Week 34: Permutations and Combinations


 Dorthy Douglas
 4 years ago
 Views:
Transcription
1 Week 34: Pemutations and Combinations Febuay 24, Two Counting Pinciples Addition Pinciple Let S 1, S 2,, S m be disjoint subsets of a finite set S If S S 1 S 2 S m, then S S 1 + S S m Multiplication Pinciple Let S 1, S 2,, S m be finite sets and S S 1 S 2 S m Then S S 1 S 2 S m Example 11 Detemine the numbe of positive integes which ae factos of the numbe The numbe 33 can be factoed into 3 11 By the unique factoization theoem of positive integes, each facto of the given numbe is of the fom 3 i 5 j 7 k 11 l 13 m, whee 0 i 8, 0 j 3, 0 k 9, 0 l 8, and 0 m 1 Thus the numbe of factos is Example 12 How many twodigit numbes have distinct and nonzeo digits? A twodigit numbe ab can be egaded as an odeed pai (a, b) whee a is the tens digit and b is the units digit The digits in the poblem ae equied to satisfy a 0, b 0, a b 1
2 The digit a has 9 choices, and fo each fixed a the digit b has 8 choices So the answe is The answe can be obtained in anothe way: Thee ae 90 twodigit numbes, ie, 10, 11, 12,, 99 Howeve, the 9 numbes 10, 20,, 90 should be excluded; anothe 9 numbes 11, 22,, 99 should be also excluded So the answe is Example 13 How many odd numbes between 1000 and 9999 have distinct digits? A numbe a 1 a 2 a 3 a 4 between 1000 and 9999 can be viewed as an odeed tuple (a 1, a 2, a 3, a 4 ) Since a 1 a 2 a 3 a and a 1 a 2 a 3 a 4 is odd, then a 1 1, 2,, 9 and a 4 1, 3, 5, 7, 9 Since a 1, a 2, a 3, a 4 ae distinct, we conclude: a 4 has 5 choices; when a 4 is fixed, a 1 has 8 ( 9 1) choices; when a 1 and a 4 ae fixed, a 2 has 8 ( 10 2) choices; and when a 1, a 2, a 4 ae fixed, a 3 has 7 ( 10 3) choices Thus the answe is Example 14 In how many ways to make a basket of fuit fom 6 oanges, 7 apples, and 8 bananas so that the basket contains at least two apples and one banana? Let a 1, a 2, a 3 be the numbe of oanges, apples, and bananas in the basket espectively Then 0 a 1 6, 2 a 2 7, and 1 a 3 8, ie, a 1 has 7 choices, a 2 has 6 choices, and a 3 has 8 choices Thus the answe is Geneal Ideas about Counting: Count the numbe of odeed aangements o odeed selections of objects (a) without epetition, (b) with epetition allowed Count the numbe of unodeed aangements o unodeed selections of objects (a) without epetition, 2
3 (b) with epetition allowed A multiset M is a collection whose membes need not be distinct Fo instance, the collection M {a, a, b, b, c, d, d, d, 1, 2, 2, 2, 3, 3, 3, 3} is a multiset; and sometimes it is convenient to wite M {2a, 2b, c, 3d, 1, 3 2, 4 3} A multiset M ove a set S can be viewed as a function v : S N fom S to the set N of nonnegative integes; each element x S is epeated v(x) times in M; we wite M (S, v) Example 15 How many integes between 0 and 10,000 have exactly one digit equal to 5? Fist Method Let S be the set of such numbes, and let S i be the set of such numbes having exactly i digits, 1 i 4 Clealy, S 1 1 Fo S 2, if the tens is 5, then the units has 9 choices; if the units is 5, then the tens has 8 choices; thus S Fo S 3, if the tens is 5, then the units has 9 choices and the hundeds has 8 choices; if the hundeds is 5, then both tens and the units have 9 choices; if the units is 5, then the tens has 9 choices and hundeds has 8 choices; thus S Fo S 4, if the thousands is 5, then each of the othe thee digits has 9 choices; if the hundeds o tens o units is 5, then the thousands has 8 choices, each of the othe two digits has 9 choices; thus S , 673 Theefoe S S 1 + S 2 + S 3 + S , 673 2, 916 Second Method Let us wite any intege with less than 5 digits in a fomal 5digit fom by adding zeos in the font Fo instance, we wite 35 as 00035, 836 as Let S i be the set of integes of S whose ith digit is 5, 1 i 4 Then S i Thus S , 916 3
4 Example 16 How many distinct 5digit numeals can be constucted out of the digits 1, 1, 1, 6, 8? The digit 6 can be located in any of the 5 positions; then 8 can be located in 4 positions Thus the answe is Pemutation of Sets Definition 21 An pemutation of n objects is a linealy odeed selection of objects fom a set of n objects The numbe of pemutations of n objects is denoted by P (n, ) An npemutation of n objects is just called a pemutation of n objects The numbe of pemutations of n objects is denoted by n!, ead n factoial Theoem 22 The numbe of pemutations of an nset equals n! P (n, ) n(n 1) (n + 1) (n )! Coollay 23 The numbe of pemutations of an nset is n! Example 21 Find the numbe of ways to put the numbes 1, 2,, 8 into the squaes of 6by6 gid so that each squae contains at most one numbe Thee ae 36 squaes in the 6by6 gid below We label the squaes by the numbes 1, 2,, 36 as follows: The filling patten on the ight can be viewed as an 8pemutation (35, 22, 7, 16, 3, 21, 11, 26) of {1, 2,, 36} Thus the answe is 36! P (36, 8) (36 8)! 36! 28! 4
