Factors of sums of powers of binomial coefficients

Size: px
Start display at page:

Download "Factors of sums of powers of binomial coefficients"

Transcription

1 ACTA ARITHMETICA LXXXVI.1 (1998) Factors of sums of powers of biomial coefficiets by Neil J. Cali (Clemso, S.C.) Dedicated to the memory of Paul Erdős 1. Itroductio. It is well ow that if ( ) a f,a = the f,0 = + 1, f,1 = 2, f,2 = ( ), ad it is possible to show (Wilf, persoal commuicatio, usig techiues i [8]) that for 3 a 9, there is o closed form for f,a as a sum of a fixed umber of hypergeometric terms. Similarly, usig asymptotic techiues, de Bruij has show [1] that if a 4, the h,a has o closed form, where ( ) a h,a = ( 1) (clearly, h +1,a = 0). I this paper we will prove that while o closed form may exist, there are iterestig divisibility properties of f,2a ad h,a. We will illustrate some of the techiues which may be applied to prove these sorts of results. Our mai results are: Theorem 1. For all positive ad a, 2 ( ( 1) ) a is divisible by ( ). Theorem 2. For all positive itegers a, m, j, [ ] a ( 1) j 1991 Mathematics Subject Classificatio: 05A10, 05A30, 11B65. [17]

2 18 N. J. Cali is divisible by t(, ), where the [ ] t(, ) is a -aalogue of the odd part of ( ). are the -biomial coefficiets, ad 2. Bacgroud. I attemptig to exted the results of previous wor [2], we were led to cosider factorizatios of sums of powers of biomial coefficiets. It uicly became clear that for eve expoets, small primes occurred as divisors i a regular fashio (Propositio 3), ad that this result could be exteded (Propositio 7) to odd expoets ad alteratig sums. Further ivestigatio revealed (Propositio 8) that for all alteratig sums, the primes dividig h,a coicided with those dividig ( ). This led us to cojecture, ad subseuetly to prove, Theorem 1; as part of our proof we obtai (Theorem 2) a correspodig result for -biomial coefficiets. 3. No-alteratig sums Propositio 3. For every iteger m 1, if p is a prime i the iterval 2a( + 1) 1 < p < = + 1 m 2ma 1 m m m(2ma 1) the p f,2a. I particular, f,2a is divisible by all primes p for which 2a( + 1) 1 < p < = a 1 2a 1. The followig lemma will eable us to covert iformatio about divisors of f,a which are greater tha ito iformatio about divisors less tha. Lemma 4. Let = s s be the expasio of i base p (ad similarly for = s s ). The f,a (mod p). i=0 f i,a P r o o f. By Lucas Theorem (see for example Graville [6]), ( ) ( ) i (mod p) i=0 i where as usual, ( i ) i 0 (mod p) if i > i. Hece all the terms i the sum over for which i > i for some i disappear, givig as claimed. f,a = ( i i=0 i =0 ) a ( i i s s 1 s =0 s 1 =0... ) a (mod p) 0 0 =0 i=0 ( i i f i,a (mod p) i=0 ) a (mod p)

3 Sums of powers of biomial coefficiets 19 Corollary 5. If l < p ad p f l,a the p f l+jp,a for all positive itegers j. We are ow i a positio to prove Propositio 3. We proceed i two stages: first, the case whe < p. Lemma 6. Let p be a prime i the iterval < p < (2a(+1) 1)/(2a 1). The p f,2a. P r o o f. Let p = + r where r > 0. The we have ( ) 2a p r ( ) 2a p r f,2a = (mod p) p r ( ) 2a r + 1 ( 1) 2a (mod p) p r ( ) 2a r + 1 (mod p) p r ( ) 2a ( + 1)( + 2)... ( + r 1) (mod p). (r 1)! If we write x (0) = 1 ad x (r) for the polyomial x(x + 1)... (x + r 1), the this last sum becomes p r ( ) ( + 2a 1)(r 1). (r 1)! We ow observe that the polyomials x (0), x (1),..., x (d) form a iteger basis for the space of all iteger polyomials of degree at most d. Hece there exist itegers c 0, c 1,..., c (r 1)(2a 1) so that Thus f,2a (( + 1) (r 1) ) 2a 1 = 1 p r (r 1)! 2a (r 1)(2a 1) j=0 (r 1)(2a 1) j=0 (r 1)(2a 1) 1 (r 1)! 2a j=0 c j ( + r) (j). c j ( + 1) (r 1) ( + r) (j) p r c j ( + 1) (r+j 1) (r 1)(2a 1) 1 (p r + 1) (r+j) (r 1)! 2a c j. r + j j=0

4 20 N. J. Cali Now, if r + (r 1)(2a 1) < p, the each of the terms i the sum is divisible by p, ad (r 1)! is ot divisible by p; hece f,2a is divisible by p. But ad r + (r 1)(2a 1) = 2ra 2a + 1 = 2pa a 2a + 1 2pa a 2a + 1 < p if ad oly if 2a( + 1) 1 p < 2a 1 completig the proof of the lemma. Now, suppose that = (m 1)p + l with l > 0 ad 2a(l + 1) 1 l < p <. 2ma 1 The, by Lemma 6, p divides f l,2a ad hece by Corollary 5, p divides f,2a. But l < p if ad oly if < mp, ad 2a(l + 1) 1 p < 2a 1 if ad oly if 2a( (m 1)p) 1 p <, 2a 1 that is, if 2a( + 1) 1 p <. 2ma 1 Thus, if 2a( + 1) 1 < p < m 2ma 1 the p divides f,2a, completig the proof of Propositio Alteratig sums. We ote that o similar result holds for the case of odd powers of biomial coefficiets (with the trivial exceptio of a = 1). Ideed, except for the power of 2 dividig f,2a+1 (which we discuss i Lemma 12), the factorizatios of sums of odd powers seem to exhibit o structure; for example, f 28,3 = However, for alteratig sums of odd powers, we have Propositio 7. p divides h,2a+1 for primes i the itervals (2a + 1)( + 1) 1 < p < = + 1 m m(2a + 1) 1 m m m(2a + 1) 1.

