Butterfly Network Analysis and The Beneˇ s Network

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Butterfly Network Analysis and The Beneˇ s Network"

Transcription

1 6.895 Theoy of Paallel Systems Lectue 17 Buttefly Netwok Analysis and The Beneˇ s Netwok Lectue: Chales Leiseson Lectue Summay 1. Netwok with N Nodes This section poves pat of the lowe bound on expected outing time fo an abitay N node netwok. 2. The Beneš Netwok This section motivates, intoduces, and analyzes the Beneˇ s connection netwok. 3. Routing on Buttefly Netwoks This section establishes that most outing poblems take only O(lg N ) time on buttefly netwoks. 1 Netwok with N Nodes In this section, we pove a lemma that completes the poof of the Netwok with N Nodes theoem given in lectue 16. Theoem 1 Fo an abitay netwok with N nodes, the expected outing time to simultaneously send a single message fom each pocesso to a andom pocesso is whee BW is the minimum bisection width. Poof E[Routing Time] =Ω(N/BW + diamete) Lemma 2 The expected outing time fo the N andom messages poblem laid out in theoem 1 is Ω(N/BW ). Poof Refe to the notes fo lectue 16 fo poof. Lemma 3 Each node v in a N node netwok chooses a destination dest(v) unifomly at andom. Then, E[Routing Time] =Ω(d), whee d is the diamete. Poof Find node x such that P{δ(x; dest(x)) d/2} 1/2 whee δ(x, y) is the shotest distance fom node x to node y. Then, E[time to oute x to its destination dest(x)] = kp {δ(x; dest(x)) = k} k kp {δ(x; dest(x)) = k} k d/2 d/2 P{δ(x; dest(x)) = k} k d/2 = d/2p{δ(x, dest(x)) d/2} = d/2 =Ω(d) 17-1

2 Figue 1: Example of a Beneˇ s Netwok. Nodes aligned vetically ae in the same level and nodes aligned hoizontally ae in the same ow. To find such an x, let δ(x, x )= d, and define S = { v : δ(x, v) < d/2} and S = { v : δ(x,v) < d/2}. We know S S = since fo all z S S, δ(x, x ) δ(x, z)+ δ(z, x ) < d/2+ d/2 < d gives a contadiction, implying no such z can exist. Theefoe, at least one of S and S has N/2 nodes. Without loss of geneality, assume that S N/2. Thus, dest(x) S with pobability 1/2, i.e., P{ δ(x, dest(x)) d/2} 1/2. Fom lemma 2 and lemma 3, we know that E[Routing Time] = Ω(N/BW ) and E[Routing Time] = Ω(diamete). Theefoe, we can conclude that E[Routing Time] = Ω(N/BW + diamete). 2 The Beneš Netwok 2.1 Wost Case Poblems fo the Buttefly Netwok We saw last lectue that the wost case congestion fo the buttefly netwok is n fo an n input netwok. This was motivated by consideing the poblem whee inputs x 1 x 2 x 3 x outputs 0000x 1 x 2 x 3 x 4. Thee ae many such bad inputs that cause n congestion fo the buttefly netwok. Fo example, conside the poblem of computing the matix tanspose whee each node epesents an element of a matix and we wish to compute (i, j) (j, i). Matix tanspose and othe opeations that elicit the wost case congestion in a buttefly netwok ae common in pogams witten fo supecomputes. It is theefoe a eal poblem to build supecomputes with buttefly netwoks because typical applications suffe seious pefomance penalties. 3 The Beneš Netwok This section intoduces the Beneš netwok, a netwok that allows povably congestion-fee communication fom inputs to a pemutation of the outputs. A Beneš netwok is constucted by ovelapping the low-ode 17-2

