Max Flow, Min Cut. Maximum Flow and Minimum Cut. Soviet Rail Network, Minimum Cut Problem

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Max Flow, Min Cut. Maximum Flow and Minimum Cut. Soviet Rail Network, 1955. Minimum Cut Problem"

Transcription

1 Maximum Flow and Minimum u Max Flow, Min u Max flow and min cu. Two very rich algorihmic problem. ornerone problem in combinaorial opimizaion. eauiful mahemaical dualiy. Minimum cu Maximum flow Max-flow min-cu heorem Ford-Fulkeron augmening pah algorihm dmond-karp heuriic iparie maching Nonrivial applicaion / reducion. Nework conneciviy. iparie maching. aa mining. Open-pi mining. irline cheduling. Image proceing. Projec elecion. aeball eliminaion. Nework reliabiliy. ecuriy of aiical daa. iribued compuing. galiarian able maching. iribued compuing. Many many more... Princeon Univeriy O lgorihm and aa rucure pring Kevin Wayne hp:// ovie Rail Nework, Minimum u Problem Nework: abracion for maerial FLOWING hrough he edge. ireced graph. apaciie on edge. ource node, ink node. Min cu problem. elee "be" e of edge o diconnec from. ource ink capaciy ource: On he hiory of he ranporaion and maximum flow problem. lexander chrijver in Mah Programming, :,.

2 u u cu i a node pariion (, T) uch ha i in and i in T. capaciy(, T) = um of weigh of edge leaving. cu i a node pariion (, T) uch ha i in and i in T. capaciy(, T) = um of weigh of edge leaving. apaciy = apaciy = Minimum u Problem Maximum Flow Problem cu i a node pariion (, T) uch ha i in and i in T. capaciy(, T) = um of weigh of edge leaving. Min cu problem. Find an - cu of minimum capaciy. Nework: abracion for maerial FLOWING hrough he edge. ireced graph. ame inpu a min cu problem apaciie on edge. ource node, ink node. Max flow problem. ign flow o edge o a o: qualize inflow and ouflow a every inermediae verex. Maximize flow en from o. ource capaciy ink apaciy =

3 Flow Flow flow f i an aignmen of weigh o edge o ha: apaciy: f(e) u(e). Flow conervaion: flow leaving v = flow enering v. excep a or flow f i an aignmen of weigh o edge o ha: apaciy: f(e) u(e). Flow conervaion: flow leaving v = flow enering v. excep a or capaciy flow Value = capaciy flow Value = Maximum Flow Problem Flow and u Max flow problem: find flow ha maximize ne flow ino ink. Obervaion. Le f be a flow, and le (, T) be any - cu. Then, he ne flow en acro he cu i equal o he amoun reaching. capaciy flow Value = Value =

4 Flow and u Flow and u Obervaion. Le f be a flow, and le (, T) be any - cu. Then, he ne flow en acro he cu i equal o he amoun reaching. Obervaion. Le f be a flow, and le (, T) be any - cu. Then, he ne flow en acro he cu i equal o he amoun reaching. Value = Value = Flow and u Max Flow and Min u Obervaion. Le f be a flow, and le (, T) be any - cu. Then he value of he flow i a mo he capaciy of he cu. Obervaion. Le f be a flow, and le (, T) be an - cu whoe capaciy equal he value of f. Then f i a max flow and (, T) i a min cu. u capaciy = Flow value u capaciy = Flow value Flow value =

5 Max-Flow Min-u Theorem Toward an lgorihm Max-flow min-cu heorem. (Ford-Fulkeron, ): In any nework, he value of max flow equal capaciy of min cu. Proof IOU: we find flow and cu uch ha Obervaion applie. Find - pah where each arc ha f(e) < u(e) and "augmen" flow along i. Min cu capaciy = Max flow value = flow Flow value = capaciy Toward an lgorihm Toward an lgorihm Find - pah where each arc ha f(e) < u(e) and "augmen" flow along i. Greedy algorihm: repea unil you ge uck. Find - pah where each arc ha f(e) < u(e) and "augmen" flow along i. Greedy algorihm: repea unil you ge uck. Fail: need o be able o "backrack." flow Flow value = flow Flow value = capaciy X X X capaciy X X X oleneck capaciy of pah = Flow value =

6 Reidual Graph ugmening Pah Original graph. Flow f(e). dge e = v-w v flow = f(e) capaciy = u(e) w ugmening pah = pah in reidual graph. Increae flow along forward edge. ecreae flow along backward edge. Reidual edge. dge e = v-w or w-v. "Undo" flow en. Reidual graph. ll he edge ha have ricly poiive reidual capaciy. v reidual capaciy = u(e) f(e) w reidual capaciy = f(e) reidual original X X X X X ugmening Pah Ford-Fulkeron ugmening Pah lgorihm Obervaion. If augmening pah, hen no ye a max flow. Q. If no augmening pah, i i a max flow? Ford-Fulkeron algorihm. Generic mehod for olving max flow. reidual while (here exi an augmening pah) { Find augmening pah P ompue boleneck capaciy of P ugmen flow along P original Flow value = X X X X X Queion. oe hi lead o a maximum flow? ye How do we find an augmening pah? - pah in reidual graph How many augmening pah doe i ake? How much effor do we pending finding a pah?

