgnals was transmtted over an arbtrary channel. We assume the sgnals are represented by a set of orthonormal
More informationProblem Set 9 Solutions
Desgn and Analyss of Algorthms May 4, 2015 Massachusetts Insttute of Technology 6.046J/18.410J Profs. Erk Demane, Srn Devadas, and Nancy Lynch Problem Set 9 Solutons Problem Set 9 Solutons Ths problem
More informationLecture 10 Support Vector Machines II
Lecture 10 Support Vector Machnes II 22 February 2016 Taylor B. Arnold Yale Statstcs STAT 365/665 1/28 Notes: Problem 3 s posted and due ths upcomng Frday There was an early bug n the faketest data; fxed
More informationSecret Communication using Artificial Noise
Secret Communcaton usng Artfcal Nose Roht Neg, Satashu Goel C Department, Carnege Mellon Unversty, PA 151, USA {neg,satashug}@ece.cmu.edu Abstract The problem of secret communcaton between two nodes over
More informationOn Network Coding of Independent and Dependent Sources in Line Networks
On Network Codng of Independent and Dependent Sources n Lne Networks Mayank Baksh, Mchelle Effros, WeHsn Gu, Ralf Koetter Department of Electrcal Engneerng Department of Electrcal Engneerng Calforna Insttute
More informationOnline Scheduling for Energy Harvesting Broadcast Channels with Finite Battery
Online Scheduling for Energy Harvesting Broadcast Channels with Finite Battery Abdulrahman Baknina Sennur Ulukus Department of Electrical and Computer Engineering University of Maryland, College Park,
More informationFundamental loopcurrent method using virtual voltage sources technique for special cases
Fundamental loopcurrent method usng vrtual voltage sources technque for specal cases George E. Chatzaraks, 1 Marna D. Tortorel 1 and Anastasos D. Tzolas 1 Electrcal and Electroncs Engneerng Departments,
More informationChapter 7 Channel Capacity and Coding
Chapter 7 Channel Capacty and Codng Contents 7. Channel models and channel capacty 7.. Channel models Bnary symmetrc channel Dscrete memoryless channels Dscretenput, contnuousoutput channel Waveform
More informationNUMERICAL DIFFERENTIATION
NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the
More information6.854J / J Advanced Algorithms Fall 2008
MIT OpenCourseWare http://ocw.mt.edu 6.854J / 18.415J Advanced Algorthms Fall 2008 For nformaton about ctng these materals or our Terms of Use, vst: http://ocw.mt.edu/terms. 18.415/6.854 Advanced Algorthms
More informationWeek3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity
Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1s tme nterval. The velocty of the partcle
More informationJoint Scheduling of Rateguaranteed and Besteffort Services over a Wireless Channel
Jont Schedulng of Rateguaranteed and Besteffort Servces over a Wreless Channel Murtaza Zafer and Eytan Modano Abstract We consder multuser schedulng over the downln channel n wreless data systems. Specfcally,
More informationCOS 521: Advanced Algorithms Game Theory and Linear Programming
COS 521: Advanced Algorthms Game Theory and Lnear Programmng Moses Charkar February 27, 2013 In these notes, we ntroduce some basc concepts n game theory and lnear programmng (LP). We show a connecton
More informationAsynchronous CSMA Policies in Multihop Wireless Networks with Primary Interference Constraints
Asynchronous CSMA Polces n Multhop Wreless Networks wth Prmary Interference Constrants Peter Marbach, Atlla Erylmaz, and Asu Ozdaglar Abstract We analyze Asynchronous Carrer Sense Multple Access (CSMA)
More informationAntenna Combining for the MIMO Downlink Channel
Antenna Combnng for the IO Downlnk Channel arxv:0704.308v [cs.it] 0 Apr 2007 Nhar Jndal Department of Electrcal and Computer Engneerng Unversty of nnesota nneapols, N 55455, USA Emal: nhar@umn.edu Abstract
More informationMAXIMIZING THE THROUGHPUT OF CDMA DATA COMMUNICATIONS THROUGH JOINT ADMISSION AND POWER CONTROL
MAXIMIZI THE THROUHPUT OF CDMA DATA COMMUICATIOS THROUH JOIT ADMISSIO AD POWER COTROL Penna Orensten, Davd oodman and Zory Marantz Department of Electrcal and Computer Engneerng Polytechnc Unversty Brooklyn,
More informationYong Joon Ryang. 1. Introduction Consider the multicommodity transportation problem with convex quadratic cost function. 1 2 (x x0 ) T Q(x x 0 )
KangweonKyungk Math. Jour. 4 1996), No. 1, pp. 7 16 AN ITERATIVE ROWACTION METHOD FOR MULTICOMMODITY TRANSPORTATION PROBLEMS Yong Joon Ryang Abstract. The optmzaton problems wth quadratc constrants often
More informationPhysics 5153 Classical Mechanics. Principle of Virtual Work1
P. Guterrez 1 Introducton Physcs 5153 Classcal Mechancs Prncple of Vrtual Work The frst varatonal prncple we encounter n mechancs s the prncple of vrtual work. It establshes the equlbrum condton of a mechancal
More informationThe optimal delay of the second test is therefore approximately 210 hours earlier than =2.