3 Figue 2: An example Beneˇ s Netwok with the middle d 2 levels eplaced with two half-sized Beneˇ s netwoks. cycles of two buttefly netwoks. An example is shown in figue 1. Theoem 4 Any n-pemutation can be outed (off-line) on an n-input Beneš netwok with node-disjoint paths. Poof The poof uses the fact that Beneˇ s netwoks ae decomposable into smalle Beneˇ s netwoks. Let n = 2 d. The poof is based on induction on d. Fo d = 1, the netwok is tivial and the theoem is obvious on examination. As the inductive step, assume that the theoem holds fo a n = 2 d input Beneš netwok. Obseve that the middle d 2 levels of the Beneˇ s netwok fom two Beneˇ s netwoks, one on top and one on bottom, each with n/2 =2 d 1 inputs, as shown in figue 2. By assumption, any input to one of the sub-beneš netwoks can be outed to any output of the same subnetwok without congestion. Theefoe, all we need to do is detemine whethe each input is to be outed though the top o the bottom subnetwok and show that the subnetwok outputs can be coectly outed to the final outputs without congestion. The only constaint we must obseve to avoid congestion is that connected pais at level 1 and at level d do not ovelap. That is, each buttefly at the fist and last level must eithe oute packets staight o cossed: butteflies cannot send both input packets to a single output node. Theefoe, by obsevation, the two inputs compising a buttefly at the fist level must be outed into diffeent subnetwoks and the two inputs compising a buttefly at the last level must come fom diffeent subnetwoks. It is this final constaint that is impotant. Define a buttefly pai as { i + n/2 (i n/2), bfly(i) = i n/2 (i>n/2). The buttefly pai is the othe node (eithe input o output) that defines a buttefly in the netwok, bfly (i) = j and bfly (j) = i. Define π(i) as the pemutation that maps inputs to outputs: π(i) = j denotes that input i should be outed to output j and π 1 (j) = i denotes that output j should be outed fom input i. We now show how to oute inputs into the uppe and lowe subnetwoks in a way that satisfies ou constaints. We stat by outing the fist input though the uppe subnetwok and connecting it to the coect output, π(1). Next, we fulfill the constaint at the last level by outing the fist path s output 17-3

4 buttefly pai, bfly (π(1)), though the lowe netwok and back to its coect input, π 1 (bfly(π(1))). We satisfy the new constaint at the input by outing the appopiate input, π 1 (bfly(π(1))), though the uppe subnetwok and to its coect output. We continue going back and foth though the uppe subnetwok when connecting an input to an output and though the lowe subnetwok when connecting an output to an input until we have defined a completed loop. Since the fist path went though the uppe subnetwok and the final etun path went though the lowe subnetwok, the input constaint on the oiginal path is satisfied. If the fist cycle of paths doesn t include all nodes, we pick an unouted node and epeat this pocess until all inputs ae outed. Following this algoithm, half of the inputs ae outed though the top subnetwok and half of the inputs ae outed though the bottom subnetwok without congestion. We have satisfied the inductive hypothesis by showing that given a n/2 =2 d 1 input congestion fee Beneˇ s netwok, we can constuct a n =2 d input congestion fee Beneˇ s netwok. By induction, we have poved the theoem. As a side note, since we always have a choice of how to oute the fist input, we can emove a switch fom each inductive level of the buttefly netwok without affecting its pefomance chaacteistics. Consequently, fo a n = 2 d input netwok, we can emove d switches fom the netwok without loss of geneality. Coollay 5 An n-input Beneš netwok can simulate any n-node, degee-d netwok in O(d lg n) time. Poof Let G be an n-node netwok with maximum degee d. Let H be an n-input Beneš netwok. We pove the coollay by showing how to simulate G on H. Identify the ith node of G with the ith input of H to simulate node computation. The ticky pat is showing how to simulate communication on edges in G using H s netwok. We do this by identifying d + 1 subsets of the edges in G such that fo each subset, we can oute packets fom input i to output j fo each edge (i, j) in the subset of G with at most O(lg n) congestion. Each subset is outed in a sepaate ound. d + 1 subsets with O(lg n) congestion each gives us a total of O(d lg n) time to simulate G. We now constuct d + 1 disjoint and spanning subsets of edges in G. Constuct a bipatite gaph Γ G =(U, V, E) whee U = {u 1,u 2,...u n }, V = {v 1,v 2,...v n },and E = {(u i,v j ) (i, j) is an edge of G}. Communication fom i to j in G is epesented by edge (u i,v j )inγ G. Since the maximum degee in G is d, the maximum degee in Γ G is also d, making Γ G a d-egula bipatite gaph. Theefoe, we can constuct an edge-coloing of Γ G using d + 1 colos (see Leighton s Intoduction to Paallel Algoithms and Achitectues fo poof). Each set of edges with a given colo foms one of the d + 1 subsets. Let S be the kth of the d + 1 subsets. Fo each edge (i, j) in S, we simulate communication on (i, j) by sending a packet fom input i to output j on H duing ound k. Since the edges incident to i and the edges incident to j ae all coloed diffeently, we know that no two packets oiginate fom o ae deliveed to the same node duing the same ound. Theefoe, S can be outed with ou O(lg n) bound. One ound fo each of the d + 1 subsets with O(lg n) outing time each completes the poof. 4 Routing on Buttefly Netwoks Even though the buttefly netwok has a outing time of Ω( n) on cetain pemutations (including many inteesting pemutations), most outing poblems take only O(lg n) time. The following theoem coves abitay N-packet outing poblems on buttefly netwoks, not just those that oute inputs to outputs. Theoem 6 Conside the N N N-packet outing poblems on an N-node (n-input, whee n = Θ(N/ lg N)) buttefly netwok. At least N N (1 1/N Ω(1) ) of these poblems can be outed in O(lg N) time. 17-4

