Dorin Andrica Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca, Romania
|
|
- Sheila Simpson
- 5 years ago
- Views:
Transcription
1 #A INTEGERS 5A (05) THE SIGNUM EQUATION FOR ERDŐS-SURÁNYI SEQUENCES Doin Andica Faculty of Mathematics and Comute Science, Babeş-Bolyai Univesity, Cluj-Naoca, Romania Eugen J. Ionascu Deatment of Mathematics, Columbus State Univesity, Columbus, Geogia Received: 3//4, Revised: 3/3/5, Acceted: 5/9/5, Published: 6/5/5 Abstact Fo an Edős-Suányi sequence it is customay to conside its signum equation. Based on some classical heuistic aguments, we conjectue the asymtotic behavio fo the numbe of solutions of this signum equation in the case of the sequence {n k } n (k ) and the sequence of imes. Suisingly, we show that this method does not aly at all fo the Fibonacci sequence. By comuting the ecise numbe of solutions, in this case, we obtain an exonential gowth, which shows, in aticula, the limitations of such an intuition.. INTRODUCTION We ecall (see [6] fo a n = n ) that a sequence of distinct ositive integes {a m } m is an Edős-Suányi sequence if evey intege k can be witten in the fom k = ±a ± a ± ± a n fo some choices of signs + and, in infinitely many ways. Using an induction agument, it iot di cult to see that {m} m, o even {m } m, is an Edős- Suányi sequence. In 979, J. Mitek oved in [8] that in fact, fo an abitay ositive intege s, {m s } m is an Edős-Suányi sequence. M. O. Dimbe gives ([4] and [5]) su cient conditions fo a sequence to be an Edős-Suányi sequence. THEOREM. Let {a m } m be a sequence of distinct ositive integes such that a = and fo evey n, a n+ ale a + + a n +. If the sequence {a m } m contains infinitely many odd integes, then it is an Edős-Suányi sequence. Honoific Membe of the Romanian Institute of Mathematics Simion Stoilow
2 INTEGERS: 5A (05) As an alication, one can check that the Fibonacci sequence satisfies the hyothesis of this theoem, since F 0 + F + + F n = F n+, n. Since F n + F n+ F n+ = 0 fo n 0, it iot suising that k = ±F ± F ± ± F n has infinitely many eesentations if it has one. The existence of at least one eesentation is elated to Zeckendof s theoem (see []) and as a cuiosity we include one such eesentation fo k = 05: 05 = 9X j F j, with j =, if and only if j {4,, 5, 7, 9}. THEOREM. Let f Q[X] be a olynomial such that fo any n Z, f(n) is an intege. If the geatest common facto of the tems of the sequence {f(n)} n is equal to, then {f(n)} n is an Edős-Suányi sequence. Using this esult, one can obtain many Edős-Suányi sequences: {(an ) k } n fo fixed a, k N, o { n+s s } n fo fixed s, ae just some ossible examles. We found that it is di cult to use an induction tye agument to show that these ae Edős-Suányi sequences. In this ae, given an Edős-Suányi sequence a = {a m } m, we ae inteested in the signum equation of a at level n, that is ±a ± a ± ± a n = 0. () Fo a fixed intege n, a solution to the signum equation is a choice of signs + and that makes () tue. We denote by S a (n) the numbe of solutions of Equation (). Clealy, if doeot divide u n, whee u n = a + + a n, then we have S a (n) = 0. We include next a few othe oeties of the sequence S a (n) (see [], []). (i) S a (n)/ n is the unique eal numbe having the oety that the function f : R! R, defined by 8 < cos a a x cos x f(x) = cos an x if x 6= 0, : if x = 0, is a deivative, i.e., it is the deivative of a di eentiable function on R. (ii) S a (n) is the tem not deending on z in the exansion of (z a + z a )(za + z ) a (zan + z an ). (iii) S a (n) is the coe cient of z un/ in the olynomial ( + z a )( + z a ) ( + z an ).
3 INTEGERS: 5A (05) 3 (iv) S a (n) can be calculated using the following integal fomula S a (n) = n Z 0 cos(a t) cos(a t) cos(a n t)dt. () (v) S a (n) is the numbe of odeed biatitions of the set {a, a,, a n } into two classes having equal sums. (vi) S a (n) is the numbe of distinct subsets of {a, a,, a n } whose elements sum to u n / if divides u n, and S a (n) = 0 othewise. (vii) S a (n) = N a (n), whee N a (n) is the numbe of eesentations of a n as a n = ±a ± a ± ± a n. In the next section, we ae inteested in the asymtotic behavio of S a (n), when n!.. Heuistic Asymtotic Behavio fo S a (n) S. Finch [7] has a vey inteesting obabilistic but heuistic aoach to aive at asymtotic esults fo S a (n). In this section, we ae going to descibe it in detail stating with an Edős-Suányi sequence a. Fo a ositive intege j, we conside the andom vaiable E j taking two values: a j o a j with obability / each. We assume that fo i 6= j, each E i is indeendent of E j. We then let the andom vaiable S n := E + E + + E n. (3) Since each of the E j has exectation zeo, we see that the exectation of S n is E(S n ) = P n E(E j) = 0. Because each of the E j has vaiation j with j = Va(E j ) = E( E j E(E j ) ) = a j, and the andom vaiables ae mutually indeendent, we conclude that s n = Va(S n ) = Va(E j ) = a j. We would like to use the Bey-Essen Theoem ([3]), which is a vey good vesion of the Cental Limit Theoem fo this situation. THEOREM. Let X, X,..., be indeendent andom vaiables with E(X j ) = 0, E(Xj ) = j > 0 and j = E( X j 3 ) <. If v u G n = (X + X + + X n )/ t j,
4 INTEGERS: 5A (05) 4 then su xr P (G n ale x) fo some univesal constant C. Z x e t dt ale C max alejalen q Pn j j j (4) THEOREM. Let a = {a n } n be a non-deceasing Edős-Suányi sequence of ositive integes, and let {S a (n)} n be the sequence counting the numbe of solutions of its signum equations. If we set s n = a + a + + a n, then n S a(n) ale C a n, (5) fo n big enough and fo some constant C (which doeot deend on a). Poof. We ae setting X j = E j in Theoem. and obseve that j = a 3 j = a n. Inequality (4) then becomes max alejalen j j su P (S n ale x ) xr In aticula fo y < x, we have P (y < S n ale x ) Z x e t a n dt ale C. Z x y e t dt ale C a n. and so Fo an abitay (0, ) and x =, y =, let us obseve that P (y < S n ale x ) = P ( < S n < ) = n S a(n) and so the inequality above can be witten as n S a(n) Z e t dt ale C a n. Since Z e t dt Z dt = letting! 0 and inceasing the constant C slightly, we get n S a(n) ale C a n.