7 Max-Flow Min-u Theorem Proof of Max-Flow Min-u Theorem ugmening pah heorem. flow f i a max flow if and only if here are no augmening pah. Max-flow min-cu heorem. The value of he max flow i equal o he capaciy of he min cu. We prove boh imulaneouly by howing he following are equivalen: (i) f i a max flow. (ii) There i no augmening pah relaive o f. (iii) There exi a cu whoe capaciy equal he value of f. (i) (ii) equivalen o no (ii) no (i), which wa Obervaion (ii) (iii) nex lide (iii) (i) hi wa Obervaion (ii) (iii). If here i no augmening pah relaive o f, hen here exi a cu whoe capaciy equal he value of f. Proof. Le f be a flow wih no augmening pah. Le be e of verice reachable from in reidual graph. conain ; ince no augmening pah, doe no conain all edge e leaving in original nework have f(e) = u(e) all edge e enering in original nework have f(e) = f e ou of e ou of f ( e) u( e) capaciy (, T) e in o f ( e) T reidual nework Max Flow Nework Implemenaion Ford-Fulkeron lgorihm: Implemenaion dge in original graph may correpond o or reidual edge. May need o ravere edge e = v-w in forward or revere direcion. Flow = f(e), capaciy = u(e). Iner wo copie of each edge, one in adjacency li of v and one in w. Ford-Fulkeron main loop. // while here exi an augmening pah, ue i while (augpah()) { public cla dge { privae in v, w; privae in cap; privae in flow; // from, o // capaciy from v o w // flow from v o w public dge(in v, in w, in cap) {... public in cap() { reurn cap; public in flow() { reurn flow; public boolean from(in v) { reurn hi.v == v; public in oher(in v) { reurn from(v)? hi.w : hi.v; public in capro(in v) { reurn from(v)? flow : cap - flow; public void addflowro(in v, in d) { flow += from(v)? -d : d; // compue boleneck capaciy in bole = INFINITY; for (in v = ; v!= ; v = T(v)) bole = Mah.min(bole, pred[v].capro(v)); // augmen flow for (in v = ; v!= ; v = T(v)) pred[v].addflowro(v, bole); // keep rack of oal flow en from o value += bole;

8 Ford-Fulkeron lgorihm: nalyi hooing Good ugmening Pah umpion: all capaciie are ineger beween and U. Ue care when elecing augmening pah. Invarian: every flow value and every reidual capaciie remain an ineger hroughou he algorihm. Theorem: he algorihm erminae in a mo f * V U ieraion. orollary: if U =, hen algorihm run in V ieraion. no polynomial in inpu ize! Inegraliy heorem: if all arc capaciie are ineger, hen here exi a max flow f for which every flow value i an ineger. Original Nework hooing Good ugmening Pah hooing Good ugmening Pah Ue care when elecing augmening pah. Ue care when elecing augmening pah. X X X Original Nework Original Nework

9 hooing Good ugmening Pah hooing Good ugmening Pah Ue care when elecing augmening pah. Ue care when elecing augmening pah. X X X Original Nework Original Nework ieraion poible! hooing Good ugmening Pah hore ugmening Pah Ue care when elecing augmening pah. ome choice lead o exponenial algorihm. lever choice lead o polynomial algorihm. Opimal choice for real world problem??? eign goal i o chooe augmening pah o ha: an find augmening pah efficienly. Few ieraion. hooe augmening pah wih: dmond-karp () Fewe number of arc. (hore pah) Max boleneck capaciy. (fae pah) hore augmening pah. ay o implemen wih F. Find augmening pah wih fewe number of arc. while (!q.impy()) { in v = q.dequeue(); InIeraor i = G.neighbor(v); while(i.hanex()) { dge e = i.nex(); in w = e.oher(v); if (e.capro(w) > ) { // i v-w a reidual edge? if (w[w] > w[v] + ) { w[w] = w[v] + ; pred[w] = e; // keep rack of hore pah q.enqueue(w); reurn (w[] < INFINITY); // i here an augmening pah?

10 hore ugmening Pah nalyi Fae ugmening Pah Lengh of hore augmening pah increae monoonically. ricly increae afer a mo augmenaion. mo V oal augmening pah. O( V) running ime. Fae augmening pah. Find augmening pah whoe boleneck capaciy i maximum. eliver mo amoun of flow o ink. olve uing ijkra-yle (PF) algorihm. X v reidual capaciy w if (w[w] < Mah.min(w[v], e.capro(w)) { w[w] = Mah.min(w[v], e.capro(w)); pred[w] = v; Finding a fae pah. O( log V) per augmenaion wih binary heap. Fac. O( log U) augmenaion if capaciie are beween and U. hooing an ugmening Pah Hiory of Wor-ae Running Time hooing an augmening pah. ny pah will do wide laiude in implemening Ford-Fulkeron. Generic prioriy fir earch. ome choice lead o good wor-cae performance. hore augmening pah fae augmening pah variaion on a heme: PF verage cae no well underood. Reearch challenge. Pracice: olve max flow problem on real nework in linear ime. Theory: prove i for wor-cae nework. Year... icoverer Mehod ympoic Time anzig implex V U Ford, Fulkeron ugmening pah V U dmond-karp hore pah V dmond-karp Max capaciy log U ( + V log V) iniz Improved hore pah V dmond-karp, iniz apaciy caling log U iniz-gabow Improved capaciy caling V log U Karzanov Preflow-puh V leaor-tarjan ynamic ree V log V Goldberg-Tarjan FIFO preflow-puh V log (V / ) Goldberg-Rao Lengh funcion / log (V / ) log U V / log (V / ) log U rc capaciie are beween and U.

