Chapter Finding Small Vertex Covers. Extending the Limits of Tractability. Coping With NP-Completeness. Vertex Cover

Size: px
Start display at page:

Download "Chapter Finding Small Vertex Covers. Extending the Limits of Tractability. Coping With NP-Completeness. Vertex Cover"

Transcription

1 Coping With NP-Compltnss Chaptr 0 Extning th Limits o Tractability Q. Suppos I n to solv an NP-complt problm. What shoul I o? A. Thory says you'r unlikly to in poly-tim algorithm. Must sacriic on o thr sir aturs. Solv problm to optimality. Solv problm in polynomial tim. Solv arbitrary instancs o th problm. This lctur. Solv som spcial cass o NP-complt problms that aris in practic. Slis by Kvin Wayn. 00 Parson-Aison Wsly. All rights rsrv. Vrtx Covr 0. Fining Small Vrtx Covrs VERTEX COVER: Givn a graph G = (V, E) an an intgr k, is thr a subst o vrtics S V such that S k, an or ach g (u, v) ithr u S, or v S, or both k = S = {, 6, 7, 0 9 0

2 Fining Small Vrtx Covrs Fining Small Vrtx Covrs Q. What i k is small? Brut orc. O(k nk+ ). Try all C(n, k) = O(n k ) substs o siz k. Taks O(k n) tim to chck whthr a subst is a vrtx covr. Goal. Limit xponntial pnncy on k,.g., to O( k k n). Ex. n =,000, k = 0. Brut. k nk+ = 0 inasibl. Bttr. k k n = 07 asibl. Rmark. I k is a constant, algorithm is poly-tim; i k is a small constant, thn it's also practical. Claim. Lt u-v b an g o G. G has a vrtx covr o siz k i at last on o G { u an G { v has a vrtx covr o siz k-. P. Suppos G has a vrtx covr S o siz k. S contains ithr u or v (or both). Assum it contains u. S { u is a vrtx covr o G { u. lt v an all incint gs P. Suppos S is a vrtx covr o G { u o siz k-. Thn S { u is a vrtx covr o G. Claim. I G has a vrtx covr o siz k, it has k(n-) gs. P. Each vrtx covrs at most n- gs. 6 Fining Small Vrtx Covrs: Algorithm Fining Small Vrtx Covrs: Rcursion Tr Claim. Th ollowing algorithm trmins i G has a vrtx covr o siz k in O( k kn) tim. boolan Vrtx-Covr(G, k) { i (G contains no gs) rturn tru i (G contains kn gs) rturn als lt (u, v) b any g o G a = Vrtx-Covr(G - {u, k-) b = Vrtx-Covr(G - {v, k-) rturn a or b $ c i k = 0 & T(n, k) " % cn i k = & ' T(n,k #)+ ckn i k > k k- ( T(n, k) " k ck n k- P. Corrctnss ollows rom prvious two claims. Thr ar k+ nos in th rcursion tr; ach invocation taks O(kn) tim. k- k- k - i k- k

3 Inpnnt St on Trs 0. Solving NP-Har Problms on Trs Inpnnt st on trs. Givn a tr, in a maximum carinality subst o nos such that no two shar an g. Fact. A tr on at last two nos has at last two la nos. gr = Ky obsrvation. I v is a la, thr xists a maximum siz inpnnt st containing v. u P. (xchang argumnt) Consir a max carinality inpnnt st S. I v S, w'r on. I u S an v S, thn S { v is inpnnt S not maximum. IF u S an v S, thn S { v { u is inpnnt. v 0 Inpnnt St on Trs: Gry Algorithm Wight Inpnnt St on Trs Thorm. Th ollowing gry algorithm ins a maximum carinality inpnnt st in orsts (an hnc trs). Inpnnt-St-In-A-Forst(F) { S φ whil (F has at last on g) { Lt = (u, v) b an g such that v is a la A v to S Dlt rom F nos u an v, an all gs incint to thm. rturn S P. Corrctnss ollows rom th prvious ky obsrvation. Rmark. Can implmnt in O(n) tim by consiring nos in postorr. Wight inpnnt st on trs. Givn a tr an no wights w v > 0, in an inpnnt st S that maximizs Σ v S w v. Obsrvation. I (u, v) is an g such that v is a la no, thn ithr OPT inclus u, or it inclus all la nos incint to u. Dynamic programming solution. Root tr at som no, say r. OPT in (u) = max wight inpnnt st o subtr root at u, containing u. OPT out (u) = max wight inpnnt st o subtr root at u, not containing u. OPT in (u) = w u + # OPT out (v) v " chilrn(u) OPT out (u) = # max { OPT in (v), OPT out (v) v " chilrn(u) v r u w x chilrn(u) = { v, w, x

4 Wight Inpnnt St on Trs: Dynamic Programming Algorithm Contxt Thorm. Th ynamic programming algorithm ins a maximum wight inpnnt st in a tr in O(n) tim. Wight-Inpnnt-St-In-A-Tr(T) { Root th tr at a no r orach (no u o T in postorr) { i (u is a la) { M in [u] = w u M out [u] = 0 ls { M in [u] = w u + Σ v chilrn(u) M out [v] M out [u] = Σ v chilrn(u) max(m in [v], M out [v]) rturn max(m in [r], M out [r]) nsurs a no is visit atr all its chilrn P. Taks O(n) tim sinc w visit nos in postorr an xamin ach g xactly onc. can also in inpnnt st itsl (not just valu) Inpnnt st on trs. This structur spcial cas is tractabl bcaus w can in a no that braks th communication among th subproblms in irnt subtrs. u s Chaptr 0., but proc with caution Graphs o boun tr with. Elgant gnralization o trs that: Capturs a rich class o graphs that aris in practic. Enabls composition into inpnnt pics. u Wavlngth-Division Multiplxing 0. Circular Arc Coloring Wavlngth-ivision multiplxing (WDM). Allows m communication strams (arcs) to shar a portion o a ibr optic cabl, provi thy ar transmitt using irnt wavlngths. Ring topology. Spcial cas is whn ntwork is a cycl on n nos. Ba nws. NP-complt, vn on rings. Brut orc. Can trmin i k colors suic in O(k m ) tim by trying all k-colorings. c b a Goal. O((k)) poly(m, n) on rings. n =, m = 6 6

