Some remarks on Kurepa s left factorial

Size: px
Start display at page:

Download "Some remarks on Kurepa s left factorial"

Transcription

1 Som rmarks on Kurpa s lft factorial arxiv:math/ v1 [math.nt] 21 Oct 2004 Brnd C. Kllnr Abstract W stablish a connction btwn th subfactorial function S(n) and th lft factorial function of Kurpa K(n). Som lmntary proprtis and congruncs of both functions ar dscribd. Finally, w giv a calculatd distribution of prims blow of K(n). Kywords: Lft factorial function, subfactorial function, drangmnts Mathmatics Subjct Classification 2000: 11B65 1 Introduction Th subfactorial function is dfind by S(n) = n! n ( 1) k, n N 0 k! which givs th numbr of prmutations of n lmnts without any fixpoints, also calld drangmnts of n lmnts, s [6, p. 195]. This was alrady provn by P. R. d Montmort [2] in L. Eulr [3] indpndntly gav a proof in 1753, s also [4]. This function has th proprtis ( is Eulr s numbr) S(n) = ns(n 1)+( 1) n, (1.1) S(n) = (n 1)(S(n 1)+S(n 2)), (1.2) { n! 0, 2 n S(n) = +δ n with δ n = 1, 2 n. (1.3) Kurpa s lft factorial function is dfind by n 1 K(0) = 0, K(n) = k!, n N. In D. Kurpa [8] introducd th lft factorial function which is dnotd by!n = K(n). Somtims th subfactorial function is also dnotd by!n, so w do not us this notation to avoid confusion. For mor dtails of th following conjctur s a ovrviw of A. Ivić and Ž. Mijajlović [7]. 1

2 Conjctur 1.1 (Kurpa s lft factorial hypothsis) Th following quivalnt statmnts hold (K(n),n!) = 2, n 2, K(n) 0 (mod n), n > 2, K(p) 0 (mod p), p odd prim. (KH) Rcntly, D. Barsky and B. Bnzaghou [1] hav givn a proof of this hypothsis. Sinc K(n) is also rlatd to Bll numbrs B n via K(p) B 1 (mod p) for any prim p, thy actually provd that B 1 (mod p) is always valid for any odd prim p. Gssl[5, Sct. 7/10] givs somrcursividntitis ofs(n), B n, andothrswithumbral calculus. Dfin symbolically S n = S(n) and B n = B n with S 0 = B 0 = 1, thn on may writ B n+1 = (B +1) n and n! = (S +1) n, n 0. (1.4) Intrstingly, both squncs hav th sam proprty as follows. Lmma 1.2 Lt p b a prim. Thn p ( 1) k B k p ( 1) k S(k) 0 (mod p) with B p 2 (mod p) and S(p) 1 (mod p). Proof. By (1.1) and Wilson s thorm, w hav S(p) 1 (p 1)! (mod p). ( Hnc, w can rwrit (1.4) by B p (B +1) and S p (S +1) (mod p). Sinc ) k ( 1) k (mod p) for 0 k < p, this provids th proposd congrunc. Now, w us a congrunc of Touchard for Bll numbrs, s [5, Sct. 10, Thorm 10.1]. Thn B n+p B n+1 B n 0 (mod p), n 0. With n = 0 and B 0 = B 1 = 1, w obtain B p 2 (mod p). First valus of K(n), S(n), and B n ar givn in th following tabl. n K(n) S(n) B n

3 2 Congruncs btwn K(n) and S(n) Lmma 2.1 Lt n b a positiv intgr, thn K(n) ( 1) n 1 S(n 1) (mod n). Proof. Cas n = 1 is trivial. Lt n 2. Thn w hav n 1 ( ) n 1 n 1 ( ) n 1 ( 1) n 1 S(n 1) = ( 1) n 1 k (n 1 k)! = ( 1) k k! k k by turning th summation. Sinc it is valid for 0 k < n ( ) n 1 ( 1) k k! = ( 1) k (n 1) (n k) k! (mod n), k this provids, trm by trm, th congrunc claimd abov. By Lmma 2.1 and (1.3), w asily obtain a gnralization, howvr, which is only notd for prims lswhr. Corollary 2.2 Lt n b a positiv intgr, thn (n 1)! K(n) ( 1) n 1 +δ n 1 (mod n). Hnc, (KH) is quivalnt to (n 1)! δ n 1 (mod n), n > 2, whil by rcursiv proprty (1.1) n! δ n 1 (mod n), n 1 is always valid. Corollary 2.3 Lt n b a positiv intgr, thn (KH) is quivalnt to n! (n 1)! 0 (mod n) n = 1,2. Lmma 2.4 Lt p b a prim, thn K(p) K(p l) S(l 1) (l 1)! (mod p), l = 1,...,p. 3