11 n pplicaion iparie Maching Jon placemen. ompanie make job offer. uden have job choice. an we fill every job? iparie maching. Inpu: undireced and biparie graph G. e of edge M i a maching if each verex appear a mo once. Max maching: find a max cardinaliy maching. an we employ every uden? lice-dobe ob-yahoo arol-hp ave-pple liza-im Frank-un Maching M -, -, - L R iparie Maching iparie Maching iparie maching. Inpu: undireced and biparie graph G. e of edge M i a maching if each verex appear a mo once. Max maching: find a max cardinaliy maching. Reduce o max flow. reae a direced graph G'. irec all arc from L o R, and give infinie (or uni) capaciy. dd ource, and uni capaciy arc from o each node in L. dd ink, and uni capaciy arc from each node in R o. L R Maching M -, -, -, - L R G G'

12 iparie Maching: Proof of orrecne iparie Maching: Proof of orrecne laim. Maching in G of cardinaliy k induce flow in G' of value k. Given maching M = { -, -, - of cardinaliy. onider flow f ha end uni along each of pah: f i a flow, and ha cardinaliy. laim. Flow f of value k in G' induce maching of cardinaliy k in G. y inegraliy heorem, here exi / valued flow f of value k. onider M = e of edge from L o R wih f(e) =. each node in L and R inciden o a mo one edge in M M = k L R L R G G' G G' Reducion Reducion. Given an inance of biparie maching. Tranform i o a max flow problem. olve max flow problem. Tranform max flow oluion o biparie maching oluion. Iue. How expenive i ranformaion? O( + V) I i beer o olve problem direcly? O( V / ) biparie maching oom line: max flow i an exremely rich problem-olving model. Many imporan pracical problem reduce o max flow. We know good algorihm for olving max flow problem.

Chapter 13. Network Flow III Applications. 13.1 Edge disjoint paths. 13.1.1 Edge-disjoint paths in a directed graphs

Chapter 13. Network Flow III Applications. 13.1 Edge disjoint paths. 13.1.1 Edge-disjoint paths in a directed graphs Chaper 13 Nework Flow III Applicaion CS 573: Algorihm, Fall 014 Ocober 9, 014 13.1 Edge dijoin pah 13.1.1 Edge-dijoin pah in a direced graph 13.1.1.1 Edge dijoin pah queiong: graph (dir/undir)., : verice.

More information

How Much Can Taxes Help Selfish Routing?

How Much Can Taxes Help Selfish Routing? How Much Can Taxe Help Selfih Rouing? Tim Roughgarden (Cornell) Join wih Richard Cole (NYU) and Yevgeniy Dodi (NYU) Selfih Rouing a direced graph G = (V,E) a ource and a deinaion one uni of raffic from

More information

On the Connection Between Multiple-Unicast Network Coding and Single-Source Single-Sink Network Error Correction

On the Connection Between Multiple-Unicast Network Coding and Single-Source Single-Sink Network Error Correction On he Connecion Beween Muliple-Unica ework Coding and Single-Source Single-Sink ework Error Correcion Jörg Kliewer JIT Join work wih Wenao Huang and Michael Langberg ework Error Correcion Problem: Adverary

More information

The Transport Equation

The Transport Equation The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be

More information

Mathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)

Mathematics in Pharmacokinetics What and Why (A second attempt to make it clearer) Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions

More information

Stock Trading with Recurrent Reinforcement Learning (RRL) CS229 Application Project Gabriel Molina, SUID 5055783

Stock Trading with Recurrent Reinforcement Learning (RRL) CS229 Application Project Gabriel Molina, SUID 5055783 Sock raing wih Recurren Reinforcemen Learning (RRL) CS9 Applicaion Projec Gabriel Molina, SUID 555783 I. INRODUCION One relaively new approach o financial raing is o use machine learning algorihms o preic

More information

Random Walk in 1-D. 3 possible paths x vs n. -5 For our random walk, we assume the probabilities p,q do not depend on time (n) - stationary

Random Walk in 1-D. 3 possible paths x vs n. -5 For our random walk, we assume the probabilities p,q do not depend on time (n) - stationary Random Walk in -D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes

More information

Optimal Investment and Consumption Decision of Family with Life Insurance

Optimal Investment and Consumption Decision of Family with Life Insurance Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker

More information

The Role of Science and Mathematics in Software Development

The Role of Science and Mathematics in Software Development The cienific mehod i eenial in applicaion of compuaion A peronal opinion formed on he bai of decade of experience a a The Role of Science and Mahemaic in Sofware Developmen CS educaor auhor algorihm deigner

More information

Fortified financial forecasting models: non-linear searching approaches

Fortified financial forecasting models: non-linear searching approaches 0 Inernaional Conference on Economic and inance Reearch IPEDR vol.4 (0 (0 IACSIT Pre, Singapore orified financial forecaing model: non-linear earching approache Mohammad R. Hamidizadeh, Ph.D. Profeor,

More information

Chapter 6: Business Valuation (Income Approach)

Chapter 6: Business Valuation (Income Approach) Chaper 6: Business Valuaion (Income Approach) Cash flow deerminaion is one of he mos criical elemens o a business valuaion. Everyhing may be secondary. If cash flow is high, hen he value is high; if he

More information

Inductance and Transient Circuits

Inductance and Transient Circuits Chaper H Inducance and Transien Circuis Blinn College - Physics 2426 - Terry Honan As a consequence of Faraday's law a changing curren hrough one coil induces an EMF in anoher coil; his is known as muual

More information

The Role of the Scientific Method in Software Development. Robert Sedgewick Princeton University

The Role of the Scientific Method in Software Development. Robert Sedgewick Princeton University The Role of he Scienific Mehod in Sofware Developmen Rober Sedgewick Princeon Univeriy The cienific mehod i neceary in algorihm deign and ofware developmen Scienific mehod creae a model decribing naural

More information

Morningstar Investor Return

Morningstar Investor Return Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion

More information

Globally-Optimal Greedy Algorithms for Tracking a Variable Number of Objects

Globally-Optimal Greedy Algorithms for Tracking a Variable Number of Objects Globally-Opimal Greedy Algorihm for Tracking a Variable Number of Objec Hamed Piriavah Deva Ramanan Charle C. Fowlke Deparmen of Compuer Science, Univeriy of California, Irvine {hpiriav,dramanan,fowlke}@ic.uci.edu

More information

Chapter 8: Regression with Lagged Explanatory Variables

Chapter 8: Regression with Lagged Explanatory Variables Chaper 8: Regression wih Lagged Explanaory Variables Time series daa: Y for =1,..,T End goal: Regression model relaing a dependen variable o explanaory variables. Wih ime series new issues arise: 1. One

