# Solutions for HW9. Bipartite: put the red vertices in V 1 and the black in V 2. Not bipartite!

Save this PDF as:

Size: px
Start display at page:

Download "Solutions for HW9. Bipartite: put the red vertices in V 1 and the black in V 2. Not bipartite!"

## Transcription

1 Solutions for HW9 Exerise 28. () Drw C 6, W 6 K 6, n K 5,3. C 6 : W 6 : K 6 : K 5,3 : () Whih of the following re iprtite? Justify your nswer. Biprtite: put the re verties in V 1 n the lk in V 2. Biprtite: put the re verties in V 1 n the lk in V 2. Not iprtite! Consier the three verties olore re. For the ske of ontrition, ssume tht it is iprtite. Pik ny one of them to e in V 1. Tht woul fore the other two to e in V 2. But they re jent, whih is ontrition. () Hyperues re iprtite.

2 (i) The following is the 4-ue: She in the verties tht hve n even numer of 0 s. Explin why the 4-ue is iprtite. None of the she verties re pirwise jent. None of the non-she verties re pirwise jent. So put ll the she verties in V 1 n ll the rest in V 2 to see tht Q 4 is iprtite. (ii) Explin why Q n is iprtite in generl. [Hint: onsier the prity of the numer of 0 s in the lel of vertex.] Any two verties with n even numer of 0 s iffer in t lest two its, n so re non-jent. Similrly, ny two verties with n o numer of 0 s iffer in t lest two its, n so re non-jent. So let V 1 = { verties with n even numer of 0 s } n V 2 = { verties with n o numer of 0 s }.

3 / Grphs In Exerises fin the jeny mtrix of the given irete multigrph with respet to the verties liste in lpheti orer. Exerise () For eh of the following pirs, list their egree sequenes. Then re they isomorphi?. If not, u 1 u 3 why? If yes, give n isomorphism. 35. u 2 v 1 (i) (ii) v 4 v u 3 u 2 u 1 v 4 eg seq: 2,2,2,1,1 eg seq: 2,2,2,2,2 Isomorphi: In Exerises rw the grph represente y the given jeny mtrix. u 1 u 1 u 2 Isomorphi: u 1 v 1, u 2, u 3 v 4, u 17 v 1, u 2 vu 3, u , v 3, Is every zero one squre mtrix tht is symmetri n (iii) hs zeros on the igonl the jeny mtrix of simple grph? u 1 u 26. Use n iniene 2 mtrix to represent the grphs in Exerises 1 n 2. v Use n iniene mtrix to represent the grphs in Exerises Wht is theusum 3 of the entries in row of the jeny mtrix for n unirete grph? Forv 4 irete grph? 29. Wht is the sum of the entries in olumn of the jeny mtrix for n unirete grph? For irete grph? Wht is the sum of the entries in row of the iniene Right eg seq: 4,3,3,2,2. mtrix for n unirete grph? 31. Wht is the sum of the entries in olumn of the iniene mtrix for n unirete grph? 32. Fin n jeny mtrix for eh of these grphs. ) K n ) C n ) W n ) K m,n e) Q n 33. Fin iniene mtries for the grphs in prts () () of (v) Exerise 32. In Exerises etermine whether the given pir of grphs u is isomorphi. Exhiit 2 v n isomorphism or provie 2 rigorous rgument tht none exists. u 1 u 3 v v 1 (iv) u 3 v u 2 v 1 u 3 v 4 v v 1 u 6 u1 u u 1 u 2 u u 1 v 1 Left eg seq: 3,3,3,3,2; Deg seq: 4,4,3,3,2 40. u 6 Isomorphi: Not isomorphi: ifferent eg seq s. u 3, u 2, u 2 v 6 u u 15 v 1, 3 v 4 v 4 u 1 u 2 u 6 u 3 v 7 v 6 v 4 v 1 v 4 v 1 v 6 u 1 u 2 u 3 4 v 4 v 4 Deg seq: 3,3,2,2,2,2 Not isomorphi: Right hs 3-yle; Left oesn t.

4 () How mny isomorphism lsses re there for simple grphs with 4 verties? Drw them. () How mny eges oes grph hve if its egree sequene is 4, 3, 3, 2, 2? Drw grph with this egree sequene. Cn you rw simple grph with this sequene? By the hnshke lemm, So there re 7 eges. sequene: 2 E = = 14. Here is n isomorphism lss of simple grphs tht hs tht egree () For whih vlues of n, m re these grphs regulr? Wht is the egree? (i) K n (ii) C n (iii) W n (iv) Q n (v) K m,n (i) K n : Regulr for ll n, of egree n 1. (ii) C n : Regulr for ll n, of egree 2. (iii) W n : Regulr only for n = 3, of egree 3. (iv) Q n : Regulr for ll n, of egree n. (v) K m,n : Regulr for n = m, n. (e) How mny verties oes regulr grph of egree four with 10 eges hve? By the hnshke theorem, so V = = V 4 (f) Show tht every non-inresing finite sequene of nonnegtive integers whose terms sum to n even numer is the egree sequene of grph (where loops re llowe). Illustrte your proof on the egree sequene 7,7,6,4,3,2,2,1,0,0. [Hint: A loops first.] For egree sequene 1, 2,..., n, rw one vertex v i for eh egree i, n tth i /2 loops tthe to v i. Then for eh i for whih i is even, v i so fr h egree i. For the i for whih i is o, v i urrently hs egree i 1. Sine the terms sum to n even numer, there must e n even numer of i for whih i is o; pir these i s up: the first with the seon, the thir with the fourth, n so on. Now rw n ege etween ll suh pire verties. The resulting grph hs the pproprite egree sequene. (g) Show tht isomorphism of simple grphs is n equivlene reltion.

