Mid-Term Examination - Spring 2014 Mathematical Programming with Applications to Economics Total Score: 45; Time: 3 hours

Size: px
Start display at page:

Download "Mid-Term Examination - Spring 2014 Mathematical Programming with Applications to Economics Total Score: 45; Time: 3 hours"

Transcription

1 Mi-Term Exmintion - Spring 0 Mthemtil Progrmming with Applitions to Eonomis Totl Sore: 5; Time: hours. Let G = (N, E) e irete grph. Define the inegree of vertex i N s the numer of eges tht re oming into i. Formlly, inegree(i) = {j N \ {i} : (j, i) E}. Similrly, efine the outegree of vertex i N s the numer of eges tht re oming out of i. Formlly, outegree(i) = {j N \ {i} : (i, j) E}. Suppose G stisfies the property tht for every vertex i N, outegree(i) = inegree(i). Tke pir of verties s, t N n suppose there re k ege-isjoint pths from s to t in G. Does G lso ontin k ege-isjoint pths from t to s? Prove this or provie ounterexmple. (5 mrks) Answer. We onsier ny s t ut (S, N \ S) of the grph G. Let C + = {(i, j) E : i S, j N \ S} n C = {(i, j) E : i N \ S, j S}. Now, inegree(k) = {i N : (i, k) E} = {i S : (i, k) E} + C. Similrly, outegree(k) = {i N : (k, i) E} = {i S : (k, i) E} + C +. Using the ft tht {i S : (i, k) E} = {i S : (k, i) E} n inegree(k) = outegree(k), we get C+ = C. Now, sine there re k ege-isjoint pths from s to t, there is ut suh tht C + = k. But for the sme ut C = k. This implies tht we n remove this set of k eges in C n there will e no pth from t to s. By Menger s theorem, the numer of isjoint pths from t to s is k.

2 . Consier the following system of inequlities. x x x x x x x x x x x x x x x x. () Construt the unerlying weighte irete grph of this system of inequlities suh tht the potentils of this grph orrespons to solution of this system. ( mrks) Answer. The unerlying irete grph hs four noes orresponing to eh of the four vriles. This is shown in Figure. Figure : Direte grph () Use this to onlue if this system of inequlities hs solution. If yes, fin solution or rgue why it oes not hve solution. (7 mrks) Answer. Sine there re no yles of negtive length in the unerlying irete grph, the system of inequlities hs solution. One solution n e foun y setting, sy, x = 0, n tking shortest pth from the orresponing to x to every other noe. In prtiulr, x =, x =, x = 5.

3 . Suppose G = (N, E, w) is weighte irete omplete grph. Further, suppose tht the weights in G stisfy the following tringle inequlity: for every i, j, k N we hve w(i, j) + w(j, k) w(i, k). Show tht G stisfies potentil equivlene if n only if w(i, j) = w(j, i) for every i, j N. (5 mrks) Answer. Fix ny pir of noes i, j N. Let shortest pth from i to j e (i, i,..., i k, j). Then, pplying tringle inequlity suessively, we get s(i, j) = w(i, i ) + w(i, i ) w(i k, i k ) + w(i k, j) w(i, i ) + w(i, i ) w(i k, i k ) + w(i k, j)... w(i, j). But w(i, j) s(i, j) implies tht w(i, j) = s(i, j). Now, potentil equivlene is equivlent to requiring s(i, j)+s(j, i) = 0 for ll i, j N. Hene, potentil equivlene hols if n only if w(i, j) + w(j, i) = 0 for ll i, j N.. Let G = (N, E) e n unirete iprtite grph, i.e., there exists prtition B H of N suh tht for every {i, j} E, we hve i B n j H. We sy iprtite grph G is k-regulr, where k is positive integer, if egree of every vertex is k. () Show tht if G is k-regulr iprtite grph, then B = H. ( mrks) Answer. Sine G is k-regulr, the sum of egree of verties on B sie is k B, whih is equl to the sum of egree of verties on H sie. Hene, k B = k H. This implies tht B = H. () Show tht if G is k-regulr iprtite grph, then the size of mximum mthing in G is B. (8 mrks) Answer. Let C e minimum vertex over. Clerly, B is vertex over. Hene, C B. Sine eh vertex hs egree k, y efinition of vertex over, k C k B. Hene, C = B. So, B is minimum vertex over. By Konig s theorem, the size of mximum mthing in G is B. 5. Let G = (N, E, w) e weighte unirete grph tht is onnete. Let C = (i, i,...,i k, i ) e yle of G. Let {i, i } e the unique mximum weight ege in yle C. Similrly, let {i, i } e the unique minimum weight ege in yle C. () Show tht {i, i } will not elong to ny MCST of G or give ounterexmple. (5 mrks)