5 INTEGERS: 5A (05) 5 Remak: In geneal, the constant C in (5) is the best that one can obtain, but in some aticula cases the constant C can be elaced by a sequence convegent to zeo. If this sequence is such that Ca n = o() then (5) becomes a genuine asymtotic fomula: S a (n) n = + o(). (6) We next discuss instances when it is vey likely that this is the case and an examle when this is clealy false... The Sequence n k If k N, and a = {n k } n, it is easy to show that fo > 0 we have j = n O(n ). This shows that in this case = k+ n k+ + O(n k ). The asymtotic fomula (6) suggests that we can fomulate the following: CONJECTURE. Fo evey ositive intege k the following elation holds: lim n! n 0 o 3 (mod 4) S k (n) n n k+ (k + ) =, (7) whee S k (n) stands fo S a (n) in this case. Fo k = the elation (7) was stated in the ae [] and it was called the Andica-Tomescu Conjectue in the ecent ae [0]. It was ecently oved to be coect by B. Sullivan [0]. Fo k, although we do not yet have a oof, we believe that unde simila estictions on n as in the case k =, (7) is still valid... The Sequence of Pimes The sequence {a} = {} of imes =, = 3,... is an Edös-Suányi sequence. This was shown in an ingenious way in the ae [4] by using the esult in Theoem.. Using the Pime Numbe Theoem, and a esult fom ([9]), we deive the following asymtotic identity: s n = j = ( + o(n)) (ln n + ln(ln n) )3 n3 = ( + o(n)) 3 ln n 3 n3 (ln n). Since the ode of gowth of iot geate than n, Equation (6) suggests that the following may be tue.
6 INTEGERS: 5A (05) 6 Figue : Distibution of S, and N (0, 88) CONJECTURE. If is the sequence of imes, then S (n) lim n!, n odd n n n ln n = 6. (8) In 830, H. F. Schek conjectued the following ecusive fomulas geneating the m-th ime fom the evious m imes: fo evey ositive intege n, and n = ± ± ± ± n + n n+ = ± ± ± ± n + n, fo some choices of signs + and. Such eesentations wee shown to exist in 98 by S. S. Pillai. Accoding to oety (viii) (age 3), the numbe of eesentations of n+ as n+ = ± ± ± ± n fo some choices of signs + and is N (n + ) = S (n + ). It is easy to see that S (n) = 0 fo evey even n and fo instance S () = 0, S (3) =, S (5) = ( = 0),..., which is ecisely half the sequence A0894 in the Online Encycloedia of Intege Sequences. Using the fist 000 tems, we calculated that S (n) n n (ln n + ln(ln n) )3/ 6 n.3897 fo n odd, n > 99, ln n which suots Conjectue. In Figue, we have included the distibution of X := ± ± ± which has zeo mean and a standad deviation of about Hee X takes even values fom 7 to 7. As we can see, this gah di es clealy fom that of the dilated nomal distibution N (0, 88) but if we account fo the half the values, the odd values whee the obability is zeo, we see an almost efect matching with N (0, 88) (Figue ) (the facto of which shows u in the oof of Theoem.3).
7 INTEGERS: 5A (05) 7 Figue : Distibution of S, and N (0, 88).3. The Fibonacci Sequence In contast, let us look at the well-known sequence F 0 = 0, F = F =, F 3 =,..., F n+ = F n + F n fo n, and at the elated sequence {S F (n)} n 0, whee S F (n) is the numbe of witings of 0 as a sum ±F 0 ± F ± F ± ± F n. Fo instance, it is easy to check that S F (0) =, S F () = 0 and S F () = 4. In fact, we have the following suising fomula in geneal. THEOREM.3 Fo n = 3(k ) +, k N, and {0,, } we have 8 >< k if = 0, S F (n) = 0 if =, >: k+ if =. (9) Poof. It is known that fo evey n N, we have F 0 + F F n = F n+. Fo a fixed n N, it is clea that S F (n + ) is twice the numbe of witings of F n+ as ±F n+ ± ± F ± F 0. Let us assume we have such a witing of F n+ : F n+ = n+ F n+ + n F n F 0, with i {, }, i = 0,,,, n+. (0) We claim that that n+ = and n =. By way of contadiction, if n+ = then F n+3 = F n+ + F n+ = n F n F 0 ale F F n = F n+ < F n+3. This contadiction shows that n+ =. Then Equation (0) becomes F n = F n+ F n+ = n F n + + F + 0 F 0.