5 () Reflexive: the ientity mp on verties is n isomorphism of grph to itself. () Symmetri: If f is n isomorphism f : G 1 G 2, then f : V 1 V 2 is ijetive, n therefore hs n inverse. Sine f preserves jeny, so oes f 1. So f 1 : G 2 G 1 is n isomorphism. () Trnsitive: If f : G 1 G 2 n g : G 2 G 3 re isomorphisms, then g f : G 1 G 3 is n isomorphism, sine the omposition of ijetive n ege-preserving mps is ijetive n ege-preserving. Exerise 30. () Consier the grph G = (i) Give n exmple of sugrph of G tht is not inue. H = (ii) How mny inue sugrphs oes G hve? List them. There re 4 verties, so there re 2 4 inue sugrphs: n (iii) How mny sugrphs oes G hve? A grph with m eges hs extly 2 m sugrphs with the sme vertex set. So, going through the inue sugrphs (the lrgest sugrph of G with eh possile vertex set), we get sugrphs of G in totl.

6 (iv) Let e e the ege onneting n. Drw G e n G/e., G e = G/e = (v) Let e e the ege onneting n. Drw G e n G/e., G e = G/e = (vi) Let e e n ege onneting n. Drw G + e. G = (vii) Drw Ḡ. G = () Show tht G = is isomorphi to its omplement. Sine Ḡ = the mp gives n isomorphism.,,,,

7 () Fin simple grph with 5 verties tht is isomorphi to its own omplement. (Strt with: how mny eges must it hve?) Sine there re 10 possile eges, G must hve 5 eges. One exmple tht will work is C 5 : G = = Ḡ = Exerise 31. () Drw the isomorphism lsses of onnete grphs on 4 verties, n give the vertex n ege onnetivity numer for eh. κ = 1 λ = 1 κ = 1 λ = 1 κ = 1 λ = 1 κ = 2 λ = 2 κ = 2 λ = 2 κ = 3 λ = 3 () Show tht if v is vertex of o egree, then there is pth from v to nother vertex of o egree. By the Hnshke Theorem, every grph hs n even numer of o egree verties. Notie tht eh onnete omponent is n inue sugrph with the sme egrees. So eh onnete omponent lso hs n even numer of o egree verties. So if onnete omponent hs n o egree vertex, it must hve two. So those two verties re onnete y wlk. () Prove tht for every simple grph, either G is onnete, or Ḡ is onnete. Suppose G is not onnete. Let H 1, H 2,..., H k e the onnete omponents of G (i.e. the sugrphs inue y eh set of verties etermine y the onnete omponents). First, onsier two verties in ifferent onnete omponents in G: u H i, v H j, i j. Sine u n v re in ifferent onnete omponents in G, they re ertinly not jent; thus u n v re jent in Ḡ, n therefore in the sme onnete omponent. Now onsier two verties in the sme onnete omponent in G: u, v H i. Sine there is more thn one onnete omponent in G, let w H j, i j. By our previous rgument, u n v re oth in the neighorhoo of w in Ḡ, n so u, w, v is pth in (G). Thus u n v re onnete in Ḡ. Thus Ḡ is onnete. () Rell tht κ(g) is the vertex onnetivity of G n λ(g) is the ege onnetivity of G. Give exmples of grphs for whih eh of the following re stisfie. Let δ = min v V eg(v).

8 (i) κ(g) = λ(g) < min v V eg(v) κ = λ = 1, δ = 2. (ii) κ(g) < λ(g) = min v V eg(v) κ = 1, λ = 2, δ = 2. (iii) κ(g) < λ(g) < min v V eg(v) κ = 1, λ = 2, δ = 3. (iv) κ(g) = λ(g) = min v V eg(v) Cyles of length 3 or more hve κ = λ = δ = 2. (e) For the following theorem, pik ny of prts (ii) (iv) n show (refully!) tht it s equivlent to prt (i). Theorem: For simple grph with t lest 3 verties, the following re equivlent. (i) G is onnete n ontins no ut vertex. (ii) Every two verties in V re ontine in some yle. (iii) Every two eges in E re ontine in some yle, n G ontins no isolte verties. (iv) For ny three verties u, v, w V, there is pth from u to v ontining w. (ii) = (i): If every two verties u n v re ontine in some yle, then there re two internlly isjoint u v pths. Thus u n v re onnetes, n the eletion of ny thir vertex will not isonnet u from v. Thus κ(g) 2. (i) = (ii): Assume κ(g) 2. Consier u, v V, n set = (u, v). If = 1, then u n v re jent. Sine V 3, there is some w V istint from u n v. Sine κ(g) 2, there is some u w pth P 1 not ontining v; similrly there is some w v pth P 2 not ontining u. Let x e the first vertex on P 1 whih is lso on P 2 (sine w is on oth pths, suh vertex exists). Then the wlk from u on P 1 to x, then on P 2 to v, n finlly k on the ege vu, is yle ontining u n v. If > 1, suppose tht for ny w V with (u, z) <, there is some yle ontining u n w. Consier miniml length u v pth, n let w e v s neighor on this pth, so tht (u, w) = 1. By our inutive hypothesis, there is yle C ontining u n w. Either C lso ontins v, in whih we lso hve yle ontining u n v, or it oesn t. If C oes not ontin v, then sine κ(g) 2, we lso hve pth P from v to u not ontining w. Let x e the first vertex on P