5 Wavlngth-Division Multiplxing Rviw: Intrval Coloring Wavlngth-ivision multiplxing (WDM). Allows m communication strams (arcs) to shar a portion o a ibr optic cabl, provi thy ar transmitt using irnt wavlngths. Intrval coloring. Gry algorithm ins coloring such that numbr o colors quals pth o schul. maximum numbr o strams at on location Ring topology. Spcial cas is whn ntwork is a cycl on n nos. c j Ba nws. NP-complt, vn on rings. a b g h i Brut orc. Can trmin i k colors suic in O(k m ) tim by trying all k-colorings. Goal. O((k)) poly(m, n) on rings. c b a Circular arc coloring. Wak uality: numbr o colors pth. Strong uality os not hol. n =, m = 6 max pth = min colors = 7 8 (Almost) Transorming Circular Arc Coloring to Intrval Coloring Circular Arc Coloring: Dynamic Programming Algorithm Circular arc coloring. Givn a st o n arcs with pth k, can th arcs b color with k colors? Equivalnt problm. Cut th ntwork btwn nos v an v n. Th arcs can b color with k colors i th intrvals can b color with k colors in such a way that "slic" arcs hav th sam color. Dynamic programming algorithm. Assign istinct color to ach intrval which bgins at cut no v 0. At ach no v i, som intrvals may inish, an othrs may bgin. Enumrat all k-colorings o th intrvals through v i that ar consistnt with th colorings o th intrvals through v i-. Th arcs ar k-colorabl i som coloring o intrvals ning at cut no v 0 is consistnt with original coloring o th sam intrvals. ys v 0 v colors o a', b', an c' must corrspon to colors o a", b", an c" c' b' b' a" b" v v a' c" c" v v 0 v v v v v 0 v 0 v v v v v 0 9 0

6 Circular Arc Coloring: Running Tim Running tim. O(k! n). n phass o th algorithm. Bottlnck in ach phas is numrating all consistnt colorings. Thr ar at most k intrvals through v i, so thr ar at most k! colorings to consir. Extra Slis Rmark. This algorithm is practical or small valus o k (say k = 0) vn i th numbr o nos n (or paths) is larg. Vrtx Covr Vrtx Covr in Bipartit Graphs Vrtx covr. Givn an unirct graph G = (V, E), a vrtx covr is a subst o vrtics S V such that or ach g (u, v) E, ithr u S or v S or both. ' ' ' S = {,,, ', ' S = ' '

7 Vrtx Covr Vrtx Covr: König-Egrváry Thorm Wak uality. Lt M b a matching, an lt S b a vrtx covr. Thn, M S. König-Egrváry Thorm. In a bipartit graph, th max carinality o a matching is qual to th min carinality o a vrtx covr. P. Each vrtx can covr at most on g in any matching. ' ' S* = {, ', ', ' S* = ' ' ' M = -', -', -' M = ' M* = -', -', -', -' M* = ' ' ' ' 6 Vrtx Covr: Proo o König-Egrváry Thorm Vrtx Covr: Proo o König-Egrváry Thorm König-Egrváry Thorm. In a bipartit graph, th max carinality o a matching is qual to th min carinality o a vrtx covr. Suics to in matching M an covr S such that M = S. Formulat max low problm as or bipartit matching. Lt M b max carinality matching an lt (A, B) b min cut. König-Egrváry Thorm. In a bipartit graph, th max carinality o a matching is qual to th min carinality o a vrtx covr. Suics to in matching M an covr S such that M = S. Formulat max low problm as or bipartit matching. Lt M b max carinality matching an lt (A, B) b min cut. Din L A = L A, L B = L B, R A = R A, R B = R B. ' Claim. S = L B R A is a vrtx covr. consir (u, v) E ' u L A, v R B impossibl sinc ininit capacity thus, ithr u L B or v R A or both s ' t Claim. S = M. max-low min-cut thorm M = cap(a, B) ' only gs o orm (s, u) or (v, t) contribut to cap(a, B) M = cap(a, B) = L B + R A = S. ' 7 8

8 Rgistr Allocation Rgistr Allocation Rgistr. On o k o high-sp mmory locations in computr's CPU. say Rgistr allocator. Part o an optimizing compilr that controls which variabls ar sav in th rgistrs as compil program xcuts. variabls or tmporaris Intrrnc graph. Nos ar "liv rangs." Eg u-v i thr xists an opration whr both u an v ar "liv" at th sam tim. Obsrvation. [Chaitin, 98] Can solv rgistr allocation problm i intrrnc graph is k-colorabl. Spilling. I graph is not k-colorabl (or w can't in a k-coloring), w "spill" crtain variabls to main mmory an swap back as n. typically inrquntly us variabls that ar not in innr loops 0 A Usul Proprty Chaitin's Algorithm Rmark. Rgistr allocation problm is NP-har. Ky act. I a no v in graph G has wr than k nighbors, G is k-colorabl i G { v is k-colorabl. lt v an all incint gs P. Dlt no v rom G an color G { v. I G { v is not k-colorabl, thn nithr is G. I G { v is k-colorabl, thn thr is at last on rmaining color lt or v. Vrtx-Color(G, k) { say, no with wst nighbors whil (G is not mpty) { Pick a no v with wr than k nighbors Push v on stack Dlt v an all its incint gs whil (stack is not mpty) { Pop nxt no v rom th stack Assign v a color irnt rom its nighboring nos which hav alray bn color v k = k = G is -colorabl vn though all nos hav gr

9 Chaitin's Algorithm Thorm. [Kmp 879, Chaitin 98] Chaitin's algorithm proucs a k-coloring o any graph with max gr k-. P. Follows rom ky act sinc ach no has wr than k nighbors. algorithm succs in k-coloring many graphs with max gr k Rmark. I algorithm nvr ncountrs a graph whr all nos hav gr k, thn it proucs a k-coloring. Practic. Chaitin's algorithm (an variants) ar xtrmly ctiv an wily us in ral compilrs or rgistr allocation.

