Online-routing on the butterfly network: probabilistic analysis

Size: px
Start display at page:

Download "Online-routing on the butterfly network: probabilistic analysis"

Transcription

1 Online-outing on the buttefly netwok: obabilistic analysis Andey Gubichev Contents 1 Intoduction: definitions 1 Aveage case behavio of the geedy algoithm 3.1 Bounds on congestion Bounds on unning time Conclusion 7 4 Bibliogahy 7 1 Intoduction: definitions In this talk we will examine the aveage-case behavio of the geedy algoithm in buttefly netwok. Let us fist intoduce some useful notions and give simle examles. Definition 1 (Buttefly). The -dimensional buttefly consists of (+1) nodes and +1 edges. A node is a ai w, i, i is the level of the node, w is the ow numbe (-bit). An edge links two nodes w, i and w, i if and only if i = i + 1 and eithe w = w, o w and w diffes in the ith bit. Figue 1 shows an examle of 3-dimensional buttefly. The acket outing oblem is the oblem of outing N ackets fom level 0 to level in a -dimensional buttefly. Each acket u, 0 has its own destination π(u), whee π : [1, N] [1, N] is a emutation. The most commonly used emutations ae the bit-evesal emutation: π(u 1 u ) = u u 1, 1

2 Figue 1: Thee-dimensional buttefly. and the tansose emutation: π(u 1 u u +1 u ) = u +1 u u 1 u We will insist that ou outing algoithms be on-line: thee is no global contolle that can ecomute outing aths, each node decides what to do with a acket that ass though it based on its local contolle and infomation fom acket. Definition. The geedy ath fom u, 0 to v, is the unique ath of length fom the fist node to the second node. The geedy algoithm is the algoithm that constains each acket to follow its geedy ath. The congestion oblem is that many ackets might ass though a single node o edge, but only one acket can use the aticula edge o node at a time. Theoem 1. The geedy algoithm will oute N ackets to thei destinations in a -buttefly in O( (N)) stes. In fact, the bit-evesal emutation and the tansosal emutation ae the wost-case emutations fo geedy outing. In the following section we will find out that in aveage case the geedy algoithm behaves much bette.

3 Aveage case behavio of the geedy algoithm We will divide ou analysis into two ats. Fist of all, we will bound the congestion. If we obtain the bound C fo congestion, we will automatically have a bound fo the unning time: (C 1). In the second at we will get a tighte bound fo the unning time. In this section we conside the outing oblem fo which each acket has a andom destination (destinations ae selected indeendently and unifomly fom among the N ossible oututs). Hee we also allow moe than one acket to stat at each inut (denote by the numbe of ackets at each inut)..1 Bounds on congestion Theoem. Fo all but at most a 1/N 3/ faction of the ossible outing oblems with ackets e inut in a -dimensional buttefly at most C ackets ass though each node duing a geedy outing whee e, if C = ( ) e / log, if Poof. Ou main aim hee is to bound the obability P (v) that o moe acket aths ass though some node v fo each > 0 and fo each node fom -dimensional buttefly. Let v be the node on ith level of the buttefly. Thee ae i inuts that can each v and i = N i choices of destinations that can cause the acket to ass though v. Since we choose destinations andomly among N destinations, the obability fo each of i ackets to ass though v is N i /N = i. A simle illustation is given on figue. If o moe ackets ass though v, then thee exists a subset of ackets and all of them must ass though v: ( ) ( ) i P (v) ( i ) i e i = ) The ue bound does not deend on v, i. Hence, the obability that o moe ackets ass though all nodes in the buttefly is at most N (e/). This bound deceases if inceases. If, let = e and we get ) ( ) e 1 N N = N 1 e 1/N 3/ In case that let = e log( ) and x = : N ) ( ) e = N x 3 e log(x/ ) = N N

4 Figue : Choices of inuts and destinations that cause the acket to ass though v The minimum fo log(x/ ) whee x occus if = e: These facts comlete the oof. e log(x/ ) N N N N e+ 1/N In fact, the faction of "bad" outing oblems can be made abitay small. Coollay 1. α the congestion fo all but 1/N α oblems is at most O(α) + o(α ) Poof. In ode to ove it we can modify the evious oof. If let = eα = O(α): ) N N α If let = e log( ) = o(α ): ) N N α Now we see that outing oblems with bit-evesal emutation and tansose emutation ae incedibly ae: fo 99% of all outing oblems at most C + O(1) ackets ass though any node duing the outing. 4

5 Let us conside two secial cases of the theoem. If = 1, we have a single N-acket outing oblem. With high obability, the maximum numbe of ackets that ass though any node is O(/ log ) with high obability. The second case is when = Θ(), and we have Θ(N ) ackets on an N - node buttefly. At most O() ackets ass though any node with high obability.. Bounds on unning time If two o moe ackets ae waiting to exit a node, we need to secify a otocol fo deciding which acket will exit the node fist. We will use a andom-ank otocol in such cases: assign a andom ioity key (P ) [1, K] to each acket P define a total ode on ackets: t(p ) is the ank of P define w(p ) = ((P ), t(p )). If P/ = P, we say that w(p < w(p ) if and only if (P ) < (P ), o (P ) = (P ) and t(p ) < t(p ). if thee is a collision, we choose the acket with minimal w Conside the outing oblem with congestion C. Let P 0 be the last acket to each its destination v 0 at time T. It was delayed at v 1, l 0 is the numbe of stes in ath v 1 v 0. P 0 was delayed duing the ste T l 0. Let P 1 be the acket esonsible fo delaying P 0. Next ecod the ath of P 1 fom the time it was last delayed befoe ste T l 0 until the ste T l 0. Let l 1 be the numbe of edges in this ath and v the node whee P 1 was last delayed at ste T l 0 l 1 1. We oceed to ecod the sequence of delays and emove eeated nodes. We get delay ath P = v 0 v 1... v s - a simle ath of length. An examle of the delay ath is given on figue 3. Each acket on the figue 3 consists of the destination (binay numbe),the name and the andom ank. It is obvious that T l 0 l 1 l s (s 1) = 1 and l l s =. Hence, T = s +. An active delay sequence consists of a delay ath P integes l 0 1, l 1 0,..., l s 1 0, l l s 1 = nodes v 0,v 1,...,v s : v i is the node of P on level l 0... l s 1 diffeent ackets P 0, P 1,..., P s : the geedy ath fo P i contains v i keys k 0, k 1,..., k s fo the ackets: k s k s 1... k 0, k i [0, K] and (P i ) = k i fo 0 i s. 5