8 INTEGERS: 5A (05) 8 Again, if n = then we obtain F n = n F n + + F + 0 F 0 ale F n+, F n ale F n ) F n < F n. So, this contadiction says that we must have n = and theefoe (0) is equivalent to 0 = n F n + F + 0 F 0, with i {, }, i = 0,,,, n. This means that S F (n + ) = S F (n ). Taking into account that S F (0) =, S F () = 0, S F () = 4, using an induction agument we see that we get the fomula in (9). It is known that fo evey n N, F0 + F + + Fn = F n F n+ and so = Fn F n+. Hence the left-hand side in (6) can be simlified to S F (n) n s n 8 8 F3k >< k 3 F 3k if n = 3k 3, = 0 if n = 3k, >: 4 F3k k F 3k if n = 3k. If we denote golden atio, as usual, by, i.e., = 5+, then since F n = n 5 +o(), we have 3k 5/ F 3k 3 F 3k = 5 +o() and F 3k F 3k = +o(). Because 3 /4.059 we see that S F (n) lim n!,n 0 o (mod 3) n =, 3k / 5 which makes (6) vey fa fom giving any asymtotic infomation about S F (n). Refeences [] Andica, D., Ionascu, E. J., Some Unexected Connections Between Analysis and Combinatoics, in Mathematics without boundaies. Toics in ue Mathematics, Th.M. Rassias and P. Padalos, Eds., Singe (04), -0 [] Andica, D., Tomescu, I., On an intege sequence elated to a oduct of tigonometic fuctions, and its combinatoial elevance, J. Intege Seq. 5(00), Aticle [3] Bey, A. C., The Accuacy of the Gaussian Aoximation to the Sum of Indeendent Vaiates, Tans. Ame. Math. Soc. 49()(94), -36 [4] Dimbe, M. O., A oblem of eesentation of integes (Romanian), G.M.-B, 0-(983), [5] Dimbe, M. O., Genealization of eesentation theoem of Edős and Suányi, Comment. Math. XXVII(988), No.,
9 INTEGERS: 5A (05) 9 [6] Edős, P., Suányi, J., Toics in the Theoy of Numbes, Singe-Velag, 003. [7] Finch, S. R., Signum equations and extemal coe cients, eole.fas.havad.edu/ sfinch/ [8] Mitek, J., Genealization of a theoem of Edős and Suányi, Comment. Math. XXI(979) [9] T. Salát, and S. Znám, On the sums of ime owes, Acta Fac. Re. Univ. Com. Math. (968), 5 [0] Sullivan, B.D., On a Conjectue of Andica and Tomescu, J. Intege Seq. 6(03), Aticle [] Zeckendof, E., Reésentation deombeatuels a une somme de nombes de Fibonacci ou deombes de Lucas, Bull. Soc. R. Sci. Liège 4(97), 79-8
556: MATHEMATICAL STATISTICS I
556: MATHEMATICAL STATISTICS I CHAPTER 5: STOCHASTIC CONVERGENCE The following efinitions ae state in tems of scala anom vaiables, but exten natually to vecto anom vaiables efine on the same obability
More informationON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION. 1. Introduction. 1 r r. r k for every set E A, E \ {0},
ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION E. J. IONASCU and A. A. STANCU Abstact. We ae inteested in constucting concete independent events in puely atomic pobability
More informationEdge Cover Time for Regular Graphs
1 2 3 47 6 23 11 Jounal of Intege Sequences, Vol. 11 (28, Aticle 8.4.4 Edge Cove Time fo Regula Gahs Robeto Tauaso Diatimento di Matematica Univesità di Roma To Vegata via della Riceca Scientifica 133
More informationBEST CONSTANTS FOR UNCENTERED MAXIMAL FUNCTIONS. Loukas Grafakos and Stephen Montgomery-Smith University of Missouri, Columbia
BEST CONSTANTS FOR UNCENTERED MAXIMAL FUNCTIONS Loukas Gafakos and Stehen Montgomey-Smith Univesity of Missoui, Columbia Abstact. We ecisely evaluate the oeato nom of the uncenteed Hady-Littlewood maximal
More informationProblem Set #10 Math 471 Real Analysis Assignment: Chapter 8 #2, 3, 6, 8
Poblem Set #0 Math 47 Real Analysis Assignment: Chate 8 #2, 3, 6, 8 Clayton J. Lungstum Decembe, 202 xecise 8.2 Pove the convese of Hölde s inequality fo = and =. Show also that fo eal-valued f / L ),
More informationProbabilistic number theory : A report on work done. What is the probability that a randomly chosen integer has no square factors?
Pobabilistic numbe theoy : A eot on wo done What is the obability that a andomly chosen intege has no squae factos? We can constuct an initial fomula to give us this value as follows: If a numbe is to
More informationk. s k=1 Part of the significance of the Riemann zeta-function stems from Theorem 9.2. If s > 1 then 1 p s
9 Pimes in aithmetic ogession Definition 9 The Riemann zeta-function ζs) is the function which assigns to a eal numbe s > the convegent seies k s k Pat of the significance of the Riemann zeta-function
More informationMethod for Approximating Irrational Numbers
Method fo Appoximating Iational Numbes Eic Reichwein Depatment of Physics Univesity of Califonia, Santa Cuz June 6, 0 Abstact I will put foth an algoithm fo poducing inceasingly accuate ational appoximations
More informationSemicanonical basis generators of the cluster algebra of type A (1)
Semicanonical basis geneatos of the cluste algeba of type A (1 1 Andei Zelevinsky Depatment of Mathematics Notheasten Univesity, Boston, USA andei@neu.edu Submitted: Jul 7, 006; Accepted: Dec 3, 006; Published:
More informationApproximating the minimum independent dominating set in perturbed graphs
Aoximating the minimum indeendent dominating set in etubed gahs Weitian Tong, Randy Goebel, Guohui Lin, Novembe 3, 013 Abstact We investigate the minimum indeendent dominating set in etubed gahs gg, )
More informationNumerical solution of the first order linear fuzzy differential equations using He0s variational iteration method
Malaya Jounal of Matematik, Vol. 6, No. 1, 80-84, 2018 htts://doi.og/16637/mjm0601/0012 Numeical solution of the fist ode linea fuzzy diffeential equations using He0s vaiational iteation method M. Ramachandan1
More informationSolution to HW 3, Ma 1a Fall 2016
Solution to HW 3, Ma a Fall 206 Section 2. Execise 2: Let C be a subset of the eal numbes consisting of those eal numbes x having the popety that evey digit in the decimal expansion of x is, 3, 5, o 7.
More informationSincere Voting and Information Aggregation with Voting Costs
Sincee Voting and Infomation Aggegation with Voting Costs Vijay Kishna y and John Mogan z August 007 Abstact We study the oeties of euilibium voting in two-altenative elections unde the majoity ule. Votes
More informationAn Estimate of Incomplete Mixed Character Sums 1 2. Mei-Chu Chang 3. Dedicated to Endre Szemerédi for his 70th birthday.