5 Sums of powers of biomial coefficiets 21 P r o o f. Ideed, by examiig the proof of Propositio 3, we see that if we defie ( ( )) a g,a = ( 1) so that g,2a = f,2a ad g,2a+1 = h,2a+1, the g,a is divisible by all primes i each of the itervals ( + 1)a 1 < p < m ma 1 so Propositios 3 ad 7 are really the same result. For all alteratig sums we have Propositio 8. If p ( ) the p h,a. P r o o f. Clearly 2 divides h,a if ad oly if 2 divides the middle term, ( ) a, ( as all of the other terms cacel (mod 2). Sice 2 divides ), 2 divides h,a. Now let p be a odd prime dividig ( ) ; we will show that p divides h,a. By Kummer s theorem, at least oe of the digits of writte i base p is odd (sice if all are eve, the there are o carries i computig + = i base p). Let the digits of i base p be () s, () s 1,..., () 1, () 0. The as i Lemma 4, 2 ( ) a ( () i ( ) a ) ( 1) ( 1) ()i i i=0 i =0 (sice p is odd, ( 1) = ( 1) s ). Now, sice p ( ), at least oe of the digits of i base p is odd; but the the correspodig term i the product is zero, ad so p h,a, completig the proof of Propositio 8. After computig some examples, it is atural to cojecture (ad the, of course, to prove!) Theorem The mai theorems. We will prove Theorem 1 by cosiderig - biomial coefficiets. Defiitios. Let be a positive iteger. Throughout we will deote the umber of 1 s i the biary expasio of by l() (so that 2 l() ( ) ). We further defie the followig polyomials i a idetermiate : θ () = 1 1 = (the -aalogue of ), φ () = d (1 d ) µ(/d) i

6 22 N. J. Cali (the th cyclotomic polyomial i ),! = θ i () (the -aalogue of!), ad (the -aalogue of ( ) ). Further, defie ad r(x, ) = j x [ ] = i=1!! ( )! (1 j ) = (1 )!, s(x, ) = t(, ) = 2j+1 x s(, ) s(/2, )s(/4, )s(/8, ).... (1 2j+1 ) Note that the apparetly ifiite product i the deomiator is i fact fiite, sice s(x, ) = 1 if x < 1. We ow mae some useful observatios about t(, ). First, so t(, ) = = = s(, ) = r(, ) r(/2, 2 ), s(, ) s(/2, ) 2 s(/2, ) s(/4, ) 2 s(/4, ) s(/8, ) 2 r(, ) r(/2, ) 2 r(/2, ) r(/4, ) 2 r(/4, ) r(/8, ) 2 r(/2, 2 ) r(/4, 2 ) 2 r(/4, 2 ) r(/8, 2 ) 2 r(, ) r(/2, ) 2 r(/2, ) r(/4, ) 2 r(/2, 2 ) r(/4, 2 ) 2 r(/8, 2 ) r(/16, 2 ) 2 r(/4, ) r(/8, ) 2 r(/4, 2 ) r(/8, 2 ) 2 s(/8, ) s(/16, ) r(/8, ) r(/16, ) 2 r(/8, 2 ) r(/16, 2 ) 2 where agai, the apparetly ifiite product is i fact fiite. Now, sice r(x, ) r(x/2, ) 2 r(x, 2 ) r(x/2, 2 ) 2 { 1 if x is eve, 1 2 if x is odd,...

7 as 1, we see that Sums of powers of biomial coefficiets 23 lim t(, ) = 1 Further, t( + 1, ) has a factor 1, so ( ) 2 l. lim t( + 1, ) = 0. 1 I other words, sice 2 l ( ), we may regard t(, ) as the -aalogue of the largest odd factor of ( ). Lemma 9. We have t(, ) = m φ m() where the product is over those odd m for which /m is odd. P r o o f. Clearly, if m is eve the φ m () does ot divide t(, ). Suppose m is odd; the φ m () divides s(, ) exactly /m /2 times, ad hece φ m () divides t(, ) m /2 2m /2 /2... 4m 2 j /2... m times. Now, by cosiderig the biary expasio of /m, it is immediate that this is 0 if /m is eve, ad 1 if /m is odd. Lemma 10. Let m,, be o-egative itegers ad write = m +, = m +, = ( ) m + ( ), where, are the least o-egative residues of, (mod m). The [ ] [ ] ( ) (mod φ m ()) where [ ] is tae to be 0 if <. P r o o f. See [3], [4] or [7]. Proof of Theorem 2. It is eough to show that if m ad = /m are odd, the [ ] a φ m () ( 1) j. But, from Lemma 10, [ ] a [ ] ( 1) j a ( ) a =0 =0 ( 1) + j (mod φ m ()) ( [ ] a )( ( ) a ) = ( 1) j ( 1) mj =0 =0 ad sice m ad are odd, the secod sum is zero, ad we are doe.