4 Proof. Lt l {1,...,p}. W thn hav K(p) K(p l) = k=p l k! = l (p k)! k=1 l k=1 ( 1) k (k 1)! = S(l 1) (l 1)! (mod p), sinc follows by Wilson s thorm. (p k)! ( 1)k (k 1)! (mod p) (2.1) Corollary 2.5 Lt p b an odd prim, thn (KH) implis for 0 l < p rspctivly K( l) S(l) l! l!k( l) (mod p) l! +δ l (mod p). Sinc (KH) is tru, w obtain, as an xampl, th following congruncs K(p) 0, K() 1, K(p 2) 0, K(p 3) 1 2, K(p 4) 1 3 (mod p). 3 Proprtis of K(n) To dscrib som intrsting proprtis of K(n), w introduc th following dfinition which w nam aftr Kurpa. Dfinition 3.1 Lt p b an odd prim. Th pair (p,n) is calld a Kurpa pair if p r K(n) with som intgr r 1. Th max. intgr r is calld th ordr of (p,n). Th indx of p is dfind by i K (p) = #{n : (p,n) is a Kurpa pair}. If i K (p) > 0, thn p is calld a Kurpa prim. W hav,.g., th Kurpa pairs (19,7), (19,12), and (19,16). If (KH) would fail at an odd prim p, thn this would imply i K (p) =. This is an asy consqunc of p K(p), p (p+m)! for m 0. Th cas p = 2 is handld sparatly. On asily ss that 2 K(n) for n 2 and K(n) 2 (mod 4) for n 4. First valus of i K (p) ar givn in th following tabl. p i K (p)

5 Thorm 3.2 Lt (p,n) b a Kurpa pair. Thn p > n > 3 is valid with K(p) ( 1) n n!s( n) (mod p) which implis p S(p 1 n). Furthrmor on has i K (p) 4. Consquntly, thr xist infinitly many Kurpa prims. Proof. For now, lt pbanoddprim. Lt(p,n) bakurpapair. Sincp K(p+m) for m 0 by validity of (KH) and firstvalus of K( ) ar 0,1,2,4, this yilds p > n > 3. W us Lmma 2.4 with n = p l, thn w hav 0 K(p) K(p) K(n) S( n) ( n)! (mod p) which provids th rsult by mans of (2.1) and also p S( n). Now, w hav to count possibl Kurpa pairs. Corollary 2.5 shows that K(p 2) 0 (mod p). If p K(n) thn p K(n+l) for l = 1,2,3. This is sn by n! 0 (mod p) and n!+(n+1)! = (n+2)n! 0, n!+(n+1)!+(n+2)! = (n+2) 2 n! 0 (mod p), sinc n p 2. On th othr sid, w hav 4 n. Thn a simpl counting argumnt provids i K (p) 4. Finally, K(n) for n and p K(n) p > n for odd prims imply infinitly many Kurpa prims. Now, th rmarkabl fact of K(n) is th finitnss of Kurpa pairs for all odd prims. In p-adic analysis, th sris K( ) = k! is an xampl of a convrgnt sris rsp. K(n) is a convrgnt squnc which lis in Z p, th ring of p-adic intgrs. Thn (KH) is quivalnt to K( ) is a unit in Z p for all odd prims p. Th bhavior (mod p r ) is illustratd by th following thorm. Not that l r is rlatd to th so-calld Smarandach function for factorials. Thorm 3.3 Lt p,r b positiv intgrs with p prim. Thn th squnc is constant for n l r p with r l r and l r = min l K(n) (mod p r ), n 0 { l : l+ l σ p(l) r whr σ p (l) givs th sum of digits of l in bas p. Proof. W hav to dtrmin a minimal l with th proprty ord p (lp)! r. Counting factors which ar divisibl by p, w obtain }, ord p (lp)! = l+ord p l! = l + l σ p(l) by mans of th p-adic valuation of factorials, s [9, Sction 3.1, p. 241]. 5