6 Figue 3: Delay ath Thee exist lots of ossible delay sequences. The obability that (P i ) = k i fo 0 i s is K (s+1). Thee ae N ossible delay aths (they ae uniquely defined by endoints). Thee ae ( ) s+ s 1 choices fo l0,..., l s : thee is one-to-one coesondence between choices fo l i and (s + )-bit binay sting t with s 1 zeos, whee l i is the numbe of "1" between (i + 1)st and (i + )nd zeos in the sting 01t0 Since we fix P and l i s, the nodes v i ae detemined. Then thee ae at most C choices fo any P. Hence, thee ae at most C s+1 choices fo all ackets. We also have ( s+k s+1) ways to choose k 0,..., k s such that k s k s 1 k 0 and k i [1, K]: thee is one-to-one coesondence between choices fo k i and (s + K)-bit binay sting u with s + 1 zeos, whee k i is the numbe of "1" to the left of the (s + 1 i)th zeo in the sting 1u. Put it all togethe: the obability that thee is an active delay sequence with s + 1 ackets is at most ( ) ( ) s + s + K N C s+1 K (s+1) s 1 s + 1 We can show that if K = s + 1 = 8eC, and C, this obability is at most ( ) K 4eC N 3 N 3 4e = o(n 7 ), K and if K = s + 1 = 8e / log ( ) C, and C, this obability is at most ( ) K 4eC N 3 o(n 1 ) K Ou esult is that with high obability thee is no active delay ath with s + 1 ackets, whee 6

7 8eC, if C s + 1 = ( ) 8e / log, if C C Since T = + s, we get O(C) +, if C T = ( ) O(/ log ), if C C This fact comletes the analysis. 3 Conclusion "Tyical" outing oblem in actice ae not at all the same as "tyical" outing oblems in a mathematical sense: while the latte ae likely to have a easonable unning time, the fome have vey bad estimation of unning time. 4 Bibliogahy 1 F. T. Leighton. Intoduction to Paallel Algoithms and Achitectues: Aays, Tees, Hyecubes. Mogan Kaufmann Publ., 199 Fiedhelm Meye auf de Heide. Kommunikation in Paallelen Rechenmodellen. 7

Data Structures and Algorithm Analysis (CSC317) Randomized algorithms (part 2)

Data Structures and Algorithm Analysis (CSC317) Randomized algorithms (part 2) Data Stuctues and Algoithm Analysis (CSC317) Randomized algoithms (at 2) Hiing oblem - eview c Cost to inteview (low i ) Cost to fie/hie (exensive ) n Total numbe candidates m Total numbe hied c h O(c

More information

Analysis of Arithmetic. Analysis of Arithmetic. Analysis of Arithmetic Round-Off Errors. Analysis of Arithmetic. Analysis of Arithmetic

Analysis of Arithmetic. Analysis of Arithmetic. Analysis of Arithmetic Round-Off Errors. Analysis of Arithmetic. Analysis of Arithmetic In the fixed-oint imlementation of a digital filte only the esult of the multilication oeation is quantied The eesentation of a actical multilie with the quantie at its outut is shown below u v Q ^v The

More information

Approximating the minimum independent dominating set in perturbed graphs

Approximating the minimum independent dominating set in perturbed graphs Aoximating the minimum indeendent dominating set in etubed gahs Weitian Tong, Randy Goebel, Guohui Lin, Novembe 3, 013 Abstact We investigate the minimum indeendent dominating set in etubed gahs gg, )

More information

c( 1) c(0) c(1) Note z 1 represents a unit interval delay Figure 85 3 Transmit equalizer functional model

c( 1) c(0) c(1) Note z 1 represents a unit interval delay Figure 85 3 Transmit equalizer functional model Relace 85.8.3.2 with the following: 85.8.3.2 Tansmitted outut wavefom The 40GBASE-CR4 and 100GBASE-CR10 tansmit function includes ogammable equalization to comensate fo the fequency-deendent loss of the

More information

HOW TO TEACH THE FUNDAMENTALS OF INFORMATION SCIENCE, CODING, DECODING AND NUMBER SYSTEMS?

HOW TO TEACH THE FUNDAMENTALS OF INFORMATION SCIENCE, CODING, DECODING AND NUMBER SYSTEMS? 6th INTERNATIONAL MULTIDISCIPLINARY CONFERENCE HOW TO TEACH THE FUNDAMENTALS OF INFORMATION SCIENCE, CODING, DECODING AND NUMBER SYSTEMS? Cecília Sitkuné Göömbei College of Nyíegyháza Hungay Abstact: The

More information

Analysis of Finite Word-Length Effects

Analysis of Finite Word-Length Effects T-6.46 Digital Signal Pocessing and Filteing 8.9.4 Intoduction Analysis of Finite Wod-Length Effects Finite wodlength effects ae caused by: Quantization of the filte coefficients ounding / tuncation of

More information

A Bijective Approach to the Permutational Power of a Priority Queue

A Bijective Approach to the Permutational Power of a Priority Queue A Bijective Appoach to the Pemutational Powe of a Pioity Queue Ia M. Gessel Kuang-Yeh Wang Depatment of Mathematics Bandeis Univesity Waltham, MA 02254-9110 Abstact A pioity queue tansfoms an input pemutation