4 Answer. Assume for ontrition tht {i, i } elongs to some MCST T of G. Hene, some ege {i j, i j+ } in C oes not elong to this MCST. Aing {i j, i j+ ) retes unique yle whih is roken y removing {i, i }. Sine the new grph ontins N eges, this is lso spnning tree. But the weight of {i j, i j+ ) is stritly less thn {i, i } implies tht T is not n MCST, ontrition. () Show tht {i, i } will elong to every MCST of G or give ounterexmple. (5 mrks) Answer. Figure gives ounterexmple, where the MCST is shown in rk eges ut the ege {, } is not prt of the MCST. Notie tht {, } is the minimum weight ege of the yle (,,, ). Figure : A ounterexmple 6. Fin mximum mthing n minimum vertex over of the unirete grph in Figure. (5 mrks) e f g Figure : An unirete grph Answer. The mximum mthing n minimum vertex over for the grph in Figure is shown in Figure in rk. Clerly, the shown eges onstitute mthing. It is

5 lso mximum mthing sine the mximum mthing nnot e of size greter thn. Further, the shown verties ( in numer) is minimum vertex over euse of Konig s theorem. f e g Figure : Mximum mthing n minimum vertex over 5

CS 491G Combinatorial Optimization Lecture Notes

CS 491G Combinatorial Optimization Lecture Notes CS 491G Comintoril Optimiztion Leture Notes Dvi Owen July 30, August 1 1 Mthings Figure 1: two possile mthings in simple grph. Definition 1 Given grph G = V, E, mthing is olletion of eges M suh tht e i,

More information

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

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

More information

CIT 596 Theory of Computation 1. Graphs and Digraphs

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

More information

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)}

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)

More information

Counting Paths Between Vertices. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs

Counting Paths Between Vertices. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs Isomorphism of Grphs Definition The simple grphs G 1 = (V 1, E 1 ) n G = (V, E ) re isomorphi if there is ijetion (n oneto-one n onto funtion) f from V 1 to V with the property tht n re jent in G 1 if

More information

Solutions to Problem Set #1

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

More information

18.06 Problem Set 4 Due Wednesday, Oct. 11, 2006 at 4:00 p.m. in 2-106

18.06 Problem Set 4 Due Wednesday, Oct. 11, 2006 at 4:00 p.m. in 2-106 8. Problem Set Due Wenesy, Ot., t : p.m. in - Problem Mony / Consier the eight vetors 5, 5, 5,..., () List ll of the one-element, linerly epenent sets forme from these. (b) Wht re the two-element, linerly

More information

Logic, Set Theory and Computability [M. Coppenbarger]

Logic, Set Theory and Computability [M. Coppenbarger] 14 Orer (Hnout) Definition 7-11: A reltion is qusi-orering (or preorer) if it is reflexive n trnsitive. A quisi-orering tht is symmetri is n equivlene reltion. A qusi-orering tht is nti-symmetri is n orer

More information

Graph Algorithms. Vertex set = { a,b,c,d } Edge set = { {a,c}, {b,c}, {c,d}, {b,d}} Figure 1: An example for a simple graph

Graph Algorithms. Vertex set = { a,b,c,d } Edge set = { {a,c}, {b,c}, {c,d}, {b,d}} Figure 1: An example for a simple graph Inin Institute of Informtion Tehnology Design n Mnufturing, Knheepurm, Chenni 00, Ini An Autonomous Institute uner MHRD, Govt of Ini http://www.iiitm..in COM 0T Design n Anlysis of Algorithms -Leture Notes

More information

1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. 1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the

More information

On a Class of Planar Graphs with Straight-Line Grid Drawings on Linear Area

On a Class of Planar Graphs with Straight-Line Grid Drawings on Linear Area Journl of Grph Algorithms n Applitions http://jg.info/ vol. 13, no. 2, pp. 153 177 (2009) On Clss of Plnr Grphs with Stright-Line Gri Drwings on Liner Are M. Rezul Krim 1,2 M. Siur Rhmn 1 1 Deprtment of

More information

Section 1.3 Triangles

Section 1.3 Triangles Se 1.3 Tringles 21 Setion 1.3 Tringles LELING TRINGLE The line segments tht form tringle re lled the sides of the tringle. Eh pir of sides forms n ngle, lled n interior ngle, nd eh tringle hs three interior

More information

Algebra 2 Semester 1 Practice Final

Algebra 2 Semester 1 Practice Final Alger 2 Semester Prtie Finl Multiple Choie Ientify the hoie tht est ompletes the sttement or nswers the question. To whih set of numers oes the numer elong?. 2 5 integers rtionl numers irrtionl numers

More information

22: Union Find. CS 473u - Algorithms - Spring April 14, We want to maintain a collection of sets, under the operations of:

22: Union Find. CS 473u - Algorithms - Spring April 14, We want to maintain a collection of sets, under the operations of: 22: Union Fin CS 473u - Algorithms - Spring 2005 April 14, 2005 1 Union-Fin We wnt to mintin olletion of sets, uner the opertions of: 1. MkeSet(x) - rete set tht ontins the single element x. 2. Fin(x)