More information

Top-K Structural Diversity Search in Large Networks

Top-K Structural Diversity Search in Large Networks Top-K Srucural Diversiy Search in Large Neworks Xin Huang, Hong Cheng, Rong-Hua Li, Lu Qin, Jeffrey Xu Yu The Chinese Universiy of Hong Kong Guangdong Province Key Laboraory of Popular High Performance

More information

Heat demand forecasting for concrete district heating system

Heat demand forecasting for concrete district heating system Hea demand forecaing for concree diric heaing yem Bronilav Chramcov Abrac Thi paper preen he reul of an inveigaion of a model for hor-erm hea demand forecaing. Foreca of hi hea demand coure i ignifican

More information

Signal Processing and Linear Systems I

Signal Processing and Linear Systems I Sanford Universiy Summer 214-215 Signal Processing and Linear Sysems I Lecure 5: Time Domain Analysis of Coninuous Time Sysems June 3, 215 EE12A:Signal Processing and Linear Sysems I; Summer 14-15, Gibbons

More information

The Application of Multi Shifts and Break Windows in Employees Scheduling

The Application of Multi Shifts and Break Windows in Employees Scheduling The Applicaion of Muli Shifs and Brea Windows in Employees Scheduling Evy Herowai Indusrial Engineering Deparmen, Universiy of Surabaya, Indonesia Absrac. One mehod for increasing company s performance

More information

Mortality Variance of the Present Value (PV) of Future Annuity Payments

Mortality Variance of the Present Value (PV) of Future Annuity Payments Morali Variance of he Presen Value (PV) of Fuure Annui Pamens Frank Y. Kang, Ph.D. Research Anals a Frank Russell Compan Absrac The variance of he presen value of fuure annui pamens plas an imporan role

More information

Single-machine Scheduling with Periodic Maintenance and both Preemptive and. Non-preemptive jobs in Remanufacturing System 1

Single-machine Scheduling with Periodic Maintenance and both Preemptive and. Non-preemptive jobs in Remanufacturing System 1 Absrac number: 05-0407 Single-machine Scheduling wih Periodic Mainenance and boh Preempive and Non-preempive jobs in Remanufacuring Sysem Liu Biyu hen Weida (School of Economics and Managemen Souheas Universiy

More information

µ r of the ferrite amounts to 1000...4000. It should be noted that the magnetic length of the + δ

µ r of the ferrite amounts to 1000...4000. It should be noted that the magnetic length of the + δ Page 9 Design of Inducors and High Frequency Transformers Inducors sore energy, ransformers ransfer energy. This is he prime difference. The magneic cores are significanly differen for inducors and high

More information

Lecture 15 Isolated DC-DC converters

Lecture 15 Isolated DC-DC converters ELEC440/940 Lecure 15 olae C-C converer Ofen, he oupu C volage fro a C-C converer u be iolae fro he inpu AC upply. C power upplie for appliance an equipen are goo exaple. i avanageou o have he iolaion

More information

A Comparative Study of Linear and Nonlinear Models for Aggregate Retail Sales Forecasting

A Comparative Study of Linear and Nonlinear Models for Aggregate Retail Sales Forecasting A Comparaive Sudy of Linear and Nonlinear Model for Aggregae Reail Sale Forecaing G. Peer Zhang Deparmen of Managemen Georgia Sae Univeriy Alana GA 30066 (404) 651-4065 Abrac: The purpoe of hi paper i

More information

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees

More information

THE FIRM'S INVESTMENT DECISION UNDER CERTAINTY: CAPITAL BUDGETING AND RANKING OF NEW INVESTMENT PROJECTS

THE FIRM'S INVESTMENT DECISION UNDER CERTAINTY: CAPITAL BUDGETING AND RANKING OF NEW INVESTMENT PROJECTS VII. THE FIRM'S INVESTMENT DECISION UNDER CERTAINTY: CAPITAL BUDGETING AND RANKING OF NEW INVESTMENT PROJECTS The mos imporan decisions for a firm's managemen are is invesmen decisions. While i is surely

More information

Optimal Path Routing in Single and Multiple Clock Domain Systems

Optimal Path Routing in Single and Multiple Clock Domain Systems IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN, TO APPEAR. 1 Opimal Pah Rouing in Single and Muliple Clock Domain Syem Soha Haoun, Senior Member, IEEE, Charle J. Alper, Senior Member, IEEE ) Abrac Shrinking

More information

Quality Assurance in Software Development

Quality Assurance in Software Development Insiue for Sofware Technology Qualiy Assurance in Sofware Developmen Qualiässicherung in der Sofwareenwicklung A.o.Univ.-Prof. Dipl.-Ing. Dr. Bernhard Aichernig Insiu für Sofwareechnologie (IST) TU Graz

More information

Multiprocessor Systems-on-Chips

Multiprocessor Systems-on-Chips Par of: Muliprocessor Sysems-on-Chips Edied by: Ahmed Amine Jerraya and Wayne Wolf Morgan Kaufmann Publishers, 2005 2 Modeling Shared Resources Conex swiching implies overhead. On a processing elemen,

More information

STABILITY OF LOAD BALANCING ALGORITHMS IN DYNAMIC ADVERSARIAL SYSTEMS

STABILITY OF LOAD BALANCING ALGORITHMS IN DYNAMIC ADVERSARIAL SYSTEMS STABILITY OF LOAD BALANCING ALGORITHMS IN DYNAMIC ADVERSARIAL SYSTEMS ELLIOT ANSHELEVICH, DAVID KEMPE, AND JON KLEINBERG Absrac. In he dynamic load balancing problem, we seek o keep he job load roughly

More information

4. International Parity Conditions

4. International Parity Conditions 4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency

More information

Better Bounds for Online Load Balancing on Unrelated Machines

Better Bounds for Online Load Balancing on Unrelated Machines Beer Bound for Online Load Balancing on Unrelaed Machine Ioanni Caragianni Abrac We udy he roblem of cheduling ermanen ob on unrelaed machine when he obecive i o minimize he L norm of he machine load.