5 Example 22 What is the numbe of ways to aange the 26 alphabets so that no two of the vowels a, e, i, o, and u occu next to each othe? We fist have the 21 consonants aanged abitaily and thee ae 21! ways to do so Fo each such 21pemutation, we aange the 5 vowels a, e, i, o, u in 22 positions between consonants; thee ae P (22, 5) ways of such aangement Thus the answe is 21! P (22, 5) 21! 22! 17! Example 23 Find the numbe of 7digit numbes in base 10 such that all digits ae nonzeo, distinct, and the digits 8 and 9 do not appea next to each othe Fist Method The numbes in question can be viewed as 7pemutations of {1, 2,, 9} with cetain estictions Such pemutations can be divided into thee types: (i) pemutations without 8 and 9; (ii) pemutations with eithe 8 o 9 but not both; and (iii) pemutations with both 8 and 9, but not next to each othe (i) Thee ae P (7, 7) 7! 5, 040 such pemutations (ii) Thee ae P (7, 6) 6pemutations of {1, 2,, 7} Thus thee ae 2 7 P (7, 6) 2 7 7! 1! 70, 560 such pemutations (iii) Fo each 5pemutation of {1, 2,, 7}, thee ae 6 ways to inset 8 in it, and then thee ae 5 ways to inset 9 Thus thee ae 6 5 P (7, 5) 75, 600 Theefoe the answe is 5, , , , 200 Second Method Let S be the set of 7pemutations of {1, 2,, 9} Let A be the subset of 7pemutations of S in the poblem Then Ā is the set of 7pemutations of S such that eithe 89 o 98 appeas somewhee We may think of 89 and 98 as a single object in whole, then Ā can be viewed as the set of 6pemutations of {1, 2, 3, 4, 5, 6, 7, 89} with 89 and 6pemutations of {1, 2, 3, 4, 5, 6, 7, 98} with 98 It follows that Ā 2(P (8, 6) P (7, 6)) Thus the answe is A P (9, 7) 2 ( P (8, 6) P (7, 6) ) 9! ( 8! 2! 2 2! 7! ) 151, 200 1! 5
6 The set Ā can be obtained by taking all 5pemutations of {1, 2,, 7} fist and then by adding 89 o 98 to one of 6 positions of the 5pemutations Then A P (9, 7) 2P (7, 5) 6 9! 2! 7! 6 151, 200 A cicula pemutation of a set S is an odeed objects of S aanged as a cicle; thee is no the beginning object and the ending object Theoem 24 The numbe of cicula pemutations of an nset equals P (n, ) n! (n )! Poof Let S be an nset Let X be the set of all pemutations of S, and let Y be the set of all cicula pemutations of S Define a function f : X Y, a 1 a 2 a a a 1 a 2 a a 1 a 2 as follows: Fo each pemutation a 1 a 2 a of S, f(a 1 a 2 a ) is the cicula pemutation such that a 1 a 2 a a 1 a 2 is counteclockwise on a cicle Clealy, f is sujective Moeove, thee ae exactly pemutations sent to one cicula pemutation In fact, the pemutations a 1 a 2 a 3 a 1 a, a 2 a 3 a 4 a a 1,, a a 1 a 2 a 2 a 1 ae sent to the same cicula pemutation Thus the answe is Y X P (n, ) Coollay 25 The numbe of cicula pemutations of an nset is (n 1)! Example 24 Twelve people, including two who do no wish to sit next to each othe, ae to be seated at a ound table How many cicula seating plans can be made? 6
7 Fist Method We may have 11 people (including one of the two unhappy pesons but not both) to sit fist; thee ae 10! such seating plans Next the second unhappy peson can sit anywhee except the left side and ight side of the fist unhappy peson; thee ae 9 choices fo the second unhappy peson Thus the answe is 9 10! Second Method Thee ae 11! seating plans fo the 12 people with no estiction We need to exclude those seating plans that the unhappy pesons a and b sit next to each othe Note that a and b can sit next to each othe in two ways: ab and ba We may view a and b as an insepaable twin; thee ae 2 10! such seating plans Thus the answe is given by 11! 2 10! 9 10! Example 25 How many diffeent pattens of necklaces with 18 beads can be made out of 25 available beads of the same size but in diffeent colos? Answe: P (25,18) ! 36 7! 3 Combinations of Sets A combination is a collection of objects (ode is immateial) fom a given set An combination of an nset S is an subset of S We denote by ( ) n the numbe of combinations of an nset, ead n choose Theoem 31 The numbe of combinations of an nset equals ( n ) n! P (n, )!(n )!! Fist Poof Let S be an nset Let X be the set of all pemutations of S, and let Y be the set of all subsets of S Conside a map f : X Y defined by f(a 1 a 2 a a +1 a n ) {a 1, a 2,, a }, a 1 a 2 a n X Clealy, f is sujective Moeove, fo any subset A {a 1, a 2,, a } Y, thee ae! pemutations of A and (n )! pemutations of the complement Ā Then f 1 (A) {στ : σ is a pemutation of A and τ is a pemutation of Ā} 7
8 Thus f 1 (A)!(n )! Theefoe ( n ) X Y!(n )! n!!(n )! Second Poof Let X be the set of all pemutations of S and let Y be the set of all subsets of S Conside a map f : X Y defined by f(a 1 a 2 a ) {a 1, a 2,, a }, a 1 a 2 a X Clealy, f is sujective Moeove, thee ae exactly! pemutations of {a 1, a 2,, a } sent to {a 1, a 2,, a } Thus ( n ) Y X! P (n, )! Example 31 How many 8lette wods can be constucted fom 26 lettes of the alphabets if each wod contains 3, 4, o 5 vowels? It is undestood that thee is no estiction on the numbe of times a lette can be used in a wod The numbe of wods with 3 vowels: Thee ae ( 8 3) ways to choose 3 vowel positions in a wod; each vowel position can be filled with one of the 5 vowels; the consonant position can be any of 21 consonants Thus thee ae ( ) wods having exactly 3 vowels The numbe of wods with 4 vowels: ( 8 4) The numbe of wods with 5 vowels: ( 8 5) Thus the answe is (8 ) ( ) ( ) Coollay 32 Fo integes n, such that n 0, ( ) ( ) n n n Theoem 33 The numbe of subsets of an nset S equals ( n ) ( n ) ( n ) ( n n) n 8