An Estimate of Incomlete Mixed Chaacte Sums 2 Mei-Chu Chang 3 Dedicated to Ende Szemeédi fo his 70th bithday. 4 In this note we conside incomlete mixed chaacte sums ove a finite field F n of the fom x
More informationOn the Poisson Approximation to the Negative Hypergeometric Distribution
BULLETIN of the Malaysian Mathematical Sciences Society http://mathusmmy/bulletin Bull Malays Math Sci Soc (2) 34(2) (2011), 331 336 On the Poisson Appoximation to the Negative Hypegeometic Distibution
More informationKOEBE DOMAINS FOR THE CLASSES OF FUNCTIONS WITH RANGES INCLUDED IN GIVEN SETS
Jounal of Applied Analysis Vol. 14, No. 1 2008), pp. 43 52 KOEBE DOMAINS FOR THE CLASSES OF FUNCTIONS WITH RANGES INCLUDED IN GIVEN SETS L. KOCZAN and P. ZAPRAWA Received Mach 12, 2007 and, in evised fom,
More informationNew problems in universal algebraic geometry illustrated by boolean equations
New poblems in univesal algebaic geomety illustated by boolean equations axiv:1611.00152v2 [math.ra] 25 Nov 2016 Atem N. Shevlyakov Novembe 28, 2016 Abstact We discuss new poblems in univesal algebaic
More informationOnline-routing on the butterfly network: probabilistic analysis
Online-outing on the buttefly netwok: obabilistic analysis Andey Gubichev 19.09.008 Contents 1 Intoduction: definitions 1 Aveage case behavio of the geedy algoithm 3.1 Bounds on congestion................................
More informationA Bijective Approach to the Permutational Power of a Priority Queue
A Bijective Appoach to the Pemutational Powe of a Pioity Queue Ia M. Gessel Kuang-Yeh Wang Depatment of Mathematics Bandeis Univesity Waltham, MA 02254-9110 Abstact A pioity queue tansfoms an input pemutation
More informationCross section dependence on ski pole sti ness
Coss section deendence on ski ole sti ness Johan Bystöm and Leonid Kuzmin Abstact Ski equiment oduce SWIX has ecently esented a new ai of ski oles, called SWIX Tiac, which di es fom conventional (ound)
More information(n 1)n(n + 1)(n + 2) + 1 = (n 1)(n + 2)n(n + 1) + 1 = ( (n 2 + n 1) 1 )( (n 2 + n 1) + 1 ) + 1 = (n 2 + n 1) 2.
Paabola Volume 5, Issue (017) Solutions 151 1540 Q151 Take any fou consecutive whole numbes, multiply them togethe and add 1. Make a conjectue and pove it! The esulting numbe can, fo instance, be expessed
More informationOn the ratio of maximum and minimum degree in maximal intersecting families
On the atio of maximum and minimum degee in maximal intesecting families Zoltán Lóánt Nagy Lale Özkahya Balázs Patkós Máté Vize Mach 6, 013 Abstact To study how balanced o unbalanced a maximal intesecting
More informationGoodness-of-fit for composite hypotheses.
Section 11 Goodness-of-fit fo composite hypotheses. Example. Let us conside a Matlab example. Let us geneate 50 obsevations fom N(1, 2): X=nomnd(1,2,50,1); Then, unning a chi-squaed goodness-of-fit test
More informationA Short Combinatorial Proof of Derangement Identity arxiv: v1 [math.co] 13 Nov Introduction
A Shot Combinatoial Poof of Deangement Identity axiv:1711.04537v1 [math.co] 13 Nov 2017 Ivica Matinjak Faculty of Science, Univesity of Zageb Bijenička cesta 32, HR-10000 Zageb, Coatia and Dajana Stanić
More information3.1 Random variables
3 Chapte III Random Vaiables 3 Random vaiables A sample space S may be difficult to descibe if the elements of S ae not numbes discuss how we can use a ule by which an element s of S may be associated
More informationBounds for the Density of Abundant Integers
Bounds fo the Density of Abundant Integes Mac Deléglise CONTENTS Intoduction. Eessing A() as a Sum 2. Tivial Bounds fo A () 3. Lowe Bound fo A() 4. Ue Bounds fo A () 5. Mean Values of f(n) and Ue Bounds
More informationCMSC 425: Lecture 5 More on Geometry and Geometric Programming
CMSC 425: Lectue 5 Moe on Geomety and Geometic Pogamming Moe Geometic Pogamming: In this lectue we continue the discussion of basic geometic ogamming fom the eious lectue. We will discuss coodinate systems
More informationof the contestants play as Falco, and 1 6
JHMT 05 Algeba Test Solutions 4 Febuay 05. In a Supe Smash Bothes tounament, of the contestants play as Fox, 3 of the contestants play as Falco, and 6 of the contestants play as Peach. Given that thee
More informationOn the ratio of maximum and minimum degree in maximal intersecting families
On the atio of maximum and minimum degee in maximal intesecting families Zoltán Lóánt Nagy Lale Özkahya Balázs Patkós Máté Vize Septembe 5, 011 Abstact To study how balanced o unbalanced a maximal intesecting
More informationMath 301: The Erdős-Stone-Simonovitz Theorem and Extremal Numbers for Bipartite Graphs
Math 30: The Edős-Stone-Simonovitz Theoem and Extemal Numbes fo Bipatite Gaphs May Radcliffe The Edős-Stone-Simonovitz Theoem Recall, in class we poved Tuán s Gaph Theoem, namely Theoem Tuán s Theoem Let
More informationSOME SOLVABILITY THEOREMS FOR NONLINEAR EQUATIONS
Fixed Point Theoy, Volume 5, No. 1, 2004, 71-80 http://www.math.ubbcluj.o/ nodeacj/sfptcj.htm SOME SOLVABILITY THEOREMS FOR NONLINEAR EQUATIONS G. ISAC 1 AND C. AVRAMESCU 2 1 Depatment of Mathematics Royal