More information

CS311 Computational Structures Regular Languages and Regular Grammars. Lecture 6

CS311 Computational Structures Regular Languages and Regular Grammars. Lecture 6 CS311 Computtionl Strutures Regulr Lnguges nd Regulr Grmmrs Leture 6 1 Wht we know so fr: RLs re losed under produt, union nd * Every RL n e written s RE, nd every RE represents RL Every RL n e reognized

More information

Section 2.3. Matrix Inverses

Section 2.3. Matrix Inverses Mtri lger Mtri nverses Setion.. Mtri nverses hree si opertions on mtries, ition, multiplition, n sutrtion, re nlogues for mtries of the sme opertions for numers. n this setion we introue the mtri nlogue

More information

Lecture 6: Coding theory

Lecture 6: Coding theory Leture 6: Coing theory Biology 429 Crl Bergstrom Ferury 4, 2008 Soures: This leture loosely follows Cover n Thoms Chpter 5 n Yeung Chpter 3. As usul, some of the text n equtions re tken iretly from those

More information

arxiv: v2 [math.co] 31 Oct 2016

arxiv: v2 [math.co] 31 Oct 2016 On exlue minors of onnetivity 2 for the lss of frme mtrois rxiv:1502.06896v2 [mth.co] 31 Ot 2016 Mtt DeVos Dryl Funk Irene Pivotto Astrt We investigte the set of exlue minors of onnetivity 2 for the lss

More information

Exercise sheet 6: Solutions

Exercise sheet 6: Solutions Eerise sheet 6: Solutions Cvet emptor: These re merel etended hints, rther thn omplete solutions. 1. If grph G hs hromti numer k > 1, prove tht its verte set n e prtitioned into two nonempt sets V 1 nd

More information

Lecture 2: Cayley Graphs

Lecture 2: Cayley Graphs Mth 137B Professor: Pri Brtlett Leture 2: Cyley Grphs Week 3 UCSB 2014 (Relevnt soure mteril: Setion VIII.1 of Bollos s Moern Grph Theory; 3.7 of Gosil n Royle s Algeri Grph Theory; vrious ppers I ve re

More information

Discrete Structures Lecture 11

Discrete Structures Lecture 11 Introdution Good morning. In this setion we study funtions. A funtion is mpping from one set to nother set or, perhps, from one set to itself. We study the properties of funtions. A mpping my not e funtion.

More information

Answers and Solutions to (Some Even Numbered) Suggested Exercises in Chapter 11 of Grimaldi s Discrete and Combinatorial Mathematics

Answers and Solutions to (Some Even Numbered) Suggested Exercises in Chapter 11 of Grimaldi s Discrete and Combinatorial Mathematics Answers n Solutions to (Some Even Numere) Suggeste Exercises in Chpter 11 o Grimli s Discrete n Comintoril Mthemtics Section 11.1 11.1.4. κ(g) = 2. Let V e = {v : v hs even numer o 1 s} n V o = {v : v

More information

On the Spectra of Bipartite Directed Subgraphs of K 4

On the Spectra of Bipartite Directed Subgraphs of K 4 On the Spetr of Biprtite Direte Sugrphs of K 4 R. C. Bunge, 1 S. I. El-Znti, 1, H. J. Fry, 1 K. S. Kruss, 2 D. P. Roerts, 3 C. A. Sullivn, 4 A. A. Unsiker, 5 N. E. Witt 6 1 Illinois Stte University, Norml,

More information

Probability. b a b. a b 32.

Probability. b a b. a b 32. Proility If n event n hppen in '' wys nd fil in '' wys, nd eh of these wys is eqully likely, then proility or the hne, or its hppening is, nd tht of its filing is eg, If in lottery there re prizes nd lnks,

More information

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

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

More information

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

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

More information

Part 4. Integration (with Proofs)

Part 4. Integration (with Proofs) Prt 4. Integrtion (with Proofs) 4.1 Definition Definition A prtition P of [, b] is finite set of points {x 0, x 1,..., x n } with = x 0 < x 1

More information

April 8, 2017 Math 9. Geometry. Solving vector problems. Problem. Prove that if vectors and satisfy, then.

April 8, 2017 Math 9. Geometry. Solving vector problems. Problem. Prove that if vectors and satisfy, then. pril 8, 2017 Mth 9 Geometry Solving vetor prolems Prolem Prove tht if vetors nd stisfy, then Solution 1 onsider the vetor ddition prllelogrm shown in the Figure Sine its digonls hve equl length,, the prllelogrm

More information

CS261: A Second Course in Algorithms Lecture #5: Minimum-Cost Bipartite Matching

CS261: A Second Course in Algorithms Lecture #5: Minimum-Cost Bipartite Matching CS261: A Seon Course in Algorithms Leture #5: Minimum-Cost Biprtite Mthing Tim Roughgren Jnury 19, 2016 1 Preliminries Figure 1: Exmple of iprtite grph. The eges {, } n {, } onstitute mthing. Lst leture

More information

Monochromatic Plane Matchings in Bicolored Point Set

Monochromatic Plane Matchings in Bicolored Point Set CCCG 2017, Ottw, Ontrio, July 26 28, 2017 Monohromti Plne Mthings in Biolore Point Set A. Krim Au-Affsh Sujoy Bhore Pz Crmi Astrt Motivte y networks interply, we stuy the prolem of omputing monohromti

More information

T b a(f) [f ] +. P b a(f) = Conclude that if f is in AC then it is the difference of two monotone absolutely continuous functions.