9 4 Pemutations of Multisets Let M be a multiset An pemutation of M is an odeed aangement of objects of M If M n, then an npemutation of M is called a pemutation of M Theoem 41 Let M be a multiset of k diffeent types whee each type has infinitely many elements Then the numbe of pemutations of M equals k Example 41 What is the numbe of tenay numeals with at most 4 digits? The question is to find the numbe of 4pemutations of the multiset { 0, 1, 2} Thus the answe is Theoem 42 Let M be a multiset of k types with epetition numbes n 1, n 2,, n k espectively Let n n 1 + n n k Then the numbe of pemutations of M equals n! n 1!n 2! n k! Poof List the elements of M as a, } {{, a }, b, } {{, } b,, d, } {{, d } n 1 n 2 n k Let S be the set consisting of the elements a 1, a 2,, a n1, b 1, b 2,, b n2,, d 1, d 2,, d nk Let X be the set of all pemutations of S, and let Y be the set of all pemutations of M Thee is a map f : X Y, sending each pemutation of S to a pemutation of M by emoving the subscipts of the elements Note that fo each pemutation π of M thee ae n 1!, n 2!,, and n k! ways to put the subscipts of the fist, the second,, and the kth type elements back, independently Thus thee ae n 1!n 2! n k! elements of X sent to π by f Theefoe the answe is Y X n 1!n 2! n k! n! n 1!n 2! n k! 9
10 Coollay 43 The numbe of 01 wods of length n with exactly ones and n zeos is equal to n! ( n )!(n )! Example 42 How many possibilities ae thee fo 8 nonattacking ooks on an 8by8 chessboad? How about having 8 diffeent colo ooks? How about having 1 ed (R) ook, 3 blue (B) ooks, and 4 yellow (Y) ooks We label each squae by an odeed pai (i, j) of coodinates, (1, 1) (i, j) (8, 8) Then the ooks must occupy 8 squaes with coodinates (1, a 1 ), (2, a 2 ),, (8, a 8 ), whee a 1, a 2,, a 8 must be distinct, ie, a 1 a 2 a 8 is a pemutation of {1, 2,, 8} Thus the answe is 8! When the 8 ooks have diffeent colos, the answe is 8!8! (8!) 2 If thee ae 1 ed ook, 2 blue ooks, and 3 yellow ooks, then we have a multiset M {R, 3B, 4Y } of ooks The numbe of pemutations of M equals 8! 1!3!4!, and the answe in question is 8! 8! 1!3!4! Theoem 44 Given n ooks of k colos with n 1 ooks of the fist colo, n 2 ooks of the second colo,, and n k ooks of the kth colo The numbe of ways to aange these ooks on an nbyn boad so that no one can attack anothe equals n! n! n 1!n 2! n k! (n!) 2 n 1!n 2! n k! Example 43 Find the numbe of 8pemutations of the multiset M {a, a, a, b, b, c, c, c, c} {3a, 2b, 4c} 8! The numbe of 8pemutations of {2a, 2b, 4c}: 2!2!4! 8! The the numbe of 8pemutations of {3a, b, 4c}: 8! The numbe of 8pemutations of {3a, 2b, 3c}: Thus the answe is 8! 2!2!4! + 8! 3!1!4! + 8! 3!2!3! 3!1!4! 3!2!3! ,
11 5 Combinations of Multisets Let M be a multiset An combination of M is an unodeed collection of objects fom M Thus an combination of M is itself an submultiset of M Fo a multiset M { a 1, a 2,, a n }, an combination of M is also called an combination with epetition allowed of the nset S {a 1, a 2,, a n } The numbe of combinations with epetition allowed of an nset is denoted by n Theoem 51 Let M { a 1, a 2,, a n } be a multiset of n types Then the numbe of combinations of M is given by ( ) ( ) n n + 1 n + 1 n 1 Poof When objects ae taken fom the multiset M, we put them into the following boxes 1st 2nd nth so that the ith type objects ae contained in the ith box, 1 i n Since the objects of the same type ae indistinguishable, we may use the symbol O to denote an object in the boxes, and the objects in diffeent boxes ae sepaated by a stick Convet the symbol O to zeo 0 and the stick to one 1, any such placement is conveted into a 01 sequence of length + n 1 with exactly zeos and n 1 ones Fo example, fo n 4 and 7, {a, a, b, c, c, c, d} {b, b, b, b, d, d, d} OO O OOO O OOOO OOO Now the poblem becomes counting the numbe of 01 wods of length +(n 1) with exactly zeos and n 1 ones Thus the answe is ( ) ( ) n n + 1 n + 1 n 1 11
12 Coollay 52 The numbe n equals the numbe of ways to place identical balls into n distinct boxes Coollay 53 The numbe n equals the numbe of nonnegative intege solutions of the equation x 1 + x x n Coollay 54 The numbe n equals the numbe of nondeceasing sequences of length whose tems ae taken fom the set {1, 2,, n} Poof Each nondeceasing sequence a 1 a 2 a with 1 a i n can be identified as an combination {a 1, a 2,, a } (an multiset) fom the nset {1, 2,, n} with epetition allowed, and vice vesa Example 51 Find the numbe of nonnegative intege solutions fo the equation x 1 + x 2 + x 3 + x 4 < 19 The poblem is equivalent to finding the numbe of nonnegative intege solutions of the equation x 1 + x 2 + x 3 + x 4 + x 5 18 So the answe is 5 18 ( ) ( 22 4 )
The Binomial Distribution