8 24 N. J. Cali We observe ow that both sides of Theorem 2 are iteger polyomials; thus whe we evaluate them at = 1, the left had side (if o-zero) will divide ( the right had side. But we have already observed that t(, 1) = ) /2 l, ad hece we have proved Corollary 11. ( ) 2 ( ) a 2 l() ( 1). To prove Theorem 1 it remais to show that 2 ( ) a 2 l() ( 1). We prove a stroger result by iductio. ad Lemma 12. For all positive itegers a ad, ( ) a 2 l() 2 l() ( ) a ( 1). P r o o f. The assertio is clearly true whe = 1. Assume ow that it holds for all values less tha. For each 1 i l(), let m = 2 i ad let,,,, ( ), ( ) be defied as i Lemma 10. Writig ( [ ] a )( ( ) a ) w i () = we have [ ] a w i () (mod φ 2 i()). By our iductio hypothesis, sice l() = l( )+l( ), 2 l() w i (1) for each i. We ow wish to combie these euivaleces modulo θ 2 l()() = φ 2 ()φ 4 ()φ 8 ()... φ 2 l()() ad evaluate them at = 1. To do this, defie π 1 = 1 2 l 1 ad π i = 1 (1 ) for i = 2, 3,..., l(). 2l i+1 The, settig u i () = φ 2 ()φ 4 ()... φ 2 i 1()π i φ 2 i+1()... φ 2 l()()

9 we have Sums of powers of biomial coefficiets 25 u 1 () = 1 2 l 1 (1 + 2 )(1 + 4 )... (1 + 2 l() 1 ) 1 (mod (1 + )) ad for i 2, u i () 1 2 l i+1 (1 2 )(1 + 2 )(1 + 4 )... (1 + 2 i 2 )(1 + 2i )... (1 + 2l() 1 ) 1 i 1 (1 2 )(1 + 2i )(1 + 2i+1 )... (1 + 2l() 1 ) 2l i+1 1 (mod (1 + 2i 1 )). Further, if i j, the u i () 0 (mod φ 2 j ()). Hece, that is, [ ] a l() w i ()u i () (mod θ 2 l()()), [ ] a i=1 l() = P ()θ 2 l()() + w i ()u i () where we wish to coclude that P () is a iteger polyomial. Observe that it is sufficiet to prove that each w i ()u i () is a iteger polyomial, sice θ 2 l() is moic. To do this, cosider w i (). First, observe that w i () is divisible by 2 l( ) by our iductive hypothesis, sice < ; further, if is odd, so is, ad hece the -biomial sum i w i () is symmetric ad its coefficiets are eve; if is eve, the l( ) i 1, ad i each case, 2 l i w i () (that is, each coefficiet of w i () is divisible by 2 l i+1 ). Thus, for each i, w i ()u i () is a iteger polyomial. We have thus prove that [ ] a i=1 l() = P ()θ 2 l()() + w i ()u i () where P () has iteger coefficiets. Now, settig = 1 i both sides, we observe that u i (1) is a iteger for each i, 2 l() w i (1) for each i (ideed, u i (1) = 0 for i 2, ad u 1 (1) = 1), ad that θ 2 l()(1) = 2 l(). Hece each term o the right is divisible by 2 l(), provig that ( ) a 2 l(). i=1

10 26 N. J. Cali To prove that 2 l() we proceed similarly, settig ( [ v i () = ] a ( ) a ( 1) )( ( ) ( 1) a ), with the oly major differece beig i the proof that l() i=1 v i()u i () is a iteger polyomial: i this case, if is eve, thigs wor as above, ad if is odd, the we have is odd, ad v i () is idetically eual to 0. Note that we eed to have already prove the lemma for o-alteratig sums to prove the alteratig case. This completes the proof of Lemma 12 ad thus of Theorem 1. We gratefully acowledge may iformative discussios with Professors Joatha M. Borwei, Ira Gessel, Adrew J. Graville ad Herbert S. Wilf. Refereces [1] N. G. de Bruij, Asymptotic Methods i Aalysis, Dover, New Yor, [2] N. J. Cali, A curious biomial idetity, Discrete Math. 131 (1994), [3] M.-D. Choi, G. A. Elliott ad N. Yui, Gauss polyomials ad the rotatio algebra, Ivet. Math. 99 (1990), [4] J. Désarméie, U aalogue des cogrueces de Kummer pour les -ombres d Euler, Europea J. Combi. 3 (1982), [5] A. J. Graville, Zaphod Beeblebrox s brai ad the fifty-ith row of Pascal s triagle, Amer. Math. Mothly 99 (1992), [6], The arithmetic properties of biomial coefficiets, i: Proceedigs of the Orgaic Mathematics Worshop, 1996, idex.html (URL verified September 10, 1997). [7] G. Olive, Geeralized powers, Amer. Math. Mothly 72 (1965), [8] M. Petovše, H. S. Wilf ad D. Zeilberger, A = B, A. K. Peters, Wellesley, Mass., Departmet of Mathematical Scieces Clemso Uiversity Clemso, South Carolia U.S.A. Received o ad i revised form o (3203)

Asymptotic Growth of Functions

Asymptotic Growth of Functions CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll

More information

Module 4: Mathematical Induction

Module 4: Mathematical Induction Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate

More information

Chapter 5 O A Cojecture Of Erdíos Proceedigs NCUR VIII è1994è, Vol II, pp 794í798 Jeærey F Gold Departmet of Mathematics, Departmet of Physics Uiversity of Utah Do H Tucker Departmet of Mathematics Uiversity

More information

Irreducible polynomials with consecutive zero coefficients

Irreducible polynomials with consecutive zero coefficients Irreducible polyomials with cosecutive zero coefficiets Theodoulos Garefalakis Departmet of Mathematics, Uiversity of Crete, 71409 Heraklio, Greece Abstract Let q be a prime power. We cosider the problem

More information

A Recursive Formula for Moments of a Binomial Distribution

A Recursive Formula for Moments of a Binomial Distribution A Recursive Formula for Momets of a Biomial Distributio Árpád Béyi beyi@mathumassedu, Uiversity of Massachusetts, Amherst, MA 01003 ad Saverio M Maago smmaago@psavymil Naval Postgraduate School, Moterey,

More information

The Euler Totient, the Möbius and the Divisor Functions

The Euler Totient, the Möbius and the Divisor Functions The Euler Totiet, the Möbius ad the Divisor Fuctios Rosica Dieva July 29, 2005 Mout Holyoke College South Hadley, MA 01075 1 Ackowledgemets This work was supported by the Mout Holyoke College fellowship