T b a(f) [f ] +. P b a(f) = Conclude that if f is in AC then it is the difference of two monotone absolutely continuous functions. Rel Vribles, Fll 2014 Problem set 5 Solution suggestions Exerise 1. Let f be bsolutely ontinuous on [, b] Show tht nd T b (f) P b (f) f (x) dx [f ] +. Conlude tht if f is in AC then it is the differene

More information

Lecture 3 ( ) (translated and slightly adapted from lecture notes by Martin Klazar)

Lecture 3 ( ) (translated and slightly adapted from lecture notes by Martin Klazar) Lecture 3 (5.3.2018) (trnslted nd slightly dpted from lecture notes by Mrtin Klzr) Riemnn integrl Now we define precisely the concept of the re, in prticulr, the re of figure U(, b, f) under the grph of

More information

Maximum size of a minimum watching system and the graphs achieving the bound

Maximum size of a minimum watching system and the graphs achieving the bound Mximum size of minimum wthing system n the grphs hieving the oun Tille mximum un système e ontrôle minimum et les grphes tteignnt l orne Dvi Auger Irène Chron Olivier Hury Antoine Lostein 00D0 Mrs 00 Déprtement

More information

Lecture 4: Graph Theory and the Four-Color Theorem

Lecture 4: Graph Theory and the Four-Color Theorem CCS Disrete II Professor: Pri Brtlett Leture 4: Grph Theory n the Four-Color Theorem Week 4 UCSB 2015 Through the rest of this lss, we re going to refer frequently to things lle grphs! If you hen t seen

More information

QUADRATIC EQUATION. Contents

QUADRATIC EQUATION. Contents 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,

More information

50 AMC Lectures Problem Book 2 (36) Substitution Method

50 AMC Lectures Problem Book 2 (36) Substitution Method 0 AMC Letures Prolem Book Sustitution Metho PROBLEMS Prolem : Solve for rel : 9 + 99 + 9 = Prolem : Solve for rel : 0 9 8 8 Prolem : Show tht if 8 Prolem : Show tht + + if rel numers,, n stisf + + = Prolem

More information

AP Calculus AB Unit 4 Assessment

AP Calculus AB Unit 4 Assessment Clss: Dte: 0-04 AP Clulus AB Unit 4 Assessment Multiple Choie Identify the hoie tht best ompletes the sttement or nswers the question. A lultor my NOT be used on this prt of the exm. (6 minutes). The slope

More information

Lecture 11 Binary Decision Diagrams (BDDs)

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

More information

6.5 Improper integrals

6.5 Improper integrals Eerpt from "Clulus" 3 AoPS In. www.rtofprolemsolving.om 6.5. IMPROPER INTEGRALS 6.5 Improper integrls As we ve seen, we use the definite integrl R f to ompute the re of the region under the grph of y =

More information

September 30, :24 WSPC/Guidelines Y4Spanner

September 30, :24 WSPC/Guidelines Y4Spanner Septemer 30, 2011 12:24 WSPC/Guielines Y4Spnner Interntionl Journl of Computtionl Geometry & Applitions Worl Sientifi Pulishing Compny π/2-angle YAO GAPHS AE SPANNES POSENJIT BOSE Shool of Computer Siene,

More information

Green s Theorem. (2x e y ) da. (2x e y ) dx dy. x 2 xe y. (1 e y ) dy. y=1. = y e y. y=0. = 2 e

Green s Theorem. (2x e y ) da. (2x e y ) dx dy. x 2 xe y. (1 e y ) dy. y=1. = y e y. y=0. = 2 e Green s Theorem. Let be the boundry of the unit squre, y, oriented ounterlokwise, nd let F be the vetor field F, y e y +, 2 y. Find F d r. Solution. Let s write P, y e y + nd Q, y 2 y, so tht F P, Q. Let

More information

MATH 409 Advanced Calculus I Lecture 22: Improper Riemann integrals.

MATH 409 Advanced Calculus I Lecture 22: Improper Riemann integrals. MATH 409 Advned Clulus I Leture 22: Improper Riemnn integrls. Improper Riemnn integrl If funtion f : [,b] R is integrble on [,b], then the funtion F(x) = x f(t)dt is well defined nd ontinuous on [,b].