10. EXTENDING TRACTABILITY

10. EXTENDING TRACTABILITY Coping with NP-compltnss 0. EXTENDING TRACTABILITY ining small vrtx covrs solving NP-har problms on trs circular arc covrings vrtx covr in bipartit graphs Q. Suppos I n to solv an NP-complt problm. What

More information

Examples and applications on SSSP and MST

Examples and applications on SSSP and MST Exampls an applications on SSSP an MST Dan (Doris) H & Junhao Gan ITEE Univrsity of Qunslan COMP3506/7505, Uni of Qunslan Exampls an applications on SSSP an MST Dijkstra s Algorithm Th algorithm solvs

More information

Analysis of Algorithms - Elementary graphs algorithms -

Analysis of Algorithms - Elementary graphs algorithms - Analysis of Algorithms - Elmntary graphs algorithms - Anras Ermahl MRTC (Mälaralns Ral-Tim Rsarch Cntr) anras.rmahl@mh.s Autumn 004 Graphs Graphs ar important mathmatical ntitis in computr scinc an nginring

More information

Homework #3. 1 x. dx. It therefore follows that a sum of the

Homework #3. 1 x. dx. It therefore follows that a sum of the Danil Cannon CS 62 / Luan March 5, 2009 Homwork # 1. Th natural logarithm is dfind by ln n = n 1 dx. It thrfor follows that a sum of th 1 x sam addnd ovr th sam intrval should b both asymptotically uppr-

More information

Analysis of Algorithms - Elementary graphs algorithms -

Analysis of Algorithms - Elementary graphs algorithms - Analysis of Algorithms - Elmntary graphs algorithms - Anras Ermahl MRTC (Mälaralns Ral-Tim Rsach Cntr) anras.rmahl@mh.s Autumn 00 Graphs Graphs ar important mathmatical ntitis in computr scinc an nginring

More information

Basic Polyhedral theory

Basic Polyhedral theory Basic Polyhdral thory Th st P = { A b} is calld a polyhdron. Lmma 1. Eithr th systm A = b, b 0, 0 has a solution or thr is a vctorπ such that π A 0, πb < 0 Thr cass, if solution in top row dos not ist

More information

Week 3: Connected Subgraphs

Week 3: Connected Subgraphs Wk 3: Connctd Subgraphs Sptmbr 19, 2016 1 Connctd Graphs Path, Distanc: A path from a vrtx x to a vrtx y in a graph G is rfrrd to an xy-path. Lt X, Y V (G). An (X, Y )-path is an xy-path with x X and y

More information

CPSC 665 : An Algorithmist s Toolkit Lecture 4 : 21 Jan Linear Programming

CPSC 665 : An Algorithmist s Toolkit Lecture 4 : 21 Jan Linear Programming CPSC 665 : An Algorithmist s Toolkit Lctur 4 : 21 Jan 2015 Lcturr: Sushant Sachdva Linar Programming Scrib: Rasmus Kyng 1. Introduction An optimization problm rquirs us to find th minimum or maximum) of

More information

The Equitable Dominating Graph

The Equitable Dominating Graph Intrnational Journal of Enginring Rsarch and Tchnology. ISSN 0974-3154 Volum 8, Numbr 1 (015), pp. 35-4 Intrnational Rsarch Publication Hous http://www.irphous.com Th Equitabl Dominating Graph P.N. Vinay

More information

cycle that does not cross any edges (including its own), then it has at least

cycle that does not cross any edges (including its own), then it has at least W prov th following thorm: Thorm If a K n is drawn in th plan in such a way that it has a hamiltonian cycl that dos not cross any dgs (including its own, thn it has at last n ( 4 48 π + O(n crossings Th

More information

SOME PARAMETERS ON EQUITABLE COLORING OF PRISM AND CIRCULANT GRAPH.

SOME PARAMETERS ON EQUITABLE COLORING OF PRISM AND CIRCULANT GRAPH. SOME PARAMETERS ON EQUITABLE COLORING OF PRISM AND CIRCULANT GRAPH. K VASUDEVAN, K. SWATHY AND K. MANIKANDAN 1 Dpartmnt of Mathmatics, Prsidncy Collg, Chnnai-05, India. E-Mail:vasu k dvan@yahoo.com. 2,

More information

Combinatorial Networks Week 1, March 11-12

Combinatorial Networks Week 1, March 11-12 1 Nots on March 11 Combinatorial Ntwors W 1, March 11-1 11 Th Pigonhol Principl Th Pigonhol Principl If n objcts ar placd in hols, whr n >, thr xists a box with mor than on objcts 11 Thorm Givn a simpl

More information

CS 491 G Combinatorial Optimization

CS 491 G Combinatorial Optimization CS 49 G Cobinatorial Optiization Lctur Nots Junhui Jia. Maiu Flow Probls Now lt us iscuss or tails on aiu low probls. Thor. A asibl low is aiu i an only i thr is no -augnting path. Proo: Lt P = A asibl

More information

Y 0. Standing Wave Interference between the incident & reflected waves Standing wave. A string with one end fixed on a wall

Y 0. Standing Wave Interference between the incident & reflected waves Standing wave. A string with one end fixed on a wall Staning Wav Intrfrnc btwn th incint & rflct wavs Staning wav A string with on n fix on a wall Incint: y, t) Y cos( t ) 1( Y 1 ( ) Y (St th incint wav s phas to b, i.., Y + ral & positiv.) Rflct: y, t)

More information

Thickness and Colorability of Geometric Graphs

Thickness and Colorability of Geometric Graphs Thicknss an Colorability o Gomtric Graphs Stphan Durochr 1 Dpartmnt o Computr Scinc, Univrsity o Manitoba, Winnipg, Canaa Elln Gthnr Dpartmnt o Computr Scinc, Univrsity o Colorao Dnvr, Colorao, USA Dbajyoti

More information

Searching Linked Lists. Perfect Skip List. Building a Skip List. Skip List Analysis (1) Assume the list is sorted, but is stored in a linked list.

Searching Linked Lists. Perfect Skip List. Building a Skip List. Skip List Analysis (1) Assume the list is sorted, but is stored in a linked list. 3 3 4 8 6 3 3 4 8 6 3 3 4 8 6 () (d) 3 Sarching Linkd Lists Sarching Linkd Lists Sarching Linkd Lists ssum th list is sortd, but is stord in a linkd list. an w us binary sarch? omparisons? Work? What if

More information

First derivative analysis

First derivative analysis Robrto s Nots on Dirntial Calculus Chaptr 8: Graphical analysis Sction First drivativ analysis What you nd to know alrady: How to us drivativs to idntiy th critical valus o a unction and its trm points

More information

CS553 Lecture Register Allocation I 3

CS553 Lecture Register Allocation I 3 Low-Lvl Issus Last ltur Intrproural analysis Toay Start low-lvl issus Rgistr alloation Latr Mor rgistr alloation Instrution shuling CS553 Ltur Rgistr Alloation I 2 Rgistr Alloation Prolm Assign an unoun

More information

1 Minimum Cut Problem

1 Minimum Cut Problem CS 6 Lctur 6 Min Cut and argr s Algorithm Scribs: Png Hui How (05), Virginia Dat: May 4, 06 Minimum Cut Problm Today, w introduc th minimum cut problm. This problm has many motivations, on of which coms