The Binomial Distibution A. It would be vey tedious if, evey time we had a slightly diffeent poblem, we had to detemine the pobability distibutions fom scatch. Luckily, thee ae enough similaities between
More informationSymmetric polynomials and partitions Eugene Mukhin
Symmetic polynomials and patitions Eugene Mukhin. Symmetic polynomials.. Definition. We will conside polynomials in n vaiables x,..., x n and use the shotcut p(x) instead of p(x,..., x n ). A pemutation
More informationUNIT CIRCLE TRIGONOMETRY
UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + =   
More informationSaturated and weakly saturated hypergraphs
Satuated and weakly satuated hypegaphs Algebaic Methods in Combinatoics, Lectues 67 Satuated hypegaphs Recall the following Definition. A family A P([n]) is said to be an antichain if we neve have A B
More informationContinuous Compounding and Annualization
Continuous Compounding and Annualization Philip A. Viton Januay 11, 2006 Contents 1 Intoduction 1 2 Continuous Compounding 2 3 Pesent Value with Continuous Compounding 4 4 Annualization 5 5 A Special Poblem
More information2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES
. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES In ode to etend the definitions of the si tigonometic functions to geneal angles, we shall make use of the following ideas: In a Catesian coodinate sstem, an
More informationFigure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360!
1. What ae angles? Last time, we looked at how the Geeks intepeted measument of lengths. Howeve, as fascinated as they wee with geomety, thee was a shape that was much moe enticing than any othe : the
More informationChapter 3 Savings, Present Value and Ricardian Equivalence
Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,
More informationModel Question Paper Mathematics Class XII
Model Question Pape Mathematics Class XII Time Allowed : 3 hous Maks: 100 Ma: Geneal Instuctions (i) The question pape consists of thee pats A, B and C. Each question of each pat is compulsoy. (ii) Pat
More informationNontrivial lower bounds for the least common multiple of some finite sequences of integers
J. Numbe Theoy, 15 (007), p. 393411. Nontivial lowe bounds fo the least common multiple of some finite sequences of integes Bai FARHI bai.fahi@gmail.com Abstact We pesent hee a method which allows to
More informationThe force between electric charges. Comparing gravity and the interaction between charges. Coulomb s Law. Forces between two charges
The foce between electic chages Coulomb s Law Two chaged objects, of chage q and Q, sepaated by a distance, exet a foce on one anothe. The magnitude of this foce is given by: kqq Coulomb s Law: F whee
More informationCRRC1 Method #1: Standard Practice for Measuring Solar Reflectance of a Flat, Opaque, and Heterogeneous Surface Using a Portable Solar Reflectometer
CRRC Method #: Standad Pactice fo Measuing Sola Reflectance of a Flat, Opaque, and Heteogeneous Suface Using a Potable Sola Reflectomete Scope This standad pactice coves a technique fo estimating the
More informationSkills Needed for Success in Calculus 1
Skills Needed fo Success in Calculus Thee is much appehension fom students taking Calculus. It seems that fo man people, "Calculus" is snonmous with "difficult." Howeve, an teache of Calculus will tell
More informationVector Calculus: Are you ready? Vectors in 2D and 3D Space: Review
Vecto Calculus: Ae you eady? Vectos in D and 3D Space: Review Pupose: Make cetain that you can define, and use in context, vecto tems, concepts and fomulas listed below: Section 7.7. find the vecto defined
More informationLesson 7 Gauss s Law and Electric Fields
Lesson 7 Gauss s Law and Electic Fields Lawence B. Rees 7. You may make a single copy of this document fo pesonal use without witten pemission. 7. Intoduction While it is impotant to gain a solid conceptual
More informationCoordinate Systems L. M. Kalnins, March 2009
Coodinate Sstems L. M. Kalnins, Mach 2009 Pupose of a Coodinate Sstem The pupose of a coodinate sstem is to uniquel detemine the position of an object o data point in space. B space we ma liteall mean
More informationValuation of Floating Rate Bonds 1
Valuation of Floating Rate onds 1 Joge uz Lopez us 316: Deivative Secuities his note explains how to value plain vanilla floating ate bonds. he pupose of this note is to link the concepts that you leaned
More informationMULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION
MULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION K.C. CHANG AND TAN ZHANG In memoy of Pofesso S.S. Chen Abstact. We combine heat flow method with Mose theoy, supe and subsolution method with
More informationFXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it.
Candidates should be able to : Descibe how a mass ceates a gavitational field in the space aound it. Define gavitational field stength as foce pe unit mass. Define and use the peiod of an object descibing
More informationA r. (Can you see that this just gives the formula we had above?)