9 whih is lso on C (sine u is on P n C, x exists). u C x w v P Then the wlk from v long P to x, then long C wy from w n towr u (if x = u, wlk in either iretion), through u n roun to w, n finlly long the ege wv, is yle ontining u n v. (iii) = (i): Consier u, v V. Sine there re no isolte verties in G, eg(u), eg(v) 1, so u n v re eh inient to t lest one ege. Sine V 3, there re then t lest two eges. Sine every pir of eges re ontine in ommon yle, vertex inient to n ege must e inient to t lest two eges. Thus there re istint eges e f with e inient to u n f inient to v. Sine e n f re ontine in some yle, u n v re ontine in tht sme yle. So there re two internlly isjoint u v pths. Thus u n v re onnete, n the eletion of ny thir vertex will not isonnet u from v. Thus κ(g) 2. (i) = (iii): With G = (V, E, φ) n e, f E, efine = (e, f) = min (u, v). u φ(e) v φ(f) Now ssume κ(g) 2. Fix e, f E istint n let = (e, f). Choose u φ(e) n v φ(f) with (u, v) =, n let u n v e the other verties inient to e n f, respetively. First suppose = 0 (so tht e n f re inient to ommon vertex). Sine κ(g) 2, there is pth P from u to v not ontining u = v. Then the wlk from u long P to v, then k long f then e is yle ontining e n f. If > 0, then ssume tht for ll e g E for whih (e, g) < is ontine in yle together with e. Tke miniml-length pth from u to v, n let w e v s neighor on P, so tht (u, w) = 1, n so (e, wv) = 1. By our inutive hypothesis, there is yle C ontining e n wv. If v is on C, then the wlk long v v, then long C wy from v n towr u, then on roun to v, is yle ontining e n v v. Otherwise, let P e pth from v to u not ontining v (whih exists sine κ(g) 2). Let x e the first vertex on P whih is lso on C. Then the wlk from v long v, then long P to x, then long C wy from v n towr u, through on to v, is yle ontining e n f. (iv) = (i): Let u, v V. For ny w V istint from u n v, ssuming (iv) gives t lest one u w pth ontining v. The initil wlk from u to v on this pth gives u v pth not ontining w. So u n v re onnete in G n in G w. Thus κ(g) 2. (i) = (iv): Assume κ(g) 2, let u, v, w V, n let = (u, w).

10 If = 1, let P e pth from w to v not ontining u (whih exists euse κ(g) 2). Then the wlk from u to w n then long P to v is pth from u to v ontining w. If > 1, ssume tht for every x V with (u, x) <, there is u v pth through x. Tke miniml-length pth from u to w, n let x e w s neighor on this pth, so tht (u, x) = 1. Then our inutive hypothesis gurntees u v pth P ontining x. If P lso ontins w, then P is u v pth ontining w s esire. Otherwise, let P e pth from w to u not ontining x. Let z e the first vertex on P whih is lso on P (whih is gurntee sine u is on oth P n P ). If z sits etween x n v on P, then the wlk from u long P to x, then long xw, then long P to z, then ontinuing long P to v, is u v pth ontining w. Otherwise, z sits etween u n x on P, so tht the wlk from u long P to z, the kwrs long P towr w, up wx, n then long P towr v is u v pth through w.

### Graph Theory. Simple Graph G = (V, E). V={a,b,c,d,e,f,g,h,k} E={(a,b),(a,g),( a,h),(a,k),(b,c),(b,k),...,(h,k)}

Grph Theory Simple Grph G = (V, E). V ={verties}, E={eges}. h k g f e V={,,,,e,f,g,h,k} E={(,),(,g),(,h),(,k),(,),(,k),...,(h,k)} E =16. 1 Grph or Multi-Grph We llow loops n multiple eges. G = (V, E.ψ)

### CIT 596 Theory of Computation 1. Graphs and Digraphs

CIT 596 Theory of Computtion 1 A grph G = (V (G), E(G)) onsists of two finite sets: V (G), the vertex set of the grph, often enote y just V, whih is nonempty set of elements lle verties, n E(G), the ege

### Connectivity in Graphs. CS311H: Discrete Mathematics. Graph Theory II. Example. Paths. Connectedness. Example

Connetiit in Grphs CSH: Disrete Mthemtis Grph Theor II Instrtor: Işıl Dillig Tpil qestion: Is it possile to get from some noe to nother noe? Emple: Trin netork if there is pth from to, possile to tke trin

### Solutions to Problem Set #1

CSE 233 Spring, 2016 Solutions to Prolem Set #1 1. The movie tse onsists of the following two reltions movie: title, iretor, tor sheule: theter, title The first reltion provies titles, iretors, n tors

### Lecture 11 Binary Decision Diagrams (BDDs)

C 474A/57A Computer-Aie Logi Design Leture Binry Deision Digrms (BDDs) C 474/575 Susn Lyseky o 3 Boolen Logi untions Representtions untion n e represente in ierent wys ruth tle, eqution, K-mp, iruit, et

### Numbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point

GCSE C Emple 7 Work out 9 Give your nswer in its simplest form Numers n inies Reiprote mens invert or turn upsie own The reiprol of is 9 9 Mke sure you only invert the frtion you re iviing y 7 You multiply

### CS 360 Exam 2 Fall 2014 Name

CS 360 Exm 2 Fll 2014 Nme 1. The lsses shown elow efine singly-linke list n stk. Write three ifferent O(n)-time versions of the reverse_print metho s speifie elow. Eh version of the metho shoul output

### 1 From NFA to regular expression

Note 1: How to convert DFA/NFA to regulr expression Version: 1.0 S/EE 374, Fll 2017 Septemer 11, 2017 In this note, we show tht ny DFA cn e converted into regulr expression. Our construction would work

### Project 6: Minigoals Towards Simplifying and Rewriting Expressions

MAT 51 Wldis Projet 6: Minigols Towrds Simplifying nd Rewriting Expressions The distriutive property nd like terms You hve proly lerned in previous lsses out dding like terms ut one prolem with the wy

### Nondeterministic Finite Automata

Nondeterministi Finite utomt The Power of Guessing Tuesdy, Otoer 4, 2 Reding: Sipser.2 (first prt); Stoughton 3.3 3.5 S235 Lnguges nd utomt eprtment of omputer Siene Wellesley ollege Finite utomton (F)

### NON-DETERMINISTIC FSA

Tw o types of non-determinism: NON-DETERMINISTIC FS () Multiple strt-sttes; strt-sttes S Q. The lnguge L(M) ={x:x tkes M from some strt-stte to some finl-stte nd ll of x is proessed}. The string x = is

### Matrix & Vector Basic Linear Algebra & Calculus

Mtrix & Vector Bsic Liner lgebr & lculus Wht is mtrix? rectngulr rry of numbers (we will concentrte on rel numbers). nxm mtrix hs n rows n m columns M x4 M M M M M M M M M M M M 4 4 4 First row Secon row

### Pre-Lie algebras, rooted trees and related algebraic structures

Pre-Lie lgers, rooted trees nd relted lgeri strutures Mrh 23, 2004 Definition 1 A pre-lie lger is vetor spe W with mp : W W W suh tht (x y) z x (y z) = (x z) y x (z y). (1) Exmple 2 All ssoitive lgers

QUADRATIC EQUATION Contents Topi Pge No. Theory 0-04 Exerise - 05-09 Exerise - 09-3 Exerise - 3 4-5 Exerise - 4 6 Answer Key 7-8 Syllus Qudrti equtions with rel oeffiients, reltions etween roots nd oeffiients,

### (a) A partition P of [a, b] is a finite subset of [a, b] containing a and b. If Q is another partition and P Q, then Q is a refinement of P.

Chpter 7: The Riemnn Integrl When the derivtive is introdued, it is not hrd to see tht the it of the differene quotient should be equl to the slope of the tngent line, or when the horizontl xis is time

### Obstructions to chordal circular-arc graphs of small independence number

Ostrutions to horl irulr-r grphs of smll inepenene numer Mthew Frnis,1 Pvol Hell,2 Jurj Stho,3 Institute of Mth. Sienes, IV Cross Ro, Trmni, Chenni 600 113, Ini Shool of Comp. Siene, Simon Frser University,

### arxiv: v1 [cs.dm] 24 Jul 2017

Some lsses of grphs tht re not PCGs 1 rxiv:1707.07436v1 [s.dm] 24 Jul 2017 Pierluigi Biohi Angelo Monti Tizin Clmoneri Rossell Petreshi Computer Siene Deprtment, Spienz University of Rome, Itly pierluigi.iohi@gmil.om,

### CS 310 (sec 20) - Winter Final Exam (solutions) SOLUTIONS

CS 310 (sec 20) - Winter 2003 - Finl Exm (solutions) SOLUTIONS 1. (Logic) Use truth tles to prove the following logicl equivlences: () p q (p p) (q q) () p q (p q) (p q) () p q p q p p q q (q q) (p p)

### Separable discrete functions: recognition and sufficient conditions

Seprle isrete funtions: reognition n suffiient onitions Enre Boros Onřej Čepek Vlimir Gurvih Novemer 21, 217 rxiv:1711.6772v1 [mth.co] 17 Nov 217 Astrt A isrete funtion of n vriles is mpping g : X 1...

### STRAND J: TRANSFORMATIONS, VECTORS and MATRICES

Mthemtics SKE: STRN J STRN J: TRNSFORMTIONS, VETORS nd MTRIES J3 Vectors Text ontents Section J3.1 Vectors nd Sclrs * J3. Vectors nd Geometry Mthemtics SKE: STRN J J3 Vectors J3.1 Vectors nd Sclrs Vectors

### USA Mathematical Talent Search Round 1 Solutions Year 21 Academic Year

1/1/21. Fill in the circles in the picture t right with the digits 1-8, one digit in ech circle with no digit repeted, so tht no two circles tht re connected by line segment contin consecutive digits.

### Introduction to Graphical Models

Introution to Grhil Moels Kenji Fukumizu The Institute of Sttistil Mthemtis Comuttionl Methoology in Sttistil Inferene II Introution n Review 2 Grhil Moels Rough Sketh Grhil moels Grh: G V E V: the set

### Non Right Angled Triangles

Non Right ngled Tringles Non Right ngled Tringles urriulum Redy www.mthletis.om Non Right ngled Tringles NON RIGHT NGLED TRINGLES sin i, os i nd tn i re lso useful in non-right ngled tringles. This unit

QUADRATIC EQUATIONS OBJECTIVE PROBLEMS +. The solution of the eqution will e (), () 0,, 5, 5. The roots of the given eqution ( p q) ( q r) ( r p) 0 + + re p q r p (), r p p q, q r p q (), (d), q r p q.

