Network Congestion Games
|
|
- Fay Shepherd
- 6 years ago
- Views:
Transcription
1 Ntwork Congstion Gams Assistant Profssor Tas A&M Univrsity Collg Station, TX TX Dallas Collg Station Austin Houston
2 Bst rout dpnds on othrs Ntwork Congstion Gams
3 Travl tim incrass with congstion Highway congstion costs wr $115 billion in Avg. commutr travls 100 minuts a day. Ntwork Congstion Gams
4 Gam modl Dirctd graph G = (V,E Multipl sourc-dst. pairs (s k,t k, dmand d k Playrs (usrs: nonatomic (infinitsimally small Stratgy st: paths P k btwn (s k,t k for all k Ρ Playrs dcisions: flow vctor R or Ρ Somtims will us f R for path flow. R E ( Edg dlay (latncy functions: typically assumd continuous and nondcrasing. Ntwork Congstion Gams
5 Outlin Wardrop Equilibrium Social Optimum Pric of Anarchy Ntwork Congstion Gams
6 Outlin Wardrop Equilibrium Social Optimum Pric of Anarchy Ntwork Congstion Gams
7 Wardrop s First Principl Travl tims on usd routs ar qual and no gratr than travl tims on unusd routs. Dfinition: A flow is a Wardrop Equilibrium (WE if for vry sourc-dst. pair k and for vry path with positiv flow btwn this pair, whr path path ( path' (, ( = ( path Also calld Usr Equilibrium or Nash Equilibrium. Equilibrium flow is calld Nash flow.. for all path ' P k Ntwork Congstion Gams
8 Outlin Wardrop Equilibrium Social Optimum Pric of Anarchy Ntwork Congstion Gams
9 Wardrop s Scond Principl Th avrag [total] journy tim is minimum. Th cost of flow is dfind as th total journy tim : Dnot C( = ( = all paths E c path ( path path path ( : = ( = all paths = E path path (. (, assumd conv. Ntwork Congstion Gams
10 Wardrop s Scond Principl Th avrag [total] journy tim is minimum. Dfinition: A flow is a Social Optimum if it minimizs total dlay: Ntwork Congstion Gams. for all 0, for all, for all s.t. ( ( min : P p f K k f d E f c p P p p k p P p p E E k = = = flow constraints
11 Social Optimum Th avrag [total] journy tim is minimum. Dfinition: A flow is a Social Optimum if it minimizs total dlay: min s.t. E c ( = E flow constraints ( Ntwork Congstion Gams
12 Social Optimum Dfinition: A flow is a Social Optimum if it solvs Lmma: A flow vctor is locally optimal if for ach path p with positiv flow and ach path p, whr c' ( min = E c ( = ( E c' ( c' ' (, p c' ( Proof sktch: p marginal bnfit of marginal cost of rducing traffic on p incrasing traffic on p p. p Ntwork Congstion Gams
13 Social Optimum (SO Dfinition: A flow is a Social Optimum if it solvs min E c ( = ( E Lmma: A flow vctor is locally optimal if for ach path p with positiv flow and ach path p, Corollary 1: If costs ar conv, local opt is a global opt, and lmma givs quivalnt dfn of SO. c' ( c' ' (. p c ( p Ntwork Congstion Gams
14 Social Optimum (SO Dfinition: A flow is a Social Optimum if it solvs min E c ( = ( E Lmma: A flow vctor is locally optimal if for ach path p with positiv flow and ach path p, Corollary 2: If costs ar conv, SO is an quilibrium with rspct to modifid latncis * c' ( c' ' (. p c ( ( = c' ( = ( + p '(. Ntwork Congstion Gams
15 Social Optimum (SO Dfinition: A flow is a Social Optimum if it solvs min c ( = ( Mchanism Dsign Intrprtation: If usrs valu tim and mony qually, E imposing E tolls '( pr unit flow on ach dg will caus slfish Lmma: A flow vctor is locally optimal if for playrs ach path to rach p with th positiv Social flow Optimum! and ach path p, c' ( c' ' (. p c ( Corollary 2: If costs ar conv, SO is an quilibrium with rspct to modifid latncis * ( = c' ( = ( + p '(. Ntwork Congstion Gams
16 Computing Social Optimum (SO Dfinition: A flow is a Social Optimum if it solvs min s.t. E c ( flow constraints c ( ( Corollary 3: If costs ar conv, SO ists and can b found fficintly by solving conv program abov. = E Ntwork Congstion Gams
17 Outlin Rvisit Wardrop Equilibrium Social Optimum Pric of Anarchy Ntwork Congstion Gams
18 Equilibrium istnc WE Dfinition: A flow is a Wardrop Equilibrium if for vry sourc-dst. pair k and for vry path with positiv flow btwn this pair, path compar with: ( '(, for all path' path SO Dfinition: A flow vctor is a Social Optimum for vry sourc-dst. pair k and for vry path with positiv flow btwn this pair, c' ( c' ' (, for all path' path path Ntwork Congstion Gams
19 Equilibrium istnc WE Dfinition: A flow is a Wardrop Equilibrium if for vry sourc-dst. pair k and for vry path with positiv flow btwn this pair, h whr path h '( h ''(, for all path' path SO Dfinition: A flow vctor is a Social Optimum for vry sourc-dst. pair k and for vry path with positiv flow btwn this pair, c' ( c' ' (, for all path' path '( : = ( path. Ntwork Congstion Gams