THE IEC 61508 FORMULAS 223 The optmal delay of the second test s therefore approxmately 210 hours earler than =2. 8.4 The IEC 61508 Formulas IEC 615086 provdes approxmaton formulas for the PF for smple
More informationDistributed Channel Probing for Efficient Transmission Scheduling in Wireless Networks
Dstrbuted Channel Probng for Effcent Transmsson Schedulng n Wreless etworks Bn L and Atlla Erylmaz Abstract It s energyconsumng and operatonally cumbersome for all users to contnuously estmate the channel
More informationOPTIMUM BEAMFORMING USING TRANSMIT ANTENNA ARRAYS. feasible points with monotonically decreasing costs).
OPTIMUM BEAMFORMING USING TRANSMIT ANTENNA ARRAYS Eugene Vsotsky Upamanyu Madhow Abstract  Transmt beamformng s a powerful means of ncreasng capacty n systems n whch the transmtter s eupped wth an antenna
More informationAffine transformations and convexity
Affne transformatons and convexty The purpose of ths document s to prove some basc propertes of affne transformatons nvolvng convex sets. Here are a few onlne references for background nformaton: http://math.ucr.edu/
More informationBounds on the Effectivelength of Optimal Codes for Interference Channel with Feedback
Bounds on the Effectvelength of Optmal Codes for Interference Channel wth Feedback Mohsen Hedar EECS Department Unversty of Mchgan Ann Arbor,USA Emal: mohsenhd@umch.edu Farhad Shran ECE Department New
More informationThe Geometry of Logit and Probit
The Geometry of Logt and Probt Ths short note s meant as a supplement to Chapters and 3 of Spatal Models of Parlamentary Votng and the notaton and reference to fgures n the text below s to those two chapters.
More information2 Finite difference basics
Numersche Methoden 1, WS 11/12 B.J.P. Kaus 2 Fnte dfference bascs Consder the one The bascs of the fnte dfference method are best understood wth an example. dmensonal transent heat conducton equaton T
More informationThe Second AntiMathima on Game Theory
The Second AntMathma on Game Theory Ath. Kehagas December 1 2006 1 Introducton In ths note we wll examne the noton of game equlbrum for three types of games 1. 2player 2acton zerosum games 2. 2player
More informationMin Cut, Fast Cut, Polynomial Identities
Randomzed Algorthms, Summer 016 Mn Cut, Fast Cut, Polynomal Identtes Instructor: Thomas Kesselhem and Kurt Mehlhorn 1 Mn Cuts n Graphs Lecture (5 pages) Throughout ths secton, G = (V, E) s a multgraph.
More informationPsychology 282 Lecture #24 Outline Regression Diagnostics: Outliers
Psychology 282 Lecture #24 Outlne Regresson Dagnostcs: Outlers In an earler lecture we studed the statstcal assumptons underlyng the regresson model, ncludng the followng ponts: Formal statement of assumptons.
More informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
More informationGames of Threats. Elon Kohlberg Abraham Neyman. Working Paper
Games of Threats Elon Kohlberg Abraham Neyman Workng Paper 18023 Games of Threats Elon Kohlberg Harvard Busness School Abraham Neyman The Hebrew Unversty of Jerusalem Workng Paper 18023 Copyrght 2017
More informationDetermining Transmission Losses Penalty Factor Using Adaptive Neuro Fuzzy Inference System (ANFIS) For Economic Dispatch Application
7 Determnng Transmsson Losses Penalty Factor Usng Adaptve Neuro Fuzzy Inference System (ANFIS) For Economc Dspatch Applcaton Rony Seto Wbowo Maurdh Hery Purnomo Dod Prastanto Electrcal Engneerng Department,
More information12. The HamiltonJacobi Equation Michael Fowler
1. The HamltonJacob Equaton Mchael Fowler Back to Confguraton Space We ve establshed that the acton, regarded as a functon of ts coordnate endponts and tme, satsfes ( ) ( ) S q, t / t+ H qpt,, = 0, and
More informationProfessor Terje Haukaas University of British Columbia, Vancouver The Q4 Element
Professor Terje Haukaas Unversty of Brtsh Columba, ancouver www.nrsk.ubc.ca The Q Element Ths document consders fnte elements that carry load only n ther plane. These elements are sometmes referred to
More informationJoint Scheduling of Rateguaranteed and Besteffort Users over a Wireless Fading Channel
Jont Schedulng of Rateguaranteed and Besteffort Users over a Wreless Fadng Channel Murtaza Zafer and Eytan Modano Massachusetts Insttute of Technology Cambrdge, MA 239, USA Emal:{murtaza@mt.edu, modano@mt.edu}
More informationarxiv: v1 [math.co] 1 Mar 2014
Unonntersectng set systems Gyula O.H. Katona and Dánel T. Nagy March 4, 014 arxv:1403.0088v1 [math.co] 1 Mar 014 Abstract Three ntersecton theorems are proved. Frst, we determne the sze of the largest
More informationLecture 5 Decoding Binary BCH Codes
Lecture 5 Decodng Bnary BCH Codes In ths class, we wll ntroduce dfferent methods for decodng BCH codes 51 Decodng the [15, 7, 5] 2 BCH Code Consder the [15, 7, 5] 2 code C we ntroduced n the last lecture
More informationOn Wireless Scheduling with Partial. Channelstate Information
On Wreless Schedulng wth Partal 1 Channelstate Informaton Adtya Gopalan, Constantne Caramans and Sanjay Shakkotta Abstract A tmeslotted queued system of multple flows wth a sngleserver s consdered,
More informationx i1 =1 for all i (the constant ).
Chapter 5 The Multple Regresson Model Consder an economc model where the dependent varable s a functon of K explanatory varables. The economc model has the form: y = f ( x,x,..., ) xk Approxmate ths by
More informationMarkov Chain Monte Carlo Lecture 6
where (x 1,..., x N ) X N, N s called the populaton sze, f(x) f (x) for at least one {1, 2,..., N}, and those dfferent from f(x) are called the tral dstrbutons n terms of mportance samplng. Dfferent ways
More informationMaxMin Criterion Still DiversityOptimal?
Energy Harvestng Cooperatve Networks: Is the 1 MaxMn Crteron Stll DverstyOptmal? Zhguo Dng, Member, IEEE and H. Vncent Poor, Fellow, IEEE arxv:143.354v1 [cs.it] 3 Mar 214 Abstract Ths paper consders
More informationMEASUREMENT OF MOMENT OF INERTIA
1. measurement MESUREMENT OF MOMENT OF INERTI The am of ths measurement s to determne the moment of nerta of the rotor of an electrc motor. 1. General relatons Rotatng moton and moment of nerta Let us
More informationUniversity of California, Davis Date: June 22, 2009 Department of Agricultural and Resource Economics. PRELIMINARY EXAMINATION FOR THE Ph.D.