More information

Nondeterministic Automata vs Deterministic Automata

Nondeterministic Automata vs Deterministic Automata Nondeterministi Automt vs Deterministi Automt We lerned tht NFA is onvenient model for showing the reltionships mong regulr grmmrs, FA, nd regulr expressions, nd designing them. However, we know tht n

More information

The vertex leafage of chordal graphs

The vertex leafage of chordal graphs The vertex lefge of horl grphs Steven Chplik, Jurj Stho b Deprtment of Physis n Computer Siene, Wilfri Lurier University, 75 University Ave. West, Wterloo, Ontrio N2L 3C5, Cn b DIMAP n Mthemtis Institute,

More information

Section 4.4. Green s Theorem

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

More information

If the numbering is a,b,c,d 1,2,3,4, then the matrix representation is as follows:

If the numbering is a,b,c,d 1,2,3,4, then the matrix representation is as follows: Reltions. Solutions 1. ) true; ) true; ) flse; ) true; e) flse; f) true; g) flse; h) true; 2. 2 A B 3. Consier ll reltions tht o not inlue the given pir s n element. Oviously, the rest of the reltions

More information

Proportions: A ratio is the quotient of two numbers. For example, 2 3

Proportions: A ratio is the quotient of two numbers. For example, 2 3 Proportions: rtio is the quotient of two numers. For exmple, 2 3 is rtio of 2 n 3. n equlity of two rtios is proportion. For exmple, 3 7 = 15 is proportion. 45 If two sets of numers (none of whih is 0)

More information

Lecture 8: Abstract Algebra

Lecture 8: Abstract Algebra Mth 94 Professor: Pri Brtlett Leture 8: Astrt Alger Week 8 UCSB 2015 This is the eighth week of the Mthemtis Sujet Test GRE prep ourse; here, we run very rough-n-tumle review of strt lger! As lwys, this

More information

Mathematical Proofs Table of Contents

Mathematical Proofs Table of Contents Mthemtil Proofs Tle of Contents Proof Stnr Pge(s) Are of Trpezoi 7MG. Geometry 8.0 Are of Cirle 6MG., 9 6MG. 7MG. Geometry 8.0 Volume of Right Cirulr Cyliner 6MG. 4 7MG. Geometry 8.0 Volume of Sphere Geometry

More information

APPENDIX. Precalculus Review D.1. Real Numbers and the Real Number Line

APPENDIX. Precalculus Review D.1. Real Numbers and the Real Number Line APPENDIX D Preclculus Review APPENDIX D.1 Rel Numers n the Rel Numer Line Rel Numers n the Rel Numer Line Orer n Inequlities Asolute Vlue n Distnce Rel Numers n the Rel Numer Line Rel numers cn e represente

More information

First Midterm Examination

First Midterm Examination 24-25 Fll Semester First Midterm Exmintion ) Give the stte digrm of DFA tht recognizes the lnguge A over lphet Σ = {, } where A = {w w contins or } 2) The following DFA recognizes the lnguge B over lphet

More information

Discrete Structures, Test 2 Monday, March 28, 2016 SOLUTIONS, VERSION α

Discrete Structures, Test 2 Monday, March 28, 2016 SOLUTIONS, VERSION α Disrete Strutures, Test 2 Mondy, Mrh 28, 2016 SOLUTIONS, VERSION α α 1. (18 pts) Short nswer. Put your nswer in the ox. No prtil redit. () Consider the reltion R on {,,, d with mtrix digrph of R.. Drw

More information

Vidyalankar S.E. Sem. III [CMPN] Discrete Structures Prelim Question Paper Solution

Vidyalankar S.E. Sem. III [CMPN] Discrete Structures Prelim Question Paper Solution S.E. Sem. III [CMPN] Discrete Structures Prelim Question Pper Solution 1. () (i) Disjoint set wo sets re si to be isjoint if they hve no elements in common. Exmple : A = {0, 4, 7, 9} n B = {3, 17, 15}

More information

Prefix-Free Regular-Expression Matching

Prefix-Free Regular-Expression Matching Prefix-Free Regulr-Expression Mthing Yo-Su Hn, Yjun Wng nd Derik Wood Deprtment of Computer Siene HKUST Prefix-Free Regulr-Expression Mthing p.1/15 Pttern Mthing Given pttern P nd text T, find ll sustrings

More information

Bisimulation, Games & Hennessy Milner logic

Bisimulation, Games & Hennessy Milner logic Bisimultion, Gmes & Hennessy Milner logi Leture 1 of Modelli Mtemtii dei Proessi Conorrenti Pweł Soboiński Univeristy of Southmpton, UK Bisimultion, Gmes & Hennessy Milner logi p.1/32 Clssil lnguge theory

More information

Project 6: Minigoals Towards Simplifying and Rewriting Expressions

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

More information

NON-DETERMINISTIC FSA

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

More information

16z z q. q( B) Max{2 z z z z B} r z r z r z r z B. John Riley 19 October Econ 401A: Microeconomic Theory. Homework 2 Answers

16z z q. q( B) Max{2 z z z z B} r z r z r z r z B. John Riley 19 October Econ 401A: Microeconomic Theory. Homework 2 Answers John Riley 9 Otober 6 Eon 4A: Miroeonomi Theory Homework Answers Constnt returns to sle prodution funtion () If (,, q) S then 6 q () 4 We need to show tht (,, q) S 6( ) ( ) ( q) q [ q ] 4 4 4 4 4 4 Appeling