More information

Math 301: The Erdős-Stone-Simonovitz Theorem and Extremal Numbers for Bipartite Graphs

Math 301: The Erdős-Stone-Simonovitz Theorem and Extremal Numbers for Bipartite Graphs Math 30: The Edős-Stone-Simonovitz Theoem and Extemal Numbes fo Bipatite Gaphs May Radcliffe The Edős-Stone-Simonovitz Theoem Recall, in class we poved Tuán s Gaph Theoem, namely Theoem Tuán s Theoem Let

More information

Lot-sizing for inventory systems with product recovery

Lot-sizing for inventory systems with product recovery Lot-sizing fo inventoy systems with oduct ecovey Ruud Teunte August 29, 2003 Econometic Institute Reot EI2003-28 Abstact We study inventoy systems with oduct ecovey. Recoveed items ae as-good-as-new and

More information

New problems in universal algebraic geometry illustrated by boolean equations

New problems in universal algebraic geometry illustrated by boolean equations New poblems in univesal algebaic geomety illustated by boolean equations axiv:1611.00152v2 [math.ra] 25 Nov 2016 Atem N. Shevlyakov Novembe 28, 2016 Abstact We discuss new poblems in univesal algebaic

More information

The Substring Search Problem

The Substring Search Problem The Substing Seach Poblem One algoithm which is used in a vaiety of applications is the family of substing seach algoithms. These algoithms allow a use to detemine if, given two chaacte stings, one is

More information

Surveillance Points in High Dimensional Spaces

Surveillance Points in High Dimensional Spaces Société de Calcul Mathématique SA Tools fo decision help since 995 Suveillance Points in High Dimensional Spaces by Benad Beauzamy Januay 06 Abstact Let us conside any compute softwae, elying upon a lage

More information

15.081J/6.251J Introduction to Mathematical Programming. Lecture 6: The Simplex Method II

15.081J/6.251J Introduction to Mathematical Programming. Lecture 6: The Simplex Method II 15081J/6251J Intoduction to Mathematical Pogamming ectue 6: The Simplex Method II 1 Outline Revised Simplex method Slide 1 The full tableau implementation Anticycling 2 Revised Simplex Initial data: A,

More information

Introduction Common Divisors. Discrete Mathematics Andrei Bulatov

Introduction Common Divisors. Discrete Mathematics Andrei Bulatov Intoduction Common Divisos Discete Mathematics Andei Bulatov Discete Mathematics Common Divisos 3- Pevious Lectue Integes Division, popeties of divisibility The division algoithm Repesentation of numbes

More information

C/CS/Phys C191 Shor s order (period) finding algorithm and factoring 11/12/14 Fall 2014 Lecture 22

C/CS/Phys C191 Shor s order (period) finding algorithm and factoring 11/12/14 Fall 2014 Lecture 22 C/CS/Phys C9 Sho s ode (peiod) finding algoithm and factoing /2/4 Fall 204 Lectue 22 With a fast algoithm fo the uantum Fouie Tansfom in hand, it is clea that many useful applications should be possible.

More information

6 Matrix Concentration Bounds

6 Matrix Concentration Bounds 6 Matix Concentation Bounds Concentation bounds ae inequalities that bound pobabilities of deviations by a andom vaiable fom some value, often its mean. Infomally, they show the pobability that a andom

More information

Recognizable Infinite Triangular Array Languages

Recognizable Infinite Triangular Array Languages IOSR Jounal of Mathematics (IOSR-JM) e-issn: 2278-5728, -ISSN: 2319-765X. Volume 14, Issue 1 Ve. I (Jan. - Feb. 2018), PP 01-10 www.iosjounals.og Recognizable Infinite iangula Aay Languages 1 V.Devi Rajaselvi,

More information

AQI: Advanced Quantum Information Lecture 2 (Module 4): Order finding and factoring algorithms February 20, 2013

AQI: Advanced Quantum Information Lecture 2 (Module 4): Order finding and factoring algorithms February 20, 2013 AQI: Advanced Quantum Infomation Lectue 2 (Module 4): Ode finding and factoing algoithms Febuay 20, 203 Lectue: D. Mak Tame (email: m.tame@impeial.ac.uk) Intoduction In the last lectue we looked at the

More information

H5 Gas meter calibration

H5 Gas meter calibration H5 Gas mete calibation Calibation: detemination of the elation between the hysical aamete to be detemined and the signal of a measuement device. Duing the calibation ocess the measuement equiment is comaed

More information

Dorin Andrica Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca, Romania

Dorin Andrica Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca, Romania #A INTEGERS 5A (05) THE SIGNUM EQUATION FOR ERDŐS-SURÁNYI SEQUENCES Doin Andica Faculty of Mathematics and Comute Science, Babeş-Bolyai Univesity, Cluj-Naoca, Romania dandica@math.ubbcluj.o Eugen J. Ionascu

More information

Stanford University CS259Q: Quantum Computing Handout 8 Luca Trevisan October 18, 2012

Stanford University CS259Q: Quantum Computing Handout 8 Luca Trevisan October 18, 2012 Stanfod Univesity CS59Q: Quantum Computing Handout 8 Luca Tevisan Octobe 8, 0 Lectue 8 In which we use the quantum Fouie tansfom to solve the peiod-finding poblem. The Peiod Finding Poblem Let f : {0,...,

More information

QIP Course 10: Quantum Factorization Algorithm (Part 3)

QIP Course 10: Quantum Factorization Algorithm (Part 3) QIP Couse 10: Quantum Factoization Algoithm (Pat 3 Ryutaoh Matsumoto Nagoya Univesity, Japan Send you comments to yutaoh.matsumoto@nagoya-u.jp Septembe 2018 @ Tokyo Tech. Matsumoto (Nagoya U. QIP Couse

More information

Edge Cover Time for Regular Graphs

Edge Cover Time for Regular Graphs 1 2 3 47 6 23 11 Jounal of Intege Sequences, Vol. 11 (28, Aticle 8.4.4 Edge Cove Time fo Regula Gahs Robeto Tauaso Diatimento di Matematica Univesità di Roma To Vegata via della Riceca Scientifica 133

More information

Errata for Edition 1 of Coding the Matrix, January 13, 2017

Errata for Edition 1 of Coding the Matrix, January 13, 2017 Eata fo Edition of Coding the Matix, Januay 3, 07 You coy might not contain some of these eos. Most do not occu in the coies cuently being sold as Ail 05. Section 0.3:... the inut is a e-image of the inut...