Unversty of Calforna, Davs Date: June 22, 29 Department of Agrcultural and Resource Economcs Department of Economcs Tme: 5 hours Mcroeconomcs Readng Tme: 2 mnutes PRELIMIARY EXAMIATIO FOR THE Ph.D. DEGREE
More informationWorkshop: Approximating energies and wave functions Quantum aspects of physical chemistry
Workshop: Approxmatng energes and wave functons Quantum aspects of physcal chemstry http://quantum.bu.edu/pltl/6/6.pdf Last updated Thursday, November 7, 25 7:9:55: Copyrght 25 Dan Dll (dan@bu.edu) Department
More informationV.C The Niemeijer van Leeuwen Cumulant Approximation
V.C The Nemejer van Leeuwen Cumulant Approxmaton Unfortunately, the decmaton procedure cannot be performed exactly n hgher dmensons. For example, the square lattce can be dvded nto two sublattces. For
More informationGoodness of fit and Wilks theorem
DRAFT 0.0 Glen Cowan 3 June, 2013 Goodness of ft and Wlks theorem Suppose we model data y wth a lkelhood L(µ) that depends on a set of N parameters µ = (µ 1,...,µ N ). Defne the statstc t µ ln L(µ) L(ˆµ),
More informationI + HH H N 0 M T H = UΣV H = [U 1 U 2 ] 0 0 E S. X if X 0 0 if X < 0 (X) + = = M T 1 + N 0. r p + 1
Homework 4 Problem Capacty wth CSI only at Recever: C = log det I + E )) s HH H N M T R SS = I) SVD of the Channel Matrx: H = UΣV H = [U 1 U ] [ Σr ] [V 1 V ] H Capacty wth CSI at both transmtter and
More informationLinköping University Post Print. Cooperative Beamforming for the MISO Interference Channel
Lnköpng Unversty Post Prnt Cooperatve Beamformng for the MISO Interference Channel Johannes Lndblom and Eleftheros Karpds N.B.: When ctng ths work, cte the orgnal artcle. 9 IEEE. Personal use of ths materal
More informationExercises of Chapter 2
Exercses of Chapter ChuangCheh Ln Department of Computer Scence and Informaton Engneerng, Natonal Chung Cheng Unversty, MngHsung, Chay 61, Tawan. Exercse.6. Suppose that we ndependently roll two standard
More informationTHE CURRENT BALANCE Physics 258/259
DSH 1988, 005 THE CURRENT BALANCE Physcs 58/59 The tme average force between two parallel conductors carryng an alternatng current s measured by balancng ths force aganst the gravtatonal force on a set
More informationElectrical double layer: revisit based on boundary conditions
Electrcal double layer: revst based on boundary condtons Jong U. Km Department of Electrcal and Computer Engneerng, Texas A&M Unversty College Staton, TX 77843318, USA Abstract The electrcal double layer
More informationWhat would be a reasonable choice of the quantization step Δ?
CE 108 HOMEWORK 4 EXERCISE 1. Suppose you are samplng the output of a sensor at 10 KHz and quantze t wth a unform quantzer at 10 ts per sample. Assume that the margnal pdf of the sgnal s Gaussan wth mean
More informationCooperative Game Theory for Distributed Spectrum Sharing
Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the ICC 2007 proceedngs. Cooperatve Game Theory for Dstrbuted Spectrum Sharng Juan
More informationEcon107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4)
I. Classcal Assumptons Econ7 Appled Econometrcs Topc 3: Classcal Model (Studenmund, Chapter 4) We have defned OLS and studed some algebrac propertes of OLS. In ths topc we wll study statstcal propertes
More informationRandom Partitions of Samples
Random Parttons of Samples Klaus Th. Hess Insttut für Mathematsche Stochastk Technsche Unverstät Dresden Abstract In the present paper we construct a decomposton of a sample nto a fnte number of subsamples
More informationIntroduction to Vapor/Liquid Equilibrium, part 2. Raoult s Law:
CE304, Sprng 2004 Lecture 4 Introducton to Vapor/Lqud Equlbrum, part 2 Raoult s Law: The smplest model that allows us do VLE calculatons s obtaned when we assume that the vapor phase s an deal gas, and
