Irreducible polynomials with consecutive zero coefficients


 William Morton
 3 years ago
 Views:
Transcription
1 Irreducible polyomials with cosecutive zero coefficiets Theodoulos Garefalakis Departmet of Mathematics, Uiversity of Crete, Heraklio, Greece Abstract Let q be a prime power. We cosider the problem of the existece of moic irreducible polyomials over F q with cosecutive coefficiets fixed to zero. We show that asymptotically, there exist moic irreducible polyomials of degree over F q with roughly /3 cosecutive coefficiets fixed to zero. Key words: Irreducible polyomials, fiite fields 1 Itroductio Let F q be a fiite field with q elemets, of characteristic p. We deote by A = F q [T] the rig of polyomials over F q. It is wellkow that asymptotically, the umber of irreducible polyomials i A of degree is approximately q /. However, much less is kow about the umber, or eve the existece of irreducibles of certai form, for istace with some coefficiets fixed to give values. Give a iteger > 1, it has bee proved idepedetly by S.D. Cohe [2] ad R. Ree [13], that for all large eough q, there always is a irreducible polyomial over F q of the form T +T +a. However, much less is kow whe q is fixed ad large. I [10], T. Hase ad G.L. Mulle cojecture that give itegers > m 0 there exists a moic irreducible polyomial over F q of degree with the coefficiet of T m fixed to ay give elemet a F q. Of course, a 0 if m = 0. By cosiderig primitive polyomials with give trace, S.D. Cohe [4] proves that the cojecture is true for m = 1. I [15], D. Wa settles address: (Theodoulos Garefalakis). Preprit submitted to Fiite Fields ad Their Applicatios 7 March 2007
2 the cojecture subject to the coditio that either q > 19 or 36, leavig a fiite umber of cases to be checked. By machie assisted computatios, K.H. Ham ad G.L. Mulle have verified the cojecture for these remaiig cases i [9]. Prior to this work, E.N. Kuz mi [12] determied the umber of moic irreducibles of degree 4 with the coefficiets of T 3, T 2, T fixed to give values. Special attetio is give to the case where the coefficiets of T 3 ad T 2 are zero. Similar results o the umber of polyomials of give degree of give factorizatio patter which satisfy a additioal property, such as that certai coefficiets are prescribed, have bee obtaied by S.D. Cohe i [3]. The aalogous problem for primitive polyomials has also attracted cosiderable attetio. The aalogue of Wa s result has bee proved to be true by S.D. Cohe [5]. A survey o the subject by S. Gao, J. Howell ad D. Paario, icludig experimetal results as well as some applicatios ca be foud i [8]. It is atural to ask, ad i fact to expect, that much more tha the above is true. Namely, oe would expect that irreducible polyomials exist with may coefficiets fixed to give values. The objective of this work is to show that moic irreducible polyomials of degree over F q exist with up to c cosecutive coefficiets fixed to zero, where 0 < c < 1 ad the coditio (1 3c) log q is satisfied. Such irreducibles are called sparse ad have may practical applicatios, see [6,7]. The proof of the mai theorem is based o a estimate of a weighted sum, which is very similar to the oe that D. Wa cosiders. The mai tool is Weil s boud for character sums. We record the followig elemetary lemma, which will be useful later. Let f(t) = i=0 a i T i F q [T] ad deote by f (T) = i=0 a i T i its reciprocal. Lemma 1 Let f(t) = i=0 a i T i F q [T] with a a 0 0. The polyomial f is irreducible over F q if ad oly if f is irreducible over F q. 2 Character sums It is well kow that Dirichlet s theorem for primes i arithmetic progressio has a aalogue i A. Let f, h A relatively prime. We reserve the letter P to deote a moic irreducible polyomial i A. Let S (h, f) = {P A P h (mod f), deg(p) = }. We deote by π (h, f) the cardiality of S (h, f). The followig asymptotic versio of Dirichlet s theorem is wellkow, see for istace [14]. 2