20 Equilibrium istnc WE Dfinition: A flow is a Wardrop Equilibrium if for vry sourc-dst. pair k and for vry path with positiv flow btwn this pair, h path '( h ''(, for all path' path whr h '( : = (. Altrnativ SO Dfinition: A flow vctor is a Social Optimum if it solvs: min E c ( Ntwork Congstion Gams
21 Equilibrium istnc Altrnativ WE Dfinition: A flow vctor is a Wardrop Equilibrium if it solvs: whr h '( min E : = ( Altrnativ SO Dfinition: A flow vctor is a Social Optimum if it solvs: min E h (,. c ( E.g., h ( = 0 ( z dz Ntwork Congstion Gams
22 Equilibrium istnc Altrnativ WE Dfinition: A flow vctor is a Wardrop Equilibrium if it solvs: whr h '( min : = E ( Altrnativ SO Dfinition: A flow vctor is a Social Optimum if it solvs: min 0 E ( z dz,. c ( Ntwork Congstion Gams
23 Equilibrium istnc Altrnativ WE Dfinition: A flow vctor is a Wardrop Equilibrium if it solvs: min s.t. E 0 ( z dz flow constraints Thorm: A Wardrop Equilibrium ists and can b computd in polynomial tim. Also, if program abov is strictly conv, quilibrium is uniqu, up to sam flow cost. Ntwork Congstion Gams
24 Outlin Wardrop Equilibrium Social Optimum Pric of Anarchy Ntwork Congstion Gams
25 Eampl: Infficincy of quilibria hours 1/2 1 hour Dlay is 1.5 hours for vrybody at th uniqu Nash quilibrium Town A Town B 1 hour hours Suppos drivrs (total 1 unit of flow lav from town A towards town B. Evry drivr wants to minimiz hr own travl tim. What is th traffic on th ntwork? 1/2 In any unbalancd traffic pattrn, all drivrs on th most loadd path hav incntiv to switch thir path. Ntwork Congstion Gams
26 Eampl: Infficincy of quilibria hours 1 1 hour Dlay is 2 hours for vrybody at th uniqu Nash quilibrium Town A 0 hours Town B 1 hour hours A bnvolnt mayor builds a suprhighway conncting th fast highways of th ntwork. What is now th traffic on th ntwork? No mattr what th othr drivrs ar doing it is always bttr for m to follow th zig-zag path. Ntwork Congstion Gams
27 Eampl: Infficincy of quilibria 1/2 1 hours 1 hour hours 1 hour A B vs A B 1 hour hours 1 hour hours 1/2 Adding a fast road on a road-ntwork is not always a good ida! Brass s parado In th RHS ntwork thr ists a traffic pattrn whr all playrs hav dlay 1.5 hours. Pric of Anarchy: Ntwork Congstion Gams masurs th loss in systm prformanc du to fr-will
28 Pric of Anarchy Cost of Flow: total usr cost Social optimum: flow minimizing total usr cost Pric of anarchy: (Koutsoupias, Papadimitriou 99 sup problm instancs Equilibrium Cost Social Optimum Cost Ntwork Congstion Gams
29 Variational Inquality rprsntation of quilibria Thorm: Equilibria in nonatomic gams ar solutions to th Variational Inquality (VI ( ( ' 0 for all fasibl flows ' Proof: whr Flow f is an quilibrium = (,..., if and only if [flow l(. vctor] <= l(.. 1 P ( = ( 1(,..., P ( [latncy vctor] Proof: (=> Equilibrium flow routs along minimum-cost paths l(. Fiing path costs at l(, any othr flow that assigns flow to highr-cost paths will rsult in highr ovrall cost l(.. VI Solution ists ovr compact conv st with l( continuous [Hartman, Stampacchia 66]. (<= Suppos is not an q. Thn thr is a flow-carrying path p VI Solution uniqu if l( is monoton: (l(-l ( (- 0. with lp( > lp (. Shifting flow from p to p will obtain [Ercis: vrify] l(. < l(., contradiction. Ntwork Congstion Gams
30 Thorm**: Th pric of anarchy (PoA is 4/3 in gnral graphs and latncis i.. whr is WE and * is SO flow. Pf: 0,,, ( + = b a b a l Pric of Anarchy with linar latncis Ntwork Congstion Gams * ( ( ( E E C =, ( 3 4 ( * C C ] 4 1 ( [ ( * 2 * * * 2 E E b a b a = + + = E E a b a 2 * * 4 1 ( **Rfrncs: Roughgardn, Tardos 02; Corra, Schulz, Stir-Moss 04, 08 QED., ( ( 4 3 * C C 2 * 2 * * 2 1 ( 0 ( 4 1 ( * C C + * ( E b a + =
31 Tak-away points Equilibrium and Social Optimum in nonatomic routing gams ist and can b found fficintly via conv programs. Social optimum is an quilibrium with rspct to modifid latncis = original latncis plus toll. Pric of anarchy: 4/3 for linar latncis, can b found similarly for mor gnral classs of latncy functions. Ntwork Congstion Gams
32 Rfrncs Wardrop 52, Bckmann t al. 56, A lot of work in AGT community and othrs. Survys of rcnt work: AGT Book Nisan t al. 07 Corra, Stir-Moss 11 Ntwork Congstion Gams
33 Som opn qustions What is th pric of anarchy with rspct to othr Social Cost functions? Dynamic (tim-changing latncy functions? Uncrtain dlays? Ntwork Congstion Gams
34 Ntwork Congstion Gams
35 Announcmnt I am looking for Ph.D. studnts and a postdoc with a strong thortical background. application to: nikolova@tamu.du Ntwork Congstion Gams
Supplementary Materials
6 Supplmntary Matrials APPENDIX A PHYSICAL INTERPRETATION OF FUEL-RATE-SPEED FUNCTION A truck running on a road with grad/slop θ positiv if moving up and ngativ if moving down facs thr rsistancs: arodynamic