More information

π d i (b i z) (n 1)π )... sin(θ + )

π d i (b i z) (n 1)π )... sin(θ + ) SOME TRIGONOMETRIC IDENTITIES RELATED TO EXACT COVERS Joh Beebee Uiversity of Alaska, Achorage Jauary 18, 1990 Sherma K Stei proves that if si π = k si π b where i the b i are itegers, the are positive

More information

ORDERS OF GROWTH KEITH CONRAD

ORDERS OF GROWTH KEITH CONRAD ORDERS OF GROWTH KEITH CONRAD Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really wat to uderstad their behavior It also helps you better grasp topics i calculus

More information

1. MATHEMATICAL INDUCTION

1. MATHEMATICAL INDUCTION 1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1

More information

A probabilistic proof of a binomial identity

A probabilistic proof of a binomial identity A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two

More information

Factoring x n 1: cyclotomic and Aurifeuillian polynomials Paul Garrett

Factoring x n 1: cyclotomic and Aurifeuillian polynomials Paul Garrett <garrett@math.umn.edu> (March 16, 004) Factorig x 1: cyclotomic ad Aurifeuillia polyomials Paul Garrett Polyomials of the form x 1, x 3 1, x 4 1 have at least oe systematic factorizatio x 1 = (x 1)(x 1

More information

SUMS OF GENERALIZED HARMONIC SERIES. Michael E. Ho man Department of Mathematics, U. S. Naval Academy, Annapolis, Maryland

SUMS OF GENERALIZED HARMONIC SERIES. Michael E. Ho man Department of Mathematics, U. S. Naval Academy, Annapolis, Maryland #A46 INTEGERS 4 (204) SUMS OF GENERALIZED HARMONIC SERIES Michael E. Ho ma Departmet of Mathematics, U. S. Naval Academy, Aapolis, Marylad meh@usa.edu Courtey Moe Departmet of Mathematics, U. S. Naval

More information

NUMBERS COMMON TO TWO POLYGONAL SEQUENCES

NUMBERS COMMON TO TWO POLYGONAL SEQUENCES NUMBERS COMMON TO TWO POLYGONAL SEQUENCES DIANNE SMITH LUCAS Chia Lake, Califoria a iteger, The polygoal sequece (or sequeces of polygoal umbers) of order r (where r is r > 3) may be defied recursively

More information

1 The Binomial Theorem: Another Approach

1 The Binomial Theorem: Another Approach The Biomial Theorem: Aother Approach Pascal s Triagle I class (ad i our text we saw that, for iteger, the biomial theorem ca be stated (a + b = c a + c a b + c a b + + c ab + c b, where the coefficiets

More information

Department of Computer Science, University of Otago

Department of Computer Science, University of Otago Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS-2006-09 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly

More information

Soving Recurrence Relations

Soving Recurrence Relations Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree

More information

SUMS OF n-th POWERS OF ROOTS OF A GIVEN QUADRATIC EQUATION. N.A. Draim, Ventura, Calif., and Marjorie Bicknell Wilcox High School, Santa Clara, Calif.

SUMS OF n-th POWERS OF ROOTS OF A GIVEN QUADRATIC EQUATION. N.A. Draim, Ventura, Calif., and Marjorie Bicknell Wilcox High School, Santa Clara, Calif. SUMS OF -th OWERS OF ROOTS OF A GIVEN QUADRATIC EQUATION N.A. Draim, Vetura, Calif., ad Marjorie Bickell Wilcox High School, Sata Clara, Calif. The quadratic equatio whose roots a r e the sum or differece

More information

A CHARACTERIZATION OF MINIMAL ZERO-SEQUENCES OF INDEX ONE IN FINITE CYCLIC GROUPS

A CHARACTERIZATION OF MINIMAL ZERO-SEQUENCES OF INDEX ONE IN FINITE CYCLIC GROUPS INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(1) (2005), #A27 A CHARACTERIZATION OF MINIMAL ZERO-SEQUENCES OF INDEX ONE IN FINITE CYCLIC GROUPS Scott T. Chapma 1 Triity Uiversity, Departmet

More information

1 n. n > dt. t < n 1 + n=1

1 n. n > dt. t < n 1 + n=1 Math 05 otes C. Pomerace The harmoic sum The harmoic sum is the sum of recirocals of the ositive itegers. We kow from calculus that it diverges, this is usually doe by the itegral test. There s a more

More information

Math Discrete Math Combinatorics MULTIPLICATION PRINCIPLE:

Math Discrete Math Combinatorics MULTIPLICATION PRINCIPLE: Math 355 - Discrete Math 4.1-4.4 Combiatorics Notes MULTIPLICATION PRINCIPLE: If there m ways to do somethig ad ways to do aother thig the there are m ways to do both. I the laguage of set theory: Let

More information

A Note on Sums of Greatest (Least) Prime Factors

A Note on Sums of Greatest (Least) Prime Factors It. J. Cotemp. Math. Scieces, Vol. 8, 203, o. 9, 423-432 HIKARI Ltd, www.m-hikari.com A Note o Sums of Greatest (Least Prime Factors Rafael Jakimczuk Divisio Matemática, Uiversidad Nacioal de Luá Bueos

More information

Lecture 7: Borel Sets and Lebesgue Measure

Lecture 7: Borel Sets and Lebesgue Measure EE50: Probability Foudatios for Electrical Egieers July-November 205 Lecture 7: Borel Sets ad Lebesgue Measure Lecturer: Dr. Krisha Jagaatha Scribes: Ravi Kolla, Aseem Sharma, Vishakh Hegde I this lecture,