3 Theorem 1 π (h, f) = 1 Φ(f) q ( ) q /2 + O. Here Φ(f) is the order of the group (A/fA), ad the degree of f is assumed to be costat (ot depedig o ). Let f = T m A ad h A with deg(h) m 1. The S (h, f) cotais all irreducibles with the lower m coefficiets fixed to those of h. For m fixed, Theorem 1 gives a estimate of the umber of irreducibles with the lower m coefficiets fixed to ay values. Lemma 1 the implies that the same estimate holds if the upper m coefficiets are fixed. It is wellkow that the truth of the Riema hypothesis for fuctio fields leads to effective versios of Theorem 1, see for istace [15]. The basic tools are bouds for character sums, sometimes referred to as Weil character sums. We recall here the mai otios ad results for future referece. Let χ be a character of the group (A/fA), that is, a homomorphism from (A/fA) to C. The character χ exteded to A by zero is called a Dirichlet character mod f. The trivial character, that maps all polyomials prime to f to 1 is deoted by χ o. Defie the sum c (χ) = d deg(p)=d dχ(p /d ), where the ier sum is over all moic irreducible polyomials of degree d. It will be coveiet to express, as i [15], c (χ) i terms of the vo Magolt fuctio Λ, which is defied o A as follows: Λ(h) = deg(p) if h = P e for some irreducible P ad a iteger e, ad is zero otherwise. It s rather easy to see that c (χ) = deg(h)= Λ(h)χ(h), where the sum is over all moic polyomials of degree. Further, we defie c (χ) = χ(p), deg(p)= where the sum is over moic irreducibles of degree. We deote by π the umber of irreducible polyomials of degree i A. It is wellkow that d dπ d = q. The Moebius iversio formula the implies that π = µ(d)q /d = q + d d,d>1 µ(d)q /d. 3
4 Sice µ(d)q /d q d 2q /2, d,d>1 0 d /2 it follows that π q 2 q/2. The followig theorem follows from the Riema hypothesis for fuctio fields, see [15]. Propositio 1 If χ χ o the (1) c (χ) (deg(f) 1)q /2, (2) c (χ) 1 (deg(f) + 1)q/2. Also, c (χ o ) = q ad c (χ o) = π. Usig Propositio 1 it is ot hard to show the followig effective versio of Dirichlet s theorem for A. Theorem 2 (Theorem 5.1 of [15]) Let f, h A, with (f, h) = 1. The π (h, f) q Φ(f) m + 1 q/2. Theorem 2, applied with f = T m, immediately implies that there always exists a moic irreducible polyomial of degree with the coefficiets of roughly /2 log q lower coefficiets fixed (provided that the costat term is ot zero). Lemma 1 the esures the existece of a moic irreducible of degree with roughly /2 log q higher coefficiets fixed. More precisely we have the followig corollary. Corollary 1 Let > m 1 be itegers satisfyig q /2 (m+1)q m. For ay β 0, β 1,...,β m 1 F q with β 0 0, there exists a moic irreducible polyomial P = T + 1 i=0 a i T i i A of degree, with a i = β i, 0 i m 1. Also, there exists a moic irreducible polyomial with a i = β i, 1 i m 1. Irreducibles of degree with roughly /2 leadig or trailig coefficiets prescribed ca by foud effectively: heuristic argumets ad experimetal results suggest that oe may prescribe up to 2 log q leadig or trailig coefficiets, ad a irreducible still exists. This set polyomials is of reasoable size polyomial i, log q ad ca therefore be searched exhaustively. This method 4
5 of course depeds o uprove assumptios. To the author s kowledge, there is o method that provably costructs the irreducibles of Corollary 1. We ote that Theorem 2 ad Corollary 1 are classical. Geeralizatios have bee obtaied by M. Car [1] ad CN. Hsu [11]. It follows from these geeralizatios that irreducible polyomials of degree exist with the leadig 1 ad trailig 2 coefficiets are fixed to give values, subject to the coditio is less tha roughly /2. Ufortuately, the above results do ot give us ay iformatio about irreducibles with some of the middle coefficiets fixed. 