More information

4.5 Minimum Spanning Tree. Chapter 4. Greedy Algorithms. Minimum Spanning Tree. Motivating application

4.5 Minimum Spanning Tree. Chapter 4. Greedy Algorithms. Minimum Spanning Tree. Motivating application 1 Chaptr. Minimum panning Tr lids by Kvin Wayn. Copyright 200 Parson-Addison Wsly. All rights rsrvd. *Adjustd by Gang Tan for C33: Algorithms at Boston Collg, Fall 0 Motivating application Minimum panning

More information

Minimum Spanning Trees

Minimum Spanning Trees Yufi Tao ITEE Univrsity of Qunslan In tis lctur, w will stuy anotr classic prolm: finin a minimum spannin tr of an unirct wit rap. Intrstinly, vn tou t prolm appars ratr iffrnt from SSSP (sinl sourc sortst

More information

Recall that by Theorems 10.3 and 10.4 together provide us the estimate o(n2 ), S(q) q 9, q=1

Recall that by Theorems 10.3 and 10.4 together provide us the estimate o(n2 ), S(q) q 9, q=1 Chaptr 11 Th singular sris Rcall that by Thorms 10 and 104 togthr provid us th stimat 9 4 n 2 111 Rn = SnΓ 2 + on2, whr th singular sris Sn was dfind in Chaptr 10 as Sn = q=1 Sq q 9, with Sq = 1 a q gcda,q=1

More information

The second condition says that a node α of the tree has exactly n children if the arity of its label is n.

The second condition says that a node α of the tree has exactly n children if the arity of its label is n. CS 6110 S14 Hanout 2 Proof of Conflunc 27 January 2014 In this supplmntary lctur w prov that th λ-calculus is conflunt. This is rsult is u to lonzo Church (1903 1995) an J. arkly Rossr (1907 1989) an is

More information

Section 6.1. Question: 2. Let H be a subgroup of a group G. Then H operates on G by left multiplication. Describe the orbits for this operation.

Section 6.1. Question: 2. Let H be a subgroup of a group G. Then H operates on G by left multiplication. Describe the orbits for this operation. MAT 444 H Barclo Spring 004 Homwork 6 Solutions Sction 6 Lt H b a subgroup of a group G Thn H oprats on G by lft multiplication Dscrib th orbits for this opration Th orbits of G ar th right costs of H

More information

Strongly Connected Components

Strongly Connected Components Strongly Connctd Componnts Lt G = (V, E) b a dirctd graph Writ if thr is a path from to in G Writ if and is an quivalnc rlation: implis and implis s quivalnc classs ar calld th strongly connctd componnts

More information

UNTYPED LAMBDA CALCULUS (II)

UNTYPED LAMBDA CALCULUS (II) 1 UNTYPED LAMBDA CALCULUS (II) RECALL: CALL-BY-VALUE O.S. Basic rul Sarch ruls: (\x.) v [v/x] 1 1 1 1 v v CALL-BY-VALUE EVALUATION EXAMPLE (\x. x x) (\y. y) x x [\y. y / x] = (\y. y) (\y. y) y [\y. y /

More information

CS 580: Algorithm Design and Analysis

CS 580: Algorithm Design and Analysis CS 580: Algorithm Design and Analysis Jeremiah Blocki Purdue University Spring 2018 Homework 5 due tonight at 11:59 PM (on Blackboard) Midterm 2 on April 4 th at 8PM (MATH 175) Practice Midterm Released

More information

Propositional Logic. Combinatorial Problem Solving (CPS) Albert Oliveras Enric Rodríguez-Carbonell. May 17, 2018

Propositional Logic. Combinatorial Problem Solving (CPS) Albert Oliveras Enric Rodríguez-Carbonell. May 17, 2018 Propositional Logic Combinatorial Problm Solving (CPS) Albrt Olivras Enric Rodríguz-Carbonll May 17, 2018 Ovrviw of th sssion Dfinition of Propositional Logic Gnral Concpts in Logic Rduction to SAT CNFs

More information

4.5 Minimum Spanning Tree. Chapter 4. Greedy Algorithms. Minimum Spanning Tree. Applications

4.5 Minimum Spanning Tree. Chapter 4. Greedy Algorithms. Minimum Spanning Tree. Applications Chaptr. Minimum panning Tr Grdy Algorithms lids by Kvin Wayn. Copyright 200 Parson-Addison Wsly. All rights rsrvd. Minimum panning Tr Applications Minimum spanning tr. Givn a connctd graph G = (V, E) with

More information

priority queue ADT heaps 1

priority queue ADT heaps 1 COMP 250 Lctur 23 priority quu ADT haps 1 Nov. 1/2, 2017 1 Priority Quu Li a quu, but now w hav a mor gnral dinition o which lmnt to rmov nxt, namly th on with highst priority..g. hospital mrgncy room

More information

ANALYSIS IN THE FREQUENCY DOMAIN

ANALYSIS IN THE FREQUENCY DOMAIN ANALYSIS IN THE FREQUENCY DOMAIN SPECTRAL DENSITY Dfinition Th spctral dnsit of a S.S.P. t also calld th spctrum of t is dfind as: + { γ }. jτ γ τ F τ τ In othr words, of th covarianc function. is dfind

More information

3) Use the average steady-state equation to determine the dose. Note that only 100 mg tablets of aminophylline are available here.

3) Use the average steady-state equation to determine the dose. Note that only 100 mg tablets of aminophylline are available here. PHA 5127 Dsigning A Dosing Rgimn Answrs provi by Jry Stark Mr. JM is to b start on aminophyllin or th tratmnt o asthma. H is a non-smokr an wighs 60 kg. Dsign an oral osing rgimn or this patint such that

More information

CSI35 Chapter 11 Review

CSI35 Chapter 11 Review 1. Which of th grphs r trs? c f c g f c x y f z p q r 1 1. Which of th grphs r trs? c f c g f c x y f z p q r . Answr th qustions out th following tr 1) Which vrtx is th root of c th tr? ) wht is th hight

More information

Derangements and Applications

Derangements and Applications 2 3 47 6 23 Journal of Intgr Squncs, Vol. 6 (2003), Articl 03..2 Drangmnts and Applications Mhdi Hassani Dpartmnt of Mathmatics Institut for Advancd Studis in Basic Scincs Zanjan, Iran mhassani@iasbs.ac.ir