More information

Quantum Fourier Transform

Quantum Fourier Transform Chapte 5 Quantum Fouie Tansfom Many poblems in physics and mathematics ae solved by tansfoming a poblem into some othe poblem with a known solution. Some notable examples ae Laplace tansfom, Legende tansfom,

More information

Topic 4a Introduction to Root Finding & Bracketing Methods

Topic 4a Introduction to Root Finding & Bracketing Methods /8/18 Couse Instucto D. Raymond C. Rumpf Office: A 337 Phone: (915) 747 6958 E Mail: cumpf@utep.edu Topic 4a Intoduction to Root Finding & Backeting Methods EE 4386/531 Computational Methods in EE Outline

More information

10/04/18. P [P(x)] 1 negl(n).

10/04/18. P [P(x)] 1 negl(n). Mastemath, Sping 208 Into to Lattice lgs & Cypto Lectue 0 0/04/8 Lectues: D. Dadush, L. Ducas Scibe: K. de Boe Intoduction In this lectue, we will teat two main pats. Duing the fist pat we continue the

More information

763620SS STATISTICAL PHYSICS Solutions 2 Autumn 2012

763620SS STATISTICAL PHYSICS Solutions 2 Autumn 2012 763620SS STATISTICAL PHYSICS Solutions 2 Autumn 2012 1. Continuous Random Walk Conside a continuous one-dimensional andom walk. Let w(s i ds i be the pobability that the length of the i th displacement

More information

6 PROBABILITY GENERATING FUNCTIONS

6 PROBABILITY GENERATING FUNCTIONS 6 PROBABILITY GENERATING FUNCTIONS Cetain deivations pesented in this couse have been somewhat heavy on algeba. Fo example, detemining the expectation of the Binomial distibution (page 5.1 tuned out to

More information

EM Boundary Value Problems

EM Boundary Value Problems EM Bounday Value Poblems 10/ 9 11/ By Ilekta chistidi & Lee, Seung-Hyun A. Geneal Desciption : Maxwell Equations & Loentz Foce We want to find the equations of motion of chaged paticles. The way to do

More information

INTRODUCTION. 2. Vectors in Physics 1

INTRODUCTION. 2. Vectors in Physics 1 INTRODUCTION Vectos ae used in physics to extend the study of motion fom one dimension to two dimensions Vectos ae indispensable when a physical quantity has a diection associated with it As an example,

More information

RATIONAL BASE NUMBER SYSTEMS FOR p-adic NUMBERS

RATIONAL BASE NUMBER SYSTEMS FOR p-adic NUMBERS RAIRO-Theo. Inf. Al. 46 (202) 87 06 DOI: 0.05/ita/204 Available online at: www.aio-ita.og RATIONAL BASE NUMBER SYSTEMS FOR -ADIC NUMBERS Chistiane Fougny and Kael Klouda 2 Abstact. This ae deals with ational

More information

A STUDY OF HAMMING CODES AS ERROR CORRECTING CODES

A STUDY OF HAMMING CODES AS ERROR CORRECTING CODES AGU Intenational Jounal of Science and Technology A STUDY OF HAMMING CODES AS ERROR CORRECTING CODES Ritu Ahuja Depatment of Mathematics Khalsa College fo Women, Civil Lines, Ludhiana-141001, Punjab, (India)

More information

k. s k=1 Part of the significance of the Riemann zeta-function stems from Theorem 9.2. If s > 1 then 1 p s

k. s k=1 Part of the significance of the Riemann zeta-function stems from Theorem 9.2. If s > 1 then 1 p s 9 Pimes in aithmetic ogession Definition 9 The Riemann zeta-function ζs) is the function which assigns to a eal numbe s > the convegent seies k s k Pat of the significance of the Riemann zeta-function

More information

arxiv: v2 [stat.ml] 25 Aug 2015

arxiv: v2 [stat.ml] 25 Aug 2015 A statistical esective on andomized sketching fo odinay least-squaes Gavesh Raskutti Michael Mahoney,3 Deatment of Statistics & Deatment of Comute Science, Univesity of Wisconsin Madison Intenational Comute

More information

Exploration of the three-person duel

Exploration of the three-person duel Exploation of the thee-peson duel Andy Paish 15 August 2006 1 The duel Pictue a duel: two shootes facing one anothe, taking tuns fiing at one anothe, each with a fixed pobability of hitting his opponent.

More information

Kepler s problem gravitational attraction

Kepler s problem gravitational attraction Kele s oblem gavitational attaction Summay of fomulas deived fo two-body motion Let the two masses be m and m. The total mass is M = m + m, the educed mass is µ = m m /(m + m ). The gavitational otential

More information

4/18/2005. Statistical Learning Theory

4/18/2005. Statistical Learning Theory Statistical Leaning Theoy Statistical Leaning Theoy A model of supevised leaning consists of: a Envionment - Supplying a vecto x with a fixed but unknown pdf F x (x b Teache. It povides a desied esponse

More information

ME 3600 Control Systems Frequency Domain Analysis

ME 3600 Control Systems Frequency Domain Analysis ME 3600 Contol Systems Fequency Domain Analysis The fequency esponse of a system is defined as the steady-state esponse of the system to a sinusoidal (hamonic) input. Fo linea systems, the esulting steady-state

More information

CSCE 478/878 Lecture 4: Experimental Design and Analysis. Stephen Scott. 3 Building a tree on the training set Introduction. Outline.