3 Irreducible polyomials with cosecutive zero coefficiets Let m > l > 1 be itegers, ad deote by H l 1 the set of moic primary polyomials of A of degree l 1. We recall that a polyomial i A is called primary if it is a power of a irreducible polyomial of A. We defie w(, m, l) = h H l 1 Λ(h) P h (mod T m ) 1, (1) where the ier sum is over moic irreducibles of degree with stated property. Provig that w(, m, l) > 0 implies that there is a moic irreducible polyomial P = T + 1 i=0 a it i of degree with the coefficiets a i = 0 for l i m 1. Theorem 3 With the above otatio, w(, m, l) ql m π q 1 < m2 1 q (+l 1)/2. PROOF. First we rewrite the sum defiig w(, m, l) as w(, m, l) = 1 Φ(T m ) χ h H l 1 Λ(h) deg(p)= χ(p) χ(h), where the first sum is over the Dirichlet characters mod T m ad the third sum is over moic irreducibles of degree. Separatig the term correspodig to χ o ad rearragig we have w(, m, l) ql 1 π Φ(T m ) = 1 Φ(T m ) χ χ o deg(p)= χ(p) Λ(h) χ(h). h H l 1 5
6 Therefore, w(, m, l) ql 1 π Φ(T m ) 1 Φ(T m χ(p) Λ(h) χ(h) ) χ χ o deg(p)= h H l 1 = 1 c Φ(T m ) (χ) c l 1 ( χ) χ χ o < m + 1 q/2 (m 1)q (l 1)/2 = m2 1 q (+l 1)/2, where we used the estimates of Propositio 1, ad the fact that there are Φ(T m ) distict characters of (A/T m A). Corollary 2 Let > m > l > 1 such that q +l 2m qm 4. The there exists a moic irreducible polyomial P = T + 1 i=0 a i T i A of degree such that a m 1 = = a l = 0. PROOF. From Theorem 3 it follows that w(, m, l) > ql m π (q 1) m2 1 q (+l 1)/2. Sice π q 2q/2, we have w(, m, l) > q+l m (q 1) 2q/2+l m (q 1) (m2 1)q (+l 1)/2. (2) It suffices to prove that uder the coditio of the corollary, the righthadside is oegative. Ideed, the righthadside is at least ( ( )) 1 q +l m 1 m 2 q (+l 1)/2 + q (+l 1)/2 2q/2+l m q 1 ( q (+l)/2 m qm 2) q(+l 2)/2 0 where the first iequality holds sice q (+l 1)/2 2q/2+l m q 1 for 1 < l < m. Corollary 2 ca be used to show that there exist irreducible polyomials with a large umber of cosecutive coefficiets fixed to zero. 6
7 Corollary 3 Let 0 < c < 1 ad a atural umber such that (1 3c) 2+8 log q. The, there exists a moic irreducible polyomial of degree over F q with ay c cosecutive coefficiets, other tha the first ad the last, fixed to zero. PROOF. We apply Corollary 2 to fix m l = c (3) coefficiets. First, we observe that it suffices to cosider the case m + l. (4) The case m + l > follows from this by a applicatio of Lemma 1. We also record that the coditios i Eq. (3) ad Eq. (4) imply that 2m + c. (5) The corollary will follow if q c m qm 4 c 1 4 log q m m. Give Eq. (5), the last coditio is satisfied if 2 2 c 2 8 log q + c 3 c log q. The last iequality is clearly satisfied, uder the assumptio of the corollary. The corollary shows that for ay ǫ > 0 there exist moic irreducible polyomials of degree with up to (1/3 ǫ) coefficiets fixed to zero, provided that is large eough. As a example, we show the followig corollary. Corollary 4 Let q be a prime power, ad a positive iteger. The there exists a moic irreducible polyomial of degree over F q with ay /4 coefficiets fixed to zero for prime powers 2 q 59 ad the rages for show i the table below, ad for q 60 ad 37. 7
8 q q q PROOF. This is a cosequece of Corollary 3 for c = 1/4. We have to show that / log q (6) for the parameters i the table. The fuctio y q (t) = t 8 32 log q t is icreasig for t 32/ logq. For various values of q, we compute the sigle zero of y q (t) umerically, ad coclude that for t greater of equal to the zero y q (t) 0. The rages of t for each value of q are show i the table above. Sice y q (t) y q (t) for q q, we coclude that Eq. (6) holds for ay q 60 ad Cocludig Remarks We have proved that there exist moic irreducible polyomials of degree over F q with roughly /3 coefficiets fixed to zero. This is oly a partial extesio of the result of D. Wa [15], which shows that (uder the mild techical coditio that either q > 19 or 36) there exist moic irreducible of degree with ay oe coefficiet ca be fixed to ay value. It would be iterestig to exted the preset result, ad show that for large eough, there exist moic irreducibles of degree with roughly /3 coefficiets fixed to ay values. Ackowledgemets I would like to thak the aoymous reviewers for providig helpful commets ad for poitig out related work. 8