More information

QUADRATIC EQUATION EXERCISE - 01 CHECK YOUR GRASP

QUADRATIC EQUATION EXERCISE - 01 CHECK YOUR GRASP 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

More information

Graph width-parameters and algorithms

Graph width-parameters and algorithms Grph width-prmeters nd lgorithms Jisu Jeong (KAIST) joint work with Sigve Hortemo Sæther nd Jn Arne Telle (University of Bergen) 2015 KMS Annul Meeting 2015.10.24. YONSEI UNIVERSITY Grph width-prmeters

More information

First Midterm Examination

First Midterm Examination Çnky University Deprtment of Computer Engineering 203-204 Fll Semester First Midterm Exmintion ) Design DFA for ll strings over the lphet Σ = {,, c} in which there is no, no nd no cc. 2) Wht lnguge does

More information

Linear choosability of graphs

Linear choosability of graphs Liner hoosility of grphs Louis Esperet, Mikel Montssier, André Rspud To ite this version: Louis Esperet, Mikel Montssier, André Rspud. Liner hoosility of grphs. Stefn Felsner. 2005 Europen Conferene on

More information

CSE 332. Sorting. Data Abstractions. CSE 332: Data Abstractions. QuickSort Cutoff 1. Where We Are 2. Bounding The MAXIMUM Problem 4

CSE 332. Sorting. Data Abstractions. CSE 332: Data Abstractions. QuickSort Cutoff 1. Where We Are 2. Bounding The MAXIMUM Problem 4 Am Blnk Leture 13 Winter 2016 CSE 332 CSE 332: Dt Astrtions Sorting Dt Astrtions QuikSort Cutoff 1 Where We Are 2 For smll n, the reursion is wste. The onstnts on quik/merge sort re higher thn the ones

More information

MAT 403 NOTES 4. f + f =

MAT 403 NOTES 4. f + f = MAT 403 NOTES 4 1. Fundmentl Theorem o Clulus We will proo more generl version o the FTC thn the textook. But just like the textook, we strt with the ollowing proposition. Let R[, ] e the set o Riemnn

More information

Comparing the Pre-image and Image of a Dilation

Comparing the Pre-image and Image of a Dilation hpter Summry Key Terms Postultes nd Theorems similr tringles (.1) inluded ngle (.2) inluded side (.2) geometri men (.) indiret mesurement (.6) ngle-ngle Similrity Theorem (.2) Side-Side-Side Similrity

More information

The University of Nottingham SCHOOL OF COMPUTER SCIENCE A LEVEL 2 MODULE, SPRING SEMESTER MACHINES AND THEIR LANGUAGES ANSWERS

The University of Nottingham SCHOOL OF COMPUTER SCIENCE A LEVEL 2 MODULE, SPRING SEMESTER MACHINES AND THEIR LANGUAGES ANSWERS The University of ottinghm SCHOOL OF COMPUTR SCIC A LVL 2 MODUL, SPRIG SMSTR 2015 2016 MACHIS AD THIR LAGUAGS ASWRS Time llowed TWO hours Cndidtes my omplete the front over of their nswer ook nd sign their

More information

New and Improved Spanning Ratios for Yao Graphs

New and Improved Spanning Ratios for Yao Graphs New n Improve Spnning Rtios for Yo Grphs Luis Br Déprtement Informtique Université Lire e Bruxelles lrfl@ul..e Rolf Fgererg Deprtment of Computer Siene University of Southern Denmrk rolf@im.su.k Anré vn

More information

MULTIPLE CHOICE QUESTIONS

MULTIPLE CHOICE QUESTIONS . Qurti Equtions C h p t e r t G l n e The generl form of qurti polynomil is + + = 0 where,, re rel numers n. Zeroes of qurti polynomil n e otine y equtions given eqution equl to zero n solution it. Methos

More information

a) Read over steps (1)- (4) below and sketch the path of the cycle on a P V plot on the graph below. Label all appropriate points.

a) Read over steps (1)- (4) below and sketch the path of the cycle on a P V plot on the graph below. Label all appropriate points. Prole 3: Crnot Cyle of n Idel Gs In this prole, the strting pressure P nd volue of n idel gs in stte, re given he rtio R = / > of the volues of the sttes nd is given Finlly onstnt γ = 5/3 is given You

More information

Subsequence Automata with Default Transitions

Subsequence Automata with Default Transitions Susequene Automt with Defult Trnsitions Philip Bille, Inge Li Gørtz, n Freerik Rye Skjoljensen Tehnil University of Denmrk {phi,inge,fskj}@tu.k Astrt. Let S e string of length n with hrters from n lphet

More information

Surds and Indices. Surds and Indices. Curriculum Ready ACMNA: 233,

Surds and Indices. Surds and Indices. Curriculum Ready ACMNA: 233, Surs n Inies Surs n Inies Curriulum Rey ACMNA:, 6 www.mthletis.om Surs SURDS & & Inies INDICES Inies n surs re very losely relte. A numer uner (squre root sign) is lle sur if the squre root n t e simplifie.