CSCE 478/878 Lecture 4: Experimental Design and Analysis. Stephen Scott. 3 Building a tree on the training set Introduction. Outline. In Homewok, you ae (supposedly) Choosing a data set 2 Extacting a test set of size > 3 3 Building a tee on the taining set 4 Testing on the test set 5 Repoting the accuacy (Adapted fom Ethem Alpaydin and

More information

On The Queuing System M/E r /1/N

On The Queuing System M/E r /1/N On The Queuing System M/E //N Namh A. Abid* Azmi. K. Al-Madi** Received 3, Mach, 2 Acceted 8, Octobe, 2 Abstact: In this ae the queuing system (M/E //N) has been consideed in equilibium. The method of

More information

Probablistically Checkable Proofs

Probablistically Checkable Proofs Lectue 12 Pobablistically Checkable Poofs May 13, 2004 Lectue: Paul Beame Notes: Chis Re 12.1 Pobablisitically Checkable Poofs Oveview We know that IP = PSPACE. This means thee is an inteactive potocol

More information

Maximal Inequalities for the Ornstein-Uhlenbeck Process

Maximal Inequalities for the Ornstein-Uhlenbeck Process Poc. Ame. Math. Soc. Vol. 28, No., 2, (335-34) Reseach Reot No. 393, 998, Det. Theoet. Statist. Aahus Maimal Ineualities fo the Onstein-Uhlenbeck Pocess S.. GRAVRSN 3 and G. PSKIR 3 Let V = (V t ) t be

More information

Encapsulation theory: radial encapsulation. Edmund Kirwan *

Encapsulation theory: radial encapsulation. Edmund Kirwan * Encapsulation theoy: adial encapsulation. Edmund Kiwan * www.edmundkiwan.com Abstact This pape intoduces the concept of adial encapsulation, wheeby dependencies ae constained to act fom subsets towads

More information

arxiv:math/ v2 [math.ag] 21 Sep 2005

arxiv:math/ v2 [math.ag] 21 Sep 2005 axiv:math/0509219v2 [math.ag] 21 Se 2005 POLYNOMIAL SYSTEMS SUPPORTED ON CIRCUITS AND DESSINS D ENFANTS FREDERIC BIHAN Abstact. We study olynomial systems whose equations have as common suot a set C of

More information

F-IF Logistic Growth Model, Abstract Version

F-IF Logistic Growth Model, Abstract Version F-IF Logistic Gowth Model, Abstact Vesion Alignments to Content Standads: F-IFB4 Task An impotant example of a model often used in biology o ecology to model population gowth is called the logistic gowth

More information

Having Fun with PageRank and MapReduce. Hadoop User Group (HUG) UK 14 th April Paolo Castagna HP Labs, Bristol, UK

Having Fun with PageRank and MapReduce. Hadoop User Group (HUG) UK 14 th April Paolo Castagna HP Labs, Bristol, UK Having Fun with PageRank and MaReduce Hadoo Use Gou (HUG) UK 4 th Ail 29 htt://huguk.og/ Paolo Castagna HP Labs, Bistol, UK PageRank 2 3 2 /3 3 /2 3 2? 2 3 2 PageRank i B i ecusive definition B i backwad

More information

Problem Set #10 Math 471 Real Analysis Assignment: Chapter 8 #2, 3, 6, 8

Problem Set #10 Math 471 Real Analysis Assignment: Chapter 8 #2, 3, 6, 8 Poblem Set #0 Math 47 Real Analysis Assignment: Chate 8 #2, 3, 6, 8 Clayton J. Lungstum Decembe, 202 xecise 8.2 Pove the convese of Hölde s inequality fo = and =. Show also that fo eal-valued f / L ),

More information

THE NUMBER OF TWO CONSECUTIVE SUCCESSES IN A HOPPE-PÓLYA URN

THE NUMBER OF TWO CONSECUTIVE SUCCESSES IN A HOPPE-PÓLYA URN TH NUMBR OF TWO CONSCUTIV SUCCSSS IN A HOPP-PÓLYA URN LARS HOLST Depatment of Mathematics, Royal Institute of Technology S 100 44 Stocholm, Sweden -mail: lholst@math.th.se Novembe 27, 2007 Abstact In a

More information

Merging to ordered sequences. Efficient (Parallel) Sorting. Merging (cont.)

Merging to ordered sequences. Efficient (Parallel) Sorting. Merging (cont.) Efficient (Paae) Soting One of the most fequent opeations pefomed by computes is oganising (soting) data The access to soted data is moe convenient/faste Thee is a constant need fo good soting agoithms

More information

MULTILAYER PERCEPTRONS

MULTILAYER PERCEPTRONS Last updated: Nov 26, 2012 MULTILAYER PERCEPTRONS Outline 2 Combining Linea Classifies Leaning Paametes Outline 3 Combining Linea Classifies Leaning Paametes Implementing Logical Relations 4 AND and OR

More information

Pascal s Triangle (mod 8)

Pascal s Triangle (mod 8) Euop. J. Combinatoics (998) 9, 45 62 Pascal s Tiangle (mod 8) JAMES G. HUARD, BLAIR K. SPEARMAN AND KENNETH S. WILLIAMS Lucas theoem gives a conguence fo a binomial coefficient modulo a pime. Davis and

More information

On the ratio of maximum and minimum degree in maximal intersecting families

On the ratio of maximum and minimum degree in maximal intersecting families On the atio of maximum and minimum degee in maximal intesecting families Zoltán Lóánt Nagy Lale Özkahya Balázs Patkós Máté Vize Septembe 5, 011 Abstact To study how balanced o unbalanced a maximal intesecting