More information

COUNTING TAMELY RAMIFIED EXTENSIONS OF LOCAL FIELDS UP TO ISOMORPHISM

COUNTING TAMELY RAMIFIED EXTENSIONS OF LOCAL FIELDS UP TO ISOMORPHISM COUNTING TAMELY RAMIFIED EXTENSIONS OF LOCAL FIELDS UP TO ISOMORPHISM Jim Brown Dpartmnt of Mathmatical Scincs, Clmson Univrsity, Clmson, SC 9634, USA jimlb@g.clmson.du Robrt Cass Dpartmnt of Mathmatics,

More information

On the irreducibility of some polynomials in two variables

On the irreducibility of some polynomials in two variables ACTA ARITHMETICA LXXXII.3 (1997) On th irrducibility of som polynomials in two variabls by B. Brindza and Á. Pintér (Dbrcn) To th mmory of Paul Erdős Lt f(x) and g(y ) b polynomials with intgral cofficints

More information

MSLC Math 151 WI09 Exam 2 Review Solutions

MSLC Math 151 WI09 Exam 2 Review Solutions Eam Rviw Solutions. Comput th following rivativs using th iffrntiation ruls: a.) cot cot cot csc cot cos 5 cos 5 cos 5 cos 5 sin 5 5 b.) c.) sin( ) sin( ) y sin( ) ln( y) ln( ) ln( y) sin( ) ln( ) y y

More information

LR(0) Analysis. LR(0) Analysis

LR(0) Analysis. LR(0) Analysis LR() Analysis LR() Conlicts: Introuction Whn constructing th LR() analysis tal scri in th prvious stps, it has not n possil to gt a trministic analysr, caus thr ar svral possil actions in th sam cll. I

More information

Superposition. Thinning

Superposition. Thinning Suprposition STAT253/317 Wintr 213 Lctur 11 Yibi Huang Fbruary 1, 213 5.3 Th Poisson Procsss 5.4 Gnralizations of th Poisson Procsss Th sum of two indpndnt Poisson procsss with rspctiv rats λ 1 and λ 2,

More information

On spanning trees and cycles of multicolored point sets with few intersections

On spanning trees and cycles of multicolored point sets with few intersections On spanning trs and cycls of multicolord point sts with fw intrsctions M. Kano, C. Mrino, and J. Urrutia April, 00 Abstract Lt P 1,..., P k b a collction of disjoint point sts in R in gnral position. W

More information

First order differential equation Linear equation; Method of integrating factors

First order differential equation Linear equation; Method of integrating factors First orr iffrntial quation Linar quation; Mtho of intgrating factors Exampl 1: Rwrit th lft han si as th rivativ of th prouct of y an som function by prouct rul irctly. Solving th first orr iffrntial

More information

CSE 373: More on graphs; DFS and BFS. Michael Lee Wednesday, Feb 14, 2018

CSE 373: More on graphs; DFS and BFS. Michael Lee Wednesday, Feb 14, 2018 CSE 373: Mor on grphs; DFS n BFS Mihl L Wnsy, F 14, 2018 1 Wrmup Wrmup: Disuss with your nighor: Rmin your nighor: wht is simpl grph? Suppos w hv simpl, irt grph with x nos. Wht is th mximum numr of gs

More information

Coupled Pendulums. Two normal modes.

Coupled Pendulums. Two normal modes. Tim Dpndnt Two Stat Problm Coupld Pndulums Wak spring Two normal mods. No friction. No air rsistanc. Prfct Spring Start Swinging Som tim latr - swings with full amplitud. stationary M +n L M +m Elctron

More information

PROBLEM SET Problem 1.

PROBLEM SET Problem 1. PROLEM SET 1 PROFESSOR PETER JOHNSTONE 1. Problm 1. 1.1. Th catgory Mat L. OK, I m not amiliar with th trminology o partially orr sts, so lt s go ovr that irst. Dinition 1.1. partial orr is a binary rlation

More information

Computing and Communications -- Network Coding

Computing and Communications -- Network Coding 89 90 98 00 Computing and Communications -- Ntwork Coding Dr. Zhiyong Chn Institut of Wirlss Communications Tchnology Shanghai Jiao Tong Univrsity China Lctur 5- Nov. 05 0 Classical Information Thory Sourc

More information

Where k is either given or determined from the data and c is an arbitrary constant.

Where k is either given or determined from the data and c is an arbitrary constant. Exponntial growth and dcay applications W wish to solv an quation that has a drivativ. dy ky k > dx This quation says that th rat of chang of th function is proportional to th function. Th solution is

More information

Solutions to Homework 5

Solutions to Homework 5 Solutions to Homwork 5 Pro. Silvia Frnánz Disrt Mathmatis Math 53A, Fall 2008. [3.4 #] (a) Thr ar x olor hois or vrtx an x or ah o th othr thr vrtis. So th hromati polynomial is P (G, x) =x (x ) 3. ()

More information

The Matrix Exponential

The Matrix Exponential Th Matrix Exponntial (with xrciss) by D. Klain Vrsion 207.0.05 Corrctions and commnts ar wlcom. Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix A A k I + A + k!

More information

64. A Conic Section from Five Elements.

64. A Conic Section from Five Elements. . onic Sction from Fiv Elmnts. To raw a conic sction of which fiv lmnts - points an tangnts - ar known. W consir th thr cass:. Fiv points ar known.. Four points an a tangnt lin ar known.. Thr points an

More information

EXST Regression Techniques Page 1

EXST Regression Techniques Page 1 EXST704 - Rgrssion Tchniqus Pag 1 Masurmnt rrors in X W hav assumd that all variation is in Y. Masurmnt rror in this variabl will not ffct th rsults, as long as thy ar uncorrlatd and unbiasd, sinc thy

More information

Square of Hamilton cycle in a random graph

Square of Hamilton cycle in a random graph Squar of Hamilton cycl in a random graph Andrzj Dudk Alan Friz Jun 28, 2016 Abstract W show that p = n is a sharp thrshold for th random graph G n,p to contain th squar of a Hamilton cycl. This improvs

More information

Addition of angular momentum

Addition of angular momentum Addition of angular momntum April, 0 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat th

More information

SCHUR S THEOREM REU SUMMER 2005

SCHUR S THEOREM REU SUMMER 2005 SCHUR S THEOREM REU SUMMER 2005 1. Combinatorial aroach Prhas th first rsult in th subjct blongs to I. Schur and dats back to 1916. On of his motivation was to study th local vrsion of th famous quation

More information

Cycles and Simple Cycles. Paths and Simple Paths. Trees. Problem: There is No Completely Standard Terminology!