More information

On the L p -conjecture for locally compact groups

On the L p -conjecture for locally compact groups Arch. Math. 89 (2007), 237 242 c 2007 Birkhäuser Verlag Basel/Switzerlad 0003/889X/030237-6, ublished olie 2007-08-0 DOI 0.007/s0003-007-993-x Archiv der Mathematik O the L -cojecture for locally comact

More information

3. Greatest Common Divisor - Least Common Multiple

3. Greatest Common Divisor - Least Common Multiple 3 Greatest Commo Divisor - Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd

More information

if A S, then X \ A S, and if (A n ) n is a sequence of sets in S, then n A n S,

if A S, then X \ A S, and if (A n ) n is a sequence of sets in S, then n A n S, Lecture 5: Borel Sets Topologically, the Borel sets i a topological space are the σ-algebra geerated by the ope sets. Oe ca build up the Borel sets from the ope sets by iteratig the operatios of complemetatio

More information

+ 1= x + 1. These 4 elements form a field.

+ 1= x + 1. These 4 elements form a field. Itroductio to fiite fields II Fiite field of p elemets F Because we are iterested i doig computer thigs it would be useful for us to costruct fields havig elemets. Let s costruct a field of elemets; we

More information

8.5 Alternating infinite series

8.5 Alternating infinite series 65 8.5 Alteratig ifiite series I the previous two sectios we cosidered oly series with positive terms. I this sectio we cosider series with both positive ad egative terms which alterate: positive, egative,

More information

arxiv:1012.1336v2 [cs.cc] 8 Dec 2010

arxiv:1012.1336v2 [cs.cc] 8 Dec 2010 Uary Subset-Sum is i Logspace arxiv:1012.1336v2 [cs.cc] 8 Dec 2010 1 Itroductio Daiel M. Kae December 9, 2010 I this paper we cosider the Uary Subset-Sum problem which is defied as follows: Give itegers

More information

Theorems About Power Series

Theorems About Power Series Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real o-egative umber R, called the radius

More information

MATH 361 Homework 9. Royden Royden Royden

MATH 361 Homework 9. Royden Royden Royden MATH 61 Homework 9 Royde..9 First, we show that for ay subset E of the real umbers, E c + y = E + y) c traslatig the complemet is equivalet to the complemet of the traslated set). Without loss of geerality,

More information

THE ABRACADABRA PROBLEM

THE ABRACADABRA PROBLEM THE ABRACADABRA PROBLEM FRANCESCO CARAVENNA Abstract. We preset a detailed solutio of Exercise E0.6 i [Wil9]: i a radom sequece of letters, draw idepedetly ad uiformly from the Eglish alphabet, the expected

More information

Sequences and Series

Sequences and Series CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their

More information

x(x 1)(x 2)... (x k + 1) = [x] k n+m 1

x(x 1)(x 2)... (x k + 1) = [x] k n+m 1 1 Coutig mappigs For every real x ad positive iteger k, let [x] k deote the fallig factorial ad x(x 1)(x 2)... (x k + 1) ( ) x = [x] k k k!, ( ) k = 1. 0 I the sequel, X = {x 1,..., x m }, Y = {y 1,...,

More information

The Field Q of Rational Numbers

The Field Q of Rational Numbers Chapter 3 The Field Q of Ratioal Numbers I this chapter we are goig to costruct the ratioal umber from the itegers. Historically, the positive ratioal umbers came first: the Babyloias, Egyptias ad Grees

More information

THE HEIGHT OF q-binary SEARCH TREES

THE HEIGHT OF q-binary SEARCH TREES THE HEIGHT OF q-binary SEARCH TREES MICHAEL DRMOTA AND HELMUT PRODINGER Abstract. q biary search trees are obtaied from words, equipped with the geometric distributio istead of permutatios. The average

More information

Properties of MLE: consistency, asymptotic normality. Fisher information.

Properties of MLE: consistency, asymptotic normality. Fisher information. Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout

More information

Math 475, Problem Set #6: Solutions

Math 475, Problem Set #6: Solutions Math 475, Problem Set #6: Solutios A (a) For each poit (a, b) with a, b o-egative itegers satisfyig ab 8, cout the paths from (0,0) to (a, b) where the legal steps from (i, j) are to (i 2, j), (i, j 2),

More information

Divergence of p 1/p. Adrian Dudek. adrian.dudek[at]anu.edu.au

Divergence of p 1/p. Adrian Dudek. adrian.dudek[at]anu.edu.au Divergece of / Adria Dudek adria.dudek[at]au.edu.au Whe I was i high school, my maths teacher cheekily told me that it s ossible to add u ifiitely may umbers ad get a fiite umber. She the illustrated this

More information

arxiv: v2 [math.nt] 5 Nov 2013

arxiv: v2 [math.nt] 5 Nov 2013 arxiv:29.64v2 [math.nt] 5 Nov 23 EULER SUMS OF HYPERHARMONIC NUMBERS Ayha Dil Departmet of Mathematics, Adeiz Uiversity, 758 Atalya Turey e-mail: adil@adeiz.edu.tr Khristo N. Boyadzhiev Departmet of Mathematics

More information

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008 I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces

More information

Infinite Sequences and Series

Infinite Sequences and Series CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...