More information

Convergence Dynamics of Resource-Homogeneous Congestion Games: Technical Report

Convergence Dynamics of Resource-Homogeneous Congestion Games: Technical Report 1 Convegence Dynamics of Resouce-Homogeneous Congestion Games: Technical Repot Richad Southwell and Jianwei Huang Abstact Many esouce shaing scenaios can be modeled using congestion games A nice popety

More information

9.1 The multiplicative group of a finite field. Theorem 9.1. The multiplicative group F of a finite field is cyclic.

9.1 The multiplicative group of a finite field. Theorem 9.1. The multiplicative group F of a finite field is cyclic. Chapte 9 Pimitive Roots 9.1 The multiplicative goup of a finite fld Theoem 9.1. The multiplicative goup F of a finite fld is cyclic. Remak: In paticula, if p is a pime then (Z/p) is cyclic. In fact, this

More information

Goodness-of-fit for composite hypotheses.

Goodness-of-fit for composite hypotheses. Section 11 Goodness-of-fit fo composite hypotheses. Example. Let us conside a Matlab example. Let us geneate 50 obsevations fom N(1, 2): X=nomnd(1,2,50,1); Then, unning a chi-squaed goodness-of-fit test

More information

Lab 10: Newton s Second Law in Rotation

Lab 10: Newton s Second Law in Rotation Lab 10: Newton s Second Law in Rotation We can descibe the motion of objects that otate (i.e. spin on an axis, like a popelle o a doo) using the same definitions, adapted fo otational motion, that we have

More information

Solutions to Problem Set 8

Solutions to Problem Set 8 Massachusetts Institute of Technology 6.042J/18.062J, Fall 05: Mathematics fo Compute Science Novembe 21 Pof. Albet R. Meye and Pof. Ronitt Rubinfeld evised Novembe 27, 2005, 858 minutes Solutions to Poblem

More information

Fractional Zero Forcing via Three-color Forcing Games

Fractional Zero Forcing via Three-color Forcing Games Factional Zeo Focing via Thee-colo Focing Games Leslie Hogben Kevin F. Palmowski David E. Robeson Michael Young May 13, 2015 Abstact An -fold analogue of the positive semidefinite zeo focing pocess that

More information

CMSC 425: Lecture 5 More on Geometry and Geometric Programming

CMSC 425: Lecture 5 More on Geometry and Geometric Programming CMSC 425: Lectue 5 Moe on Geomety and Geometic Pogamming Moe Geometic Pogamming: In this lectue we continue the discussion of basic geometic ogamming fom the eious lectue. We will discuss coodinate systems

More information

Multiple Criteria Secretary Problem: A New Approach

Multiple Criteria Secretary Problem: A New Approach J. Stat. Appl. Po. 3, o., 9-38 (04 9 Jounal of Statistics Applications & Pobability An Intenational Jounal http://dx.doi.og/0.785/jsap/0303 Multiple Citeia Secetay Poblem: A ew Appoach Alaka Padhye, and

More information

On the ratio of maximum and minimum degree in maximal intersecting families

On the ratio of maximum and minimum degree in maximal intersecting families On the atio of maximum and minimum degee in maximal intesecting families Zoltán Lóánt Nagy Lale Özkahya Balázs Patkós Máté Vize Mach 6, 013 Abstact To study how balanced o unbalanced a maximal intesecting

More information

Chapter Eight Notes N P U1C8S4-6

Chapter Eight Notes N P U1C8S4-6 Chapte Eight Notes N P UC8S-6 Name Peiod Section 8.: Tigonometic Identities An identit is, b definition, an equation that is alwas tue thoughout its domain. B tue thoughout its domain, that is to sa that

More information

Divisibility. c = bf = (ae)f = a(ef) EXAMPLE: Since 7 56 and , the Theorem above tells us that

Divisibility. c = bf = (ae)f = a(ef) EXAMPLE: Since 7 56 and , the Theorem above tells us that Divisibility DEFINITION: If a and b ae integes with a 0, we say that a divides b if thee is an intege c such that b = ac. If a divides b, we also say that a is a diviso o facto of b. NOTATION: d n means

More information

Notes on McCall s Model of Job Search. Timothy J. Kehoe March if job offer has been accepted. b if searching

Notes on McCall s Model of Job Search. Timothy J. Kehoe March if job offer has been accepted. b if searching Notes on McCall s Model of Job Seach Timothy J Kehoe Mach Fv ( ) pob( v), [, ] Choice: accept age offe o eceive b and seach again next peiod An unemployed oke solves hee max E t t y t y t if job offe has

More information

working pages for Paul Richards class notes; do not copy or circulate without permission from PGR 2004/11/3 10:50

working pages for Paul Richards class notes; do not copy or circulate without permission from PGR 2004/11/3 10:50 woking pages fo Paul Richads class notes; do not copy o ciculate without pemission fom PGR 2004/11/3 10:50 CHAPTER7 Solid angle, 3D integals, Gauss s Theoem, and a Delta Function We define the solid angle,

More information

Multiple Experts with Binary Features

Multiple Experts with Binary Features Multiple Expets with Binay Featues Ye Jin & Lingen Zhang Decembe 9, 2010 1 Intoduction Ou intuition fo the poect comes fom the pape Supevised Leaning fom Multiple Expets: Whom to tust when eveyone lies

More information

Encapsulation theory: the transformation equations of absolute information hiding.

Encapsulation theory: the transformation equations of absolute information hiding. 1 Encapsulation theoy: the tansfomation equations of absolute infomation hiding. Edmund Kiwan * www.edmundkiwan.com Abstact This pape descibes how the potential coupling of a set vaies as the set is tansfomed,

More information

Numerical Integration

Numerical Integration MCEN 473/573 Chapte 0 Numeical Integation Fall, 2006 Textbook, 0.4 and 0.5 Isopaametic Fomula Numeical Integation [] e [ ] T k = h B [ D][ B] e B Jdsdt In pactice, the element stiffness is calculated numeically.