5 Figue 3: Depiction of a buttefly netwok. hoizontally ae in the same ow. Nodes aligned vetically ae in the same level and nodes aligned Poof We will see the poof of a weake esult than is needed to establish the theoem. The poof establishes a congestion bound that leads to an O(lg 2 n) time esult. WLOG, we oute packets in thee phases (figue 3 shows the ows and levels in a buttefly netwok): 1. Route packets staight acoss ows in the netwok to the coesponding outputs (the ightmost node in each ow). 2. Use geedy input to output outing, to get each packet to the coect ow in the netwok. 3. Route packets staight acoss to the final destination. We analyze the congestion at the output nodes at each ow of the netwok in Phase 1. All packets fom a given ow end up at the output node of that ow at the end of the phase. Each ow contains lg n nodes, and lg n = O(lg N ), so the congestion in Phase 1 is O(lg n). Fo Phase 2, we conside a node x at the kth level of the netwok (whee the inputs ae the 0th level and the outputs ae the (lg n)th level). By symmety, the x is equivalent to any othe node at the kth level. The numbe of packets that can each x duing Phase 2 is 2 k lg n since thee is a path to each node at the kth level of a buttefly netwok fom 2 k input nodes and each input node has at most lg n packets at the beginning of Phase 2. Thus, wost case congestion at a node x at the kth level = 2 k lg n. The pobability that a given packet passes though node x is at most 2 k. This esult follows fom the binay tee popety of buttefly netwoks and the fact that the destination of each packet is chosen independently and unifomly at andom fom all the nodes in the netwok. (See figue 4 fo a depiction of the binaay tee popety). Thus, P [a given packet passes though node x] 2 k. Conside any set of specific packets. Since the destinations of the packets ae independent of one anothe, the pobability that all the packets pass though x is given by P [all packets pass though x] (2 k ) =2 k. 17-5

6 Figue 4: Binay tee popety of a buttefly netwok. Each choice on a path fom an input node to an output node is the head of a binay tee. The pobability that at least packets pass though x is bounded by the numbe of ways to choose packets fom the total numbe of packets multiplied by the pobability that all packets pass though x: 2 k lg n P [ packets pass though x] 2 k. + Note: This agument ovecounts. If + packets pass though x, this event is counted times within ) the ( 2 k lg n ways. a ea Using the fomula b b,wehave If we choose = 2e lg N, then we obtain b ( e2 k lg n ) P [ packets pass though x] 2 k ( e lg n ) =. P [ packets pass though x] ( 1 ) 2e lg N 2 N 2e 1. N 5.4 By Boole s inequality, the pobability that any node has moe than =2e lg N packets is N times the pobability that a given node has moe than packets: 1 P [ packets pass though some node] N N 5.4 = N

7 Theefoe, N N (1 1/N 4.4 ) outing poblems see 2e lg N congestion. Since thee ae O(lg N ) levels and each node at a level has at most O(lg N ) congestion, the total congestion fo Phase 2 is O(lg 2 N ). At the end of Phase 2, thee ae O(lg N ) packets at each output node with high pobability. Theefoe, at each node in Phase 3 thee is O(lg N ) congestion with high pobability. Thus, the oveall time bound is O(lg 2 N ) foatleast N N (1 1/N Ω(1) ) outing poblems. Coollay 7 E[outing time] = O(lg N )(1 1/N 4.4 )+ O(N )(1/N 4.4 ) = O(lg N ). 7

Saturated and weakly saturated hypergraphs

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

More information

Week 3-4: Permutations and Combinations

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

More information

Symmetric polynomials and partitions Eugene Mukhin

Symmetric polynomials and partitions Eugene Mukhin Symmetic polynomials and patitions Eugene Mukhin. Symmetic polynomials.. Definition. We will conside polynomials in n vaiables x,..., x n and use the shotcut p(x) instead of p(x,..., x n ). A pemutation

More information

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

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

More information

Fast FPT-algorithms for cleaning grids

Fast FPT-algorithms for cleaning grids Fast FPT-algoithms fo cleaning gids Josep Diaz Dimitios M. Thilikos Abstact We conside the poblem that given a gaph G and a paamete k asks whethe the edit distance of G and a ectangula gid is at most k.