More informationarxiv: v3 [cs.gt] 1 Jan 2019
Pric of Anarchy in Ntworks with Htrognous Latncy Functions Sanjiv Kapoor and Junghwan Shin arxiv:407.299v3 [cs.gt] Jan 209 Abstract W addrss th prformanc of slfish ntwork routing in multi-commodity flows
More information1973 AP Calculus AB: Section I
97 AP Calculus AB: Sction I 9 Minuts No Calculator Not: In this amination, ln dnots th natural logarithm of (that is, logarithm to th bas ).. ( ) d= + C 6 + C + C + C + C. If f ( ) = + + + and ( ), g=
More informationBasic Polyhedral theory
Basic Polyhdral thory Th st P = { A b} is calld a polyhdron. Lmma 1. Eithr th systm A = b, b 0, 0 has a solution or thr is a vctorπ such that π A 0, πb < 0 Thr cass, if solution in top row dos not ist
More informationStackelberg strategies for selfish routing in general multicommodity networks
Stacklbrg stratgis for slfish routing in gnral multicommodity ntworks Gorg Karakostas Stavros G. Kolliopoulos Abstract A natural gnralization of th slfish routing stting ariss whn som of th usrs oby a
More informationAltruism, Selfishness, and Spite in Traffic Routing
Altruism, Slfishnss, and Spit in Traffic Routing Po-An Chn Univrsity of Southrn California poanchn@usc.du David Kmp Univrsity of Southrn California dkmp@usc.du ABSTRACT In this papr, w study th pric of
More informationCPSC 665 : An Algorithmist s Toolkit Lecture 4 : 21 Jan Linear Programming
CPSC 665 : An Algorithmist s Toolkit Lctur 4 : 21 Jan 2015 Lcturr: Sushant Sachdva Linar Programming Scrib: Rasmus Kyng 1. Introduction An optimization problm rquirs us to find th minimum or maximum) of
More informationStrongly Connected Components
Strongly Connctd Componnts Lt G = (V, E) b a dirctd graph Writ if thr is a path from to in G Writ if and is an quivalnc rlation: implis and implis s quivalnc classs ar calld th strongly connctd componnts
More information2008 AP Calculus BC Multiple Choice Exam
008 AP Multipl Choic Eam Nam 008 AP Calculus BC Multipl Choic Eam Sction No Calculator Activ AP Calculus 008 BC Multipl Choic. At tim t 0, a particl moving in th -plan is th acclration vctor of th particl
More informationAbstract Interpretation: concrete and abstract semantics
Abstract Intrprtation: concrt and abstract smantics Concrt smantics W considr a vry tiny languag that manags arithmtic oprations on intgrs valus. Th (concrt) smantics of th languags cab b dfind by th funzcion
More informationTraffic Optimization For a Mixture of Self-interested and Compliant Agents
In Procdings of th 32nd Confrnc on Artificial Intllignc (AAAI 2018), Nw Orlans, Lousiana, USA. Fbruary 2018 Traffic Optimization For a Mixtur of Slf-intrstd and Compliant Agnts Guni Sharon 1, Michal Albrt
More informationRandom Access Techniques: ALOHA (cont.)
Random Accss Tchniqus: ALOHA (cont.) 1 Exampl [ Aloha avoiding collision ] A pur ALOHA ntwork transmits a 200-bit fram on a shard channl Of 200 kbps at tim. What is th rquirmnt to mak this fram collision
More informationComputing and Communications -- Network Coding
89 90 98 00 Computing and Communications -- Ntwork Coding Dr. Zhiyong Chn Institut of Wirlss Communications Tchnology Shanghai Jiao Tong Univrsity China Lctur 5- Nov. 05 0 Classical Information Thory Sourc
More informationPropositional Logic. Combinatorial Problem Solving (CPS) Albert Oliveras Enric Rodríguez-Carbonell. May 17, 2018
Propositional Logic Combinatorial Problm Solving (CPS) Albrt Olivras Enric Rodríguz-Carbonll May 17, 2018 Ovrviw of th sssion Dfinition of Propositional Logic Gnral Concpts in Logic Rduction to SAT CNFs
More informationDynamic Equilibria in Fluid Queueing Networks
OPERATIONS RESEARCH Vol. 63, No. 1, January Fbruary 215, pp. 21 34 ISSN 3-364 print) ó ISSN 1526-5463 onlin) http://dx.doi.org/1.1287/opr.215.1348 215 INFORMS Dynamic Equilibria in Fluid Quuing Ntworks
More informationBifurcation Theory. , a stationary point, depends on the value of α. At certain values
Dnamic Macroconomic Thor Prof. Thomas Lux Bifurcation Thor Bifurcation: qualitativ chang in th natur of th solution occurs if a paramtr passs through a critical point bifurcation or branch valu. Local
More informationMathematics. Complex Number rectangular form. Quadratic equation. Quadratic equation. Complex number Functions: sinusoids. Differentiation Integration
Mathmatics Compl numbr Functions: sinusoids Sin function, cosin function Diffrntiation Intgration Quadratic quation Quadratic quations: a b c 0 Solution: b b 4ac a Eampl: 1 0 a= b=- c=1 4 1 1or 1 1 Quadratic
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by D. Klain Vrsion 207.0.05 Corrctions and commnts ar wlcom. Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix A A k I + A + k!