6 At th nd, w giv som rsults of calculatd Kurpapairs. Thr ar N = π(10000) 1 = 1228 odd prims blow Lt N r b th numbr of odd prims with indx i K (p) = r in this rang. Th following tabl shows th distribution of th indx i K. r N r N r /N Th calculatd Kurpa pairs with indx i K (p) = 5 ar as follows. (2203,277) (2203,788) (2203,837) (2203,1246) (2203,1927) (5227,850) (5227,1752) (5227,3451) (5227,4363) (5227,4716) (6689,1716) (6689,2404) (6689,3641) (6689,3969) (6689,6601) All prims blow appar with a simpl powr in K(n), xcpt K(3) = 4. On th othr sid, th occurrnc of highr powrs p r in K(n) sms to b vry rar. M. Zivkovic [10] givs th first xampl K(26541). Thr ar two Kurpa pairs (54503,26541) and (54503,49783), but only th first of thm has ordr two. On may ask whthr th distribution of Kurpa pairs rsp. th indx i K can b asymptotically dtrmind and vn provn. Ar thr infinitly many non-kurpa prims p with i K (p) = 0? It sms that this subjct of K(n) and its distribution of prims will b much simplr to attack as, for xampl, th mor complicatd but in a sns similar cas of th distribution of irrgular prims of Brnoulli numbrs. Acknowldgmnt Th author wishs to thank Prof. Ivić for informing about th problm of Kurpa. Brnd C. Kllnr addrss: Ritstallstr. 7, Göttingn, Grmany mail: bk@brnoulli.org Rfrncs [1] D. Barsky and B. Bnzaghou. Nombrs d Bll t somm d factorills. Journal d Théori ds Nombrs d Bordaux, 16:1 17, [2] P. R. d Montmort. Essai d analys sur ls jux d hasard. 2nd Edition of 1713, Paris, (rprintd in Annotatd Radings in th History of Statistics, Springr- Vrlag, 2001, 25 29), , [3] L. Eulr. Calcul d la probabilité dans l ju d rncontr. Mmoirs d l académi ds scincs d Brlin, 7: , [4] L. Eulr. Solutio quastionis curiosa x doctrina combinationum. Mmoirs d l académi ds scincs d St.-Ptrsbourg, 3:57 64,

7 [5] I. M. Gssl. Applications of th classical umbral calculus. Algbra Univrsalis, 49: , arxiv:math.co/ [6] R. L. Graham, D. E. Knuth, and O. Patashnik. Concrt Mathmatics. Addison- Wsly, Rading, MA, USA, [7] A. Ivić and Ž. Mijajlović. On Kurpa s Problms in Numbr Thory. Publ. Inst. Math., 57(71):19 28, arxiv:math.nt/ [8] -D. Kurpa. On th lft factorial function!n. Math. Balkan., 1: , [9] A. M. Robrt. A Cours in p-adic Analysis, volum 198 of Graduat Txts in Mathmatics. Springr-Vrlag, [10] M. Zivkovic. Thnumbrof prims n i=1 ( 1)n i i! is finit. Math. Comp., 68: ,

Derangements and Applications

Derangements and Applications 2 3 47 6 23 Journal of Intgr Squncs, Vol. 6 (2003), Articl 03..2 Drangmnts and Applications Mhdi Hassani Dpartmnt of Mathmatics Institut for Advancd Studis in Basic Scincs Zanjan, Iran mhassani@iasbs.ac.ir

More information

BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES

BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES DONALD M. DAVIS Abstract. If p is a prim (implicit in notation and n a positiv intgr, lt ν(n dnot th xponnt of p in n, and U(n n/p ν(n, th unit

More information

BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES. 1. Statement of results

BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES. 1. Statement of results BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES DONALD M. DAVIS Abstract. If p is a prim and n a positiv intgr, lt ν p (n dnot th xponnt of p in n, and u p (n n/p νp(n th unit part of n. If α