More information

The Binomial Distribution

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

More information

Chapter 3 Savings, Present Value and Ricardian Equivalence

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

More information

Coordinate Systems L. M. Kalnins, March 2009

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

More information

Approximation Algorithms for Data Management in Networks

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

More information

Top K Nearest Keyword Search on Large Graphs

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

More information

NURBS Drawing Week 5, Lecture 10

NURBS Drawing Week 5, Lecture 10 CS 43/585 Compute Gaphics I NURBS Dawing Week 5, Lectue 1 David Been, William Regli and Maim Pesakhov Geometic and Intelligent Computing Laboato Depatment of Compute Science Deel Univesit http://gicl.cs.deel.edu

More information

UNIT CIRCLE TRIGONOMETRY

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

More information

Transformations in Homogeneous Coordinates

Transformations in Homogeneous Coordinates Tansfomations in Homogeneous Coodinates (Com S 4/ Notes) Yan-Bin Jia Aug, 6 Homogeneous Tansfomations A pojective tansfomation of the pojective plane is a mapping L : P P defined as u a b c u au + bv +

More information

Nontrivial lower bounds for the least common multiple of some finite sequences of integers

Nontrivial lower bounds for the least common multiple of some finite sequences of integers J. Numbe Theoy, 15 (007), p. 393-411. Nontivial lowe bounds fo the least common multiple of some finite sequences of integes Bai FARHI bai.fahi@gmail.com Abstact We pesent hee a method which allows to

More information

Separation probabilities for products of permutations

Separation probabilities for products of permutations Sepaation pobabilities fo poducts of pemutations Olivie Benadi, Rosena R. X. Du, Alejando H. Moales and Richad P. Stanley Mach 1, 2012 Abstact We study the mixing popeties of pemutations obtained as a

More information

An Immunological Approach to Change Detection: Algorithms, Analysis and Implications

An Immunological Approach to Change Detection: Algorithms, Analysis and Implications An Immunological Appoach to Change Detection: Algoithms, Analysis and Implications Patik D haeselee Dept. of Compute Science Univesity of New Mexico Albuqueque, NM, 87131 patik@cs.unm.edu Stephanie Foest

More information

Software Engineering and Development

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

More information

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

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

More information

Carter-Penrose diagrams and black holes

Carter-Penrose diagrams and black holes Cate-Penose diagams and black holes Ewa Felinska The basic intoduction to the method of building Penose diagams has been pesented, stating with obtaining a Penose diagam fom Minkowski space. An example

More information

On Correlation Coefficient. The correlation coefficient indicates the degree of linear dependence of two random variables.

On Correlation Coefficient. The correlation coefficient indicates the degree of linear dependence of two random variables. C.Candan EE3/53-METU On Coelation Coefficient The coelation coefficient indicates the degee of linea dependence of two andom vaiables. It is defined as ( )( )} σ σ Popeties: 1. 1. (See appendi fo the poof

More information

Explicit, analytical solution of scaling quantum graphs. Abstract

Explicit, analytical solution of scaling quantum graphs. Abstract Explicit, analytical solution of scaling quantum gaphs Yu. Dabaghian and R. Blümel Depatment of Physics, Wesleyan Univesity, Middletown, CT 06459-0155, USA E-mail: ydabaghian@wesleyan.edu (Januay 6, 2003)

More information

CHAPTER 10 Aggregate Demand I

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

More information

1.4 Phase Line and Bifurcation Diag

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

More information

Theory and practise of the g-index

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

More information

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

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

More information

Model Question Paper Mathematics Class XII

Model Question Paper Mathematics Class XII Model Question Pape Mathematics Class XII Time Allowed : 3 hous Maks: 100 Ma: Geneal Instuctions (i) The question pape consists of thee pats A, B and C. Each question of each pat is compulsoy. (ii) Pat

More information

Questions for Review. By buying bonds This period you save s, next period you get s(1+r)

Questions for Review. By buying bonds This period you save s, next period you get s(1+r) MACROECONOMICS 2006 Week 5 Semina Questions Questions fo Review 1. How do consumes save in the two-peiod model? By buying bonds This peiod you save s, next peiod you get s() 2. What is the slope of a consume

More information

Seshadri constants and surfaces of minimal degree

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

More information

Algebra and Trig. I. A point is a location or position that has no size or dimension.