More informationLoad Balancing Without Regret in the Bulletin Board Model
Nonam manuscript No will b insrtd by th ditor Load Balancing Without Rgrt in th Bulltin Board Modl Robrt Klinbrg Gorgios Piliouras Éva Tardos Rcivd: dat / Accptd: dat Abstract W analyz th prformanc of
More informationWeek 3: Connected Subgraphs
Wk 3: Connctd Subgraphs Sptmbr 19, 2016 1 Connctd Graphs Path, Distanc: A path from a vrtx x to a vrtx y in a graph G is rfrrd to an xy-path. Lt X, Y V (G). An (X, Y )-path is an xy-path with x X and y
More informationSearching Linked Lists. Perfect Skip List. Building a Skip List. Skip List Analysis (1) Assume the list is sorted, but is stored in a linked list.
3 3 4 8 6 3 3 4 8 6 3 3 4 8 6 () (d) 3 Sarching Linkd Lists Sarching Linkd Lists Sarching Linkd Lists ssum th list is sortd, but is stord in a linkd list. an w us binary sarch? omparisons? Work? What if
More informationWhen Do Potential Functions Exist in Heterogeneous Routing Games? 1
Whn Do Potntial Functions Exist in Htrognous Routing Gams? 1 Farhad Farokhi 2, Walid Krichn 3,4, Alxandr M. Bayn 4,5, and Karl H. Johansson 2 Abstract W study a htrognous routing gam in which vhicls might
More information1 Minimum Cut Problem
CS 6 Lctur 6 Min Cut and argr s Algorithm Scribs: Png Hui How (05), Virginia Dat: May 4, 06 Minimum Cut Problm Today, w introduc th minimum cut problm. This problm has many motivations, on of which coms
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by Dan Klain Vrsion 28928 Corrctions and commnts ar wlcom Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix () A A k I + A + k!
More informationExamples and applications on SSSP and MST
Exampls an applications on SSSP an MST Dan (Doris) H & Junhao Gan ITEE Univrsity of Qunslan COMP3506/7505, Uni of Qunslan Exampls an applications on SSSP an MST Dijkstra s Algorithm Th algorithm solvs
More informationINTEGRATION BY PARTS
Mathmatics Rvision Guids Intgration by Parts Pag of 7 MK HOME TUITION Mathmatics Rvision Guids Lvl: AS / A Lvl AQA : C Edcl: C OCR: C OCR MEI: C INTEGRATION BY PARTS Vrsion : Dat: --5 Eampls - 6 ar copyrightd
More informationEconomics 201b Spring 2010 Solutions to Problem Set 3 John Zhu
Economics 20b Spring 200 Solutions to Problm St 3 John Zhu. Not in th 200 vrsion of Profssor Andrson s ctur 4 Nots, th charactrization of th firm in a Robinson Cruso conomy is that it maximizs profit ovr
More informationDerangements and Applications
2 3 47 6 23 Journal of Intgr Squncs, Vol. 6 (2003), Articl 03..2 Drangmnts and Applications Mhdi Hassani Dpartmnt of Mathmatics Institut for Advancd Studis in Basic Scincs Zanjan, Iran mhassani@iasbs.ac.ir
More informationData Assimilation 1. Alan O Neill National Centre for Earth Observation UK
Data Assimilation 1 Alan O Nill National Cntr for Earth Obsrvation UK Plan Motivation & basic idas Univariat (scalar) data assimilation Multivariat (vctor) data assimilation 3d-Variational Mthod (& optimal
More informationu 3 = u 3 (x 1, x 2, x 3 )
Lctur 23: Curvilinar Coordinats (RHB 8.0 It is oftn convnint to work with variabls othr than th Cartsian coordinats x i ( = x, y, z. For xampl in Lctur 5 w mt sphrical polar and cylindrical polar coordinats.
More informationMath 34A. Final Review
Math A Final Rviw 1) Us th graph of y10 to find approimat valus: a) 50 0. b) y (0.65) solution for part a) first writ an quation: 50 0. now tak th logarithm of both sids: log() log(50 0. ) pand th right
More informationare given in the table below. t (hours)
CALCULUS WORKSHEET ON INTEGRATION WITH DATA Work th following on notbook papr. Giv dcimal answrs corrct to thr dcimal placs.. A tank contains gallons of oil at tim t = hours. Oil is bing pumpd into th
More information10. The Discrete-Time Fourier Transform (DTFT)
Th Discrt-Tim Fourir Transform (DTFT Dfinition of th discrt-tim Fourir transform Th Fourir rprsntation of signals plays an important rol in both continuous and discrt signal procssing In this sction w
More informationAS 5850 Finite Element Analysis
AS 5850 Finit Elmnt Analysis Two-Dimnsional Linar Elasticity Instructor Prof. IIT Madras Equations of Plan Elasticity - 1 displacmnt fild strain- displacmnt rlations (infinitsimal strain) in matrix form
More informationHigher order derivatives
Robrto s Nots on Diffrntial Calculus Chaptr 4: Basic diffrntiation ruls Sction 7 Highr ordr drivativs What you nd to know alrady: Basic diffrntiation ruls. What you can larn hr: How to rpat th procss of
More informationRecall that by Theorems 10.3 and 10.4 together provide us the estimate o(n2 ), S(q) q 9, q=1
Chaptr 11 Th singular sris Rcall that by Thorms 10 and 104 togthr provid us th stimat 9 4 n 2 111 Rn = SnΓ 2 + on2, whr th singular sris Sn was dfind in Chaptr 10 as Sn = q=1 Sq q 9, with Sq = 1 a q gcda,q=1
More informationPartial Derivatives: Suppose that z = f(x, y) is a function of two variables.