More information

Analogue and Digital Signal Processing. First Term Third Year CS Engineering By Dr Mukhtiar Ali Unar

Analogue and Digital Signal Processing. First Term Third Year CS Engineering By Dr Mukhtiar Ali Unar Analogue and Digial Signal Processing Firs Term Third Year CS Engineering By Dr Mukhiar Ali Unar Recommended Books Haykin S. and Van Veen B.; Signals and Sysems, John Wiley& Sons Inc. ISBN: 0-7-380-7 Ifeachor

More information

Smooth Priorities for Multi-Product Inventory Control

Smooth Priorities for Multi-Product Inventory Control Smooh rioriies for Muli-roduc Invenory Conrol Francisco José.A.V. Mendonça*. Carlos F. Bispo** *Insiuo Superior Técnico - Universidade Técnica de Lisboa (email:favm@mega.is.ul.p) ** Insiuo de Sisemas e

More information

Diagnostic Examination

Diagnostic Examination Diagnosic Examinaion TOPIC XV: ENGINEERING ECONOMICS TIME LIMIT: 45 MINUTES 1. Approximaely how many years will i ake o double an invesmen a a 6% effecive annual rae? (A) 10 yr (B) 12 yr (C) 15 yr (D)

More information

Online Convex Programming and Generalized Infinitesimal Gradient Ascent

Online Convex Programming and Generalized Infinitesimal Gradient Ascent Online Convex Programming and Generalized Infiniesimal Gradien Ascen Marin Zinkevich Carnegie Mellon Universiy, 5000 Forbes Avenue, Pisburgh, PA 1513 USA maz@cs.cmu.edu Absrac Convex programming involves

More information

Task is a schedulable entity, i.e., a thread

Task is a schedulable entity, i.e., a thread Real-Time Scheduling Sysem Model Task is a schedulable eniy, i.e., a hread Time consrains of periodic ask T: - s: saring poin - e: processing ime of T - d: deadline of T - p: period of T Periodic ask T

More information

INTRODUCTION TO EMAIL MARKETING PERSONALIZATION. How to increase your sales with personalized triggered emails

INTRODUCTION TO EMAIL MARKETING PERSONALIZATION. How to increase your sales with personalized triggered emails INTRODUCTION TO EMAIL MARKETING PERSONALIZATION How o increase your sales wih personalized riggered emails ECOMMERCE TRIGGERED EMAILS BEST PRACTICES Triggered emails are generaed in real ime based on each

More information

Chapter 6 Interest Rates and Bond Valuation

Chapter 6 Interest Rates and Bond Valuation Chaper 6 Ineres Raes and Bond Valuaion Definiion and Descripion of Bonds Long-erm deb-loosely, bonds wih a mauriy of one year or more Shor-erm deb-less han a year o mauriy, also called unfunded deb Bond-sricly

More information

AP Calculus AB 2010 Scoring Guidelines

AP Calculus AB 2010 Scoring Guidelines AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in 1, he College

More information

Economics Honors Exam 2008 Solutions Question 5

Economics Honors Exam 2008 Solutions Question 5 Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I

More information

INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES

INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchange-raded ineres rae fuures and heir opions are described. The fuure opions include hose paying

More information

Name: Algebra II Review for Quiz #13 Exponential and Logarithmic Functions including Modeling

Name: Algebra II Review for Quiz #13 Exponential and Logarithmic Functions including Modeling Name: Algebra II Review for Quiz #13 Exponenial and Logarihmic Funcions including Modeling TOPICS: -Solving Exponenial Equaions (The Mehod of Common Bases) -Solving Exponenial Equaions (Using Logarihms)

More information

Physical Topology Discovery for Large Multi-Subnet Networks

Physical Topology Discovery for Large Multi-Subnet Networks Phyical Topology Dicovery for Large Muli-Subne Nework Yigal Bejerano, Yuri Breibar, Mino Garofalaki, Rajeev Raogi Bell Lab, Lucen Technologie 600 Mounain Ave., Murray Hill, NJ 07974. {bej,mino,raogi}@reearch.bell-lab.com

More information

Empirical heuristics for improving Intermittent Demand Forecasting

Empirical heuristics for improving Intermittent Demand Forecasting Empirical heuriic for improving Inermien Demand Forecaing Foio Peropoulo 1,*, Konanino Nikolopoulo 2, Georgio P. Spihouraki 1, Vailio Aimakopoulo 1 1 Forecaing & Sraegy Uni, School of Elecrical and Compuer

More information

Making a Faster Cryptanalytic Time-Memory Trade-Off

Making a Faster Cryptanalytic Time-Memory Trade-Off Making a Faser Crypanalyic Time-Memory Trade-Off Philippe Oechslin Laboraoire de Securié e de Crypographie (LASEC) Ecole Polyechnique Fédérale de Lausanne Faculé I&C, 1015 Lausanne, Swizerland philippe.oechslin@epfl.ch

More information

Fair Stateless Model Checking

Fair Stateless Model Checking Fair Saeless Model Checking Madanlal Musuvahi Shaz Qadeer Microsof Research {madanm,qadeer@microsof.com Absrac Saeless model checking is a useful sae-space exploraion echnique for sysemaically esing complex

More information

Dividend taxation, share repurchases and the equity trap

Dividend taxation, share repurchases and the equity trap Working Paper 2009:7 Deparmen of Economic Dividend axaion, hare repurchae and he equiy rap Tobia Lindhe and Jan Söderen Deparmen of Economic Working paper 2009:7 Uppala Univeriy May 2009 P.O. Box 53 ISSN

More information

Answer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1

Answer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1 Answer, Key Homework 2 Daid McInyre 4123 Mar 2, 2004 1 This prin-ou should hae 1 quesions. Muliple-choice quesions may coninue on he ne column or page find all choices before making your selecion. The

More information

Principal components of stock market dynamics. Methodology and applications in brief (to be updated ) Andrei Bouzaev, bouzaev@ya.