Algebra and Trig. I. A point is a location or position that has no size or dimension. Algeba and Tig. I 4.1 Angles and Radian Measues A Point A A B Line AB AB A point is a location o position that has no size o dimension. A line extends indefinitely in both diections and contains an infinite

More information

Integer sequences from walks in graphs

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

More information

Lecture 16: Color and Intensity. and he made him a coat of many colours. Genesis 37:3

Lecture 16: Color and Intensity. and he made him a coat of many colours. Genesis 37:3 Lectue 16: Colo and Intensity and he made him a coat of many colous. Genesis 37:3 1. Intoduction To display a pictue using Compute Gaphics, we need to compute the colo and intensity of the light at each

More information

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

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

More information

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

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

More information

Functions of a Random Variable: Density. Math 425 Intro to Probability Lecture 30. Definition Nice Transformations. Problem

Functions of a Random Variable: Density. Math 425 Intro to Probability Lecture 30. Definition Nice Transformations. Problem Intoduction One Function of Random Vaiables Functions of a Random Vaiable: Density Math 45 Into to Pobability Lectue 30 Let gx) = y be a one-to-one function whose deiatie is nonzeo on some egion A of the

More information

Continuous Compounding and Annualization

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

More information

An Introduction to Omega

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

More information

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

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

More information

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

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

More information

Questions & Answers Chapter 10 Software Reliability Prediction, Allocation and Demonstration Testing

Questions & Answers Chapter 10 Software Reliability Prediction, Allocation and Demonstration Testing M13914 Questions & Answes Chapte 10 Softwae Reliability Pediction, Allocation and Demonstation Testing 1. Homewok: How to deive the fomula of failue ate estimate. λ = χ α,+ t When the failue times follow

More information

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

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

More information

Performance Analysis of an Inverse Notch Filter and Its Application to F 0 Estimation

Performance Analysis of an Inverse Notch Filter and Its Application to F 0 Estimation Cicuits and Systems, 013, 4, 117-1 http://dx.doi.og/10.436/cs.013.41017 Published Online Januay 013 (http://www.scip.og/jounal/cs) Pefomance Analysis of an Invese Notch Filte and Its Application to F 0

More information

2. SCALARS, VECTORS, TENSORS, AND DYADS

2. SCALARS, VECTORS, TENSORS, AND DYADS 2. SCALARS, VECTORS, TENSORS, AND DYADS This section is a eview of the popeties of scalas, vectos, and tensos. We also intoduce the concept of a dyad, which is useful in MHD. A scala is a quantity that

More information

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

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

More information

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

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

More information

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

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

More information

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

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

More information

Financing Terms in the EOQ Model

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

More information

Reduced Pattern Training Based on Task Decomposition Using Pattern Distributor

Reduced Pattern Training Based on Task Decomposition Using Pattern Distributor > PNN05-P762 < Reduced Patten Taining Based on Task Decomposition Using Patten Distibuto Sheng-Uei Guan, Chunyu Bao, and TseNgee Neo Abstact Task Decomposition with Patten Distibuto (PD) is a new task

More information

Things to Remember. r Complete all of the sections on the Retirement Benefit Options form that apply to your request.

Things to Remember. r Complete all of the sections on the Retirement Benefit Options form that apply to your request. Retiement Benefit 1 Things to Remembe Complete all of the sections on the Retiement Benefit fom that apply to you equest. If this is an initial equest, and not a change in a cuent distibution, emembe to

More information

92.131 Calculus 1 Optimization Problems

92.131 Calculus 1 Optimization Problems 9 Calculus Optimization Poblems ) A Noman window has the outline of a semicicle on top of a ectangle as shown in the figue Suppose thee is 8 + π feet of wood tim available fo all 4 sides of the ectangle

More information

Mechanics 1: Work, Power and Kinetic Energy

Mechanics 1: Work, Power and Kinetic Energy Mechanics 1: Wok, Powe and Kinetic Eneg We fist intoduce the ideas of wok and powe. The notion of wok can be viewed as the bidge between Newton s second law, and eneg (which we have et to define and discuss).

More information

Research Article A Reputation-Based Identity Management Model for Cloud Computing

Research Article A Reputation-Based Identity Management Model for Cloud Computing Mathematical Poblems in Engineeing Volume 2015, Aticle ID 238245, 15 pages http://dx.doi.og/10.1155/2015/238245 Reseach Aticle A Reputation-Based Identity Management Model fo Cloud Computing Lifa Wu, 1

More information

MULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION

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

More information

SUPPORT VECTOR MACHINE FOR BANDWIDTH ANALYSIS OF SLOTTED MICROSTRIP ANTENNA

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

More information

LINES AND TANGENTS IN POLAR COORDINATES

LINES AND TANGENTS IN POLAR COORDINATES LINES AND TANGENTS IN POLAR COORDINATES ROGER ALEXANDER DEPARTMENT OF MATHEMATICS 1. Pola-coodinate equations fo lines A pola coodinate system in the plane is detemined by a point P, called the pole, and