9 Refereces [1] M. Car. Distributio des polyomes irreductibles das F[t]. Acta Arith., 88: , [2] S.D. Cohe. The distributio of polyomials over fiite fields. Acta Arith., 17: , [3] S.D. Cohe. Uiform distributio of polyomials over fiite fields. J. Lodo Math. Soc., 2(6):93 102, [4] S.D. Cohe. Primitive elemets ad polyomials with arbitrary trace. J. America Math. Soc., 83:1 7, [5] S.D. Cohe. Primitive polyomials with a prescribed coefficiet. Fiite Fields ad Applicatios, 12(3): , [6] D. Coppersmith. Fast evaluatio of logarithms i fields of characteristic two. IEEE Tras. Iform. Theory, IT30: , [7] S. Gao. Elemets of provable high orders i fiite fields. Proc. America Math. Soc., 127: , [8] S. Gao, J. Howell, ad D. Paario. Irreducible polyomials of give forms. Cotemporary Mathematics, 225:43 53, [9] K.H. Ham ad G.L. Mulle. Distributio of irreducible polyomials of small degrees over fiite fields. Math. Comp., 67(221): , [10] T. Hase ad G.L. Mulle. Primitive polyomials over fiite fields. Math. Comp., 59: , [11] CN. Hsu. The distributio of irreducible polyomials i F q [t]. J. Number Theory, 61(1):85 96, [12] E.N. Kuz mi. Irreducible polyomials over fiite fields i. Algebra ad Logic, 33(4): , [13] R. Ree. Proof of a cojecture of S. Chowla. J. Number Theory, 3(2): , [14] M. Rose. Number theory i fuctio fields. Spriger Verlag, [15] D. Wa. Geerators ad irreducible polyomials over fiite fields. Math. Comp., 66(219): ,
Factors of sums of powers of binomial coefficients
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
More informationAsymptotic 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 informationModule 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 informationProperties 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 informationA 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 informationIn 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 informationA Note on Sums of Greatest (Least) Prime Factors
It. J. Cotemp. Math. Scieces, Vol. 8, 203, o. 9, 423432 HIKARI Ltd, www.mhikari.com A Note o Sums of Greatest (Least Prime Factors Rafael Jakimczuk Divisio Matemática, Uiversidad Nacioal de Luá Bueos
More informationOn the L p conjecture for locally compact groups
Arch. Math. 89 (2007), 237 242 c 2007 Birkhäuser Verlag Basel/Switzerlad 0003/889X/0302376, ublished olie 2007080 DOI 0.007/s0003007993x Archiv der Mathematik O the L cojecture for locally comact
More information4 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 pseries (the Compariso Test), but of
More informationChapter 6: Variance, the law of large numbers and the MonteCarlo method
Chapter 6: Variace, the law of large umbers ad the MoteCarlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS200609 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More informationOur aim is to show that under reasonable assumptions a given 2πperiodic 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 informationThe 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 informationTHE 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 informationI. Chisquared Distributions
1 M 358K Supplemet to Chapter 23: CHISQUARED DISTRIBUTIONS, TDISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad tdistributios, we first eed to look at aother family of distributios, the chisquared distributios.
More information1. 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 informationON THE DENSE TRAJECTORY OF LASOTA EQUATION
UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XLIII 2005 ON THE DENSE TRAJECTORY OF LASOTA EQUATION by Atoi Leo Dawidowicz ad Najemedi Haribash Abstract. I preseted paper the dese trajectory
More informationInfinite 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 informationWHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER?
WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? JÖRG JAHNEL 1. My Motivatio Some Sort of a Itroductio Last term I tought Topological Groups at the Göttige Georg August Uiversity. This
More information8.1 Arithmetic Sequences
MCR3U Uit 8: Sequeces & Series Page 1 of 1 8.1 Arithmetic Sequeces Defiitio: A sequece is a comma separated list of ordered terms that follow a patter. Examples: 1, 2, 3, 4, 5 : a sequece of the first
More informationSAMPLE 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 informationConvexity, Inequalities, and Norms
Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for
More informationChapter 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 informationSoving 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 informationPerfect Packing Theorems and the AverageCase 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 AverageCase Behavior of Optimal ad Olie Bi Packig E. G. Coffma, Jr. C. Courcoubetis
More informationAn example of nonquenched convergence in the conditional central limit theorem for partial sums of a linear process
A example of oqueched covergece i the coditioal cetral limit theorem for partial sums of a liear process Dalibor Volý ad Michael Woodroofe Abstract A causal liear processes X,X 0,X is costructed for which
More informationMARTINGALES AND A BASIC APPLICATION
MARTINGALES AND A BASIC APPLICATION TURNER SMITH Abstract. This paper will develop the measuretheoretic approach to probability i order to preset the defiitio of martigales. From there we will apply this
More informationClass Meeting # 16: The Fourier Transform on R n
MATH 18.152 COUSE NOTES  CLASS MEETING # 16 18.152 Itroductio to PDEs, Fall 2011 Professor: Jared Speck Class Meetig # 16: The Fourier Trasform o 1. Itroductio to the Fourier Trasform Earlier i the course,
More informationTHE 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 informationLecture 4: Cauchy sequences, BolzanoWeierstrass, and the Squeeze theorem
Lecture 4: Cauchy sequeces, BolzaoWeierstrass, 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 informationFactoring 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 informationTHIN SEQUENCES AND THE GRAM MATRIX PAMELA GORKIN, JOHN E. MCCARTHY, SANDRA POTT, AND BRETT D. WICK
THIN SEQUENCES AND THE GRAM MATRIX PAMELA GORKIN, JOHN E MCCARTHY, SANDRA POTT, AND BRETT D WICK Abstract We provide a ew proof of Volberg s Theorem characterizig thi iterpolatig sequeces as those for
More informationChapter 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 informationarxiv:1012.1336v2 [cs.cc] 8 Dec 2010
Uary SubsetSum 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 SubsetSum problem which is defied as follows: Give itegers
More informationTHE HEIGHT OF qbinary SEARCH TREES
THE HEIGHT OF qbinary 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 informationA 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 informationTHE 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 information7. 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 informationTaking DCOP to the Real World: Efficient Complete Solutions for Distributed MultiEvent Scheduling
Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed MultiEvet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria
More information3. 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 informationFIBONACCI 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 informationLecture 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 BruMikowski iequality for boxes. Today we ll go over the
More informationSequences 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 informationON AN INTEGRAL OPERATOR WHICH PRESERVE THE UNIVALENCE
Proceedigs of the Iteratioal Coferece o Theory ad Applicatios of Mathematics ad Iformatics ICTAMI 3, Alba Iulia ON AN INTEGRAL OPERATOR WHICH PRESERVE THE UNIVALENCE by Maria E Gageoea ad Silvia Moldoveau
More informationSampling Distribution And Central Limit Theorem
() Samplig Distributio & Cetral Limit Samplig Distributio Ad Cetral Limit Samplig distributio of the sample mea If we sample a umber of samples (say k samples where k is very large umber) each of size,
More informationTheorems 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 oegative umber R, called the radius
More information4.1 Sigma Notation and Riemann Sums
0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas
More informationEntropy of bicapacities
Etropy of bicapacities Iva Kojadiovic LINA CNRS FRE 2729 Site école polytechique de l uiv. de Nates Rue Christia Pauc 44306 Nates, Frace iva.kojadiovic@uivates.fr JeaLuc Marichal Applied Mathematics
More informationA Faster ClauseShortening Algorithm for SAT with No Restriction on Clause Length
Joural o Satisfiability, Boolea Modelig ad Computatio 1 2005) 4960 A Faster ClauseShorteig Algorithm for SAT with No Restrictio o Clause Legth Evgey Datsi Alexader Wolpert Departmet of Computer Sciece
More informationLecture 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 informationThe 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 informationLehmer 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 informationNUMBERS 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 informationIncremental calculation of weighted mean and variance
Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically
More information3. Covariance and Correlation
Virtual Laboratories > 3. Expected Value > 1 2 3 4 5 6 3. Covariace ad Correlatio Recall that by takig the expected value of various trasformatios of a radom variable, we ca measure may iterestig characteristics
More information2.7 Sequences, Sequences of Sets
2.7. SEQUENCES, SEQUENCES OF SETS 67 2.7 Sequeces, Sequeces of Sets 2.7.1 Sequeces Defiitio 190 (sequece Let S be some set. 1. A sequece i S is a fuctio f : K S where K = { N : 0 for some 0 N}. 2. For
More information5 Boolean Decision Trees (February 11)
5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected
More informationx(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 informationCS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations
CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad
More informationDiscrete 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 informationRamseytype theorems with forbidden subgraphs
Ramseytype theorems with forbidde subgraphs Noga Alo Jáos Pach József Solymosi Abstract A graph is called Hfree if it cotais o iduced copy of H. We discuss the followig questio raised by Erdős ad Hajal.