Principal components of stock market dynamics. Methodology and applications in brief (to be updated ) Andrei Bouzaev, bouzaev@ya. Principal componens of sock marke dynamics Mehodology and applicaions in brief o be updaed Andrei Bouzaev, bouzaev@ya.ru Why principal componens are needed Objecives undersand he evidence of more han one

More information

1 HALF-LIFE EQUATIONS

1 HALF-LIFE EQUATIONS R.L. Hanna Page HALF-LIFE EQUATIONS The basic equaion ; he saring poin ; : wrien for ime: x / where fracion of original maerial and / number of half-lives, and / log / o calculae he age (# ears): age (half-life)

More information

Dynamic programming models and algorithms for the mutual fund cash balance problem

Dynamic programming models and algorithms for the mutual fund cash balance problem Submied o Managemen Science manuscrip Dynamic programming models and algorihms for he muual fund cash balance problem Juliana Nascimeno Deparmen of Operaions Research and Financial Engineering, Princeon

More information

On Certain Properties of Random Apollonian Networks

On Certain Properties of Random Apollonian Networks On Cerain Properies of Random Apollonian Neworks Alan Frieze, Charalampos E. Tsourakakis Deparmen of Mahemaical Sciences, Carnegie Mellon Universiy, USA af1p@random.mah.cmu.edu, csourak@mah.cmu.edu Absrac.

More information

Chapter 2 Problems. 3600s = 25m / s d = s t = 25m / s 0.5s = 12.5m. Δx = x(4) x(0) =12m 0m =12m

Chapter 2 Problems. 3600s = 25m / s d = s t = 25m / s 0.5s = 12.5m. Δx = x(4) x(0) =12m 0m =12m Chaper 2 Problems 2.1 During a hard sneeze, your eyes migh shu for 0.5s. If you are driving a car a 90km/h during such a sneeze, how far does he car move during ha ime s = 90km 1000m h 1km 1h 3600s = 25m

More information

CALCULATION OF OMX TALLINN

CALCULATION OF OMX TALLINN CALCULATION OF OMX TALLINN CALCULATION OF OMX TALLINN 1. OMX Tallinn index...3 2. Terms in use...3 3. Comuaion rules of OMX Tallinn...3 3.1. Oening, real-ime and closing value of he Index...3 3.2. Index

More information

Individual Health Insurance April 30, 2008 Pages 167-170

Individual Health Insurance April 30, 2008 Pages 167-170 Individual Healh Insurance April 30, 2008 Pages 167-170 We have received feedback ha his secion of he e is confusing because some of he defined noaion is inconsisen wih comparable life insurance reserve

More information

Planning Demand and Supply in a Supply Chain. Forecasting and Aggregate Planning

Planning Demand and Supply in a Supply Chain. Forecasting and Aggregate Planning Planning Demand and Supply in a Supply Chain Forecasing and Aggregae Planning 1 Learning Objecives Overview of forecasing Forecas errors Aggregae planning in he supply chain Managing demand Managing capaciy

More information

Stochastic Optimal Control Problem for Life Insurance

Stochastic Optimal Control Problem for Life Insurance Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian

More information

NASDAQ-100 Futures Index SM Methodology

NASDAQ-100 Futures Index SM Methodology NASDAQ-100 Fuures Index SM Mehodology Index Descripion The NASDAQ-100 Fuures Index (The Fuures Index ) is designed o rack he performance of a hypoheical porfolio holding he CME NASDAQ-100 E-mini Index

More information

Sampling Time-Based Sliding Windows in Bounded Space

Sampling Time-Based Sliding Windows in Bounded Space Sampling Time-Based Sliding Windows in Bounded Space Rainer Gemulla Technische Universiä Dresden 01062 Dresden, Germany gemulla@inf.u-dresden.de Wolfgang Lehner Technische Universiä Dresden 01062 Dresden,

More information

WHAT ARE OPTION CONTRACTS?

WHAT ARE OPTION CONTRACTS? WHAT ARE OTION CONTRACTS? By rof. Ashok anekar An oion conrac is a derivaive which gives he righ o he holder of he conrac o do 'Somehing' bu wihou he obligaion o do ha 'Somehing'. The 'Somehing' can be

More information

Statistical Analysis with Little s Law. Supplementary Material: More on the Call Center Data. by Song-Hee Kim and Ward Whitt

Statistical Analysis with Little s Law. Supplementary Material: More on the Call Center Data. by Song-Hee Kim and Ward Whitt Saisical Analysis wih Lile s Law Supplemenary Maerial: More on he Call Cener Daa by Song-Hee Kim and Ward Whi Deparmen of Indusrial Engineering and Operaions Research Columbia Universiy, New York, NY 17-99

More information

ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS

ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,

More information

Table of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities

Table of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities Table of conens Chaper 1 Ineres raes and facors 1 1.1 Ineres 2 1.2 Simple ineres 4 1.3 Compound ineres 6 1.4 Accumulaed value 10 1.5 Presen value 11 1.6 Rae of discoun 13 1.7 Consan force of ineres 17

More information

TEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS

TEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS TEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS RICHARD J. POVINELLI AND XIN FENG Deparmen of Elecrical and Compuer Engineering Marquee Universiy, P.O.