More information

CIRCUITS LABORATORY EXPERIMENT 7

CIRCUITS LABORATORY EXPERIMENT 7 CIRCUITS LABORATORY EXPERIMENT 7 Design of a Single Tansisto Amplifie 7. OBJECTIVES The objectives of this laboatoy ae to: (a) Gain expeience in the analysis and design of an elementay, single tansisto

More information

Economics 326: Input Demands. Ethan Kaplan

Economics 326: Input Demands. Ethan Kaplan Economics 326: Input Demands Ethan Kaplan Octobe 24, 202 Outline. Tems 2. Input Demands Tems Labo Poductivity: Output pe unit of labo. Y (K; L) L What is the labo poductivity of the US? Output is ouhgly

More information

Deflection of Electrons by Electric and Magnetic Fields

Deflection of Electrons by Electric and Magnetic Fields Physics 233 Expeiment 42 Deflection of Electons by Electic and Magnetic Fields Refeences Loain, P. and D.R. Coson, Electomagnetism, Pinciples and Applications, 2nd ed., W.H. Feeman, 199. Intoduction An

More information

Life Insurance Purchasing to Reach a Bequest. Erhan Bayraktar Department of Mathematics, University of Michigan Ann Arbor, Michigan, USA, 48109

Life Insurance Purchasing to Reach a Bequest. Erhan Bayraktar Department of Mathematics, University of Michigan Ann Arbor, Michigan, USA, 48109 Life Insuance Puchasing to Reach a Bequest Ehan Bayakta Depatment of Mathematics, Univesity of Michigan Ann Abo, Michigan, USA, 48109 S. David Pomislow Depatment of Mathematics, Yok Univesity Toonto, Ontaio,

More information

Vector Calculus: Are you ready? Vectors in 2D and 3D Space: Review

Vector Calculus: Are you ready? Vectors in 2D and 3D Space: Review Vecto Calculus: Ae you eady? Vectos in D and 3D Space: Review Pupose: Make cetain that you can define, and use in context, vecto tems, concepts and fomulas listed below: Section 7.-7. find the vecto defined

More information

Efficient Redundancy Techniques for Latency Reduction in Cloud Systems

Efficient Redundancy Techniques for Latency Reduction in Cloud Systems Efficient Redundancy Techniques fo Latency Reduction in Cloud Systems 1 Gaui Joshi, Emina Soljanin, and Gegoy Wonell Abstact In cloud computing systems, assigning a task to multiple seves and waiting fo

More information

STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION

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

More information

The transport performance evaluation system building of logistics enterprises

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

More information

Experiment 6: Centripetal Force

Experiment 6: Centripetal Force Name Section Date Intoduction Expeiment 6: Centipetal oce This expeiment is concened with the foce necessay to keep an object moving in a constant cicula path. Accoding to Newton s fist law of motion thee

More information

How to create RAID 1 mirroring with a hard disk that already has data or an operating system on it

How to create RAID 1 mirroring with a hard disk that already has data or an operating system on it AnswesThatWok TM How to set up a RAID1 mio with a dive which aleady has Windows installed How to ceate RAID 1 mioing with a had disk that aleady has data o an opeating system on it Date Company PC / Seve

More information

Online Competitive Algorithms for Ad Allocation in Cellular Networks (AdCell)

Online Competitive Algorithms for Ad Allocation in Cellular Networks (AdCell) Online Competitive Algoithms fo Ad Allocation in Cellula Netwoks (AdCell) Saeed Alaei 1, Mohammad T. Hajiaghayi 12, Vahid Liaghat 1, Dan Pei 2, and Bana Saha 1 {saeed, hajiagha, vliaghat, bana}@cs.umd.edu,

More information

QoS-Constrained Resource Allocation for a Grid-Based Multiple Source Electrocardiogram Application

QoS-Constrained Resource Allocation for a Grid-Based Multiple Source Electrocardiogram Application QoS-Constained Resouce Allocation fo a Gid-Based Multiple Souce Electocadiogam Application Dong Su Nam 1,5, Chan-Hyun Youn 1,3, Bong Hwan Lee 2, Gai Cliffod 3, and Jennife Healey 4 1 School of Engineeing,

More information

Programming Assignment #1

Programming Assignment #1 Due: Nov 3 (11:59pm). Pogamming Assignment #1 CMSC 351 Fall 2014 Rules 1) You may only use C/C++, Java. 2) You pogam should use the standad input/output. Fo example C/C++ uses should use scanf/pintf/cin/cout