### Homework 4. 0 ε 0. (00) ε 0 ε 0 (00) (11) CS 341: Foundations of Computer Science II Prof. Marvin Nakayama

CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 4 1. UsetheproceduredescriedinLemm1.55toconverttheregulrexpression(((00) (11)) 01) into n NFA. Answer: 0 0 1 1 00 0 0 11 1 1 01 0 1 (00)

### Homework 3 Solutions

CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.

### Lesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem.

27 Lesson 2: The Pythgoren Theorem nd Similr Tringles A Brief Review of the Pythgoren Theorem. Rell tht n ngle whih mesures 90º is lled right ngle. If one of the ngles of tringle is right ngle, then we

### Section 4.4. Green s Theorem

The Clulus of Funtions of Severl Vriles Setion 4.4 Green s Theorem Green s theorem is n exmple from fmily of theorems whih onnet line integrls (nd their higher-dimensionl nlogues) with the definite integrls

### The Word Problem in Quandles

The Word Prolem in Qundles Benjmin Fish Advisor: Ren Levitt April 5, 2013 1 1 Introdution A word over n lger A is finite sequene of elements of A, prentheses, nd opertions of A defined reursively: Given

### for all x in [a,b], then the area of the region bounded by the graphs of f and g and the vertical lines x = a and x = b is b [ ( ) ( )] A= f x g x dx

Applitions of Integrtion Are of Region Between Two Curves Ojetive: Fin the re of region etween two urves using integrtion. Fin the re of region etween interseting urves using integrtion. Desrie integrtion

### State Minimization for DFAs

Stte Minimiztion for DFAs Red K & S 2.7 Do Homework 10. Consider: Stte Minimiztion 4 5 Is this miniml mchine? Step (1): Get rid of unrechle sttes. Stte Minimiztion 6, Stte is unrechle. Step (2): Get rid

QUADRATIC EQUATION EXERCISE - 0 CHECK YOUR GRASP. Sine sum of oeffiients 0. Hint : It's one root is nd other root is 8 nd 5 5. tn other root 9. q 4p 0 q p q p, q 4 p,,, 4 Hene 7 vlues of (p, q) 7 equtions

### APPROXIMATION AND ESTIMATION MATHEMATICAL LANGUAGE THE FUNDAMENTAL THEOREM OF ARITHMETIC LAWS OF ALGEBRA ORDER OF OPERATIONS

TOPIC 2: MATHEMATICAL LANGUAGE NUMBER AND ALGEBRA You shoul unerstn these mthemtil terms, n e le to use them ppropritely: ² ition, sutrtion, multiplition, ivision ² sum, ifferene, prout, quotient ² inex

### Grammar. Languages. Content 5/10/16. Automata and Languages. Regular Languages. Regular Languages

5//6 Grmmr Automt nd Lnguges Regulr Grmmr Context-free Grmmr Context-sensitive Grmmr Prof. Mohmed Hmd Softwre Engineering L. The University of Aizu Jpn Regulr Lnguges Context Free Lnguges Context Sensitive

### 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. {, } {, } {, } {, } {, } {,

### Farey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University

U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions

### Chapter 2 Finite Automata

Chpter 2 Finite Automt 28 2.1 Introduction Finite utomt: first model of the notion of effective procedure. (They lso hve mny other pplictions). The concept of finite utomton cn e derived y exmining wht

### CS 373, Spring Solutions to Mock midterm 1 (Based on first midterm in CS 273, Fall 2008.)

CS 373, Spring 29. Solutions to Mock midterm (sed on first midterm in CS 273, Fll 28.) Prolem : Short nswer (8 points) The nswers to these prolems should e short nd not complicted. () If n NF M ccepts

### Identifying and Classifying 2-D Shapes

Ientifying n Clssifying -D Shpes Wht is your sign? The shpe n olour of trffi signs let motorists know importnt informtion suh s: when to stop onstrution res. Some si shpes use in trffi signs re illustrte

### Data Structures LECTURE 10. Huffman coding. Example. Coding: problem definition

Dt Strutures, Spring 24 L. Joskowiz Dt Strutures LEURE Humn oing Motivtion Uniquel eipherle oes Prei oes Humn oe onstrution Etensions n pplitions hpter 6.3 pp 385 392 in tetook Motivtion Suppose we wnt

### Math 211A Homework. Edward Burkard. = tan (2x + z)

Mth A Homework Ewr Burkr Eercises 5-C Eercise 8 Show tht the utonomous system: 5 Plne Autonomous Systems = e sin 3y + sin cos + e z, y = sin ( + 3y, z = tn ( + z hs n unstble criticl point t = y = z =

### Solutions for HW11. Exercise 34. (a) Use the recurrence relation t(g) = t(g e) + t(g/e) to count the number of spanning trees of v 1

Solutions for HW Exris. () Us th rurrn rltion t(g) = t(g ) + t(g/) to ount th numr of spnning trs of v v v u u u Rmmr to kp multipl gs!! First rrw G so tht non of th gs ross: v u v Rursing on = (v, u ):