More information

Q-SAC: Toward QoS Optimized Service Automatic Composition *

Q-SAC: Toward QoS Optimized Service Automatic Composition * Q-SAC: Toward QoS Opimized Service Auomaic Composiion * Hanhua Chen, Hai Jin, Xiaoming Ning, Zhipeng Lü Cluser and Grid Compuing Lab Huazhong Universiy of Science and Technology, Wuhan, 4374, China Email:

More information

The Roos of Lisp paul graham Draf, January 18, 2002. In 1960, John McCarhy published a remarkable paper in which he did for programming somehing like wha Euclid did for geomery. 1 He showed how, given

More information

CHAPTER FIVE. Solutions for Section 5.1

CHAPTER FIVE. Solutions for Section 5.1 CHAPTER FIVE 5. SOLUTIONS 87 Soluions for Secion 5.. (a) The velociy is 3 miles/hour for he firs hours, 4 miles/hour for he ne / hour, and miles/hour for he las 4 hours. The enire rip lass + / + 4 = 6.5

More information

Quality-Of-Service Class Specific Traffic Matrices in IP/MPLS Networks

Quality-Of-Service Class Specific Traffic Matrices in IP/MPLS Networks ualiy-of-service Class Specific Traffic Marices in IP/MPLS Neworks Sefan Schnier Deusche Telekom, T-Sysems D-4 Darmsad +4 sefan.schnier@-sysems.com Franz Harleb Deusche Telekom, T-Sysems D-4 Darmsad +4

More information

Quality-Of-Service Class Specific Traffic Matrices in IP/MPLS Networks

Quality-Of-Service Class Specific Traffic Matrices in IP/MPLS Networks ualiy-of-service Class Specific Traffic Marices in IP/MPLS Neworks Sefan Schnier Deusche Telekom, T-Sysems D-4 Darmsad +4 sefan.schnier@-sysems.com Franz Harleb Deusche Telekom, T-Sysems D-4 Darmsad +4

More information

Duration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.

Duration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613. Graduae School of Business Adminisraion Universiy of Virginia UVA-F-38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised

More information

Algorithm Design and Analysis

Algorithm Design and Analysis Algorithm Design and Analysis LECTURE 27 Approximation Algorithms Load Balancing Weighted Vertex Cover Reminder: Fill out SRTEs online Don t forget to click submit Sofya Raskhodnikova 12/6/2011 S. Raskhodnikova;

More information

Nanocubes for Real-Time Exploration of Spatiotemporal Datasets

Nanocubes for Real-Time Exploration of Spatiotemporal Datasets Nanocube for RealTime Exploraion of Spaioemporal Daae Lauro Lin, Jame T Kloowki, and arlo Scheidegger Fig 1 Example viualizaion of 210 million public geolocaed Twier po over he coure of a year The daa

More information

The option pricing framework

The option pricing framework Chaper 2 The opion pricing framework The opion markes based on swap raes or he LIBOR have become he larges fixed income markes, and caps (floors) and swapions are he mos imporan derivaives wihin hese markes.

More information

Premium Income of Indian Life Insurance Industry

Premium Income of Indian Life Insurance Industry Premium Income of Indian Life Insurance Indusry A Toal Facor Produciviy Approach Ram Praap Sinha* Subsequen o he passage of he Insurance Regulaory and Developmen Auhoriy (IRDA) Ac, 1999, he life insurance

More information

Genetic Algorithm Search for Predictive Patterns in Multidimensional Time Series

Genetic Algorithm Search for Predictive Patterns in Multidimensional Time Series Geneic Algorihm Search for Predicive Paerns in Mulidimensional Time Series Arnold Polanski School of Managemen and Economics Queen s Universiy of Belfas 25 Universiy Square Belfas BT7 1NN, Unied Kingdom

More information

Performance Center Overview. Performance Center Overview 1

Performance Center Overview. Performance Center Overview 1 Performance Cener Overview Performance Cener Overview 1 ODJFS Performance Cener ce Cener New Performance Cener Model Performance Cener Projec Meeings Performance Cener Execuive Meeings Performance Cener

More information

On the degrees of irreducible factors of higher order Bernoulli polynomials

On the degrees of irreducible factors of higher order Bernoulli polynomials ACTA ARITHMETICA LXII.4 (1992 On he degrees of irreducible facors of higher order Bernoulli polynomials by Arnold Adelberg (Grinnell, Ia. 1. Inroducion. In his paper, we generalize he curren resuls on

More information

Improvement of a TCP Incast Avoidance Method for Data Center Networks

Improvement of a TCP Incast Avoidance Method for Data Center Networks Improvemen of a Incas Avoidance Mehod for Daa Cener Neworks Kazuoshi Kajia, Shigeyuki Osada, Yukinobu Fukushima and Tokumi Yokohira The Graduae School of Naural Science and Technology, Okayama Universiy

More information

13. a. If the one-year discount factor is.905, what is the one-year interest rate?

13. a. If the one-year discount factor is.905, what is the one-year interest rate? CHAPTER 3: Pracice quesions 3. a. If he one-year discoun facor is.905, wha is he one-year ineres rae? = DF = + r 0.905 r = 0.050 = 0.50% b. If he wo-year ineres rae is 0.5 percen, wha is he wo-year discoun

More information

Trends in TCP/IP Retransmissions and Resets

Trends in TCP/IP Retransmissions and Resets Trends in TCP/IP Reransmissions and Reses Absrac Concordia Chen, Mrunal Mangrulkar, Naomi Ramos, and Mahaswea Sarkar {cychen, mkulkarn, msarkar,naramos}@cs.ucsd.edu As he Inerne grows larger, measuring

More information

Infrastructure and Evolution in Division of Labour

Infrastructure and Evolution in Division of Labour Infrarucure and Evoluion in Diviion of Labour Mei Wen Monah Univery (Thi paper ha been publihed in RDE. (), 9-06) April 997 Abrac Thi paper udie he relaionhip beween infrarucure ependure and endogenou

More information

Information Theoretic Approaches for Predictive Models: Results and Analysis

Information Theoretic Approaches for Predictive Models: Results and Analysis Informaion Theoreic Approaches for Predicive Models: Resuls and Analysis Monica Dinculescu Supervised by Doina Precup Absrac Learning he inernal represenaion of parially observable environmens has proven

More information

17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides

17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides 7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion

More information

The Equivalent Loan Principle and the Value of Corporate Promised Cash Flows. David C. Nachman*

The Equivalent Loan Principle and the Value of Corporate Promised Cash Flows. David C. Nachman* he Equivalen Loan Principle and he Value of Corporae Promied Cah Flow by David C. Nachman* Revied February, 2002 *J. Mack Robinon College of Buine, Georgia Sae Univeriy, 35 Broad Sree, Alana, GA 30303-3083.