More information

Gauss Law. Physics 231 Lecture 2-1

Gauss Law. Physics 231 Lecture 2-1 Gauss Law Physics 31 Lectue -1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing

More information

Semipartial (Part) and Partial Correlation

Semipartial (Part) and Partial Correlation Semipatial (Pat) and Patial Coelation his discussion boows heavily fom Applied Multiple egession/coelation Analysis fo the Behavioal Sciences, by Jacob and Paticia Cohen (975 edition; thee is also an updated

More information

Strength Analysis and Optimization Design about the key parts of the Robot

Strength Analysis and Optimization Design about the key parts of the Robot Intenational Jounal of Reseach in Engineeing and Science (IJRES) ISSN (Online): 2320-9364, ISSN (Pint): 2320-9356 www.ijes.og Volume 3 Issue 3 ǁ Mach 2015 ǁ PP.25-29 Stength Analysis and Optimization Design

More information

Secure Smartcard-Based Fingerprint Authentication

Secure Smartcard-Based Fingerprint Authentication Secue Smatcad-Based Fingepint Authentication [full vesion] T. Chales Clancy Compute Science Univesity of Mayland, College Pak tcc@umd.edu Nega Kiyavash, Dennis J. Lin Electical and Compute Engineeing Univesity

More information

Experiment MF Magnetic Force

Experiment MF Magnetic Force Expeiment MF Magnetic Foce Intoduction The magnetic foce on a cuent-caying conducto is basic to evey electic moto -- tuning the hands of electic watches and clocks, tanspoting tape in Walkmans, stating

More information

2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES

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

More information

INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS

INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS Vesion:.0 Date: June 0 Disclaime This document is solely intended as infomation fo cleaing membes and othes who ae inteested in

More information

High Availability Replication Strategy for Deduplication Storage System

High Availability Replication Strategy for Deduplication Storage System Zhengda Zhou, Jingli Zhou College of Compute Science and Technology, Huazhong Univesity of Science and Technology, *, zhouzd@smail.hust.edu.cn jlzhou@mail.hust.edu.cn Abstact As the amount of digital data

More information

Financial Derivatives for Computer Network Capacity Markets with Quality-of-Service Guarantees

Financial Derivatives for Computer Network Capacity Markets with Quality-of-Service Guarantees Financial Deivatives fo Compute Netwok Capacity Makets with Quality-of-Sevice Guaantees Pette Pettesson pp@kth.se Febuay 2003 SICS Technical Repot T2003:03 Keywods Netwoking and Intenet Achitectue. Abstact

More information

FI3300 Corporate Finance

FI3300 Corporate Finance Leaning Objectives FI00 Copoate Finance Sping Semeste 2010 D. Isabel Tkatch Assistant Pofesso of Finance Calculate the PV and FV in multi-peiod multi-cf time-value-of-money poblems: Geneal case Pepetuity

More information

BIOS American Megatrends Inc (AMI) v02.61 BIOS setup guide and manual for AM2/AM2+/AM3 motherboards

BIOS American Megatrends Inc (AMI) v02.61 BIOS setup guide and manual for AM2/AM2+/AM3 motherboards BIOS Ameican Megatends Inc (AMI) v02.61 BIOS setup guide and manual fo AM2/AM2+/AM3 motheboads The BIOS setup, also called CMOS setup, is a cucial pat of the pope setting up of a PC the BIOS (Basic Input

More information

How to recover your Exchange 2003/2007 mailboxes and emails if all you have available are your PRIV1.EDB and PRIV1.STM Information Store database

How to recover your Exchange 2003/2007 mailboxes and emails if all you have available are your PRIV1.EDB and PRIV1.STM Information Store database AnswesThatWok TM Recoveing Emails and Mailboxes fom a PRIV1.EDB Exchange 2003 IS database How to ecove you Exchange 2003/2007 mailboxes and emails if all you have available ae you PRIV1.EDB and PRIV1.STM

More information

Do Vibrations Make Sound?