More information

I 3 2 = I I 4 = 2A

I 3 2 = I I 4 = 2A ECE 210 Eletril Ciruit Anlysis University of llinois t Chigo 2.13 We re ske to use KCL to fin urrents 1 4. The key point in pplying KCL in this prolem is to strt with noe where only one of the urrents

More information

y = c 2 MULTIPLE CHOICE QUESTIONS (MCQ's) (Each question carries one mark) is...

y = c 2 MULTIPLE CHOICE QUESTIONS (MCQ's) (Each question carries one mark) is... . Liner Equtions in Two Vriles C h p t e r t G l n e. Generl form of liner eqution in two vriles is x + y + 0, where 0. When we onsier system of two liner equtions in two vriles, then suh equtions re lle

More information

System Validation (IN4387) November 2, 2012, 14:00-17:00

System Validation (IN4387) November 2, 2012, 14:00-17:00 System Vlidtion (IN4387) Novemer 2, 2012, 14:00-17:00 Importnt Notes. The exmintion omprises 5 question in 4 pges. Give omplete explntion nd do not onfine yourself to giving the finl nswer. Good luk! Exerise

More information

Introduction to Olympiad Inequalities

Introduction to Olympiad Inequalities Introdution to Olympid Inequlities Edutionl Studies Progrm HSSP Msshusetts Institute of Tehnology Snj Simonovikj Spring 207 Contents Wrm up nd Am-Gm inequlity 2. Elementry inequlities......................

More information

A Lower Bound for the Length of a Partial Transversal in a Latin Square, Revised Version

A Lower Bound for the Length of a Partial Transversal in a Latin Square, Revised Version A Lower Bound for the Length of Prtil Trnsversl in Ltin Squre, Revised Version Pooy Htmi nd Peter W. Shor Deprtment of Mthemtil Sienes, Shrif University of Tehnology, P.O.Bo 11365-9415, Tehrn, Irn Deprtment

More information

Graph States EPIT Mehdi Mhalla (Calgary, Canada) Simon Perdrix (Grenoble, France)

Graph States EPIT Mehdi Mhalla (Calgary, Canada) Simon Perdrix (Grenoble, France) Grph Sttes EPIT 2005 Mehdi Mhll (Clgry, Cnd) Simon Perdrix (Grenole, Frne) simon.perdrix@img.fr Grph Stte: Introdution A grph-sed representtion of the entnglement of some (lrge) quntum stte. Verties: quits

More information

CHAPTER 4: DETERMINANTS

CHAPTER 4: DETERMINANTS CHAPTER 4: DETERMINANTS MARKS WEIGHTAGE 0 mrks NCERT Importnt Questions & Answers 6. If, then find the vlue of. 8 8 6 6 Given tht 8 8 6 On epnding both determinnts, we get 8 = 6 6 8 36 = 36 36 36 = 0 =

More information

SOME COPLANAR POINTS IN TETRAHEDRON

SOME COPLANAR POINTS IN TETRAHEDRON Journl of Pure n Applie Mthemtis: Avnes n Applitions Volume 16, Numer 2, 2016, Pges 109-114 Aville t http://sientifivnes.o.in DOI: http://x.oi.org/10.18642/jpm_7100121752 SOME COPLANAR POINTS IN TETRAHEDRON

More information

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

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

More information

A Primer on Continuous-time Economic Dynamics

A Primer on Continuous-time Economic Dynamics Eonomis 205A Fll 2008 K Kletzer A Primer on Continuous-time Eonomi Dnmis A Liner Differentil Eqution Sstems (i) Simplest se We egin with the simple liner first-orer ifferentil eqution The generl solution

More information

CS 330 Formal Methods and Models Dana Richards, George Mason University, Spring 2016 Quiz Solutions

CS 330 Formal Methods and Models Dana Richards, George Mason University, Spring 2016 Quiz Solutions CS 330 Forml Methods nd Models Dn Richrds, George Mson University, Spring 2016 Quiz Solutions Quiz 1, Propositionl Logic Dte: Ferury 9 1. (4pts) ((p q) (q r)) (p r), prove tutology using truth tles. p

More information

Directed acyclic graphs with the unique dipath property

Directed acyclic graphs with the unique dipath property Direte yli grphs with the unique ipth property Jen-Clue Bermon, Mihel Cosnr, Stéphne érennes To ite this version: Jen-Clue Bermon, Mihel Cosnr, Stéphne érennes. Direte yli grphs with the unique ipth property.