Chaptr Functions o Two Variabls Applid Calculus 61 Sction : Calculus o Functions o Two Variabls Now that ou hav som amiliarit with unctions o two variabls it s tim to start appling calculus to hlp us solv
More informationCramér-Rao Inequality: Let f(x; θ) be a probability density function with continuous parameter
WHEN THE CRAMÉR-RAO INEQUALITY PROVIDES NO INFORMATION STEVEN J. MILLER Abstract. W invstigat a on-paramtr family of probability dnsitis (rlatd to th Parto distribution, which dscribs many natural phnomna)
More informationA. Limits and Horizontal Asymptotes ( ) f x f x. f x. x "±# ( ).
A. Limits and Horizontal Asymptots What you ar finding: You can b askd to find lim x "a H.A.) problm is asking you find lim x "# and lim x "$#. or lim x "±#. Typically, a horizontal asymptot algbraically,
More informationGradebook & Midterm & Office Hours
Your commnts So what do w do whn on of th r's is 0 in th quation GmM(1/r-1/r)? Do w nd to driv all of ths potntial nrgy formulas? I don't undrstand springs This was th first lctur I actually larnd somthing
More informationLimiting value of higher Mahler measure
Limiting valu of highr Mahlr masur Arunabha Biswas a, Chris Monico a, a Dpartmnt of Mathmatics & Statistics, Txas Tch Univrsity, Lubbock, TX 7949, USA Abstract W considr th k-highr Mahlr masur m k P )
More informationFinding low cost TSP and 2-matching solutions using certain half integer subtour vertices
Finding low cost TSP and 2-matching solutions using crtain half intgr subtour vrtics Sylvia Boyd and Robrt Carr Novmbr 996 Introduction Givn th complt graph K n = (V, E) on n nods with dg costs c R E,
More informationcycle that does not cross any edges (including its own), then it has at least
W prov th following thorm: Thorm If a K n is drawn in th plan in such a way that it has a hamiltonian cycl that dos not cross any dgs (including its own, thn it has at last n ( 4 48 π + O(n crossings Th
More informationA Heterogeneous Routing Game
A Htrognous Routing Gam Farhad Farohi Walid Krichn Alxandr M. Bayn and Karl H. Johansson Abstract Most litratur on routing gams ma th assumtion that drivrs or vhicls ar of th sam ty and hnc xrinc th sam
More informationChapter 13 GMM for Linear Factor Models in Discount Factor form. GMM on the pricing errors gives a crosssectional
Chaptr 13 GMM for Linar Factor Modls in Discount Factor form GMM on th pricing rrors givs a crosssctional rgrssion h cas of xcss rturns Hors rac sting for charactristic sting for pricd factors: lambdas
More informationDefinition1: The ratio of the radiation intensity in a given direction from the antenna to the radiation intensity averaged over all directions.
Dirctivity or Dirctiv Gain. 1 Dfinition1: Dirctivity Th ratio of th radiation intnsity in a givn dirction from th antnna to th radiation intnsity avragd ovr all dirctions. Dfinition2: Th avg U is obtaind
More informationFirst derivative analysis
Robrto s Nots on Dirntial Calculus Chaptr 8: Graphical analysis Sction First drivativ analysis What you nd to know alrady: How to us drivativs to idntiy th critical valus o a unction and its trm points
More informationProblem Set 6 Solutions
6.04/18.06J Mathmatics for Computr Scinc March 15, 005 Srini Dvadas and Eric Lhman Problm St 6 Solutions Du: Monday, March 8 at 9 PM in Room 3-044 Problm 1. Sammy th Shark is a financial srvic providr
More informationCOHORT MBA. Exponential function. MATH review (part2) by Lucian Mitroiu. The LOG and EXP functions. Properties: e e. lim.