More information

A Converse to Low-Rank Matrix Completion

A Converse to Low-Rank Matrix Completion A Convese to Low-Rank Matix Completion Daniel L. Pimentel-Alacón, Robet D. Nowak Univesity of Wisconsin-Madison Abstact In many pactical applications, one is given a subset Ω of the enties in a d N data

More information

Probabilistic number theory : A report on work done. What is the probability that a randomly chosen integer has no square factors?

Probabilistic number theory : A report on work done. What is the probability that a randomly chosen integer has no square factors? Pobabilistic numbe theoy : A eot on wo done What is the obability that a andomly chosen intege has no squae factos? We can constuct an initial fomula to give us this value as follows: If a numbe is to

More information

Pushdown Automata (PDAs)

Pushdown Automata (PDAs) CHAPTER 2 Context-Fee Languages Contents Context-Fee Gammas definitions, examples, designing, ambiguity, Chomsky nomal fom Pushdown Automata definitions, examples, euivalence with context-fee gammas Non-Context-Fee

More information

Momentum is conserved if no external force

Momentum is conserved if no external force Goals: Lectue 13 Chapte 9 v Employ consevation of momentum in 1 D & 2D v Examine foces ove time (aka Impulse) Chapte 10 v Undestand the elationship between motion and enegy Assignments: l HW5, due tomoow

More information

Nuclear and Particle Physics - Lecture 20 The shell model

Nuclear and Particle Physics - Lecture 20 The shell model 1 Intoduction Nuclea and Paticle Physics - Lectue 0 The shell model It is appaent that the semi-empiical mass fomula does a good job of descibing tends but not the non-smooth behaviou of the binding enegy.

More information

A Statistical Perspective on Randomized Sketching for Ordinary Least-Squares

A Statistical Perspective on Randomized Sketching for Ordinary Least-Squares Jounal of Machine Leaning Reseach 7 (206) -3 Submitted 8/5; Published /6 A Statistical Pesective on Randomized Sketching fo Odinay Least-Squaes Gavesh Raskutti Deatment of Statistics Univesity of Wisconsin

More information

arxiv: v1 [math.co] 1 Apr 2011

arxiv: v1 [math.co] 1 Apr 2011 Weight enumeation of codes fom finite spaces Relinde Juius Octobe 23, 2018 axiv:1104.0172v1 [math.co] 1 Ap 2011 Abstact We study the genealized and extended weight enumeato of the - ay Simplex code and

More information

Physics 2B Chapter 22 Notes - Magnetic Field Spring 2018

Physics 2B Chapter 22 Notes - Magnetic Field Spring 2018 Physics B Chapte Notes - Magnetic Field Sping 018 Magnetic Field fom a Long Staight Cuent-Caying Wie In Chapte 11 we looked at Isaac Newton s Law of Gavitation, which established that a gavitational field

More information

3.1 Random variables

3.1 Random variables 3 Chapte III Random Vaiables 3 Random vaiables A sample space S may be difficult to descibe if the elements of S ae not numbes discuss how we can use a ule by which an element s of S may be associated

More information

Suggested Solutions to Homework #4 Econ 511b (Part I), Spring 2004

Suggested Solutions to Homework #4 Econ 511b (Part I), Spring 2004 Suggested Solutions to Homewok #4 Econ 5b (Pat I), Sping 2004. Conside a neoclassical gowth model with valued leisue. The (epesentative) consume values steams of consumption and leisue accoding to P t=0

More information

Solution to HW 3, Ma 1a Fall 2016

Solution to HW 3, Ma 1a Fall 2016 Solution to HW 3, Ma a Fall 206 Section 2. Execise 2: Let C be a subset of the eal numbes consisting of those eal numbes x having the popety that evey digit in the decimal expansion of x is, 3, 5, o 7.

More information

Lecture 28: Convergence of Random Variables and Related Theorems

Lecture 28: Convergence of Random Variables and Related Theorems EE50: Pobability Foundations fo Electical Enginees July-Novembe 205 Lectue 28: Convegence of Random Vaiables and Related Theoems Lectue:. Kishna Jagannathan Scibe: Gopal, Sudhasan, Ajay, Swamy, Kolla An

More information

Fuzzy economic order quantity model with imperfect items, shortages and inspection errors

Fuzzy economic order quantity model with imperfect items, shortages and inspection errors Available online at www.sciencediect.com Sstems Engineeing Pocedia 4 (0) 8 89 The nd Intenational Confeence on Comleit Science & Infomation Engineeing Fuzz economic ode quantit model with imefect items,

More information

Auchmuty High School Mathematics Department Advanced Higher Notes Teacher Version

Auchmuty High School Mathematics Department Advanced Higher Notes Teacher Version The Binomial Theoem Factoials Auchmuty High School Mathematics Depatment The calculations,, 6 etc. often appea in mathematics. They ae called factoials and have been given the notation n!. e.g. 6! 6!!!!!

More information

Journal of Inequalities in Pure and Applied Mathematics

Journal of Inequalities in Pure and Applied Mathematics Jounal of Inequalities in Pue and Applied Mathematics COEFFICIENT INEQUALITY FOR A FUNCTION WHOSE DERIVATIVE HAS A POSITIVE REAL PART S. ABRAMOVICH, M. KLARIČIĆ BAKULA AND S. BANIĆ Depatment of Mathematics

More information

Unobserved Correlation in Ascending Auctions: Example And Extensions

Unobserved Correlation in Ascending Auctions: Example And Extensions Unobseved Coelation in Ascending Auctions: Example And Extensions Daniel Quint Univesity of Wisconsin Novembe 2009 Intoduction In pivate-value ascending auctions, the winning bidde s willingness to pay

More information

Section 11. Timescales Radiation transport in stars

Section 11. Timescales Radiation transport in stars Section 11 Timescales 11.1 Radiation tanspot in stas Deep inside stas the adiation eld is vey close to black body. Fo a black-body distibution the photon numbe density at tempeatue T is given by n = 2

More information

What molecular weight polymer is necessary to provide steric stabilization? = [1]

What molecular weight polymer is necessary to provide steric stabilization? = [1] 1/7 What molecula weight polyme is necessay to povide steic stabilization? The fist step is to estimate the thickness of adsobed polyme laye necessay fo steic stabilization. An appoximation is: 1 t A d

More information

Mathematisch-Naturwissenschaftliche Fakultät I Humboldt-Universität zu Berlin Institut für Physik Physikalisches Grundpraktikum.