More informationA generalization of the Bernstein polynomials
A genealization of the Benstein polynomials Halil Ouç and Geoge M Phillips Mathematical Institute, Univesity of St Andews, Noth Haugh, St Andews, Fife KY16 9SS, Scotland Dedicated to Philip J Davis This
More informationCONGRUENCES INVOLVING ( )
Jounal of Numbe Theoy 101, 157-1595 CONGRUENCES INVOLVING ZHI-Hong Sun School of Mathematical Sciences, Huaiyin Nomal Univesity, Huaian, Jiangsu 001, PR China Email: zhihongsun@yahoocom Homeage: htt://wwwhytceducn/xsjl/szh
More informationA proof of the binomial theorem
A poof of the binomial theoem If n is a natual numbe, let n! denote the poduct of the numbes,2,3,,n. So! =, 2! = 2 = 2, 3! = 2 3 = 6, 4! = 2 3 4 = 24 and so on. We also let 0! =. If n is a non-negative
More informationPascal s Triangle (mod 8)
Euop. J. Combinatoics (998) 9, 45 62 Pascal s Tiangle (mod 8) JAMES G. HUARD, BLAIR K. SPEARMAN AND KENNETH S. WILLIAMS Lucas theoem gives a conguence fo a binomial coefficient modulo a pime. Davis and
More informationOn weak exponential expansiveness of skew-evolution semiflows in Banach spaces
Yue et al. Jounal of Inequalities and Alications 014 014:165 htt://www.jounalofinequalitiesandalications.com/content/014/1/165 R E S E A R C H Oen Access On weak exonential exansiveness of skew-evolution
More informationKepler s problem gravitational attraction
Kele s oblem gavitational attaction Summay of fomulas deived fo two-body motion Let the two masses be m and m. The total mass is M = m + m, the educed mass is µ = m m /(m + m ). The gavitational otential
More informationNOTE. Some New Bounds for Cover-Free Families
Jounal of Combinatoial Theoy, Seies A 90, 224234 (2000) doi:10.1006jcta.1999.3036, available online at http:.idealibay.com on NOTE Some Ne Bounds fo Cove-Fee Families D. R. Stinson 1 and R. Wei Depatment
More informationSTUDY OF SOLUTIONS OF LOGARITHMIC ORDER TO HIGHER ORDER LINEAR DIFFERENTIAL-DIFFERENCE EQUATIONS WITH COEFFICIENTS HAVING THE SAME LOGARITHMIC ORDER
UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA doi: 104467/20843828AM170027078 542017, 15 32 STUDY OF SOLUTIONS OF LOGARITHMIC ORDER TO HIGHER ORDER LINEAR DIFFERENTIAL-DIFFERENCE EQUATIONS WITH COEFFICIENTS
More informationPearson s Chi-Square Test Modifications for Comparison of Unweighted and Weighted Histograms and Two Weighted Histograms
Peason s Chi-Squae Test Modifications fo Compaison of Unweighted and Weighted Histogams and Two Weighted Histogams Univesity of Akueyi, Bogi, v/noduslód, IS-6 Akueyi, Iceland E-mail: nikolai@unak.is Two
More informationLecture 28: Convergence of Random Variables and Related Theorems
EE50: Pobability Foundations fo Electical Enginees July-Novembe 205 Lectue 28: Convegence of Random Vaiables and Related Theoems Lectue:. Kishna Jagannathan Scibe: Gopal, Sudhasan, Ajay, Swamy, Kolla An
More informationH.W.GOULD West Virginia University, Morgan town, West Virginia 26506
A F I B O N A C C I F O R M U L A OF LUCAS A N D ITS SUBSEQUENT M A N I F E S T A T I O N S A N D R E D I S C O V E R I E S H.W.GOULD West Viginia Univesity, Mogan town, West Viginia 26506 Almost eveyone
More informationAnalysis of Arithmetic. Analysis of Arithmetic. Analysis of Arithmetic Round-Off Errors. Analysis of Arithmetic. Analysis of Arithmetic
In the fixed-oint imlementation of a digital filte only the esult of the multilication oeation is quantied The eesentation of a actical multilie with the quantie at its outut is shown below u v Q ^v The
More information9.1 The multiplicative group of a finite field. Theorem 9.1. The multiplicative group F of a finite field is cyclic.
Chapte 9 Pimitive Roots 9.1 The multiplicative goup of a finite fld Theoem 9.1. The multiplicative goup F of a finite fld is cyclic. Remak: In paticula, if p is a pime then (Z/p) is cyclic. In fact, this
More informationQuasi-Randomness and the Distribution of Copies of a Fixed Graph
Quasi-Randomness and the Distibution of Copies of a Fixed Gaph Asaf Shapia Abstact We show that if a gaph G has the popety that all subsets of vetices of size n/4 contain the coect numbe of tiangles one
More informationJournal of Inequalities in Pure and Applied Mathematics
Jounal of Inequalities in Pue and Applied Mathematics COEFFICIENT INEQUALITY FOR A FUNCTION WHOSE DERIVATIVE HAS A POSITIVE REAL PART S. ABRAMOVICH, M. KLARIČIĆ BAKULA AND S. BANIĆ Depatment of Mathematics
More informationIntegral operator defined by q-analogue of Liu-Srivastava operator
Stud. Univ. Babeş-Bolyai Math. 582013, No. 4, 529 537 Integal opeato defined by q-analogue of Liu-Sivastava opeato Huda Aldweby and Maslina Daus Abstact. In this pape, we shall give an application of q-analogues
More informationThe Congestion of n-cube Layout on a Rectangular Grid S.L. Bezrukov J.D. Chavez y L.H. Harper z M. Rottger U.-P. Schroeder Abstract We consider the pr