More information

Introduction to Arithmetic Geometry Fall 2013 Lecture #20 11/14/2013

Introduction to Arithmetic Geometry Fall 2013 Lecture #20 11/14/2013 18.782 Introduction to Arithmtic Gomtry Fall 2013 Lctur #20 11/14/2013 20.1 Dgr thorm for morphisms of curvs Lt us rstat th thorm givn at th nd of th last lctur, which w will now prov. Thorm 20.1. Lt φ:

More information

COUNTING TAMELY RAMIFIED EXTENSIONS OF LOCAL FIELDS UP TO ISOMORPHISM

COUNTING TAMELY RAMIFIED EXTENSIONS OF LOCAL FIELDS UP TO ISOMORPHISM COUNTING TAMELY RAMIFIED EXTENSIONS OF LOCAL FIELDS UP TO ISOMORPHISM Jim Brown Dpartmnt of Mathmatical Scincs, Clmson Univrsity, Clmson, SC 9634, USA jimlb@g.clmson.du Robrt Cass Dpartmnt of Mathmatics,

More information

Recall that by Theorems 10.3 and 10.4 together provide us the estimate o(n2 ), S(q) q 9, q=1

Recall that by Theorems 10.3 and 10.4 together provide us the estimate o(n2 ), S(q) q 9, q=1 Chaptr 11 Th singular sris Rcall that by Thorms 10 and 104 togthr provid us th stimat 9 4 n 2 111 Rn = SnΓ 2 + on2, whr th singular sris Sn was dfind in Chaptr 10 as Sn = q=1 Sq q 9, with Sq = 1 a q gcda,q=1

More information

The Matrix Exponential

The Matrix Exponential Th Matrix Exponntial (with xrciss) by Dan Klain Vrsion 28928 Corrctions and commnts ar wlcom Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix () A A k I + A + k!

More information

The Matrix Exponential

The Matrix Exponential Th Matrix Exponntial (with xrciss) by D. Klain Vrsion 207.0.05 Corrctions and commnts ar wlcom. Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix A A k I + A + k!

More information

ON RIGHT(LEFT) DUO PO-SEMIGROUPS. S. K. Lee and K. Y. Park

ON RIGHT(LEFT) DUO PO-SEMIGROUPS. S. K. Lee and K. Y. Park Kangwon-Kyungki Math. Jour. 11 (2003), No. 2, pp. 147 153 ON RIGHT(LEFT) DUO PO-SEMIGROUPS S. K. L and K. Y. Park Abstract. W invstigat som proprtis on right(rsp. lft) duo po-smigroups. 1. Introduction

More information

Section 6.1. Question: 2. Let H be a subgroup of a group G. Then H operates on G by left multiplication. Describe the orbits for this operation.

Section 6.1. Question: 2. Let H be a subgroup of a group G. Then H operates on G by left multiplication. Describe the orbits for this operation. MAT 444 H Barclo Spring 004 Homwork 6 Solutions Sction 6 Lt H b a subgroup of a group G Thn H oprats on G by lft multiplication Dscrib th orbits for this opration Th orbits of G ar th right costs of H

More information

SCHUR S THEOREM REU SUMMER 2005

SCHUR S THEOREM REU SUMMER 2005 SCHUR S THEOREM REU SUMMER 2005 1. Combinatorial aroach Prhas th first rsult in th subjct blongs to I. Schur and dats back to 1916. On of his motivation was to study th local vrsion of th famous quation

More information

Hardy-Littlewood Conjecture and Exceptional real Zero. JinHua Fei. ChangLing Company of Electronic Technology Baoji Shannxi P.R.

Hardy-Littlewood Conjecture and Exceptional real Zero. JinHua Fei. ChangLing Company of Electronic Technology Baoji Shannxi P.R. Hardy-Littlwood Conjctur and Excptional ral Zro JinHua Fi ChangLing Company of Elctronic Tchnology Baoji Shannxi P.R.China E-mail: fijinhuayoujian@msn.com Abstract. In this papr, w assum that Hardy-Littlwood

More information

The Equitable Dominating Graph

The Equitable Dominating Graph Intrnational Journal of Enginring Rsarch and Tchnology. ISSN 0974-3154 Volum 8, Numbr 1 (015), pp. 35-4 Intrnational Rsarch Publication Hous http://www.irphous.com Th Equitabl Dominating Graph P.N. Vinay