More informationFeasibility and Optimization of Delay Guarantees for Nonhomogeneous Flows in IEEE WLANs
Feasblty and Optmzaton of Delay Guarantees for Nonhomogeneous Flows n IEEE 8. WLANs Yan Gao,CheeWeTan, Yng Huang, Zheng Zeng,P.R.Kumar Department of Computer Scence & CSL, Unversty of Illnos at UrbanaChampagn
More informationThe (Q, r) Inventory Policy in Production Inventory Systems
Proceedngs of SKISE Fall Conference, Nov 1415, 2008 The ( Inventory Polcy n Producton Inventory Systems JoonSeok Km Sejong Unversty Seoul, Korea Abstract We examne the effectveness of the conventonal
More informationPopClick Noise Detection Using InterFrame Correlation for Improved Portable Auditory Sensing
Advanced Scence and Technology Letters, pp.164168 http://dx.do.org/10.14257/astl.2013 PopClc Nose Detecton Usng InterFrame Correlaton for Improved Portable Audtory Sensng Dong Yun Lee, Kwang Myung Jeon,
More informationTimeVarying Systems and Computations Lecture 6
TmeVaryng Systems and Computatons Lecture 6 Klaus Depold 14. Januar 2014 The Kalman Flter The Kalman estmaton flter attempts to estmate the actual state of an unknown dscrete dynamcal system, gven nosy
More informationFACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP
C O L L O Q U I U M M A T H E M A T I C U M VOL. 80 1999 NO. 1 FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP BY FLORIAN K A I N R A T H (GRAZ) Abstract. Let H be a Krull monod wth nfnte class
More information8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 493 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces you have studed thus far n the text are real vector spaces because the scalars
More informationP exp(tx) = 1 + t 2k M 2k. k N
1. Subgaussan tals Defnton. Say that a random varable X has a subgaussan dstrbuton wth scale factor σ< f P exp(tx) exp(σ 2 t 2 /2) for all real t. For example, f X s dstrbuted N(,σ 2 ) then t s subgaussan.
More informationGraphical Analysis of a BJT Amplifier
4/6/2011 A Graphcal Analyss of a BJT Amplfer lecture 1/18 Graphcal Analyss of a BJT Amplfer onsder agan ths smple BJT amplfer: ( t) = + ( t) O O o B + We note that for ths amplfer, the output oltage s
More informationChapter 8 Indicator Variables
Chapter 8 Indcator Varables In general, e explanatory varables n any regresson analyss are assumed to be quanttatve n nature. For example, e varables lke temperature, dstance, age etc. are quanttatve n
More informationSIMPLE REACTION TIME AS A FUNCTION OF TIME UNCERTAINTY 1
Journal of Expermental Vol. 5, No. 3, 1957 Psychology SIMPLE REACTION TIME AS A FUNCTION OF TIME UNCERTAINTY 1 EDMUND T. KLEMMER Operatonal Applcatons Laboratory, Ar Force Cambrdge Research Center An earler
More informationMultiuser Cognitive Access of Continuous Time Markov Channels: Maximum Throughput and Effective Bandwidth Regions
Multuser Cogntve Access of Contnuous Tme Markov Channels: Maxmum Throughput and Effectve Bandwdth Regons Shyao Chen, and Lang Tong School of Electrcal and Computer Engneerng Cornell Unversty, Ithaca, NY
More informationIterative Multiuser Receiver Utilizing Soft Decoding Information
teratve Multuser Recever Utlzng Soft Decodng nformaton Kmmo Kettunen and Tmo Laaso Helsn Unversty of Technology Laboratory of Telecommuncatons Technology emal: Kmmo.Kettunen@hut.f, Tmo.Laaso@hut.f Abstract
More informationA New Algorithm for Finding a Fuzzy Optimal. Solution for Fuzzy Transportation Problems
Appled Mathematcal Scences, Vol. 4, 200, no. 2, 7990 A New Algorthm for Fndng a Fuzzy Optmal Soluton for Fuzzy Transportaton Problems P. Pandan and G. Nataraan Department of Mathematcs, School of Scence
More informationSingular Value Decomposition: Theory and Applications
Sngular Value Decomposton: Theory and Applcatons Danel Khashab Sprng 2015 Last Update: March 2, 2015 1 Introducton A = UDV where columns of U and V are orthonormal and matrx D s dagonal wth postve real