More information

Necessary and sucient conditions for some two. Abstract. Further we show that the necessary conditions for the existence of an OD(44 s 1 s 2 )

Necessary and sucient conditions for some two. Abstract. Further we show that the necessary conditions for the existence of an OD(44 s 1 s 2 ) Neessry n suient onitions for some two vrile orthogonl esigns in orer 44 C. Koukouvinos, M. Mitrouli y, n Jennifer Seerry z Deite to Professor Anne Penfol Street Astrt We give new lgorithm whih llows us

More information

arxiv: v1 [cs.cg] 28 Apr 2009

arxiv: v1 [cs.cg] 28 Apr 2009 Orienttion-Constrine Retngulr Lyouts Dvi Eppstein 1 n Elen Mumfor 2 1 Deprtment of Computer Siene, University of Cliforni, Irvine, USA 2 Deprtment of Mthemtis n Computer Siene, TU Einhoven, The Netherlns

More information

Can one hear the shape of a drum?

Can one hear the shape of a drum? Cn one her the shpe of drum? After M. K, C. Gordon, D. We, nd S. Wolpert Corentin Lén Università Degli Studi di Torino Diprtimento di Mtemti Giuseppe Peno UNITO Mthemtis Ph.D Seminrs Mondy 23 My 2016 Motivtion:

More information

A Study on the Properties of Rational Triangles

A Study on the Properties of Rational Triangles Interntionl Journl of Mthemtis Reserh. ISSN 0976-5840 Volume 6, Numer (04), pp. 8-9 Interntionl Reserh Pulition House http://www.irphouse.om Study on the Properties of Rtionl Tringles M. Q. lm, M.R. Hssn

More information

Boolean Algebra cont. The digital abstraction

Boolean Algebra cont. The digital abstraction Boolen Alger ont The igitl strtion Theorem: Asorption Lw For every pir o elements B. + =. ( + ) = Proo: () Ientity Distriutivity Commuttivity Theorem: For ny B + = Ientity () ulity. Theorem: Assoitive

More information

Harvard University Computer Science 121 Midterm October 23, 2012

Harvard University Computer Science 121 Midterm October 23, 2012 Hrvrd University Computer Science 121 Midterm Octoer 23, 2012 This is closed-ook exmintion. You my use ny result from lecture, Sipser, prolem sets, or section, s long s you quote it clerly. The lphet is

More information

Technische Universität München Winter term 2009/10 I7 Prof. J. Esparza / J. Křetínský / M. Luttenberger 11. Februar Solution

Technische Universität München Winter term 2009/10 I7 Prof. J. Esparza / J. Křetínský / M. Luttenberger 11. Februar Solution Tehnishe Universität Münhen Winter term 29/ I7 Prof. J. Esprz / J. Křetínský / M. Luttenerger. Ferur 2 Solution Automt nd Forml Lnguges Homework 2 Due 5..29. Exerise 2. Let A e the following finite utomton:

More information

Graph Theory. Dr. Saad El-Zanati, Faculty Mentor Ryan Bunge Graduate Assistant Illinois State University REU. Graph Theory

Graph Theory. Dr. Saad El-Zanati, Faculty Mentor Ryan Bunge Graduate Assistant Illinois State University REU. Graph Theory Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion Grph Theory Dniel Gibson, Concordi University Jckelyn Ngel, Dominicn University Benjmin Stnley, New Mexico Stte University Allison

More information

CHENG Chun Chor Litwin The Hong Kong Institute of Education

CHENG Chun Chor Litwin The Hong Kong Institute of Education PE-hing Mi terntionl onferene IV: novtion of Mthemtis Tehing nd Lerning through Lesson Study- onnetion etween ssessment nd Sujet Mtter HENG hun hor Litwin The Hong Kong stitute of Edution Report on using

More information

n f(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as n f(x i ) x; f(x) dx = lim i=1

n f(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as n f(x i ) x; f(x) dx = lim i=1 The Fundmentl Theorem of Clculus As we continue to study the re problem, let s think bck to wht we know bout computing res of regions enclosed by curves. If we wnt to find the re of the region below the

More information

Solutions to Assignment 1

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

More information

CS 573 Automata Theory and Formal Languages

CS 573 Automata Theory and Formal Languages Non-determinism Automt Theory nd Forml Lnguges Professor Leslie Lnder Leture # 3 Septemer 6, 2 To hieve our gol, we need the onept of Non-deterministi Finite Automton with -moves (NFA) An NFA is tuple

More information

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

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

More information

Section 6.1 Definite Integral

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

More information

On the properties of the two-sided Laplace transform and the Riemann hypothesis

On the properties of the two-sided Laplace transform and the Riemann hypothesis On the properties of the two-sie Lple trnsform n the Riemnn hypothesis Seong Won Ch, Ph.D. swh@gu.eu Astrt We will show interesting properties of two-sie Lple trnsform, minly of positive even funtions.

More information

Linear Algebra Introduction

Linear Algebra Introduction Introdution Wht is Liner Alger out? Liner Alger is rnh of mthemtis whih emerged yers k nd ws one of the pioneer rnhes of mthemtis Though, initilly it strted with solving of the simple liner eqution x +

More information

Suppose we want to find the area under the parabola and above the x axis, between the lines x = 2 and x = -2.

Suppose we want to find the area under the parabola and above the x axis, between the lines x = 2 and x = -2. Mth 43 Section 6. Section 6.: 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

More information