241 (SJP, Phys 1120) lectic flux, and Gauss' law Finding the lectic field due to a bunch of chages is KY! Once you know, you know the foce on any chage you put down  you can pedict (o contol) motion
More information4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first nonzero digit to
. Simplify: 0 4 ( 8) 0 64 ( 8) 0 ( 8) = (Ode of opeations fom left to ight: Paenthesis, Exponents, Multiplication, Division, Addition Subtaction). Simplify: (a 4) + (a ) (a+) = a 4 + a 0 a = a 7. Evaluate
More informationExplicit, analytical solution of scaling quantum graphs. Abstract
Explicit, analytical solution of scaling quantum gaphs Yu. Dabaghian and R. Blümel Depatment of Physics, Wesleyan Univesity, Middletown, CT 064590155, USA Email: ydabaghian@wesleyan.edu (Januay 6, 2003)
More informationSemipartial (Part) and Partial Correlation
Semipatial (Pat) and Patial Coelation his discussion boows heavily fom Applied Multiple egession/coelation Analysis fo the Behavioal Sciences, by Jacob and Paticia Cohen (975 edition; thee is also an updated
More informationOn Some Functions Involving the lcm and gcd of Integer Tuples
SCIENTIFIC PUBLICATIONS OF THE STATE UNIVERSITY OF NOVI PAZAR SER. A: APPL. MATH. INFORM. AND MECH. vol. 6, 2 (2014), 91100. On Some Functions Involving the lcm and gcd of Intege Tuples O. Bagdasa Abstact:
More informationTop K Nearest Keyword Search on Large Graphs
Top K Neaest Keywod Seach on Lage Gaphs Miao Qiao, Lu Qin, Hong Cheng, Jeffey Xu Yu, Wentao Tian The Chinese Univesity of Hong Kong, Hong Kong, China {mqiao,lqin,hcheng,yu,wttian}@se.cuhk.edu.hk ABSTRACT
More information1240 ev nm 2.5 ev. (4) r 2 or mv 2 = ke2
Chapte 5 Example The helium atom has 2 electonic enegy levels: E 3p = 23.1 ev and E 2s = 20.6 ev whee the gound state is E = 0. If an electon makes a tansition fom 3p to 2s, what is the wavelength of the
More informationSolution Derivations for Capa #8
Solution Deivations fo Capa #8 1) A ass spectoete applies a voltage of 2.00 kv to acceleate a singly chaged ion (+e). A 0.400 T field then bends the ion into a cicula path of adius 0.305. What is the ass
More informationAn Efficient Group Key Agreement Protocol for Ad hoc Networks
An Efficient Goup Key Ageement Potocol fo Ad hoc Netwoks Daniel Augot, Raghav haska, Valéie Issany and Daniele Sacchetti INRIA Rocquencout 78153 Le Chesnay Fance {Daniel.Augot, Raghav.haska, Valéie.Issany,
More informationMechanics 1: Work, Power and Kinetic Energy
Mechanics 1: Wok, Powe and Kinetic Eneg We fist intoduce the ideas of wok and powe. The notion of wok can be viewed as the bidge between Newton s second law, and eneg (which we have et to define and discuss).
More informationSTUDENT RESPONSE TO ANNUITY FORMULA DERIVATION
Page 1 STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION C. Alan Blaylock, Hendeson State Univesity ABSTRACT This pape pesents an intuitive appoach to deiving annuity fomulas fo classoom use and attempts
More informationRisk Sensitive Portfolio Management With CoxIngersollRoss Interest Rates: the HJB Equation
Risk Sensitive Potfolio Management With CoxIngesollRoss Inteest Rates: the HJB Equation Tomasz R. Bielecki Depatment of Mathematics, The Notheasten Illinois Univesity 55 Noth St. Louis Avenue, Chicago,
More informationVoltage ( = Electric Potential )
V1 Voltage ( = Electic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage is
More informationVoltage ( = Electric Potential )
V1 of 9 Voltage ( = lectic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage
More informationAn Introduction to Omega
An Intoduction to Omega Con Keating and William F. Shadwick These distibutions have the same mean and vaiance. Ae you indiffeent to thei iskewad chaacteistics? The Finance Development Cente 2002 1 Fom
More informationEpisode 401: Newton s law of universal gravitation
Episode 401: Newton s law of univesal gavitation This episode intoduces Newton s law of univesal gavitation fo point masses, and fo spheical masses, and gets students pactising calculations of the foce
More informationChapter 22. Outside a uniformly charged sphere, the field looks like that of a point charge at the center of the sphere.
Chapte.3 What is the magnitude of a point chage whose electic field 5 cm away has the magnitude of.n/c. E E 5.56 1 11 C.5 An atom of plutonium39 has a nuclea adius of 6.64 fm and atomic numbe Z94. Assuming
More informationApproximation Algorithms for Data Management in Networks
Appoximation Algoithms fo Data Management in Netwoks Chistof Kick Heinz Nixdof Institute and Depatment of Mathematics & Compute Science adebon Univesity Gemany kueke@upb.de Haald Räcke Heinz Nixdof Institute
More informationGauss Law. Physics 231 Lecture 21
Gauss Law Physics 31 Lectue 1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing
More informationExperiment 6: Centripetal Force
Name Section Date Intoduction Expeiment 6: Centipetal oce This expeiment is concened with the foce necessay to keep an object moving in a constant cicula path. Accoding to Newton s fist law of motion thee
More informationPhysics 235 Chapter 5. Chapter 5 Gravitation
Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus
More informationFunctions of a Random Variable: Density. Math 425 Intro to Probability Lecture 30. Definition Nice Transformations. Problem
Intoduction One Function of Random Vaiables Functions of a Random Vaiable: Density Math 45 Into to Pobability Lectue 30 Let gx) = y be a onetoone function whose deiatie is nonzeo on some egion A of the
More informationAFFILIATE MEMBERSHIP APPLICATION
Califonia Constuction Tucking Association AFFILIATE MEMBERSHIP APPLICATION Reach and Netwok with the Lagest Concentation of Constuction Tucking Fims in the U.S. Affiliate Dues  $500 Annual CCTA 334 N.
More informationINITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS
INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS Vesion:.0 Date: June 0 Disclaime This document is solely intended as infomation fo cleaing membes and othes who ae inteested in
More informationPY1052 Problem Set 8 Autumn 2004 Solutions
PY052 Poblem Set 8 Autumn 2004 Solutions H h () A solid ball stats fom est at the uppe end of the tack shown and olls without slipping until it olls off the ighthand end. If H 6.0 m and h 2.0 m, what
More informationFast FPTalgorithms for cleaning grids
Fast FPTalgoithms fo cleaning gids Josep Diaz Dimitios M. Thilikos Abstact We conside the poblem that given a gaph G and a paamete k asks whethe the edit distance of G and a ectangula gid is at most k.
More informationGraphs of Equations. A coordinate system is a way to graphically show the relationship between 2 quantities.
Gaphs of Equations CHAT PeCalculus A coodinate sstem is a wa to gaphicall show the elationship between quantities. Definition: A solution of an equation in two vaiables and is an odeed pai (a, b) such
More informationThings to Remember. r Complete all of the sections on the Retirement Benefit Options form that apply to your request.
Retiement Benefit 1 Things to Remembe Complete all of the sections on the Retiement Benefit fom that apply to you equest. If this is an initial equest, and not a change in a cuent distibution, emembe to
More informationQuestions & Answers Chapter 10 Software Reliability Prediction, Allocation and Demonstration Testing
M13914 Questions & Answes Chapte 10 Softwae Reliability Pediction, Allocation and Demonstation Testing 1. Homewok: How to deive the fomula of failue ate estimate. λ = χ α,+ t When the failue times follow
More informationSeshadri constants and surfaces of minimal degree
Seshadi constants and sufaces of minimal degee Wioletta Syzdek and Tomasz Szembeg Septembe 29, 2007 Abstact In [] we showed that if the multiple point Seshadi constants of an ample line bundle on a smooth
More informationHow To Find The Optimal Stategy For Buying Life Insuance
Life Insuance Puchasing to Reach a Bequest Ehan Bayakta Depatment of Mathematics, Univesity of Michigan Ann Abo, Michigan, USA, 48109 S. David Pomislow Depatment of Mathematics, Yok Univesity Toonto, Ontaio,
More informationQuantity Formula Meaning of variables. 5 C 1 32 F 5 degrees Fahrenheit, 1 bh A 5 area, b 5 base, h 5 height. P 5 2l 1 2w
1.4 Rewite Fomulas and Equations Befoe You solved equations. Now You will ewite and evaluate fomulas and equations. Why? So you can apply geometic fomulas, as in Ex. 36. Key Vocabulay fomula solve fo a
More informationInstructions to help you complete your enrollment form for HPHC's Medicare Supplemental Plan
Instuctions to help you complete you enollment fom fo HPHC's Medicae Supplemental Plan Thank you fo applying fo membeship to HPHC s Medicae Supplement plan. Pio to submitting you enollment fom fo pocessing,
More information12. Rolling, Torque, and Angular Momentum
12. olling, Toque, and Angula Momentum 1 olling Motion: A motion that is a combination of otational and tanslational motion, e.g. a wheel olling down the oad. Will only conside olling with out slipping.
More informationDisplacement, Velocity And Acceleration
Displacement, Velocity And Acceleation Vectos and Scalas Position Vectos Displacement Speed and Velocity Acceleation Complete Motion Diagams Outline Scala vs. Vecto Scalas vs. vectos Scala : a eal numbe,
More informationCHAPTER 10 Aggregate Demand I
CHAPTR 10 Aggegate Demand I Questions fo Review 1. The Keynesian coss tells us that fiscal policy has a multiplied effect on income. The eason is that accoding to the consumption function, highe income
More informationLecture 16: Color and Intensity. and he made him a coat of many colours. Genesis 37:3
Lectue 16: Colo and Intensity and he made him a coat of many colous. Genesis 37:3 1. Intoduction To display a pictue using Compute Gaphics, we need to compute the colo and intensity of the light at each
More informationCarterPenrose diagrams and black holes
CatePenose diagams and black holes Ewa Felinska The basic intoduction to the method of building Penose diagams has been pesented, stating with obtaining a Penose diagam fom Minkowski space. An example
More informationIntro to Circle Geometry By Raymond Cheong
Into to Cicle Geomety By Rymond Cheong Mny poblems involving cicles cn be solved by constucting ight tingles then using the Pythgoen Theoem. The min chllenge is identifying whee to constuct the ight tingle.
More informationExperiment MF Magnetic Force
Expeiment MF Magnetic Foce Intoduction The magnetic foce on a cuentcaying conducto is basic to evey electic moto  tuning the hands of electic watches and clocks, tanspoting tape in Walkmans, stating
More informationSpirotechnics! September 7, 2011. Amanda Zeringue, Michael Spannuth and Amanda Zeringue Dierential Geometry Project
Spiotechnics! Septembe 7, 2011 Amanda Zeingue, Michael Spannuth and Amanda Zeingue Dieential Geomety Poject 1 The Beginning The geneal consensus of ou goup began with one thought: Spiogaphs ae awesome.