MTH rviw part b Lucian Mitroiu Th LOG and EXP functions Th ponntial function p : R, dfind as Proprtis: lim > lim p Eponntial function Y 8 6 - -8-6 - - X Th natural logarithm function ln in US- log: function
More informationCalculus concepts derivatives
All rasonabl fforts hav bn mad to mak sur th nots ar accurat. Th author cannot b hld rsponsibl for any damags arising from th us of ths nots in any fashion. Calculus concpts drivativs Concpts involving
More informationOn the irreducibility of some polynomials in two variables
ACTA ARITHMETICA LXXXII.3 (1997) On th irrducibility of som polynomials in two variabls by B. Brindza and Á. Pintér (Dbrcn) To th mmory of Paul Erdős Lt f(x) and g(y ) b polynomials with intgral cofficints
More informationEquidistribution and Weyl s criterion
Euidistribution and Wyl s critrion by Brad Hannigan-Daly W introduc th ida of a sunc of numbrs bing uidistributd (mod ), and w stat and prov a thorm of Hrmann Wyl which charactrizs such suncs. W also discuss
More informationBINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES. 1. Statement of results
BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES DONALD M. DAVIS Abstract. If p is a prim and n a positiv intgr, lt ν p (n dnot th xponnt of p in n, and u p (n n/p νp(n th unit part of n. If α
More informationME 321 Kinematics and Dynamics of Machines S. Lambert Winter 2002
3.4 Forc Analysis of Linkas An undrstandin of forc analysis of linkas is rquird to: Dtrmin th raction forcs on pins, tc. as a consqunc of a spcifid motion (don t undrstimat th sinificanc of dynamic or
More informationGeneral Notes About 2007 AP Physics Scoring Guidelines
AP PHYSICS C: ELECTRICITY AND MAGNETISM 2007 SCORING GUIDELINES Gnral Nots About 2007 AP Physics Scoring Guidlins 1. Th solutions contain th most common mthod of solving th fr-rspons qustions and th allocation
More informationSECTION where P (cos θ, sin θ) and Q(cos θ, sin θ) are polynomials in cos θ and sin θ, provided Q is never equal to zero.
SETION 6. 57 6. Evaluation of Dfinit Intgrals Exampl 6.6 W hav usd dfinit intgrals to valuat contour intgrals. It may com as a surpris to larn that contour intgrals and rsidus can b usd to valuat crtain
More informationChapter 14 Aggregate Supply and the Short-run Tradeoff Between Inflation and Unemployment
Chaptr 14 Aggrgat Supply and th Short-run Tradoff Btwn Inflation and Unmploymnt Modifid by Yun Wang Eco 3203 Intrmdiat Macroconomics Florida Intrnational Univrsity Summr 2017 2016 Worth Publishrs, all
More informationOn spanning trees and cycles of multicolored point sets with few intersections
On spanning trs and cycls of multicolord point sts with fw intrsctions M. Kano, C. Mrino, and J. Urrutia April, 00 Abstract Lt P 1,..., P k b a collction of disjoint point sts in R in gnral position. W
More informationA Prey-Predator Model with an Alternative Food for the Predator, Harvesting of Both the Species and with A Gestation Period for Interaction
Int. J. Opn Problms Compt. Math., Vol., o., Jun 008 A Pry-Prdator Modl with an Altrnativ Food for th Prdator, Harvsting of Both th Spcis and with A Gstation Priod for Intraction K. L. arayan and. CH. P.
More informationSCHUR S THEOREM REU SUMMER 2005
SCHUR S THEOREM REU SUMMER 2005 1. Combinatorial aroach Prhas th first rsult in th subjct blongs to I. Schur and dats back to 1916. On of his motivation was to study th local vrsion of th famous quation
More informationHospital Readmission Reduction Strategies Using a Penalty-Incentive Model
Procdings of th 2016 Industrial and Systms Enginring Rsarch Confrnc H. Yang, Z. Kong, and MD Sardr, ds. Hospital Radmission Rduction Stratgis Using a Pnalty-Incntiv Modl Michll M. Alvarado Txas A&M Univrsity
More informationDISTRIBUTION OF DIFFERENCE BETWEEN INVERSES OF CONSECUTIVE INTEGERS MODULO P
DISTRIBUTION OF DIFFERENCE BETWEEN INVERSES OF CONSECUTIVE INTEGERS MODULO P Tsz Ho Chan Dartmnt of Mathmatics, Cas Wstrn Rsrv Univrsity, Clvland, OH 4406, USA txc50@cwru.du Rcivd: /9/03, Rvisd: /9/04,
More informationOptimizing Product Launches in the Presence of Strategic Consumers Appendix
Optimizing Product Launchs in th Prsnc of Stratgic Consumrs Appndix Ilan Lobl Jigar Patl Gustavo Vulcano Jiawi Zhang Lonard N. Strn School of Businss, Nw York Univrsity, 44 Wst Fourth St., Nw York, NY
More informationRENT, LEASE OR BUY: RANDOMIZED ALGORITHMS FOR MULTISLOPE SKI RENTAL
RENT, LEASE OR BUY: RANDOMIZED ALGORITHMS FOR MULTISLOPE SKI RENTAL ZVI LOTKER 1, BOAZ PATT-SHAMIR 2, AND DROR RAWITZ 3 1 Dpt. of Communication Systms Enginring, Bn Gurion Univrsity, Br Shva 84105, Isral.
More informationMA 262, Spring 2018, Final exam Version 01 (Green)
MA 262, Spring 218, Final xam Vrsion 1 (Grn) INSTRUCTIONS 1. Switch off your phon upon ntring th xam room. 2. Do not opn th xam booklt until you ar instructd to do so. 3. Bfor you opn th booklt, fill in
More informationLinear Non-Gaussian Structural Equation Models
IMPS 8, Durham, NH Linar Non-Gaussian Structural Equation Modls Shohi Shimizu, Patrik Hoyr and Aapo Hyvarinn Osaka Univrsity, Japan Univrsity of Hlsinki, Finland Abstract Linar Structural Equation Modling
More informationCombinatorial Networks Week 1, March 11-12
1 Nots on March 11 Combinatorial Ntwors W 1, March 11-1 11 Th Pigonhol Principl Th Pigonhol Principl If n objcts ar placd in hols, whr n >, thr xists a box with mor than on objcts 11 Thorm Givn a simpl
More informationHardy-Littlewood Conjecture and Exceptional real Zero. JinHua Fei. ChangLing Company of Electronic Technology Baoji Shannxi P.R.