More information

Fourier Series and the Wave Equation Part 2

Fourier Series and the Wave Equation Part 2 Fourier Series ad the Wave Equatio Part There are two big ideas i our work this week. The first is the use of liearity to break complicated problems ito simple pieces. The secod is the use of the symmetries

More information

1.3 Binomial Coefficients

1.3 Binomial Coefficients 18 CHAPTER 1. COUNTING 1. Biomial Coefficiets I this sectio, we will explore various properties of biomial coefficiets. Pascal s Triagle Table 1 cotais the values of the biomial coefficiets ( ) for 0to

More information

Section 11.3: The Integral Test

Section 11.3: The Integral Test Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult

More information

7 b) 0. Guided Notes for lesson P.2 Properties of Exponents. If a, b, x, y and a, b, 0, and m, n Z then the following properties hold: 1 n b

7 b) 0. Guided Notes for lesson P.2 Properties of Exponents. If a, b, x, y and a, b, 0, and m, n Z then the following properties hold: 1 n b Guided Notes for lesso P. Properties of Expoets If a, b, x, y ad a, b, 0, ad m, Z the the followig properties hold:. Negative Expoet Rule: b ad b b b Aswers must ever cotai egative expoets. Examples: 5

More information

A Gentle Introduction to Algorithms: Part II

A Gentle Introduction to Algorithms: Part II A Getle Itroductio to Algorithms: Part II Cotets of Part I:. Merge: (to merge two sorted lists ito a sigle sorted list.) 2. Bubble Sort 3. Merge Sort: 4. The Big-O, Big-Θ, Big-Ω otatios: asymptotic bouds

More information

Page 2 of 14 = T(-2) + 2 = [ T(-3)+1 ] + 2 Substitute T(-3)+1 for T(-2) = T(-3) + 3 = [ T(-4)+1 ] + 3 Substitute T(-4)+1 for T(-3) = T(-4) + 4 After i

Page 2 of 14 = T(-2) + 2 = [ T(-3)+1 ] + 2 Substitute T(-3)+1 for T(-2) = T(-3) + 3 = [ T(-4)+1 ] + 3 Substitute T(-4)+1 for T(-3) = T(-4) + 4 After i Page 1 of 14 Search C455 Chapter 4 - Recursio Tree Documet last modified: 02/09/2012 18:42:34 Uses: Use recursio tree to determie a good asymptotic boud o the recurrece T() = Sum the costs withi each level

More information

CS103X: Discrete Structures Homework 4 Solutions

CS103X: Discrete Structures Homework 4 Solutions CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible six-figure salaries i whole dollar amouts are there that cotai at least

More information

Alternatives To Pearson s and Spearman s Correlation Coefficients

Alternatives To Pearson s and Spearman s Correlation Coefficients Alteratives To Pearso s ad Spearma s Correlatio Coefficiets Floreti Smaradache Chair of Math & Scieces Departmet Uiversity of New Mexico Gallup, NM 8730, USA Abstract. This article presets several alteratives

More information

Lecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)

Lecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009) 18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the Bru-Mikowski iequality for boxes. Today we ll go over the

More information

THE UNLIKELY UNION OF PARTITIONS AND DIVISORS

THE UNLIKELY UNION OF PARTITIONS AND DIVISORS THE UNLIKELY UNION OF PARTITIONS AND DIVISORS Abdulkadir Hasse, Thomas J. Osler, Mathematics Departmet ad Tirupathi R. Chadrupatla, Mechaical Egieerig Rowa Uiversity Glassboro, NJ 828 I the multiplicative

More information

3.2 Introduction to Infinite Series

3.2 Introduction to Infinite Series 3.2 Itroductio to Ifiite Series May of our ifiite sequeces, for the remaider of the course, will be defied by sums. For example, the sequece S m := 2. () is defied by a sum. Its terms (partial sums) are

More information

The Field of Complex Numbers

The Field of Complex Numbers The Field of Complex Numbers S. F. Ellermeyer The costructio of the system of complex umbers begis by appedig to the system of real umbers a umber which we call i with the property that i = 1. (Note that

More information

2.3. GEOMETRIC SERIES

2.3. GEOMETRIC SERIES 6 CHAPTER INFINITE SERIES GEOMETRIC SERIES Oe of the most importat types of ifiite series are geometric series A geometric series is simply the sum of a geometric sequece, Fortuately, geometric series

More information

Sequences II. Chapter 3. 3.1 Convergent Sequences

Sequences II. Chapter 3. 3.1 Convergent Sequences Chapter 3 Sequeces II 3. Coverget Sequeces Plot a graph of the sequece a ) = 2, 3 2, 4 3, 5 + 4,...,,... To what limit do you thik this sequece teds? What ca you say about the sequece a )? For ǫ = 0.,

More information

Perfect Packing Theorems and the Average-Case Behavior of Optimal and Online Bin Packing

Perfect Packing Theorems and the Average-Case Behavior of Optimal and Online Bin Packing SIAM REVIEW Vol. 44, No. 1, pp. 95 108 c 2002 Society for Idustrial ad Applied Mathematics Perfect Packig Theorems ad the Average-Case Behavior of Optimal ad Olie Bi Packig E. G. Coffma, Jr. C. Courcoubetis

More information

The second difference is the sequence of differences of the first difference sequence, 2

The second difference is the sequence of differences of the first difference sequence, 2 Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for

More information

Our aim is to show that under reasonable assumptions a given 2π-periodic function f can be represented as convergent series

Our aim is to show that under reasonable assumptions a given 2π-periodic function f can be represented as convergent series 8 Fourier Series Our aim is to show that uder reasoable assumptios a give -periodic fuctio f ca be represeted as coverget series f(x) = a + (a cos x + b si x). (8.) By defiitio, the covergece of the series

More information

THE LEAST SQUARES REGRESSION LINE and R 2

THE LEAST SQUARES REGRESSION LINE and R 2 THE LEAST SQUARES REGRESSION LINE ad R M358K I. Recall from p. 36 that the least squares regressio lie of y o x is the lie that makes the sum of the squares of the vertical distaces of the data poits from

More information

Approximating the Sum of a Convergent Series

Approximating the Sum of a Convergent Series Approximatig the Sum of a Coverget Series Larry Riddle Ages Scott College Decatur, GA 30030 lriddle@agesscott.edu The BC Calculus Course Descriptio metios how techology ca be used to explore covergece