More informationForces & Magnetic Dipoles. r r τ = μ B r
Foces & Magnetic Dipoles x θ F θ F. = AI τ = U = Fist electic moto invented by Faaday, 1821 Wie with cuent flow (in cup of Hg) otates aound a a magnet Faaday s moto Wie with cuent otates aound a Pemanent
More informationExam 3: Equation Summary
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Depatment of Physics Physics 8.1 TEAL Fall Tem 4 Momentum: p = mv, F t = p, Fext ave t= t f t= Exam 3: Equation Summay total = Impulse: I F( t ) = p Toque: τ = S S,P
More informationFinancing Terms in the EOQ Model
Financing Tems in the EOQ Model Habone W. Stuat, J. Columbia Business School New Yok, NY 1007 hws7@columbia.edu August 6, 004 1 Intoduction This note discusses two tems that ae often omitted fom the standad
More informationIt is required to solve the heatcondition equation for the excesstemperature function:
Jounal of Engineeing Physics and Themophysics. Vol. 73. No. 5. 2 METHOD OF PAIED INTEGAL EQUATIONS WITH LPAAMETE IN POBLEMS OF NONSTATIONAY HEAT CONDUCTION WITH MIXED BOUNDAY CONDITIONS FO AN INFINITE
More informationInteger sequences from walks in graphs
otes on umbe Theoy and Discete Mathematics Vol. 9, 3, o. 3, 78 84 Intege seuences fom walks in gahs Enesto Estada, and José A. de la Peña Deatment of Mathematics and Statistics, Univesity of Stathclyde
More informationChapter 4: Matrix Norms
EE448/58 Vesion.0 John Stensby Chate 4: Matix Noms The analysis of matixbased algoithms often equies use of matix noms. These algoithms need a way to quantify the "size" of a matix o the "distance" between
More informationMultiple choice questions [60 points]
1 Multiple choice questions [60 points] Answe all o the ollowing questions. Read each question caeully. Fill the coect bubble on you scanton sheet. Each question has exactly one coect answe. All questions
More informationNURBS Drawing Week 5, Lecture 10
CS 43/585 Compute Gaphics I NURBS Dawing Week 5, Lectue 1 David Been, William Regli and Maim Pesakhov Geometic and Intelligent Computing Laboato Depatment of Compute Science Deel Univesit http://gicl.cs.deel.edu
More informationDetermining solar characteristics using planetary data
Detemining sola chaacteistics using planetay data Intoduction The Sun is a G type main sequence sta at the cente of the Sola System aound which the planets, including ou Eath, obit. In this inestigation
More informationDatabase Management Systems
Contents Database Management Systems (COP 5725) D. Makus Schneide Depatment of Compute & Infomation Science & Engineeing (CISE) Database Systems Reseach & Development Cente Couse Syllabus 1 Sping 2012
More informationCHAPTER 5 GRAVITATIONAL FIELD AND POTENTIAL
CHATER 5 GRAVITATIONAL FIELD AND OTENTIAL 5. Intoduction. This chapte deals with the calculation of gavitational fields and potentials in the vicinity of vaious shapes and sizes of massive bodies. The
More informationThe LCOE is defined as the energy price ($ per unit of energy output) for which the Net Present Value of the investment is zero.
Poject Decision Metics: Levelized Cost of Enegy (LCOE) Let s etun to ou wind powe and natual gas powe plant example fom ealie in this lesson. Suppose that both powe plants wee selling electicity into the
More informationGravitation. AP Physics C
Gavitation AP Physics C Newton s Law of Gavitation What causes YOU to be pulled down? THE EARTH.o moe specifically the EARTH S MASS. Anything that has MASS has a gavitational pull towads it. F α Mm g What
More informationQuestions for Review. By buying bonds This period you save s, next period you get s(1+r)
MACROECONOMICS 2006 Week 5 Semina Questions Questions fo Review 1. How do consumes save in the twopeiod model? By buying bonds This peiod you save s, next peiod you get s() 2. What is the slope of a consume
More information12.1. FÖRSTER RESONANCE ENERGY TRANSFER
ndei Tokmakoff, MIT epatment of Chemisty, 3/5/8 11 1.1. FÖRSTER RESONNCE ENERGY TRNSFER Föste esonance enegy tansfe (FR) efes to the nonadiative tansfe of an electonic excitation fom a dono molecule to
More informationHow to create a default user profile in Windows 7
AnswesThatWok TM How to ceate a default use pofile in Windows 7 (Win 7) How to ceate a default use pofile in Windows 7 When to use this document Use this document wheneve you want to ceate a default use
More informationThank you for participating in Teach It First!
Thank you fo paticipating in Teach It Fist! This Teach It Fist Kit contains a Common Coe Suppot Coach, Foundational Mathematics teache lesson followed by the coesponding student lesson. We ae confident
More informationChapter 2. Electrostatics
Chapte. Electostatics.. The Electostatic Field To calculate the foce exeted by some electic chages,,, 3,... (the souce chages) on anothe chage Q (the test chage) we can use the pinciple of supeposition.
More informationON THE (Q, R) POLICY IN PRODUCTIONINVENTORY SYSTEMS
ON THE R POLICY IN PRODUCTIONINVENTORY SYSTEMS Saifallah Benjaafa and JoonSeok Kim Depatment of Mechanical Engineeing Univesity of Minnesota Minneapolis MN 55455 Abstact We conside a poductioninventoy
More informationMultiple choice questions [70 points]
Multiple choice questions [70 points] Answe all of the following questions. Read each question caefull. Fill the coect bubble on ou scanton sheet. Each question has exactl one coect answe. All questions
More information9.5 Amortization. Objectives
9.5 Aotization Objectives 1. Calculate the payent to pay off an aotized loan. 2. Constuct an aotization schedule. 3. Find the pesent value of an annuity. 4. Calculate the unpaid balance on a loan. Congatulations!