More information3 Basic Definitions of Probability Theory
3 Basic Defiitios of Probability Theory 3defprob.tex: Feb 10, 2003 Classical probability Frequecy probability axiomatic probability Historical developemet: Classical Frequecy Axiomatic The Axiomatic defiitio
More informationChair for Network Architectures and Services Institute of Informatics TU München Prof. Carle. Network Security. Chapter 2 Basics
Chair for Network Architectures ad Services Istitute of Iformatics TU Müche Prof. Carle Network Security Chapter 2 Basics 2.4 Radom Number Geeratio for Cryptographic Protocols Motivatio It is crucial to
More informationChapter 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 informationA note on the boundary behavior for a modiﬁed Green function in the upperhalf space
Zhag ad Pisarev Boudary Value Problems (015) 015:114 DOI 10.1186/s136610150363z RESEARCH Ope Access A ote o the boudary behavior for a modiﬁed Gree fuctio i the upperhalf space Yulia Zhag1 ad Valery
More informationAnnuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL.
Auities Uder Radom Rates of Iterest II By Abraham Zas Techio I.I.T. Haifa ISRAEL ad Haifa Uiversity Haifa ISRAEL Departmet of Mathematics, Techio  Israel Istitute of Techology, 3000, Haifa, Israel I memory
More informationNPTEL STRUCTURAL RELIABILITY
NPTEL Course O STRUCTURAL RELIABILITY Module # 0 Lecture 1 Course Format: Web Istructor: Dr. Aruasis Chakraborty Departmet of Civil Egieerig Idia Istitute of Techology Guwahati 1. Lecture 01: Basic Statistics
More informationSequences 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 information1 Itroductio Let A be a complex matrix ad let C (A) be its th compoud. It was show i [10, Formula (12)] that the imal row sum (of moduli) of elemets o
Bouds o orms of compoud matrices ad o products of eigevalues Ludwig Elser Faultat fur Mathemati Uiversitat Bielefeld Postfach 100131 D33615 Bielefeld Germay Daiel Hershowitz Departmet of Mathematics Techio
More informationSection 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 informationLearning outcomes. Algorithms and Data Structures. Time Complexity Analysis. Time Complexity Analysis How fast is the algorithm? Prof. Dr.
Algorithms ad Data Structures Algorithm efficiecy Learig outcomes Able to carry out simple asymptotic aalysisof algorithms Prof. Dr. Qi Xi 2 Time Complexity Aalysis How fast is the algorithm? Code the
More informationThe 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 informationLecture 2: Karger s Min Cut Algorithm
priceto uiv. F 3 cos 5: Advaced Algorithm Desig Lecture : Karger s Mi Cut Algorithm Lecturer: Sajeev Arora Scribe:Sajeev Today s topic is simple but gorgeous: Karger s mi cut algorithm ad its extesio.
More informationA NOTE ON BLOCK SEQUENCES IN HILBERT SPACES. S. K. Kaushik, Ghanshyam Singh and Virender University of Delhi and M.L.S.
GLASNIK MATEMATIČKI Vol. 43(63)(2008), 387 395 A NOTE ON BLOCK SEQUENCES IN HILBERT SPACES S. K. Kaushik, Ghashyam Sigh ad Vireder Uiversity of Delhi ad M.L.S. Uiversity, Idia Abstract. Block sequeces
More informationSECTION 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 informationApproximating 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 informationD I S C U S S I O N P A P E R
I N S T I T U T D E S T A T I S T I Q U E B I O S T A T I S T I Q U E E T S C I E N C E S A C T U A R I E L L E S ( I S B A UNIVERSITÉ CATHOLIQUE DE LOUVAIN D I S C U S S I O N P A P E R 2012/27 Worstcase
More information0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5
Sectio 13 KolmogorovSmirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.
More informationTrigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is
0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values
More informationElementary 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 informationDistributions of Order Statistics
Chapter 2 Distributios of Order Statistics We give some importat formulae for distributios of order statistics. For example, where F k: (x)=p{x k, x} = I F(x) (k, k + 1), I x (a,b)= 1 x t a 1 (1 t) b 1
More informationSUMS OF nth 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 information23 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 informationA RANDOM PERMUTATION MODEL ARISING IN CHEMISTRY
J. Appl. Prob. 45, 060 070 2008 Prited i Eglad Applied Probability Trust 2008 A RANDOM PERMUTATION MODEL ARISING IN CHEMISTRY MARK BROWN, The City College of New York EROL A. PEKÖZ, Bosto Uiversity SHELDON
More informationNonlife insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring
Nolife isurace mathematics Nils F. Haavardsso, Uiversity of Oslo ad DNB Skadeforsikrig Mai issues so far Why does isurace work? How is risk premium defied ad why is it importat? How ca claim frequecy
More informationBuilding Blocks Problem Related to Harmonic Series
TMME, vol3, o, p.76 Buildig Blocks Problem Related to Harmoic Series Yutaka Nishiyama Osaka Uiversity of Ecoomics, Japa Abstract: I this discussio I give a eplaatio of the divergece ad covergece of ifiite
More informationRunning Time ( 3.1) Analysis of Algorithms. Experimental Studies ( 3.1.1) Limitations of Experiments. Pseudocode ( 3.1.2) Theoretical Analysis
Ruig Time ( 3.) Aalysis of Algorithms Iput Algorithm Output A algorithm is a stepbystep procedure for solvig a problem i a fiite amout of time. Most algorithms trasform iput objects ito output objects.
More informationCS85: You Can t Do That (Lower Bounds in Computer Science) Lecture Notes, Spring 2008. Amit Chakrabarti Dartmouth College
CS85: You Ca t Do That () Lecture Notes, Sprig 2008 Amit Chakrabarti Dartmouth College Latest Update: May 9, 2008 Lecture 1 Compariso Trees: Sortig ad Selectio Scribe: William Che 1.1 Sortig Defiitio 1.1.1
More informationArithmetic of Triangular Fuzzy Variable from Credibility Theory
Vol., Issue 3, August 0 Arithmetic of Triagular Fuzzy Variable from Credibility Theory Ritupara Chutia (Correspodig Author) Departmet of Mathematics Gauhati Uiversity, Guwahati, Assam, Idia. Rituparachutia7@rediffmail.com
More informationSection 1.6: Proof by Mathematical Induction
Sectio.6 Proof by Iductio Sectio.6: Proof by Mathematical Iductio Purpose of Sectio: To itroduce the Priciple of Mathematical Iductio, both weak ad the strog versios, ad show how certai types of theorems
More informationModified Line Search Method for Global Optimization
Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o
More informationHypothesis testing. Null and alternative hypotheses
Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate
More informationUnit 20 Hypotheses Testing
Uit 2 Hypotheses Testig Objectives: To uderstad how to formulate a ull hypothesis ad a alterative hypothesis about a populatio proportio, ad how to choose a sigificace level To uderstad how to collect
More informationOn Formula to Compute Primes. and the n th Prime
Applied Mathematical cieces, Vol., 0, o., 3535 O Formula to Compute Primes ad the th Prime Issam Kaddoura Lebaese Iteratioal Uiversity Faculty of Arts ad cieces, Lebao issam.kaddoura@liu.edu.lb amih AbdulNabi
More information2. Degree Sequences. 2.1 Degree Sequences
2. Degree Sequeces The cocept of degrees i graphs has provided a framewor for the study of various structural properties of graphs ad has therefore attracted the attetio of may graph theorists. Here we
More informationInteger Factorization Algorithms
Iteger Factorizatio Algorithms Coelly Bares Departmet of Physics, Orego State Uiversity December 7, 004 This documet has bee placed i the public domai. Cotets I. Itroductio 3 1. Termiology 3. Fudametal
More informationConfidence Intervals for One Mean
Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a
More informationTAYLOR 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 informationTHE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n
We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample
More informationLECTURE 13: Crossvalidation
LECTURE 3: Crossvalidatio Resampli methods Cross Validatio Bootstrap Bias ad variace estimatio with the Bootstrap Threeway data partitioi Itroductio to Patter Aalysis Ricardo GutierrezOsua Texas A&M
More information