The Congestion of n-cube Layout on a Rectangula Gid S.L. Bezukov J.D. Chavez y L.H. Hape z M. Rottge U.-P. Schoede Abstact We conside the poblem of embedding the n-dimensional cube into a ectangula gid
More informationarxiv: v1 [math.nt] 12 May 2017
SEQUENCES OF CONSECUTIVE HAPPY NUMBERS IN NEGATIVE BASES HELEN G. GRUNDMAN AND PAMELA E. HARRIS axiv:1705.04648v1 [math.nt] 12 May 2017 ABSTRACT. Fo b 2 and e 2, let S e,b : Z Z 0 be the function taking
More informationThe least common multiple of a quadratic sequence
The least common multile of a quadatic sequence Javie Cilleuelo Abstact Fo any ieducible quadatic olynomial fx in Z[x] we obtain the estimate log l.c.m. f1,..., fn n log n Bn on whee B is a constant deending
More informationONE-POINT CODES USING PLACES OF HIGHER DEGREE
ONE-POINT CODES USING PLACES OF HIGHER DEGREE GRETCHEN L. MATTHEWS AND TODD W. MICHEL DEPARTMENT OF MATHEMATICAL SCIENCES CLEMSON UNIVERSITY CLEMSON, SC 29634-0975 U.S.A. E-MAIL: GMATTHE@CLEMSON.EDU, TMICHEL@CLEMSON.EDU
More informationMaximal Inequalities for the Ornstein-Uhlenbeck Process
Poc. Ame. Math. Soc. Vol. 28, No., 2, (335-34) Reseach Reot No. 393, 998, Det. Theoet. Statist. Aahus Maimal Ineualities fo the Onstein-Uhlenbeck Pocess S.. GRAVRSN 3 and G. PSKIR 3 Let V = (V t ) t be
More informationChapter 3: Theory of Modular Arithmetic 38
Chapte 3: Theoy of Modula Aithmetic 38 Section D Chinese Remainde Theoem By the end of this section you will be able to pove the Chinese Remainde Theoem apply this theoem to solve simultaneous linea conguences
More informationarxiv: v1 [math.co] 6 Mar 2008
An uppe bound fo the numbe of pefect matchings in gaphs Shmuel Fiedland axiv:0803.0864v [math.co] 6 Ma 2008 Depatment of Mathematics, Statistics, and Compute Science, Univesity of Illinois at Chicago Chicago,
More informationMultiple Criteria Secretary Problem: A New Approach
J. Stat. Appl. Po. 3, o., 9-38 (04 9 Jounal of Statistics Applications & Pobability An Intenational Jounal http://dx.doi.og/0.785/jsap/0303 Multiple Citeia Secetay Poblem: A ew Appoach Alaka Padhye, and
More informationSurveillance Points in High Dimensional Spaces
Société de Calcul Mathématique SA Tools fo decision help since 995 Suveillance Points in High Dimensional Spaces by Benad Beauzamy Januay 06 Abstact Let us conside any compute softwae, elying upon a lage
More information( ) F α. a. Sketch! r as a function of r for fixed θ. For the sketch, assume that θ is roughly the same ( )
. An acoustic a eflecting off a wav bounda (such as the sea suface) will see onl that pat of the bounda inclined towad the a. Conside a a with inclination to the hoizontal θ (whee θ is necessail positive,
More informationStanford University CS259Q: Quantum Computing Handout 8 Luca Trevisan October 18, 2012
Stanfod Univesity CS59Q: Quantum Computing Handout 8 Luca Tevisan Octobe 8, 0 Lectue 8 In which we use the quantum Fouie tansfom to solve the peiod-finding poblem. The Peiod Finding Poblem Let f : {0,...,
More informationf h = u, h g = v, we have u + v = f g. So, we wish
Answes to Homewok 4, Math 4111 (1) Pove that the following examples fom class ae indeed metic spaces. You only need to veify the tiangle inequality. (a) Let C be the set of continuous functions fom [0,
More informationON LACUNARY INVARIANT SEQUENCE SPACES DEFINED BY A SEQUENCE OF MODULUS FUNCTIONS
STUDIA UNIV BABEŞ BOLYAI, MATHEMATICA, Volume XLVIII, Numbe 4, Decembe 2003 ON LACUNARY INVARIANT SEQUENCE SPACES DEFINED BY A SEQUENCE OF MODULUS FUNCTIONS VATAN KARAKAYA AND NECIP SIMSEK Abstact The
More informationarxiv: v1 [math.nt] 28 Oct 2017
ON th COEFFICIENT OF DIVISORS OF x n axiv:70049v [mathnt] 28 Oct 207 SAI TEJA SOMU Abstact Let,n be two natual numbes and let H(,n denote the maximal absolute value of th coefficient of divisos of x n
More informationLacunary I-Convergent Sequences
KYUNGPOOK Math. J. 52(2012), 473-482 http://dx.doi.og/10.5666/kmj.2012.52.4.473 Lacunay I-Convegent Sequences Binod Chanda Tipathy Mathematical Sciences Division, Institute of Advanced Study in Science
More informationKirby-Melvin s τ r and Ohtsuki s τ for Lens Spaces
Kiby-Melvin s τ and Ohtsuki s τ fo Lens Saces Bang-He Li & Tian-Jun Li axiv:math/9807155v1 [mathqa] 27 Jul 1998 Abstact Exlicit fomulae fo τ (L(,q)) and τ(l(,q)) ae obtained fo all L(,q) Thee ae thee systems
More informationNumerical approximation to ζ(2n+1)
Illinois Wesleyan Univesity Fom the SelectedWoks of Tian-Xiao He 6 Numeical appoximation to ζ(n+1) Tian-Xiao He, Illinois Wesleyan Univesity Michael J. Dancs Available at: https://woks.bepess.com/tian_xiao_he/6/
More informationq i i=1 p i ln p i Another measure, which proves a useful benchmark in our analysis, is the chi squared divergence of p, q, which is defined by
CSISZÁR f DIVERGENCE, OSTROWSKI S INEQUALITY AND MUTUAL INFORMATION S. S. DRAGOMIR, V. GLUŠČEVIĆ, AND C. E. M. PEARCE Abstact. The Ostowski integal inequality fo an absolutely continuous function is used
More informationRATIONAL BASE NUMBER SYSTEMS FOR p-adic NUMBERS