### Problem Set 9. Figure 1: Diagram. This picture is a rough sketch of the 4 parabolas that give us the area that we need to find. The equations are:

(x + y ) = y + (x + y ) = x + Problem Set 9 Discussion: Nov., Nov. 8, Nov. (on probbility nd binomil coefficients) The nme fter the problem is the designted writer of the solution of tht problem. (No one

### 8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers.

8. Complex Numers The rel numer system is dequte for solving mny mthemticl prolems. But it is necessry to extend the rel numer system to solve numer of importnt prolems. Complex numers do not chnge the

### Line Integrals and Entire Functions

Line Integrls nd Entire Funtions Defining n Integrl for omplex Vlued Funtions In the following setions, our min gol is to show tht every entire funtion n be represented s n everywhere onvergent power series

### MTH 505: Number Theory Spring 2017

MTH 505: Numer Theory Spring 207 Homework 2 Drew Armstrong The Froenius Coin Prolem. Consider the eqution x ` y c where,, c, x, y re nturl numers. We cn think of \$ nd \$ s two denomintions of coins nd \$c

### 1 Nondeterministic Finite Automata

1 Nondeterministic Finite Automt Suppose in life, whenever you hd choice, you could try oth possiilities nd live your life. At the end, you would go ck nd choose the one tht worked out the est. Then you

### Solutions to Assignment 1

MTHE 237 Fll 2015 Solutions to Assignment 1 Problem 1 Find the order of the differentil eqution: t d3 y dt 3 +t2 y = os(t. Is the differentil eqution liner? Is the eqution homogeneous? b Repet the bove

### dx dt dy = G(t, x, y), dt where the functions are defined on I Ω, and are locally Lipschitz w.r.t. variable (x, y) Ω.

Chpter 8 Stility theory We discuss properties of solutions of first order two dimensionl system, nd stility theory for specil clss of liner systems. We denote the independent vrile y t in plce of x, nd

### The master ring problem

2005 Interntionl Conferene on Anlysis of Algorithms DMTCS pro. AD, 2005, 287 296 The mster ring problem Hs Shhni 1 n Lis Zhng 2 1 Computer Siene Dept., Tehnion, Hif 32000, Isrel. 2 Bell Lbs, Luent Tehnologies,

### CS 330 Formal Methods and Models

CS 330 Forml Methods nd Models Dn Richrds, section 003, George Mson University, Fll 2017 Quiz Solutions Quiz 1, Propositionl Logic Dte: Septemer 7 1. Prove (p q) (p q), () (5pts) using truth tles. p q

### Continuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom

Lerning Gols Continuous Rndom Vriles Clss 5, 8.05 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf) nd cumultive

### Lexical Analysis Finite Automate

Lexicl Anlysis Finite Automte CMPSC 470 Lecture 04 Topics: Deterministic Finite Automt (DFA) Nondeterministic Finite Automt (NFA) Regulr Expression NFA DFA A. Finite Automt (FA) FA re grph, like trnsition

### Convex Sets and Functions

B Convex Sets nd Functions Definition B1 Let L, +, ) be rel liner spce nd let C be subset of L The set C is convex if, for ll x,y C nd ll [, 1], we hve 1 )x+y C In other words, every point on the line

### Data Compression Techniques (Spring 2012) Model Solutions for Exercise 4

58487 Dt Compressio Tehiques (Sprig 0) Moel Solutios for Exerise 4 If you hve y fee or orretios, plese ott jro.lo t s.helsii.fi.. Prolem: Let T = Σ = {,,, }. Eoe T usig ptive Huffm oig. Solutio: R 4 U

### LIP. Laboratoire de l Informatique du Parallélisme. Ecole Normale Supérieure de Lyon

LIP Lortoire de l Informtique du Prllélisme Eole Normle Supérieure de Lyon Institut IMAG Unité de reherhe ssoiée u CNRS n 1398 One-wy Cellulr Automt on Cyley Grphs Zsuzsnn Rok Mrs 1993 Reserh Report N

### CS375: Logic and Theory of Computing

CS375: Logic nd Theory of Computing Fuhu (Frnk) Cheng Deprtment of Computer Science University of Kentucky 1 Tle of Contents: Week 1: Preliminries (set lger, reltions, functions) (red Chpters 1-4) Weeks

### Section 6.1 Definite Integral

Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined

### 4 VECTORS. 4.0 Introduction. Objectives. Activity 1

4 VECTRS Chpter 4 Vectors jectives fter studying this chpter you should understnd the difference etween vectors nd sclrs; e le to find the mgnitude nd direction of vector; e le to dd vectors, nd multiply

### Euler and Hamilton Paths

Euler an Hamilton Paths The town of Königserg, Prussia (now know as Kaliningra an part of the Russian repuli), was ivie into four setion y ranhes of the Pregel River. These four setions C A D B Figure:

### Thermodynamics. Question 1. Question 2. Question 3 3/10/2010. Practice Questions PV TR PV T R

/10/010 Question 1 1 mole of idel gs is rought to finl stte F y one of three proesses tht hve different initil sttes s shown in the figure. Wht is true for the temperture hnge etween initil nd finl sttes?