Do Vibrations Make Sound? Do Vibations Make Sound? Gade 1: Sound Pobe Aligned with National Standads oveview Students will lean about sound and vibations. This activity will allow students to see and hea how vibations do in fact

More information

Uncertain Version Control in Open Collaborative Editing of Tree-Structured Documents

Uncertain Version Control in Open Collaborative Editing of Tree-Structured Documents Uncetain Vesion Contol in Open Collaboative Editing of Tee-Stuctued Documents M. Lamine Ba Institut Mines Télécom; Télécom PaisTech; LTCI Pais, Fance mouhamadou.ba@ telecom-paistech.f Talel Abdessalem

More information

Cloud Service Reliability: Modeling and Analysis

Cloud Service Reliability: Modeling and Analysis Cloud Sevice eliability: Modeling and Analysis Yuan-Shun Dai * a c, Bo Yang b, Jack Dongaa a, Gewei Zhang c a Innovative Computing Laboatoy, Depatment of Electical Engineeing & Compute Science, Univesity

More information

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

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

More information

An Efficient Group Key Agreement Protocol for Ad hoc Networks

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

More information

BINOMIAL THEOREM. 1. Introduction. 2. The Binomial Coefficients. ( x + 1), we get. and. When we expand

BINOMIAL THEOREM. 1. Introduction. 2. The Binomial Coefficients. ( x + 1), we get. and. When we expand BINOMIAL THEOREM Itoductio Whe we epad ( + ) ad ( + ), we get ad ( + ) = ( + )( + ) = + + + = + + ( + ) = ( + )( + ) = ( + )( + + ) = + + + + + = + + + 4 5 espectively Howeve, whe we ty to epad ( + ) ad

More information

A Comparative Analysis of Data Center Network Architectures

A Comparative Analysis of Data Center Network Architectures A Compaative Analysis of Data Cente Netwok Achitectues Fan Yao, Jingxin Wu, Guu Venkataamani, Suesh Subamaniam Depatment of Electical and Compute Engineeing, The Geoge Washington Univesity, Washington,

More information

Load Balancing in Processor Sharing Systems

Load Balancing in Processor Sharing Systems Load Balancing in ocesso Shaing Systems Eitan Altman INRIA Sophia Antipolis 2004, oute des Lucioles 06902 Sophia Antipolis, Fance altman@sophia.inia.f Utzi Ayesta LAAS-CNRS Univesité de Toulouse 7, Avenue

More information

Load Balancing in Processor Sharing Systems

Load Balancing in Processor Sharing Systems Load Balancing in ocesso Shaing Systems Eitan Altman INRIA Sophia Antipolis 2004, oute des Lucioles 06902 Sophia Antipolis, Fance altman@sophia.inia.f Utzi Ayesta LAAS-CNRS Univesité de Toulouse 7, Avenue

More information

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

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

More information

The Role of Gravity in Orbital Motion

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

More information

Mining Relatedness Graphs for Data Integration

Mining Relatedness Graphs for Data Integration Mining Relatedness Gaphs fo Data Integation Jeemy T. Engle (jtengle@indiana.edu) Ying Feng (yingfeng@indiana.edu) Robet L. Goldstone (goldsto@indiana.edu) Indiana Univesity Bloomington, IN. 47405 USA Abstact

More information

Episode 401: Newton s law of universal gravitation

Episode 401: Newton s law of universal gravitation Episode 401: Newton s law of univesal gavitation This episode intoduces Newton s law of univesal gavitation fo point masses, and fo spheical masses, and gets students pactising calculations of the foce

More information

How to create a default user profile in Windows 7

How to create a default user profile in Windows 7 AnswesThatWok TM How to ceate a default use pofile in Windows 7 (Win 7) How to ceate a default use pofile in Windows 7 When to use this document Use this document wheneve you want to ceate a default use

More information

Skills Needed for Success in Calculus 1

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

More information

Physics 235 Chapter 5. Chapter 5 Gravitation

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

More information

Database Management Systems

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

More information

The LCOE is defined as the energy price ($ per unit of energy output) for which the Net Present Value of the investment is zero.

The LCOE is defined as the energy price ($ per unit of energy output) for which the Net Present Value of the investment is zero. Poject Decision Metics: Levelized Cost of Enegy (LCOE) Let s etun to ou wind powe and natual gas powe plant example fom ealie in this lesson. Suppose that both powe plants wee selling electicity into the

More information

Graphs of Equations. A coordinate system is a way to graphically show the relationship between 2 quantities.

Graphs of Equations. A coordinate system is a way to graphically show the relationship between 2 quantities. Gaphs of Equations CHAT Pe-Calculus A coodinate sstem is a wa to gaphicall show the elationship between quantities. Definition: A solution of an equation in two vaiables and is an odeed pai (a, b) such

More information

An Approach to Optimized Resource Allocation for Cloud Simulation Platform

An Approach to Optimized Resource Allocation for Cloud Simulation Platform An Appoach to Optimized Resouce Allocation fo Cloud Simulation Platfom Haitao Yuan 1, Jing Bi 2, Bo Hu Li 1,3, Xudong Chai 3 1 School of Automation Science and Electical Engineeing, Beihang Univesity,

More information