Hardy-Littlwood Conjctur and Excptional ral Zro JinHua Fi ChangLing Company of Elctronic Tchnology Baoji Shannxi P.R.China E-mail: fijinhuayoujian@msn.com Abstract. In this papr, w assum that Hardy-Littlwood
More informationConstruction of asymmetric orthogonal arrays of strength three via a replacement method
isid/ms/26/2 Fbruary, 26 http://www.isid.ac.in/ statmath/indx.php?modul=prprint Construction of asymmtric orthogonal arrays of strngth thr via a rplacmnt mthod Tian-fang Zhang, Qiaoling Dng and Alok Dy
More informationa 1and x is any real number.
Calcls Nots Eponnts an Logarithms Eponntial Fnction: Has th form y a, whr a 0, a an is any ral nmbr. Graph y, Graph y ln y y Th Natral Bas (Elr s nmbr): An irrational nmbr, symboliz by th lttr, appars
More informationON RIGHT(LEFT) DUO PO-SEMIGROUPS. S. K. Lee and K. Y. Park
Kangwon-Kyungki Math. Jour. 11 (2003), No. 2, pp. 147 153 ON RIGHT(LEFT) DUO PO-SEMIGROUPS S. K. L and K. Y. Park Abstract. W invstigat som proprtis on right(rsp. lft) duo po-smigroups. 1. Introduction
More informationElectrical Flows, Laplacian Systems, and Faster Approximation of Maximum Flow in Undirected Graphs
Elctrical Flows, Laplacian Systms, and Fastr Approximation of Maximum Flow in Undirctd Graphs Paul Christiano MIT Jonathan A. Klnr MIT Alksandr Mądry MIT Shang-Hua Tng Univrsity of Southrn California Octobr
More informationExam 1. It is important that you clearly show your work and mark the final answer clearly, closed book, closed notes, no calculator.
Exam N a m : _ S O L U T I O N P U I D : I n s t r u c t i o n s : It is important that you clarly show your work and mark th final answr clarly, closd book, closd nots, no calculator. T i m : h o u r
More informationBrief Introduction to Statistical Mechanics
Brif Introduction to Statistical Mchanics. Purpos: Ths nots ar intndd to provid a vry quick introduction to Statistical Mchanics. Th fild is of cours far mor vast than could b containd in ths fw pags.
More information(Upside-Down o Direct Rotation) β - Numbers
Amrican Journal of Mathmatics and Statistics 014, 4(): 58-64 DOI: 10593/jajms0140400 (Upsid-Down o Dirct Rotation) β - Numbrs Ammar Sddiq Mahmood 1, Shukriyah Sabir Ali,* 1 Dpartmnt of Mathmatics, Collg
More informationA Penalized Best-Response Algorithm for Non-Linear Single-Path Routing Problems
A Pnalizd Bst-Rspons Algorithm for Non-Linar Singl-Path Routing Problms Olivir Brun, Balakrishna Prabhu, Josslin Vallt To cit this vrsion: Olivir Brun, Balakrishna Prabhu, Josslin Vallt. A Pnalizd Bst-Rspons
More information1997 AP Calculus AB: Section I, Part A
997 AP Calculus AB: Sction I, Part A 50 Minuts No Calculator Not: Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs for which f () is a ral numbr.. (4 6 ) d= 4 6 6
More informationBINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES
BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES DONALD M. DAVIS Abstract. If p is a prim (implicit in notation and n a positiv intgr, lt ν(n dnot th xponnt of p in n, and U(n n/p ν(n, th unit
More informationIntroduction to Arithmetic Geometry Fall 2013 Lecture #20 11/14/2013
18.782 Introduction to Arithmtic Gomtry Fall 2013 Lctur #20 11/14/2013 20.1 Dgr thorm for morphisms of curvs Lt us rstat th thorm givn at th nd of th last lctur, which w will now prov. Thorm 20.1. Lt φ:
More informationRandom Process Part 1
Random Procss Part A random procss t (, ζ is a signal or wavform in tim. t : tim ζ : outcom in th sampl spac Each tim w rapat th xprimnt, a nw wavform is gnratd. ( W will adopt t for short. Tim sampls
More informationFourier Transforms and the Wave Equation. Key Mathematics: More Fourier transform theory, especially as applied to solving the wave equation.