9/6/7 Mustf Jrrr: Leture Notes in Disrete Mthemtis. Birzeit University Plestine 05 Funtions 7.. Introdution to Funtions 7. One-to-One Onto Inverse funtions mjrrr 05 Wth this leture nd downlod the slides

### MATH 573 FINAL EXAM. May 30, 2007

MATH 573 FINAL EXAM My 30, 007 NAME: Solutions 1. This exm is due Wednesdy, June 6 efore the 1:30 pm. After 1:30 pm I will NOT ccept the exm.. This exm hs 1 pges including this cover. There re 10 prolems.

### Improper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral.

Improper Integrls Introduction When we defined the definite integrl f d we ssumed tht f ws continuous on [, ] where [, ] ws finite, closed intervl There re t lest two wys this definition cn fil to e stisfied:

### Geometry of the Circle - Chords and Angles. Geometry of the Circle. Chord and Angles. Curriculum Ready ACMMG: 272.

Geometry of the irle - hords nd ngles Geometry of the irle hord nd ngles urriulum Redy MMG: 272 www.mthletis.om hords nd ngles HRS N NGLES The irle is si shpe nd so it n e found lmost nywhere. This setion

### 20 MATHEMATICS POLYNOMIALS

0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of

### PART 1 MULTIPLE CHOICE Circle the appropriate response to each of the questions below. Each question has a value of 1 point.

PART MULTIPLE CHOICE Circle the pproprite response to ech of the questions below. Ech question hs vlue of point.. If in sequence the second level difference is constnt, thn the sequence is:. rithmetic

### Math 32B Discussion Session Week 8 Notes February 28 and March 2, f(b) f(a) = f (t)dt (1)

Green s Theorem Mth 3B isussion Session Week 8 Notes Februry 8 nd Mrh, 7 Very shortly fter you lerned how to integrte single-vrible funtions, you lerned the Fundmentl Theorem of lulus the wy most integrtion

### Linear Systems with Constant Coefficients

Liner Systems with Constnt Coefficients 4-3-05 Here is system of n differentil equtions in n unknowns: x x + + n x n, x x + + n x n, x n n x + + nn x n This is constnt coefficient liner homogeneous system

### A CLASS OF GENERAL SUPERTREE METHODS FOR NESTED TAXA

A CLASS OF GENERAL SUPERTREE METHODS FOR NESTED TAXA PHILIP DANIEL AND CHARLES SEMPLE Astrt. Amlgmting smller evolutionry trees into single prent tree is n importnt tsk in evolutionry iology. Trditionlly,

### Chapter 4 Regular Grammar and Regular Sets. (Solutions / Hints)

C K Ngpl Forml Lnguges nd utomt Theory Chpter 4 Regulr Grmmr nd Regulr ets (olutions / Hints) ol. (),,,,,,,,,,,,,,,,,,,,,,,,,, (),, (c) c c, c c, c, c, c c, c, c, c, c, c, c, c c,c, c, c, c, c, c, c, c,

### MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.

MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.

### a,b a 1 a 2 a 3 a,b 1 a,b a,b 2 3 a,b a,b a 2 a,b CS Determinisitic Finite Automata 1

CS4 45- Determinisitic Finite Automt -: Genertors vs. Checkers Regulr expressions re one wy to specify forml lnguge String Genertor Genertes strings in the lnguge Deterministic Finite Automt (DFA) re nother

### CS 311 Homework 3 due 16:30, Thursday, 14 th October 2010

CS 311 Homework 3 due 16:30, Thursdy, 14 th Octoer 2010 Homework must e sumitted on pper, in clss. Question 1. [15 pts.; 5 pts. ech] Drw stte digrms for NFAs recognizing the following lnguges:. L = {w

### Automata and Languages

Automt nd Lnguges Prof. Mohmed Hmd Softwre Engineering Lb. The University of Aizu Jpn Grmmr Regulr Grmmr Context-free Grmmr Context-sensitive Grmmr Regulr Lnguges Context Free Lnguges Context Sensitive

### Homework Solution - Set 5 Due: Friday 10/03/08

CE 96 Introduction to the Theory of Computtion ll 2008 Homework olution - et 5 Due: ridy 10/0/08 1. Textook, Pge 86, Exercise 1.21. () 1 2 Add new strt stte nd finl stte. Mke originl finl stte non-finl.

### 3 Regular expressions

3 Regulr expressions Given n lphet Σ lnguge is set of words L Σ. So fr we were le to descrie lnguges either y using set theory (i.e. enumertion or comprehension) or y n utomton. In this section we shll

### Chapter Five - Eigenvalues, Eigenfunctions, and All That

Chpter Five - Eigenvlues, Eigenfunctions, n All Tht The prtil ifferentil eqution methos escrie in the previous chpter is specil cse of more generl setting in which we hve n eqution of the form L 1 xux,tl

### 1.3 SCALARS AND VECTORS

Bridge Course Phy I PUC 24 1.3 SCLRS ND VECTORS Introdution: Physis is the study of nturl phenomen. The study of ny nturl phenomenon involves mesurements. For exmple, the distne etween the plnet erth nd

### nd edges. Eh edge hs either one endpoint: end(e) = fxg in whih se e is termed loop t vertex x, or two endpoints: end(e) = fx; yg in whih se e is terme

Theory of Regions Eri Bdouel nd Philippe Drondeu Iris, Cmpus de Beulieu, F-35042 Rennes Cedex, Frne E-mil : feri.bdouel,philippe.drondeug@iris.fr Astrt. The synthesis prolem for nets onsists in deiding