More information

(Upside-Down o Direct Rotation) β - Numbers

(Upside-Down o Direct Rotation) β - Numbers Amrican Journal of Mathmatics and Statistics 014, 4(): 58-64 DOI: 10593/jajms0140400 (Upsid-Down o Dirct Rotation) β - Numbrs Ammar Sddiq Mahmood 1, Shukriyah Sabir Ali,* 1 Dpartmnt of Mathmatics, Collg

More information

cycle that does not cross any edges (including its own), then it has at least

cycle that does not cross any edges (including its own), then it has at least W prov th following thorm: Thorm If a K n is drawn in th plan in such a way that it has a hamiltonian cycl that dos not cross any dgs (including its own, thn it has at last n ( 4 48 π + O(n crossings Th

More information

INCOMPLETE KLOOSTERMAN SUMS AND MULTIPLICATIVE INVERSES IN SHORT INTERVALS. xy 1 (mod p), (x, y) I (j)

INCOMPLETE KLOOSTERMAN SUMS AND MULTIPLICATIVE INVERSES IN SHORT INTERVALS. xy 1 (mod p), (x, y) I (j) INCOMPLETE KLOOSTERMAN SUMS AND MULTIPLICATIVE INVERSES IN SHORT INTERVALS T D BROWNING AND A HAYNES Abstract W invstigat th solubility of th congrunc xy (mod ), whr is a rim and x, y ar rstrictd to li

More information

Limiting value of higher Mahler measure

Limiting value of higher Mahler measure Limiting valu of highr Mahlr masur Arunabha Biswas a, Chris Monico a, a Dpartmnt of Mathmatics & Statistics, Txas Tch Univrsity, Lubbock, TX 7949, USA Abstract W considr th k-highr Mahlr masur m k P )

More information

Chapter 10. The singular integral Introducing S(n) and J(n)

Chapter 10. The singular integral Introducing S(n) and J(n) Chaptr Th singular intgral Our aim in this chaptr is to rplac th functions S (n) and J (n) by mor convnint xprssions; ths will b calld th singular sris S(n) and th singular intgral J(n). This will b don

More information

Construction of asymmetric orthogonal arrays of strength three via a replacement method

Construction of asymmetric orthogonal arrays of strength three via a replacement method isid/ms/26/2 Fbruary, 26 http://www.isid.ac.in/ statmath/indx.php?modul=prprint Construction of asymmtric orthogonal arrays of strngth thr via a rplacmnt mthod Tian-fang Zhang, Qiaoling Dng and Alok Dy

More information

On the irreducibility of some polynomials in two variables

On the irreducibility of some polynomials in two variables ACTA ARITHMETICA LXXXII.3 (1997) On th irrducibility of som polynomials in two variabls by B. Brindza and Á. Pintér (Dbrcn) To th mmory of Paul Erdős Lt f(x) and g(y ) b polynomials with intgral cofficints

More information

Einstein Equations for Tetrad Fields

Einstein Equations for Tetrad Fields Apiron, Vol 13, No, Octobr 006 6 Einstin Equations for Ttrad Filds Ali Rıza ŞAHİN, R T L Istanbul (Turky) Evry mtric tnsor can b xprssd by th innr product of ttrad filds W prov that Einstin quations for

More information

Propositional Logic. Combinatorial Problem Solving (CPS) Albert Oliveras Enric Rodríguez-Carbonell. May 17, 2018

Propositional Logic. Combinatorial Problem Solving (CPS) Albert Oliveras Enric Rodríguez-Carbonell. May 17, 2018 Propositional Logic Combinatorial Problm Solving (CPS) Albrt Olivras Enric Rodríguz-Carbonll May 17, 2018 Ovrviw of th sssion Dfinition of Propositional Logic Gnral Concpts in Logic Rduction to SAT CNFs

More information

On the number of pairs of positive integers x,y H such that x 2 +y 2 +1, x 2 +y 2 +2 are square-free

On the number of pairs of positive integers x,y H such that x 2 +y 2 +1, x 2 +y 2 +2 are square-free arxiv:90.04838v [math.nt] 5 Jan 09 On th numbr of pairs of positiv intgrs x,y H such that x +y +, x +y + ar squar-fr S. I. Dimitrov Abstract In th prsnt papr w show that thr xist infinitly many conscutiv

More information

Deift/Zhou Steepest descent, Part I

Deift/Zhou Steepest descent, Part I Lctur 9 Dift/Zhou Stpst dscnt, Part I W now focus on th cas of orthogonal polynomials for th wight w(x) = NV (x), V (x) = t x2 2 + x4 4. Sinc th wight dpnds on th paramtr N N w will writ π n,n, a n,n,

More information

Fourier Transforms and the Wave Equation. Key Mathematics: More Fourier transform theory, especially as applied to solving the wave equation.