More information

Measurable Functions

Measurable Functions Measurable Fuctios Dug Le 1 1 Defiitio It is ecessary to determie the class of fuctios that will be cosidered for the Lebesgue itegratio. We wat to guaratee that the sets which arise whe workig with these

More information

4 n. n 1. You shold think of the Ratio Test as a generalization of the Geometric Series Test. For example, if a n ar n is a geometric sequence then

4 n. n 1. You shold think of the Ratio Test as a generalization of the Geometric Series Test. For example, if a n ar n is a geometric sequence then SECTION 2.6 THE RATIO TEST 79 2.6. THE RATIO TEST We ow kow how to hadle series which we ca itegrate (the Itegral Test), ad series which are similar to geometric or p-series (the Compariso Test), but of

More information

THE ARITHMETIC OF INTEGERS. - multiplication, exponentiation, division, addition, and subtraction

THE ARITHMETIC OF INTEGERS. - multiplication, exponentiation, division, addition, and subtraction THE ARITHMETIC OF INTEGERS - multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,

More information

FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix

FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. Powers of a matrix We begi with a propositio which illustrates the usefuless of the diagoalizatio. Recall that a square matrix A is diogaalizable if

More information

5.3. Generalized Permutations and Combinations

5.3. Generalized Permutations and Combinations 53 GENERALIZED PERMUTATIONS AND COMBINATIONS 73 53 Geeralized Permutatios ad Combiatios 53 Permutatios with Repeated Elemets Assume that we have a alphabet with letters ad we wat to write all possible

More information

TAYLOR SERIES, POWER SERIES

TAYLOR SERIES, POWER SERIES TAYLOR SERIES, POWER SERIES The followig represets a (icomplete) collectio of thigs that we covered o the subject of Taylor series ad power series. Warig. Be prepared to prove ay of these thigs durig the

More information

BINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients

BINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients 652 (12-26) Chapter 12 Sequeces ad Series 12.5 BINOMIAL EXPANSIONS I this sectio Some Examples Otaiig the Coefficiets The Biomial Theorem I Chapter 5 you leared how to square a iomial. I this sectio you

More information

I. Chi-squared Distributions

I. Chi-squared Distributions 1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.

More information

Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13

Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13 EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may

More information

13 Fast Fourier Transform (FFT)

13 Fast Fourier Transform (FFT) 13 Fast Fourier Trasform FFT) The fast Fourier trasform FFT) is a algorithm for the efficiet implemetatio of the discrete Fourier trasform. We begi our discussio oce more with the cotiuous Fourier trasform.

More information

Recursion and Recurrences

Recursion and Recurrences Chapter 5 Recursio ad Recurreces 5.1 Growth Rates of Solutios to Recurreces Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer. Cosider, for example,

More information

AN ASYMPTOTIC ROBIN INEQUALITY. Patrick Solé CNRS/LAGA, Université Paris 8, Saint-Denis, France.

AN ASYMPTOTIC ROBIN INEQUALITY. Patrick Solé CNRS/LAGA, Université Paris 8, Saint-Denis, France. #A8 INTEGERS 6 (206) AN ASYMPTOTIC ROBIN INEQUALITY Patrick Solé CNRS/LAGA, Uiversité Paris 8, Sait-Deis, Frace. sole@est.fr Yuyag Zhu Departmet of Math ad Physics, Hefei Uiversity, Hefei, Chia zhuyy@hfuu.edu.c

More information

THE COMPLETENESS OF CONVERGENT SEQUENCES SPACE OF FUZZY NUMBERS. Hee Chan Choi

THE COMPLETENESS OF CONVERGENT SEQUENCES SPACE OF FUZZY NUMBERS. Hee Chan Choi Kagweo-Kyugki Math. Jour. 4 (1996), No. 2, pp. 117 124 THE COMPLETENESS OF CONVERGENT SEQUENCES SPACE OF FUZZY NUMBERS Hee Cha Choi Abstract. I this paper we defie a ew fuzzy metric θ of fuzzy umber sequeces,

More information

Chapter 5: Inner Product Spaces

Chapter 5: Inner Product Spaces Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples

More information

Lecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem

Lecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem Lecture 4: Cauchy sequeces, Bolzao-Weierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits

More information

Lehmer s problem for polynomials with odd coefficients

Lehmer s problem for polynomials with odd coefficients Aals of Mathematics, 166 (2007), 347 366 Lehmer s problem for polyomials with odd coefficiets By Peter Borwei, Edward Dobrowolski, ad Michael J. Mossighoff* Abstract We prove that if f(x) = 1 k=0 a kx

More information

Chapter 7 Methods of Finding Estimators

Chapter 7 Methods of Finding Estimators Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of

More information

Repeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern.

Repeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern. 5.5 Fractios ad Decimals Steps for Chagig a Fractio to a Decimal. Simplify the fractio, if possible. 2. Divide the umerator by the deomiator. d d Repeatig Decimals Repeatig Decimals are decimal umbers

More information

Section 8.3 : De Moivre s Theorem and Applications

Section 8.3 : De Moivre s Theorem and Applications The Sectio 8 : De Moivre s Theorem ad Applicatios Let z 1 ad z be complex umbers, where z 1 = r 1, z = r, arg(z 1 ) = θ 1, arg(z ) = θ z 1 = r 1 (cos θ 1 + i si θ 1 ) z = r (cos θ + i si θ ) ad z 1 z =

More information

THE LEAST COMMON MULTIPLE OF A QUADRATIC SEQUENCE

THE LEAST COMMON MULTIPLE OF A QUADRATIC SEQUENCE THE LEAST COMMON MULTIPLE OF A QUADRATIC SEQUENCE JAVIER CILLERUELO Abstract. We obtai, for ay irreducible quadratic olyomial f(x = ax 2 + bx + c, the asymtotic estimate log l.c.m. {f(1,..., f(} log. Whe