Mathematisch-Naturwissenschaftliche Fakultät I Humboldt-Universität zu Berlin Institut für Physik Physikalisches Grundpraktikum. Mathematisch-Natuwissenschaftliche Fakultät I Humboldt-Univesität zu Belin Institut fü Physik Physikalisches Gundpaktikum Vesuchspotokoll Polaisation duch Reflexion (O11) duchgefüht am 10.11.2009 mit Vesuchspatne

More information

Multihop MIMO Relay Networks with ARQ

Multihop MIMO Relay Networks with ARQ Multihop MIMO Relay Netwoks with ARQ Yao Xie Deniz Gündüz Andea Goldsmith Depatment of Electical Engineeing Stanfod Univesity Stanfod CA Depatment of Electical Engineeing Pinceton Univesity Pinceton NJ

More information

COORDINATE TRANSFORMATIONS - THE JACOBIAN DETERMINANT

COORDINATE TRANSFORMATIONS - THE JACOBIAN DETERMINANT COORDINATE TRANSFORMATIONS - THE JACOBIAN DETERMINANT Link to: phsicspages home page. To leave a comment o epot an eo, please use the auilia blog. Refeence: d Inveno, Ra, Intoducing Einstein s Relativit

More information

LINEAR AND NONLINEAR ANALYSES OF A WIND-TUNNEL BALANCE

LINEAR AND NONLINEAR ANALYSES OF A WIND-TUNNEL BALANCE LINEAR AND NONLINEAR ANALYSES O A WIND-TUNNEL INTRODUCTION BALANCE R. Kakehabadi and R. D. Rhew NASA LaRC, Hampton, VA The NASA Langley Reseach Cente (LaRC) has been designing stain-gauge balances fo utilization

More information

BEST CONSTANTS FOR UNCENTERED MAXIMAL FUNCTIONS. Loukas Grafakos and Stephen Montgomery-Smith University of Missouri, Columbia

BEST CONSTANTS FOR UNCENTERED MAXIMAL FUNCTIONS. Loukas Grafakos and Stephen Montgomery-Smith University of Missouri, Columbia BEST CONSTANTS FOR UNCENTERED MAXIMAL FUNCTIONS Loukas Gafakos and Stehen Montgomey-Smith Univesity of Missoui, Columbia Abstact. We ecisely evaluate the oeato nom of the uncenteed Hady-Littlewood maximal

More information

Chapter 1: Introduction to Polar Coordinates

Chapter 1: Introduction to Polar Coordinates Habeman MTH Section III: ola Coodinates and Comple Numbes Chapte : Intoduction to ola Coodinates We ae all comfotable using ectangula (i.e., Catesian coodinates to descibe points on the plane. Fo eample,

More information

The geometric construction of Ewald sphere and Bragg condition:

The geometric construction of Ewald sphere and Bragg condition: The geometic constuction of Ewald sphee and Bagg condition: The constuction of Ewald sphee must be done such that the Bagg condition is satisfied. This can be done as follows: i) Daw a wave vecto k in

More information

Lecture 8 - Gauss s Law

Lecture 8 - Gauss s Law Lectue 8 - Gauss s Law A Puzzle... Example Calculate the potential enegy, pe ion, fo an infinite 1D ionic cystal with sepaation a; that is, a ow of equally spaced chages of magnitude e and altenating sign.

More information

ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION. 1. Introduction. 1 r r. r k for every set E A, E \ {0},

ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION. 1. Introduction. 1 r r. r k for every set E A, E \ {0}, ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION E. J. IONASCU and A. A. STANCU Abstact. We ae inteested in constucting concete independent events in puely atomic pobability

More information

MATH 415, WEEK 3: Parameter-Dependence and Bifurcations

MATH 415, WEEK 3: Parameter-Dependence and Bifurcations MATH 415, WEEK 3: Paamete-Dependence and Bifucations 1 A Note on Paamete Dependence We should pause to make a bief note about the ole played in the study of dynamical systems by the system s paametes.

More information

FZX: Personal Lecture Notes from Daniel W. Koon St. Lawrence University Physics Department CHAPTER 7

FZX: Personal Lecture Notes from Daniel W. Koon St. Lawrence University Physics Department CHAPTER 7 FZX: Pesonal Lectue Notes fom Daniel W. Koon St. Lawence Univesity Physics Depatment CHAPTER 7 Please epot any glitches, bugs o eos to the autho: dkoon at stlawu.edu. 7. Momentum and Impulse Impulse page

More information

Splay Trees Handout. Last time we discussed amortized analysis of data structures

Splay Trees Handout. Last time we discussed amortized analysis of data structures Spla Tees Handout Amotied Analsis Last time we discussed amotied analsis of data stuctues A wa of epessing that even though the wost-case pefomance of an opeation can be bad, the total pefomance of a sequence

More information

Sincere Voting and Information Aggregation with Voting Costs

Sincere Voting and Information Aggregation with Voting Costs Sincee Voting and Infomation Aggegation with Voting Costs Vijay Kishna y and John Mogan z August 007 Abstact We study the oeties of euilibium voting in two-altenative elections unde the majoity ule. Votes

More information