Cycles and Simple Cycles. Paths and Simple Paths. Trees. Problem: There is No Completely Standard Terminology! Outlin Computr Sin 331, Spnnin, n Surphs Mik Joson Dprtmnt o Computr Sin Univrsity o Clry Ltur #30 1 Introution 2 3 Dinition 4 Spnnin 5 6 Mik Joson (Univrsity o Clry) Computr Sin 331 Ltur #30 1 / 20 Mik

More information

Figure 1: Closed surface, surface with boundary, or not a surface?

Figure 1: Closed surface, surface with boundary, or not a surface? QUESTION 1 (10 marks) Two o th topological spacs shown in Figur 1 ar closd suracs, two ar suracs with boundary, and two ar not suracs. Dtrmin which is which. You ar not rquird to justiy your answr, but,

More information

Higher order derivatives

Higher order derivatives Robrto s Nots on Diffrntial Calculus Chaptr 4: Basic diffrntiation ruls Sction 7 Highr ordr drivativs What you nd to know alrady: Basic diffrntiation ruls. What you can larn hr: How to rpat th procss of

More information

The Matrix Exponential

The Matrix Exponential Th Matrix Exponntial (with xrciss) by Dan Klain Vrsion 28928 Corrctions and commnts ar wlcom Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix () A A k I + A + k!

More information

From Elimination to Belief Propagation

From Elimination to Belief Propagation School of omputr Scinc Th lif Propagation (Sum-Product lgorithm Probabilistic Graphical Modls (10-708 Lctur 5, Sp 31, 2007 Rcptor Kinas Rcptor Kinas Kinas X 5 ric Xing Gn G T X 6 X 7 Gn H X 8 Rading: J-hap

More information

Introduction to Arithmetic Geometry Fall 2013 Lecture #20 11/14/2013

Introduction to Arithmetic Geometry Fall 2013 Lecture #20 11/14/2013 18.782 Introduction to Arithmtic Gomtry Fall 2013 Lctur #20 11/14/2013 20.1 Dgr thorm for morphisms of curvs Lt us rstat th thorm givn at th nd of th last lctur, which w will now prov. Thorm 20.1. Lt φ:

More information

Hardy-Littlewood Conjecture and Exceptional real Zero. JinHua Fei. ChangLing Company of Electronic Technology Baoji Shannxi P.R.

Hardy-Littlewood Conjecture and Exceptional real Zero. JinHua Fei. ChangLing Company of Electronic Technology Baoji Shannxi P.R. Hardy-Littlwood Conjctur and Excptional ral Zro JinHua Fi ChangLing Company of Elctronic Tchnology Baoji Shannxi P.R.China E-mail: fijinhuayoujian@msn.com Abstract. In this papr, w assum that Hardy-Littlwood

More information

Abstract Interpretation: concrete and abstract semantics

Abstract Interpretation: concrete and abstract semantics Abstract Intrprtation: concrt and abstract smantics Concrt smantics W considr a vry tiny languag that manags arithmtic oprations on intgrs valus. Th (concrt) smantics of th languags cab b dfind by th funzcion

More information

BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES. 1. Statement of results

BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES. 1. Statement of results BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES DONALD M. DAVIS Abstract. If p is a prim and n a positiv intgr, lt ν p (n dnot th xponnt of p in n, and u p (n n/p νp(n th unit part of n. If α

More information

Function Spaces. a x 3. (Letting x = 1 =)) a(0) + b + c (1) = 0. Row reducing the matrix. b 1. e 4 3. e 9. >: (x = 1 =)) a(0) + b + c (1) = 0

Function Spaces. a x 3. (Letting x = 1 =)) a(0) + b + c (1) = 0. Row reducing the matrix. b 1. e 4 3. e 9. >: (x = 1 =)) a(0) + b + c (1) = 0 unction Spacs Prrquisit: Sction 4.7, Coordinatization n this sction, w apply th tchniqus of Chaptr 4 to vctor spacs whos lmnts ar functions. Th vctor spacs P n and P ar familiar xampls of such spacs. Othr

More information

Kernels. ffl A kernel K is a function of two objects, for example, two sentence/tree pairs (x1; y1) and (x2; y2)

Kernels. ffl A kernel K is a function of two objects, for example, two sentence/tree pairs (x1; y1) and (x2; y2) Krnls krnl K is a function of two ojcts, for xampl, two sntnc/tr pairs (x1; y1) an (x2; y2) K((x1; y1); (x2; y2)) Intuition: K((x1; y1); (x2; y2)) is a masur of th similarity (x1; y1) twn (x2; y2) an ormally:

More information

Addition of angular momentum

Addition of angular momentum Addition of angular momntum April, 07 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat

More information

Quasi-Classical States of the Simple Harmonic Oscillator

Quasi-Classical States of the Simple Harmonic Oscillator Quasi-Classical Stats of th Simpl Harmonic Oscillator (Draft Vrsion) Introduction: Why Look for Eignstats of th Annihilation Oprator? Excpt for th ground stat, th corrspondnc btwn th quantum nrgy ignstats

More information

Weak Unit Disk and Interval Representation of Graphs

Weak Unit Disk and Interval Representation of Graphs Wak Unit Disk and Intrval Rprsntation o Graphs M. J. Alam, S. G. Kobourov, S. Pupyrv, and J. Toniskottr Dpartmnt o Computr Scinc, Univrsity o Arizona, Tucson, USA Abstract. W study a variant o intrsction

More information

(Upside-Down o Direct Rotation) β - Numbers

(Upside-Down o Direct Rotation) β - Numbers Amrican Journal of Mathmatics and Statistics 014, 4(): 58-64 DOI: 10593/jajms0140400 (Upsid-Down o Dirct Rotation) β - Numbrs Ammar Sddiq Mahmood 1, Shukriyah Sabir Ali,* 1 Dpartmnt of Mathmatics, Collg

More information

Bifurcation Theory. , a stationary point, depends on the value of α. At certain values

Bifurcation Theory. , a stationary point, depends on the value of α. At certain values Dnamic Macroconomic Thor Prof. Thomas Lux Bifurcation Thor Bifurcation: qualitativ chang in th natur of th solution occurs if a paramtr passs through a critical point bifurcation or branch valu. Local

More information

Chapter 10. The singular integral Introducing S(n) and J(n)

Chapter 10. The singular integral Introducing S(n) and J(n) Chaptr Th singular intgral Our aim in this chaptr is to rplac th functions S (n) and J (n) by mor convnint xprssions; ths will b calld th singular sris S(n) and th singular intgral J(n). This will b don