More information

Lecture 5: Span, linear independence, bases, and dimension

Lecture 5: Span, linear independence, bases, and dimension Lecture 5: Spa, liear idepedece, bases, ad dimesio Travis Schedler Thurs, Sep 23, 2010 (versio: 9/21 9:55 PM) 1 Motivatio Motivatio To uderstad what it meas that R has dimesio oe, R 2 dimesio 2, etc.;

More information

Lesson 12. Sequences and Series

Lesson 12. Sequences and Series Retur to List of Lessos Lesso. Sequeces ad Series A ifiite sequece { a, a, a,... a,...} ca be thought of as a list of umbers writte i defiite order ad certai patter. It is usually deoted by { a } =, or

More information

ARITHMETIC AND GEOMETRIC PROGRESSIONS

ARITHMETIC AND GEOMETRIC PROGRESSIONS Arithmetic Ad Geometric Progressios Sequeces Ad ARITHMETIC AND GEOMETRIC PROGRESSIONS Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives

More information

Matrix Transforms of A-statistically Convergent Sequences with Speed

Matrix Transforms of A-statistically Convergent Sequences with Speed Filomat 27:8 2013, 1385 1392 DOI 10.2298/FIL1308385 Published by Faculty of Scieces ad Mathematics, Uiversity of Niš, Serbia vailable at: http://www.pmf.i.ac.rs/filomat Matrix Trasforms of -statistically

More information

7. Sample Covariance and Correlation

7. Sample Covariance and Correlation 1 of 8 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 7. Sample Covariace ad Correlatio The Bivariate Model Suppose agai that we have a basic radom experimet, ad that X ad Y

More information

arxiv: v1 [math.co] 31 Oct 2008

arxiv: v1 [math.co] 31 Oct 2008 SUMMATION OF HYPERHARMONIC SERIES ISTVÁN MEZŐ arxiv:08.004v [math.co] 3 Oct 008 Abstract. We shall show that the sum of the series formed by the so-called hyperharmoic umbers ca be expressed i terms of

More information

SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx

SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval

More information

1 Introduction to reducing variance in Monte Carlo simulations

1 Introduction to reducing variance in Monte Carlo simulations Copyright c 007 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a uow mea µ = E(X) of a distributio by

More information

2-3 The Remainder and Factor Theorems

2-3 The Remainder and Factor Theorems - The Remaider ad Factor Theorems Factor each polyomial completely usig the give factor ad log divisio 1 x + x x 60; x + So, x + x x 60 = (x + )(x x 15) Factorig the quadratic expressio yields x + x x

More information

Divide and Conquer. Maximum/minimum. Integer Multiplication. CS125 Lecture 4 Fall 2015

Divide and Conquer. Maximum/minimum. Integer Multiplication. CS125 Lecture 4 Fall 2015 CS125 Lecture 4 Fall 2015 Divide ad Coquer We have see oe geeral paradigm for fidig algorithms: the greedy approach. We ow cosider aother geeral paradigm, kow as divide ad coquer. We have already see a

More information

8.3 POLAR FORM AND DEMOIVRE S THEOREM

8.3 POLAR FORM AND DEMOIVRE S THEOREM SECTION 8. POLAR FORM AND DEMOIVRE S THEOREM 48 8. POLAR FORM AND DEMOIVRE S THEOREM Figure 8.6 (a, b) b r a 0 θ Complex Number: a + bi Rectagular Form: (a, b) Polar Form: (r, θ) At this poit you ca add,

More information

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,

More information

Fast Fourier Transform

Fast Fourier Transform 18.310 lecture otes November 18, 2013 Fast Fourier Trasform Lecturer: Michel Goemas I these otes we defie the Discrete Fourier Trasform, ad give a method for computig it fast: the Fast Fourier Trasform.

More information

Elementary Theory of Russian Roulette

Elementary Theory of Russian Roulette Elemetary Theory of Russia Roulette -iterestig patters of fractios- Satoshi Hashiba Daisuke Miematsu Ryohei Miyadera Itroductio. Today we are goig to study mathematical theory of Russia roulette. If some

More information

Week 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable

Week 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5

More information

{{1}, {2, 4}, {3}} {{1, 3, 4}, {2}} {{1}, {2}, {3, 4}} 5.4 Stirling Numbers

{{1}, {2, 4}, {3}} {{1, 3, 4}, {2}} {{1}, {2}, {3, 4}} 5.4 Stirling Numbers . Stirlig Numbers Whe coutig various types of fuctios from., we quicly discovered that eumeratig the umber of oto fuctios was a difficult problem. For a domai of five elemets ad a rage of four elemets,

More information

B1. Fourier Analysis of Discrete Time Signals

B1. Fourier Analysis of Discrete Time Signals B. Fourier Aalysis of Discrete Time Sigals Objectives Itroduce discrete time periodic sigals Defie the Discrete Fourier Series (DFS) expasio of periodic sigals Defie the Discrete Fourier Trasform (DFT)

More information

Chapter One BASIC MATHEMATICAL TOOLS

Chapter One BASIC MATHEMATICAL TOOLS Chapter Oe BAIC MATHEMATICAL TOOL As the reader will see, the study of the time value of moey ivolves substatial use of variables ad umbers that are raised to a power. The power to which a variable is

More information

when n = 1, 2, 3, 4, 5, 6, This list represents the amount of dollars you have after n days. Note: The use of is read as and so on.

when n = 1, 2, 3, 4, 5, 6, This list represents the amount of dollars you have after n days. Note: The use of is read as and so on. Geometric eries Before we defie what is meat by a series, we eed to itroduce a related topic, that of sequeces. Formally, a sequece is a fuctio that computes a ordered list. uppose that o day 1, you have

More information

Taking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling

Taking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed Multi-Evet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria

More information