RAIRO-Theo. Inf. Al. 46 (202) 87 06 DOI: 0.05/ita/204 Available online at: www.aio-ita.og RATIONAL BASE NUMBER SYSTEMS FOR -ADIC NUMBERS Chistiane Fougny and Kael Klouda 2 Abstact. This ae deals with ational
More informationON THE INVERSE SIGNED TOTAL DOMINATION NUMBER IN GRAPHS. D.A. Mojdeh and B. Samadi
Opuscula Math. 37, no. 3 (017), 447 456 http://dx.doi.og/10.7494/opmath.017.37.3.447 Opuscula Mathematica ON THE INVERSE SIGNED TOTAL DOMINATION NUMBER IN GRAPHS D.A. Mojdeh and B. Samadi Communicated
More informationInternet Appendix for A Bayesian Approach to Real Options: The Case of Distinguishing Between Temporary and Permanent Shocks
Intenet Appendix fo A Bayesian Appoach to Real Options: The Case of Distinguishing Between Tempoay and Pemanent Shocks Steven R. Genadie Gaduate School of Business, Stanfod Univesity Andey Malenko Gaduate
More informationA Multivariate Normal Law for Turing s Formulae
A Multivaiate Nomal Law fo Tuing s Fomulae Zhiyi Zhang Depatment of Mathematics and Statistics Univesity of Noth Caolina at Chalotte Chalotte, NC 28223 Abstact This pape establishes a sufficient condition
More informationarxiv: v1 [math.co] 4 May 2017
On The Numbe Of Unlabeled Bipatite Gaphs Abdullah Atmaca and A Yavuz Ouç axiv:7050800v [mathco] 4 May 207 Abstact This pape solves a poblem that was stated by M A Haison in 973 [] This poblem, that has
More informationAn upper bound on the number of high-dimensional permutations
An uppe bound on the numbe of high-dimensional pemutations Nathan Linial Zu Luia Abstact What is the highe-dimensional analog of a pemutation? If we think of a pemutation as given by a pemutation matix,
More informationAsymptotically Lacunary Statistical Equivalent Sequence Spaces Defined by Ideal Convergence and an Orlicz Function
"Science Stays Tue Hee" Jounal of Mathematics and Statistical Science, 335-35 Science Signpost Publishing Asymptotically Lacunay Statistical Equivalent Sequence Spaces Defined by Ideal Convegence and an
More information6 PROBABILITY GENERATING FUNCTIONS
6 PROBABILITY GENERATING FUNCTIONS Cetain deivations pesented in this couse have been somewhat heavy on algeba. Fo example, detemining the expectation of the Binomial distibution (page 5.1 tuned out to
More informationON SPARSELY SCHEMMEL TOTIENT NUMBERS. Colin Defant 1 Department of Mathematics, University of Florida, Gainesville, Florida
#A8 INTEGERS 5 (205) ON SPARSEL SCHEMMEL TOTIENT NUMBERS Colin Defant Depatment of Mathematics, Univesity of Floida, Gainesville, Floida cdefant@ufl.edu Received: 7/30/4, Revised: 2/23/4, Accepted: 4/26/5,
More informationBrief summary of functional analysis APPM 5440 Fall 2014 Applied Analysis
Bief summay of functional analysis APPM 5440 Fall 014 Applied Analysis Stephen Becke, stephen.becke@coloado.edu Standad theoems. When necessay, I used Royden s and Keyzsig s books as a efeence. Vesion
More informationSuggested Solutions to Homework #4 Econ 511b (Part I), Spring 2004
Suggested Solutions to Homewok #4 Econ 5b (Pat I), Sping 2004. Conside a neoclassical gowth model with valued leisue. The (epesentative) consume values steams of consumption and leisue accoding to P t=0
More informationC/CS/Phys C191 Shor s order (period) finding algorithm and factoring 11/12/14 Fall 2014 Lecture 22
C/CS/Phys C9 Sho s ode (peiod) finding algoithm and factoing /2/4 Fall 204 Lectue 22 With a fast algoithm fo the uantum Fouie Tansfom in hand, it is clea that many useful applications should be possible.
More informationOn a quantity that is analogous to potential and a theorem that relates to it
Su une quantité analogue au potential et su un théoème y elatif C R Acad Sci 7 (87) 34-39 On a quantity that is analogous to potential and a theoem that elates to it By R CLAUSIUS Tanslated by D H Delphenich
More informationSUFFICIENT CONDITIONS FOR MAXIMALLY EDGE-CONNECTED AND SUPER-EDGE-CONNECTED GRAPHS DEPENDING ON THE CLIQUE NUMBER
Discussiones Mathematicae Gaph Theoy 39 (019) 567 573 doi:10.7151/dmgt.096 SUFFICIENT CONDITIONS FOR MAXIMALLY EDGE-CONNECTED AND SUPER-EDGE-CONNECTED GRAPHS DEPENDING ON THE CLIQUE NUMBER Lutz Volkmann
More information( x 0 U0 0 x>0 U wavenumbe k deceases by a facto fom left hand Hence to ight hand egion. Thus we should see wavelength of egion again E,U ositive but
4 Sing 99 Poblem Set 3 Solutions Physics May 6, 999 Handout nal exam will given on Tuesday, May 8 fom 9:00 to :30 in Baton Hall. It will be witten as a 90 minute exam, but all am, will have full eiod fom
More informationTHE PARITY OF THE PERIOD OF THE CONTINUED FRACTION OF d
THE PARITY OF THE PERIOD OF THE CONTINUED FRACTION OF d ÉTIENNE FOUVRY AND JÜRGEN KLÜNERS Abstact. We ove that asymtotically in the set of squaefee integes d not divisible by imes conguent to 3 mod the
More informationOn The Queuing System M/E r /1/N
On The Queuing System M/E //N Namh A. Abid* Azmi. K. Al-Madi** Received 3, Mach, 2 Acceted 8, Octobe, 2 Abstact: In this ae the queuing system (M/E //N) has been consideed in equilibium. The method of
More informationIs the general form of Renyi s entropy a contrast for source separation?