More information

Supplementary Materials

Supplementary Materials 6 Supplmntary Matrials APPENDIX A PHYSICAL INTERPRETATION OF FUEL-RATE-SPEED FUNCTION A truck running on a road with grad/slop θ positiv if moving up and ngativ if moving down facs thr rsistancs: arodynamic

More information

a g f 8 e 11 Also: Minimum degree, maximum degree, vertex of degree d 1 adjacent to vertex of degree d 2,...

a g f 8 e 11 Also: Minimum degree, maximum degree, vertex of degree d 1 adjacent to vertex of degree d 2,... Warmup: Lt b 2 c 3 d 1 12 6 4 5 10 9 7 a 8 11 (i) Vriy tat G is connctd by ivin an xampl o a walk rom vrtx a to ac o t vrtics b. (ii) Wat is t sortst pat rom a to c? to? (iii) Wat is t lonst pat rom a

More information

Basic Logic Review. Rules. Lecture Roadmap Combinational Logic. Textbook References. Basic Logic Gates (2-input versions)

Basic Logic Review. Rules. Lecture Roadmap Combinational Logic. Textbook References. Basic Logic Gates (2-input versions) Lctur Roadmap ombinational Logic EE 55 Digital Systm Dsign with VHDL Lctur Digital Logic Rrshr Part ombinational Logic Building Blocks Basic Logic Rviw Basic Gats D Morgan s Law ombinational Logic Building

More information

Finding low cost TSP and 2-matching solutions using certain half integer subtour vertices

Finding low cost TSP and 2-matching solutions using certain half integer subtour vertices Finding low cost TSP and 2-matching solutions using crtain half intgr subtour vrtics Sylvia Boyd and Robrt Carr Novmbr 996 Introduction Givn th complt graph K n = (V, E) on n nods with dg costs c R E,

More information

Deift/Zhou Steepest descent, Part I

Deift/Zhou Steepest descent, Part I Lctur 9 Dift/Zhou Stpst dscnt, Part I W now focus on th cas of orthogonal polynomials for th wight w(x) = NV (x), V (x) = t x2 2 + x4 4. Sinc th wight dpnds on th paramtr N N w will writ π n,n, a n,n,

More information

y = 2xe x + x 2 e x at (0, 3). solution: Since y is implicitly related to x we have to use implicit differentiation: 3 6y = 0 y = 1 2 x ln(b) ln(b)

y = 2xe x + x 2 e x at (0, 3). solution: Since y is implicitly related to x we have to use implicit differentiation: 3 6y = 0 y = 1 2 x ln(b) ln(b) 4. y = y = + 5. Find th quation of th tangnt lin for th function y = ( + ) 3 whn = 0. solution: First not that whn = 0, y = (1 + 1) 3 = 8, so th lin gos through (0, 8) and thrfor its y-intrcpt is 8. y

More information

Random Access Techniques: ALOHA (cont.)

Random Access Techniques: ALOHA (cont.) Random Accss Tchniqus: ALOHA (cont.) 1 Exampl [ Aloha avoiding collision ] A pur ALOHA ntwork transmits a 200-bit fram on a shard channl Of 200 kbps at tim. What is th rquirmnt to mak this fram collision

More information

A Propagating Wave Packet Group Velocity Dispersion

A Propagating Wave Packet Group Velocity Dispersion Lctur 8 Phys 375 A Propagating Wav Packt Group Vlocity Disprsion Ovrviw and Motivation: In th last lctur w lookd at a localizd solution t) to th 1D fr-particl Schrödingr quation (SE) that corrsponds to

More information

General Caching Is Hard: Even with Small Pages

General Caching Is Hard: Even with Small Pages Algorithmica manuscript No. (will b insrtd by th ditor) Gnral Caching Is Hard: Evn with Small Pags Luká² Folwarczný Ji í Sgall August 1, 2016 Abstract Caching (also known as paging) is a classical problm

More information

Final Exam Solutions

Final Exam Solutions CS 2 Advancd Data Structurs and Algorithms Final Exam Solutions Jonathan Turnr /8/20. (0 points) Suppos that r is a root of som tr in a Fionacci hap. Assum that just for a dltmin opration, r has no childrn

More information

Text: WMM, Chapter 5. Sections , ,

Text: WMM, Chapter 5. Sections , , Lcturs 6 - Continuous Probabilit Distributions Tt: WMM, Chaptr 5. Sctions 6.-6.4, 6.6-6.8, 7.-7. In th prvious sction, w introduc som of th common probabilit distribution functions (PDFs) for discrt sampl

More information

ON RIGHT(LEFT) DUO PO-SEMIGROUPS. S. K. Lee and K. Y. Park

ON RIGHT(LEFT) DUO PO-SEMIGROUPS. S. K. Lee and K. Y. Park Kangwon-Kyungki Math. Jour. 11 (2003), No. 2, pp. 147 153 ON RIGHT(LEFT) DUO PO-SEMIGROUPS S. K. L and K. Y. Park Abstract. W invstigat som proprtis on right(rsp. lft) duo po-smigroups. 1. Introduction

More information

Linked-List Implementation. Linked-lists for two sets. Multiple Operations. UNION Implementation. An Application of Disjoint-Set 1/9/2014

Linked-List Implementation. Linked-lists for two sets. Multiple Operations. UNION Implementation. An Application of Disjoint-Set 1/9/2014 Disjoint Sts Data Strutur (Chap. 21) A disjoint-st is a olltion ={S 1, S 2,, S k } o distint dynami sts. Eah st is idntiid by a mmbr o th st, alld rprsntativ. Disjoint st oprations: MAKE-SET(x): rat a

More information

CSE 373. Graphs 1: Concepts, Depth/Breadth-First Search reading: Weiss Ch. 9. slides created by Marty Stepp

CSE 373. Graphs 1: Concepts, Depth/Breadth-First Search reading: Weiss Ch. 9. slides created by Marty Stepp CSE 373 Grphs 1: Conpts, Dpth/Brth-First Srh ring: Wiss Ch. 9 slis rt y Mrty Stpp http://www.s.wshington.u/373/ Univrsity o Wshington, ll rights rsrv. 1 Wht is grph? 56 Tokyo Sttl Soul 128 16 30 181 140

More information

b. How many ternary words of length 23 with eight 0 s, nine 1 s and six 2 s?

b. How many ternary words of length 23 with eight 0 s, nine 1 s and six 2 s? MATH 3012 Finl Exm, My 4, 2006, WTT Stunt Nm n ID Numr 1. All our prts o this prolm r onrn with trnry strings o lngth n, i.., wors o lngth n with lttrs rom th lpht {0, 1, 2}.. How mny trnry wors o lngth