More informationSum Capacity of Multiuser MIMO Broadcast Channels with Block Diagonalization
Sum Capacty of Multuser MIMO Broadcast Channels wth Block Dagonalzaton Zukang Shen, Runhua Chen, Jeffrey G. Andrews, Robert W. Heath, Jr., and Bran L. Evans The Unversty of Texas at Austn, Austn, Texas
More informationNetwork Utility Maximization in Adversarial Environments
echncal Report Network Utlty Maxmzaton n Adversaral Envronments Qngka Lang and Eytan Modano Laboratory for Informaton and Decson Systems Massachusetts Insttute of echnology, Cambrdge, MA arxv:1712.08672v2
More informationA General Power Allocation Scheme to Guarantee Quality of Service in Downlink and Uplink NOMA Systems
A General Power Allocaton Scheme to Guarantee Qualty of Servce n Downlnk and Uplnk N Systems Zheng Yang, Student Member, IEEE, Zhguo Dng, Senor Member, IEEE, Pngzh Fan, Fellow, IEEE, and Naofal AlDhahr,
More informationAssuming that the transmission delay is negligible, we have
Baseband Transmsson of Bnary Sgnals Let g(t), =,, be a sgnal transmtted over an AWG channel. Consder the followng recever g (t) + + Σ x(t) LTI flter h(t) y(t) t = nt y(nt) threshold comparator Decson ˆ
More informationLecture 10: Euler s Equations for Multivariable
Lecture 0: Euler s Equatons for Multvarable Problems Let s say we re tryng to mnmze an ntegral of the form: {,,,,,, ; } J f y y y y y y d We can start by wrtng each of the y s as we dd before: y (, ) (
More informationA New Refinement of Jacobi Method for Solution of Linear System Equations AX=b
Int J Contemp Math Scences, Vol 3, 28, no 17, 819827 A New Refnement of Jacob Method for Soluton of Lnear System Equatons AX=b F Naem Dafchah Department of Mathematcs, Faculty of Scences Unversty of Gulan,
More informationCONJUGACY IN THOMPSON S GROUP F. 1. Introduction
CONJUGACY IN THOMPSON S GROUP F NICK GILL AND IAN SHORT Abstract. We complete the program begun by Brn and Squer of charactersng conjugacy n Thompson s group F usng the standard acton of F as a group of
More informationTHE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens
THE CHINESE REMAINDER THEOREM KEITH CONRAD We should thank the Chnese for ther wonderful remander theorem. Glenn Stevens 1. Introducton The Chnese remander theorem says we can unquely solve any par of
More informationCHAPTER 7 STOCHASTIC ECONOMIC EMISSION DISPATCHMODELED USING WEIGHTING METHOD
90 CHAPTER 7 STOCHASTIC ECOOMIC EMISSIO DISPATCHMODELED USIG WEIGHTIG METHOD 7.1 ITRODUCTIO early 70% of electrc power produced n the world s by means of thermal plants. Thermal power statons are the
More informationThe Finite Element Method
The Fnte Element Method GENERAL INTRODUCTION Read: Chapters 1 and 2 CONTENTS Engneerng and analyss Smulaton of a physcal process Examples mathematcal model development Approxmate solutons and methods of
More informationC4B Machine Learning Answers II. = σ(z) (1 σ(z)) 1 1 e z. e z = σ(1 σ) (1 + e z )
C4B Machne Learnng Answers II.(a) Show that for the logstc sgmod functon dσ(z) dz = σ(z) ( σ(z)) A. Zsserman, Hlary Term 20 Start from the defnton of σ(z) Note that Then σ(z) = σ = dσ(z) dz = + e z e z
More informationCOMPLETE BUFFER SHARING IN ATM NETWORKS UNDER BURSTY ARRIVALS
COMPLETE BUFFER SHARING WITH PUSHOUT THRESHOLDS IN ATM NETWORKS UNDER BURSTY ARRIVALS Ozgur Aras and Tugrul Dayar Abstract. Broadband Integrated Servces Dgtal Networks (B{ISDNs) are to support multple
More informationNatural Language Processing and Information Retrieval
Natural Language Processng and Informaton Retreval Support Vector Machnes Alessandro Moschtt Department of nformaton and communcaton technology Unversty of Trento Emal: moschtt@ds.untn.t Summary Support
More informationCase A. P k = Ni ( 2L i k 1 ) + (# big cells) 10d 2 P k.