Lur 7 Fourir Transforms and th Wav Euation Ovrviw and Motivation: W first discuss a fw faturs of th Fourir transform (FT), and thn w solv th initial-valu problm for th wav uation using th Fourir transform
More informationDiploma Macro Paper 2
Diploma Macro Papr 2 Montary Macroconomics Lctur 6 Aggrgat supply and putting AD and AS togthr Mark Hays 1 Exognous: M, G, T, i*, π Goods markt KX and IS (Y, C, I) Mony markt (LM) (i, Y) Labour markt (P,
More informationEinstein Equations for Tetrad Fields
Apiron, Vol 13, No, Octobr 006 6 Einstin Equations for Ttrad Filds Ali Rıza ŞAHİN, R T L Istanbul (Turky) Evry mtric tnsor can b xprssd by th innr product of ttrad filds W prov that Einstin quations for
More informationDeift/Zhou Steepest descent, Part I
Lctur 9 Dift/Zhou Stpst dscnt, Part I W now focus on th cas of orthogonal polynomials for th wight w(x) = NV (x), V (x) = t x2 2 + x4 4. Sinc th wight dpnds on th paramtr N N w will writ π n,n, a n,n,
More informationInefficiency of Standard Multi-Unit Auctions
Infficincy of Standard Multi-Unit Auctions Bart d Kijzr 1 Evanglos Markakis 3 Guido Schäfr 1,2 Orstis Tllis 3 1 CWI Amstrdam, Th Nthrlands 2 VU Amstrdam, Th Nthrlands {b.d.kijzr,g.schafr}@cwi.nl 3 Athns
More informationMor Tutorial at www.dumblittldoctor.com Work th problms without a calculator, but us a calculator to chck rsults. And try diffrntiating your answrs in part III as a usful chck. I. Applications of Intgration
More informationECE602 Exam 1 April 5, You must show ALL of your work for full credit.
ECE62 Exam April 5, 27 Nam: Solution Scor: / This xam is closd-book. You must show ALL of your work for full crdit. Plas rad th qustions carfully. Plas chck your answrs carfully. Calculators may NOT b
More informationSearch sequence databases 3 10/25/2016
Sarch squnc databass 3 10/25/2016 Etrm valu distribution Ø Suppos X is a random variabl with probability dnsity function p(, w sampl a larg numbr S of indpndnt valus of X from this distribution for an
More informationSOME PARAMETERS ON EQUITABLE COLORING OF PRISM AND CIRCULANT GRAPH.
SOME PARAMETERS ON EQUITABLE COLORING OF PRISM AND CIRCULANT GRAPH. K VASUDEVAN, K. SWATHY AND K. MANIKANDAN 1 Dpartmnt of Mathmatics, Prsidncy Collg, Chnnai-05, India. E-Mail:vasu k dvan@yahoo.com. 2,
More informationDavisson Germer experiment
Announcmnts: Davisson Grmr xprimnt Homwork st 5 is today. Homwork st 6 will b postd latr today. Mad a good guss about th Nobl Priz for 2013 Clinton Davisson and Lstr Grmr. Davisson won Nobl Priz in 1937.
More informationAn Application of Hardy-Littlewood Conjecture. JinHua Fei. ChangLing Company of Electronic Technology Baoji Shannxi P.R.China
An Application of Hardy-Littlwood Conjctur JinHua Fi ChangLing Company of Elctronic Tchnology Baoji Shannxi P.R.China E-mail: fijinhuayoujian@msn.com Abstract. In this papr, w assum that wakr Hardy-Littlwood
More informationChapter 3 Exponential and Logarithmic Functions. Section a. In the exponential decay model A. Check Point Exercises
Chaptr Eponntial and Logarithmic Functions Sction. Chck Point Erciss. a. A 87. Sinc is yars aftr, whn t, A. b. A A 87 k() k 87 k 87 k 87 87 k.4 Thus, th growth function is A 87 87.4t.4t.4t A 87..4t 87.4t
More informationAP Calculus Multiple-Choice Question Collection connect to college success
AP Calculus Multipl-Choic Qustion Collction 969 998 connct to collg succss www.collgboard.com Th Collg Board: Conncting Studnts to Collg Succss Th Collg Board is a not-for-profit mmbrship association whos
More informationSCALING OF SYNCHROTRON RADIATION WITH MULTIPOLE ORDER. J. C. Sprott
SCALING OF SYNCHROTRON RADIATION WITH MULTIPOLE ORDER J. C. Sprott PLP 821 Novmbr 1979 Plasma Studis Univrsity of Wisconsin Ths PLP Rports ar informal and prliminary and as such may contain rrors not yt
More informationChapter 13 Aggregate Supply
Chaptr 13 Aggrgat Supply 0 1 Larning Objctivs thr modls of aggrgat supply in which output dpnds positivly on th pric lvl in th short run th short-run tradoff btwn inflation and unmploymnt known as th Phillips
More informationSolution of Assignment #2
olution of Assignmnt #2 Instructor: Alirza imchi Qustion #: For simplicity, assum that th distribution function of T is continuous. Th distribution function of R is: F R ( r = P( R r = P( log ( T r = P(log
More informationElements of Statistical Thermodynamics
24 Elmnts of Statistical Thrmodynamics Statistical thrmodynamics is a branch of knowldg that has its own postulats and tchniqus. W do not attmpt to giv hr vn an introduction to th fild. In this chaptr,
More informationAddition of angular momentum
Addition of angular momntum April, 0 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat th
More informationRent, Lease or Buy: Randomized Algorithms for Multislope Ski Rental
Rnt, Las or Buy: Randomizd Algorithms for Multislop Ski Rntal Zvi Lotkr, Boaz Patt-Shamir, Dror Rawitz To cit this vrsion: Zvi Lotkr, Boaz Patt-Shamir, Dror Rawitz. Rnt, Las or Buy: Randomizd Algorithms
More informationSliding Mode Flow Rate Observer Design
Sliding Mod Flow Rat Obsrvr Dsign Song Liu and Bin Yao School of Mchanical Enginring, Purdu Univrsity, Wst Lafaytt, IN797, USA liu(byao)@purdudu Abstract Dynamic flow rat information is ndd in a lot of
More information