More information

EEO 401 Digital Signal Processing Prof. Mark Fowler

EEO 401 Digital Signal Processing Prof. Mark Fowler EEO 401 Digital Signal Procssing Prof. Mark Fowlr Dtails of th ot St #19 Rading Assignmnt: Sct. 7.1.2, 7.1.3, & 7.2 of Proakis & Manolakis Dfinition of th So Givn signal data points x[n] for n = 0,, -1

More information

Random Process Part 1

Random Process Part 1 Random Procss Part A random procss t (, ζ is a signal or wavform in tim. t : tim ζ : outcom in th sampl spac Each tim w rapat th xprimnt, a nw wavform is gnratd. ( W will adopt t for short. Tim sampls

More information

Differentiation of Exponential Functions

Differentiation of Exponential Functions Calculus Modul C Diffrntiation of Eponntial Functions Copyright This publication Th Northrn Albrta Institut of Tchnology 007. All Rights Rsrvd. LAST REVISED March, 009 Introduction to Diffrntiation of

More information

Roadmap. XML Indexing. DataGuide example. DataGuides. Strong DataGuides. Multiple DataGuides for same data. CPS Topics in Database Systems

Roadmap. XML Indexing. DataGuide example. DataGuides. Strong DataGuides. Multiple DataGuides for same data. CPS Topics in Database Systems Roadmap XML Indxing CPS 296.1 Topics in Databas Systms Indx fabric Coopr t al. A Fast Indx for Smistructurd Data. VLDB, 2001 DataGuid Goldman and Widom. DataGuids: Enabling Qury Formulation and Optimization

More information

Aim To manage files and directories using Linux commands. 1. file Examines the type of the given file or directory

Aim To manage files and directories using Linux commands. 1. file Examines the type of the given file or directory m E x. N o. 3 F I L E M A N A G E M E N T Aim To manag ils and dirctoris using Linux commands. I. F i l M a n a g m n t 1. il Examins th typ o th givn il or dirctory i l i l n a m > ( o r ) < d i r c t

More information

Multiple Short Term Infusion Homework # 5 PHA 5127

Multiple Short Term Infusion Homework # 5 PHA 5127 Multipl Short rm Infusion Homwork # 5 PHA 527 A rug is aministr as a short trm infusion. h avrag pharmacokintic paramtrs for this rug ar: k 0.40 hr - V 28 L his rug follows a on-compartmnt boy mol. A 300

More information

Solution of Assignment #2

Solution of Assignment #2 olution of Assignmnt #2 Instructor: Alirza imchi Qustion #: For simplicity, assum that th distribution function of T is continuous. Th distribution function of R is: F R ( r = P( R r = P( log ( T r = P(log

More information

Outlines: Graphs Part-4. Applications of Depth-First Search. Directed Acyclic Graph (DAG) Generic scheduling problem.

Outlines: Graphs Part-4. Applications of Depth-First Search. Directed Acyclic Graph (DAG) Generic scheduling problem. Outlins: Graps Part-4 Applications o DFS Elmntary Grap Aloritms Topoloical Sort o Dirctd Acyclic Grap Stronly Connctd Componnts PART-4 1 2 Applications o Dpt-First Sarc Topoloical Sort: Usin dpt-irst sarc

More information

International Journal of Scientific & Engineering Research, Volume 6, Issue 7, July ISSN

International Journal of Scientific & Engineering Research, Volume 6, Issue 7, July ISSN Intrnational Journal of Scintific & Enginring Rsarch, Volum 6, Issu 7, July-25 64 ISSN 2229-558 HARATERISTIS OF EDGE UTSET MATRIX OF PETERSON GRAPH WITH ALGEBRAI GRAPH THEORY Dr. G. Nirmala M. Murugan

More information

Indeterminate Forms and L Hôpital s Rule. Indeterminate Forms

Indeterminate Forms and L Hôpital s Rule. Indeterminate Forms SECTION 87 Intrminat Forms an L Hôpital s Rul 567 Sction 87 Intrminat Forms an L Hôpital s Rul Rcogniz its that prouc intrminat forms Apply L Hôpital s Rul to valuat a it Intrminat Forms Rcall from Chaptrs

More information

MATHEMATICS (B) 2 log (D) ( 1) = where z =

MATHEMATICS (B) 2 log (D) ( 1) = where z = MATHEMATICS SECTION- I STRAIGHT OBJECTIVE TYPE This sction contains 9 multipl choic qustions numbrd to 9. Each qustion has choic (A), (B), (C) and (D), out of which ONLY-ONE is corrct. Lt I d + +, J +

More information

The graph of y = x (or y = ) consists of two branches, As x 0, y + ; as x 0, y +. x = 0 is the

The graph of y = x (or y = ) consists of two branches, As x 0, y + ; as x 0, y +. x = 0 is the Copyright itutcom 005 Fr download & print from wwwitutcom Do not rproduc by othr mans Functions and graphs Powr functions Th graph of n y, for n Q (st of rational numbrs) y is a straight lin through th

More information

Mathematics 1110H Calculus I: Limits, derivatives, and Integrals Trent University, Summer 2018 Solutions to the Actual Final Examination

Mathematics 1110H Calculus I: Limits, derivatives, and Integrals Trent University, Summer 2018 Solutions to the Actual Final Examination Mathmatics H Calculus I: Limits, rivativs, an Intgrals Trnt Univrsity, Summr 8 Solutions to th Actual Final Eamination Tim-spac: 9:-: in FPHL 7. Brought to you by Stfan B lan k. Instructions: Do parts

More information

Economics 201b Spring 2010 Solutions to Problem Set 3 John Zhu

Economics 201b Spring 2010 Solutions to Problem Set 3 John Zhu Economics 20b Spring 200 Solutions to Problm St 3 John Zhu. Not in th 200 vrsion of Profssor Andrson s ctur 4 Nots, th charactrization of th firm in a Robinson Cruso conomy is that it maximizs profit ovr

More information

Paths. Connectivity. Euler and Hamilton Paths. Planar graphs.

Paths. Connectivity. Euler and Hamilton Paths. Planar graphs. Pths.. Eulr n Hmilton Pths.. Pth D. A pth rom s to t is squn o gs {x 0, x 1 }, {x 1, x 2 },... {x n 1, x n }, whr x 0 = s, n x n = t. D. Th lngth o pth is th numr o gs in it. {, } {, } {, } {, } {, } {,

More information