THE CELLULAR METHOD In ths lecture, we ntroduce the cellular method as an approach to ncdence geometry theorems lke the SzemerédTrotter theorem. The method was ntroduced n the paper Combnatoral complexty
More informationF statistic = s2 1 s 2 ( F for Fisher )
Stat 4 ANOVA Analyss of Varance /6/04 Comparng Two varances: F dstrbuton Typcal Data Sets One way analyss of varance : example Notaton for one way ANOVA Comparng Two varances: F dstrbuton We saw that the
More informationRELIABILITY ASSESSMENT
CHAPTER Rsk Analyss n Engneerng and Economcs RELIABILITY ASSESSMENT A. J. Clark School of Engneerng Department of Cvl and Envronmental Engneerng 4a CHAPMAN HALL/CRC Rsk Analyss for Engneerng Department
More informationA Simple Inventory System
A Smple Inventory System Lawrence M. Leems and Stephen K. Park, DscreteEvent Smulaton: A Frst Course, Prentce Hall, 2006 Hu Chen Computer Scence Vrgna State Unversty Petersburg, Vrgna February 8, 2017
More informationCalculus of Variations Basics
Chapter 1 Calculus of Varatons Bascs 1.1 Varaton of a General Functonal In ths chapter, we derve the general formula for the varaton of a functonal of the form J [y 1,y 2,,y n ] F x,y 1,y 2,,y n,y 1,y
More informationDESIGN OPTIMIZATION OF CFRP RECTANGULAR BOX SUBJECTED TO ARBITRARY LOADINGS
Munch, Germany, 2630 th June 2016 1 DESIGN OPTIMIZATION OF CFRP RECTANGULAR BOX SUBJECTED TO ARBITRARY LOADINGS Q.T. Guo 1*, Z.Y. L 1, T. Ohor 1 and J. Takahash 1 1 Department of Systems Innovaton, School
More informationLecture 6/7 (February 10/12, 2014) DIRAC EQUATION. The nonrelativistic Schrödinger equation was obtained by noting that the Hamiltonian 2
P470 Lecture 6/7 (February 10/1, 014) DIRAC EQUATION The nonrelatvstc Schrödnger equaton was obtaned by notng that the Hamltonan H = P (1) m can be transformed nto an operator form wth the substtutons
More informationA PROCEDURE FOR SIMULATING THE NONLINEAR CONDUCTION HEAT TRANSFER IN A BODY WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY.
Proceedngs of the th Brazlan Congress of Thermal Scences and Engneerng  ENCIT 006 Braz. Soc. of Mechancal Scences and Engneerng  ABCM, Curtba, Brazl, Dec. 58, 006 A PROCEDURE FOR SIMULATING THE NONLINEAR
More informationPHYS 705: Classical Mechanics. Canonical Transformation II
1 PHYS 705: Classcal Mechancs Canoncal Transformaton II Example: Harmonc Oscllator f ( x) x m 0 x U( x) x mx x LT U m Defne or L p p mx x x m mx x H px L px p m p x m m H p 1 x m p m 1 m H x p m x m m
More informationLearning Theory: Lecture Notes
Learnng Theory: Lecture Notes Lecturer: Kamalka Chaudhur Scrbe: Qush Wang October 27, 2012 1 The Agnostc PAC Model Recall that one of the constrants of the PAC model s that the data dstrbuton has to be
More informationPricing Problems under the Nested Logit Model with a Quality Consistency Constraint
Prcng Problems under the Nested Logt Model wth a Qualty Consstency Constrant James M. Davs, Huseyn Topaloglu, Davd P. Wllamson 1 Aprl 28, 2015 Abstract We consder prcng problems when customers choose among
More information