More information

Efficient One-time Signature Schemes for Stream Authentication *

Efficient One-time Signature Schemes for Stream Authentication * JOURNAL OF INFORMATION SCIENCE AND ENGINEERING, 611-64 (006) Efficien One-ime Signaure Schemes for Sream Auhenicaion * YONGSU PARK AND YOOKUN CHO + College of Informaion and Communicaions Hanyang Universiy

More information

Energy and Performance Management of Green Data Centers: A Profit Maximization Approach

Energy and Performance Management of Green Data Centers: A Profit Maximization Approach Energy and Performance Managemen of Green Daa Ceners: A Profi Maximizaion Approach Mahdi Ghamkhari, Suden Member, IEEE, and Hamed Mohsenian-Rad, Member, IEEE Absrac While a large body of work has recenly

More information

A Load Balancing Method in Downlink LTE Network based on Load Vector Minimization

A Load Balancing Method in Downlink LTE Network based on Load Vector Minimization A Load Balancing Mehod in Downlink LTE Nework based on Load Vecor Minimizaion Fanqin Zhou, Lei Feng, Peng Yu, and Wenjing Li Sae Key Laboraory of Neworking and Swiching Technology, Beijing Universiy of

More information

Journal Of Business & Economics Research September 2005 Volume 3, Number 9

Journal Of Business & Economics Research September 2005 Volume 3, Number 9 Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy Yi-Kang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo

More information

The Binary Blocking Flow Algorithm. Andrew V. Goldberg Microsoft Research Silicon Valley www.research.microsoft.com/ goldberg/

The Binary Blocking Flow Algorithm. Andrew V. Goldberg Microsoft Research Silicon Valley www.research.microsoft.com/ goldberg/ The Binary Blocking Flow Algorithm Andrew V. Goldberg Microsoft Research Silicon Valley www.research.microsoft.com/ goldberg/ Why this Max-Flow Talk? The result: O(min(n 2/3, m 1/2 )mlog(n 2 /m)log(u))

More information

Robust Bandwidth Allocation Strategies

Robust Bandwidth Allocation Strategies Robu Bandwidh Allocaion Sraegie Oliver Heckmann, Jen Schmi, Ralf Seinmez Mulimedia Communicaion Lab (KOM), Darmad Univeriy of Technology Merckr. 25 D-64283 Darmad Germany {Heckmann, Schmi, Seinmez}@kom.u-darmad.de

More information

PATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM

PATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM PATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM WILLIAM L. COOPER Deparmen of Mechanical Engineering, Universiy of Minnesoa, 111 Church Sree S.E., Minneapolis, MN 55455 billcoop@me.umn.edu

More information

Module 3 Design for Strength. Version 2 ME, IIT Kharagpur

Module 3 Design for Strength. Version 2 ME, IIT Kharagpur Module 3 Design for Srengh Lesson 2 Sress Concenraion Insrucional Objecives A he end of his lesson, he sudens should be able o undersand Sress concenraion and he facors responsible. Deerminaion of sress

More information

E0 370 Statistical Learning Theory Lecture 20 (Nov 17, 2011)

E0 370 Statistical Learning Theory Lecture 20 (Nov 17, 2011) E0 370 Saisical Learning Theory Lecure 0 (ov 7, 0 Online Learning from Expers: Weighed Majoriy and Hedge Lecurer: Shivani Agarwal Scribe: Saradha R Inroducion In his lecure, we will look a he problem of

More information

Present Value Methodology

Present Value Methodology Presen Value Mehodology Econ 422 Invesmen, Capial & Finance Universiy of Washingon Eric Zivo Las updaed: April 11, 2010 Presen Value Concep Wealh in Fisher Model: W = Y 0 + Y 1 /(1+r) The consumer/producer

More information

ARCH 2013.1 Proceedings

ARCH 2013.1 Proceedings Aricle from: ARCH 213.1 Proceedings Augus 1-4, 212 Ghislain Leveille, Emmanuel Hamel A renewal model for medical malpracice Ghislain Léveillé École d acuaria Universié Laval, Québec, Canada 47h ARC Conference

More information

How has globalisation affected inflation dynamics in the United Kingdom?

How has globalisation affected inflation dynamics in the United Kingdom? 292 Quarerly Bullein 2008 Q3 How ha globaliaion affeced inflaion dynamic in he Unied Kingdom? By Jennifer Greenlade and Sephen Millard of he Bank Srucural Economic Analyi Diviion and Chri Peacock of he

More information

Stock option grants have become an. Final Approval Copy. Valuation of Stock Option Grants Under Multiple Severance Risks GURUPDESH S.

Stock option grants have become an. Final Approval Copy. Valuation of Stock Option Grants Under Multiple Severance Risks GURUPDESH S. Valuaion of Sock Opion Gran Under Muliple Severance Rik GURUPDESH S. PANDHER i an aian profeor in he deparmen of finance a DePaul Univeriy in Chicago, IL. gpandher@depaul.edu GURUPDESH S. PANDHER Execuive

More information

Calculation of variable annuity market sensitivities using a pathwise methodology

Calculation of variable annuity market sensitivities using a pathwise methodology cuing edge Variable annuiie Calculaion of variable annuiy marke eniiviie uing a pahwie mehodology Under radiional finie difference mehod, he calculaion of variable annuiy eniiviie can involve muliple Mone

More information