### Lecture 3. Limits of Functions and Continuity

Lecture 3 Limits of Functions nd Continuity Audrey Terrs April 26, 21 1 Limits of Functions Notes I m skipping the lst section of Chpter 6 of Lng; the section bout open nd closed sets We cn probbly live

### Resources. Introduction: Binding. Resource Types. Resource Sharing. The type of a resource denotes its ability to perform different operations

Introduction: Binding Prt of 4-lecture introduction Scheduling Resource inding Are nd performnce estimtion Control unit synthesis This lecture covers Resources nd resource types Resource shring nd inding

### 13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS

33 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS As simple ppliction of the results we hve obtined on lgebric extensions, nd in prticulr on the multiplictivity of extension degrees, we cn nswer (in

### 4.1. Probability Density Functions

STT 1 4.1-4. 4.1. Proility Density Functions Ojectives. Continuous rndom vrile - vers - discrete rndom vrile. Proility density function. Uniform distriution nd its properties. Expected vlue nd vrince of

### Nondeterminism. Nondeterministic Finite Automata. Example: Moves on a Chessboard. Nondeterminism (2) Example: Chessboard (2) Formal NFA

Nondeterminism Nondeterministic Finite Automt Nondeterminism Subset Construction A nondeterministic finite utomton hs the bility to be in severl sttes t once. Trnsitions from stte on n input symbol cn

### Vectors. Chapter14. Syllabus reference: 4.1, 4.2, 4.5 Contents:

hpter Vetors Syllus referene:.,.,.5 ontents: D E F G H I J K Vetors nd slrs Geometri opertions with vetors Vetors in the plne The mgnitude of vetor Opertions with plne vetors The vetor etween two points

### MATH 174A: PROBLEM SET 5. Suggested Solution

MATH 174A: PROBLEM SET 5 Suggested Solution Problem 1. Suppose tht I [, b] is n intervl. Let f 1 b f() d for f C(I; R) (i.e. f is continuous rel-vlued function on I), nd let L 1 (I) denote the completion

### 3.1 Review of Sine, Cosine and Tangent for Right Angles

Foundtions of Mth 11 Section 3.1 Review of Sine, osine nd Tngent for Right Tringles 125 3.1 Review of Sine, osine nd Tngent for Right ngles The word trigonometry is derived from the Greek words trigon,

### MATH 101A: ALGEBRA I PART B: RINGS AND MODULES 35

MATH 101A: ALGEBRA I PART B: RINGS AND MODULES 35 9. Modules over PID This week we re proving the fundmentl theorem for finitely generted modules over PID, nmely tht they re ll direct sums of cyclic modules.

### Graph Isomorphism. Graphs - II. Cayley s Formula. Planar Graphs. Outline. Is K 5 planar? The number of labeled trees on n nodes is n n-2

Grt Thortil Is In Computr Sin Vitor Amhik CS 15-251 Ltur 9 Grphs - II Crngi Mllon Univrsity Grph Isomorphism finition. Two simpl grphs G n H r isomorphi G H if thr is vrtx ijtion V H ->V G tht prsrvs jny

### V={A,B,C,D,E} E={ (A,D),(A,E),(B,D), (B,E),(C,D),(C,E)}

Introution Computr Sin & Enginring 423/823 Dsign n Anlysis of Algorithms Ltur 03 Elmntry Grph Algorithms (Chptr 22) Stphn Sott (Apt from Vinohnrn N. Vriym) I Grphs r strt t typs tht r pplil to numrous

### Triangles The following examples explore aspects of triangles:

Tringles The following exmples explore spects of tringles: xmple 1: ltitude of right ngled tringle + xmple : tringle ltitude of the symmetricl ltitude of n isosceles x x - 4 +x xmple 3: ltitude of the

### Automata and Regular Languages

Chpter 9 Automt n Regulr Lnguges 9. Introution This hpter looks t mthemtil moels of omputtion n lnguges tht esrie them. The moel-lnguge reltionship hs multiple levels. We shll explore the simplest level,

### SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics

SCHOOL OF ENGINEERING & BUIL ENVIRONMEN Mthemtics An Introduction to Mtrices Definition of Mtri Size of Mtri Rows nd Columns of Mtri Mtri Addition Sclr Multipliction of Mtri Mtri Multipliction 7 rnspose

### Greedoid polynomial, chip-firing, and G-parking function for directed graphs. Connections in Discrete Mathematics

Greedoid polynomil, hip-firing, nd G-prking funtion for direted grphs Swee Hong Chn Cornell University Connetions in Disrete Mthemtis June 15, 2015 Motivtion Tutte polynomil [Tut54] is polynomil defined

### What else can you do?

Wht else cn you do? ngle sums The size of specil ngle types lernt erlier cn e used to find unknown ngles. tht form stright line dd to 180c. lculte the size of + M, if L is stright line M + L = 180c( stright

### The Riemann-Stieltjes Integral

Chpter 6 The Riemnn-Stieltjes Integrl 6.1. Definition nd Eistene of the Integrl Definition 6.1. Let, b R nd < b. ( A prtition P of intervl [, b] is finite set of points P = { 0, 1,..., n } suh tht = 0

### Transition systems (motivation)

Trnsition systems (motivtion) Course Modelling of Conurrent Systems ( Modellierung neenläufiger Systeme ) Winter Semester 2009/0 University of Duisurg-Essen Brr König Tehing ssistnt: Christoph Blume In