More informationHow To Write A Theory Of The Concept Of The Mind In A Quey
Jounal of Atificial Intelligence Reseach 31 (2008) 157204 Submitted 06/07; published 01/08 Conjunctive Quey Answeing fo the Desciption Logic SHIQ Bite Glimm Ian Hoocks Oxfod Univesity Computing Laboatoy,
More informationPHYSICS 111 HOMEWORK SOLUTION #13. May 1, 2013
PHYSICS 111 HOMEWORK SOLUTION #13 May 1, 2013 0.1 In intoductoy physics laboatoies, a typical Cavendish balance fo measuing the gavitational constant G uses lead sphees with masses of 2.10 kg and 21.0
More informationThe Supply of Loanable Funds: A Comment on the Misconception and Its Implications
JOURNL OF ECONOMICS ND FINNCE EDUCTION Volume 7 Numbe 2 Winte 2008 39 The Supply of Loanable Funds: Comment on the Misconception and Its Implications. Wahhab Khandke and mena Khandke* STRCT Recently FieldsHat
More informationVISCOSITY OF BIODIESEL FUELS
VISCOSITY OF BIODIESEL FUELS One of the key assumptions fo ideal gases is that the motion of a given paticle is independent of any othe paticles in the system. With this assumption in place, one can use
More informationAP Physics Electromagnetic Wrap Up
AP Physics Electomagnetic Wap Up Hee ae the gloious equations fo this wondeful section. F qsin This is the equation fo the magnetic foce acting on a moing chaged paticle in a magnetic field. The angle
More informationest using the formula I = Prt, where I is the interest earned, P is the principal, r is the interest rate, and t is the time in years.
9.2 Inteest Objectives 1. Undestand the simple inteest fomula. 2. Use the compound inteest fomula to find futue value. 3. Solve the compound inteest fomula fo diffeent unknowns, such as the pesent value,
More informationChapter 3. Distribution Problems. 3.1 The idea of a distribution. 3.1.1 The twentyfold way
Chapter 3 Distribution Problems 3.1 The idea of a distribution Many of the problems we solved in Chapter 1 may be thought of as problems of distributing objects (such as pieces of fruit or pingpong balls)
More informationRandomized MultiChannel Interrogation Algorithm for LargeScale RFID Systems
andomized MultiChannel Inteogation Algoithm fo LageScale FID Systems AmiHamed Mohsenianad, Vahid ShahMansoui, Vincent W.S. Wong, and obet Schobe Depatment of Electical and Compute Engineeing, The
More informationSeparation probabilities for products of permutations
Sepaation pobabilities fo poducts of pemutations Olivie Benadi, Rosena R. X. Du, Alejando H. Moales and Richad P. Stanley Mach 1, 2012 Abstact We study the mixing popeties of pemutations obtained as a
More informationwww.sakshieducation.com
Viscosity. The popety of viscosity in gas is due to ) Cohesive foces between the moecues ) Coisions between the moecues ) Not having a definite voume ) Not having a definite size. When tempeatue is inceased
More informationGravitation and Kepler s Laws Newton s Law of Universal Gravitation in vectorial. Gm 1 m 2. r 2
F Gm Gavitation and Keple s Laws Newton s Law of Univesal Gavitation in vectoial fom: F 12 21 Gm 1 m 2 12 2 ˆ 12 whee the hat (ˆ) denotes a unit vecto as usual. Gavity obeys the supeposition pinciple,
More informationDeflection of Electrons by Electric and Magnetic Fields
Physics 233 Expeiment 42 Deflection of Electons by Electic and Magnetic Fields Refeences Loain, P. and D.R. Coson, Electomagnetism, Pinciples and Applications, 2nd ed., W.H. Feeman, 199. Intoduction An
More informationLATIN SQUARE DESIGN (LS) With the Latin Square design you are able to control variation in two directions.
Facts about the LS Design LATIN SQUARE DESIGN (LS) With the Latin Squae design you ae able to contol vaiation in two diections. Teatments ae aanged in ows and columns Each ow contains evey teatment.
More informationAPPLICATION AND AGREEMENT FORM FOR TELECOMMUNICATION SERVICES BUSINESS APPLICATION
Application Fom SECTION 1 COMPANY DETAILS New Company Yes No Company Name Tading As Pevious Company Name Email Addess Contact Numbe Tel Cell Fax Registeed Numbe Natue of Business Yea of Incopoation Yea
More informationChapter 17 The Kepler Problem: Planetary Mechanics and the Bohr Atom
Chapte 7 The Keple Poblem: Planetay Mechanics and the Boh Atom Keple s Laws: Each planet moves in an ellipse with the sun at one focus. The adius vecto fom the sun to a planet sweeps out equal aeas in
More informationCapital Investment and Liquidity Management with collateralized debt.
TSE 54 Novembe 14 Capital Investment and Liquidity Management with collatealized debt. Ewan Piee, Stéphane Villeneuve and Xavie Wain 7 Capital Investment and Liquidity Management with collatealized debt.
More informationSummary: Vectors. This theorem is used to find any points (or position vectors) on a given line (direction vector). Two ways RT can be applied:
Summ: Vectos ) Rtio Theoem (RT) This theoem is used to find n points (o position vectos) on given line (diection vecto). Two ws RT cn e pplied: Cse : If the point lies BETWEEN two known position vectos
More informationThe transport performance evaluation system building of logistics enterprises
Jounal of Industial Engineeing and Management JIEM, 213 6(4): 194114 Online ISSN: 213953 Pint ISSN: 2138423 http://dx.doi.og/1.3926/jiem.784 The tanspot pefomance evaluation system building of logistics
More informationConcept and Experiences on using a Wikibased System for Softwarerelated Seminar Papers
Concept and Expeiences on using a Wikibased System fo Softwaeelated Semina Papes Dominik Fanke and Stefan Kowalewski RWTH Aachen Univesity, 52074 Aachen, Gemany, {fanke, kowalewski}@embedded.wthaachen.de,
More information