Is the geneal fom of Renyi s entoy a contast fo souce seaation? Fédéic Vins 1, Dinh-Tuan Pham 2, and Michel Veleysen 1 1 Machine Leaning Gou Univesité catholique de Louvain Louvain-la-Neuve, Belgium {vins,veleysen}@dice.ucl.ac.be
More informationCENTRAL INDEX BASED SOME COMPARATIVE GROWTH ANALYSIS OF COMPOSITE ENTIRE FUNCTIONS FROM THE VIEW POINT OF L -ORDER. Tanmay Biswas
J Koean Soc Math Educ Se B: Pue Appl Math ISSNPint 16-0657 https://doiog/107468/jksmeb01853193 ISSNOnline 87-6081 Volume 5, Numbe 3 August 018, Pages 193 01 CENTRAL INDEX BASED SOME COMPARATIVE GROWTH
More informationVanishing lines in generalized Adams spectral sequences are generic
ISSN 364-0380 (on line) 465-3060 (pinted) 55 Geomety & Topology Volume 3 (999) 55 65 Published: 2 July 999 G G G G T T T G T T T G T G T GG TT G G G G GG T T T TT Vanishing lines in genealized Adams spectal
More informationRecognizable Infinite Triangular Array Languages
IOSR Jounal of Mathematics (IOSR-JM) e-issn: 2278-5728, -ISSN: 2319-765X. Volume 14, Issue 1 Ve. I (Jan. - Feb. 2018), PP 01-10 www.iosjounals.og Recognizable Infinite iangula Aay Languages 1 V.Devi Rajaselvi,
More informationFixed Point Results for Multivalued Maps
Int. J. Contemp. Math. Sciences, Vol., 007, no. 3, 119-1136 Fixed Point Results fo Multivalued Maps Abdul Latif Depatment of Mathematics King Abdulaziz Univesity P.O. Box 8003, Jeddah 1589 Saudi Aabia
More informationRECIPROCAL POWER SUMS. Anthony Sofo Victoria University, Melbourne City, Australia.
#A39 INTEGERS () RECIPROCAL POWER SUMS Athoy Sofo Victoia Uivesity, Melboue City, Austalia. athoy.sofo@vu.edu.au Received: /8/, Acceted: 6//, Published: 6/5/ Abstact I this ae we give a alteative oof ad
More informationESSENTIAL NORM OF AN INTEGRAL-TYPE OPERATOR ON THE UNIT BALL. Juntao Du and Xiangling Zhu
Opuscula Math. 38, no. 6 (8), 89 839 https://doi.og/.7494/opmath.8.38.6.89 Opuscula Mathematica ESSENTIAL NORM OF AN INTEGRAL-TYPE OPERATOR FROM ω-bloch SPACES TO µ-zygmund SPACES ON THE UNIT BALL Juntao
More informationProduct Rule and Chain Rule Estimates for Hajlasz Gradients on Doubling Metric Measure Spaces
Poduct Rule and Chain Rule Estimates fo Hajlasz Gadients on Doubling Metic Measue Saces A Eduado Gatto and Calos Segovia Fenández Ail 9, 2004 Abstact In this ae we intoduced Maximal Functions Nf, x) of
More informationEM Boundary Value Problems
EM Bounday Value Poblems 10/ 9 11/ By Ilekta chistidi & Lee, Seung-Hyun A. Geneal Desciption : Maxwell Equations & Loentz Foce We want to find the equations of motion of chaged paticles. The way to do
More informationMitscherlich s Law: Sum of two exponential Processes; Conclusions 2009, 1 st July
Mitschelich s Law: Sum of two exponential Pocesses; Conclusions 29, st July Hans Schneebege Institute of Statistics, Univesity of Elangen-Nünbeg, Gemany Summay It will be shown, that Mitschelich s fomula,
More informationGROWTH ESTIMATES THROUGH SCALING FOR QUASILINEAR PARTIAL DIFFERENTIAL EQUATIONS
Annales Academiæ Scientiaum Fennicæ Mathematica Volumen 32, 2007, 595 599 GROWTH ESTIMATES THROUGH SCALING FOR QUASILINEAR PARTIAL DIFFERENTIAL EQUATIONS Teo Kilpeläinen, Henik Shahgholian and Xiao Zhong
More informationChapter 5 Linear Equations: Basic Theory and Practice
Chapte 5 inea Equations: Basic Theoy and actice In this chapte and the next, we ae inteested in the linea algebaic equation AX = b, (5-1) whee A is an m n matix, X is an n 1 vecto to be solved fo, and
More informationRandom Variables and Probability Distribution Random Variable
Random Vaiables and Pobability Distibution Random Vaiable Random vaiable: If S is the sample space P(S) is the powe set of the sample space, P is the pobability of the function then (S, P(S), P) is called
More informationPledge: Signature:
S 202, Sing 2005 Midtem 1: 24 eb 2005 Page 1/8 Name: KEY E-mail D: @viginia.edu Pledge: Signatue: Thee ae 75 minutes fo this exam and 100 oints on the test; don t send too long on any one uestion! The
More informationTHE JEU DE TAQUIN ON THE SHIFTED RIM HOOK TABLEAUX. Jaejin Lee
Koean J. Math. 23 (2015), No. 3, pp. 427 438 http://dx.doi.og/10.11568/kjm.2015.23.3.427 THE JEU DE TAQUIN ON THE SHIFTED RIM HOOK TABLEAUX Jaejin Lee Abstact. The Schensted algoithm fist descibed by Robinson
More informationEncapsulation theory: the transformation equations of absolute information hiding.
1 Encapsulation theoy: the tansfomation equations of absolute infomation hiding. Edmund Kiwan * www.edmundkiwan.com Abstact This pape descibes how the potential coupling of a set vaies as the set is tansfomed,
More information10/04/18. P [P(x)] 1 negl(n).
Mastemath, Sping 208 Into to Lattice lgs & Cypto Lectue 0 0/04/8 Lectues: D. Dadush, L. Ducas Scibe: K. de Boe Intoduction In this lectue, we will teat two main pats. Duing the fist pat we continue the
More informationBasic propositional and. The fundamentals of deduction
Baic ooitional and edicate logic The fundamental of deduction 1 Logic and it alication Logic i the tudy of the atten of deduction Logic lay two main ole in comutation: Modeling : logical entence ae the
More information