Fourier Transforms and the Wave Equation. Key Mathematics: More Fourier transform theory, especially as applied to solving the wave equation. Lur 7 Fourir Transforms and th Wav Euation Ovrviw and Motivation: W first discuss a fw faturs of th Fourir transform (FT), and thn w solv th initial-valu problm for th wav uation using th Fourir transform

More information

SOME PARAMETERS ON EQUITABLE COLORING OF PRISM AND CIRCULANT GRAPH.

SOME PARAMETERS ON EQUITABLE COLORING OF PRISM AND CIRCULANT GRAPH. SOME PARAMETERS ON EQUITABLE COLORING OF PRISM AND CIRCULANT GRAPH. K VASUDEVAN, K. SWATHY AND K. MANIKANDAN 1 Dpartmnt of Mathmatics, Prsidncy Collg, Chnnai-05, India. E-Mail:vasu k dvan@yahoo.com. 2,

More information

Abstract Interpretation: concrete and abstract semantics

Abstract Interpretation: concrete and abstract semantics Abstract Intrprtation: concrt and abstract smantics Concrt smantics W considr a vry tiny languag that manags arithmtic oprations on intgrs valus. Th (concrt) smantics of th languags cab b dfind by th funzcion

More information

NEW APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA

NEW APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA NE APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA Mirca I CÎRNU Ph Dp o Mathmatics III Faculty o Applid Scincs Univrsity Polithnica o Bucharst Cirnumirca @yahoocom Abstract In a rcnt papr [] 5 th indinit intgrals

More information

Combinatorial Networks Week 1, March 11-12

Combinatorial Networks Week 1, March 11-12 1 Nots on March 11 Combinatorial Ntwors W 1, March 11-1 11 Th Pigonhol Principl Th Pigonhol Principl If n objcts ar placd in hols, whr n >, thr xists a box with mor than on objcts 11 Thorm Givn a simpl

More information

An Application of Hardy-Littlewood Conjecture. JinHua Fei. ChangLing Company of Electronic Technology Baoji Shannxi P.R.China

An Application of Hardy-Littlewood Conjecture. JinHua Fei. ChangLing Company of Electronic Technology Baoji Shannxi P.R.China An Application of Hardy-Littlwood Conjctur JinHua Fi ChangLing Company of Elctronic Tchnology Baoji Shannxi P.R.China E-mail: fijinhuayoujian@msn.com Abstract. In this papr, w assum that wakr Hardy-Littlwood

More information

Cramér-Rao Inequality: Let f(x; θ) be a probability density function with continuous parameter

Cramér-Rao Inequality: Let f(x; θ) be a probability density function with continuous parameter WHEN THE CRAMÉR-RAO INEQUALITY PROVIDES NO INFORMATION STEVEN J. MILLER Abstract. W invstigat a on-paramtr family of probability dnsitis (rlatd to th Parto distribution, which dscribs many natural phnomna)

More information

Higher order derivatives

Higher order derivatives Robrto s Nots on Diffrntial Calculus Chaptr 4: Basic diffrntiation ruls Sction 7 Highr ordr drivativs What you nd to know alrady: Basic diffrntiation ruls. What you can larn hr: How to rpat th procss of

More information

EEO 401 Digital Signal Processing Prof. Mark Fowler

EEO 401 Digital Signal Processing Prof. Mark Fowler EEO 401 Digital Signal Procssing Prof. Mark Fowlr Dtails of th ot St #19 Rading Assignmnt: Sct. 7.1.2, 7.1.3, & 7.2 of Proakis & Manolakis Dfinition of th So Givn signal data points x[n] for n = 0,, -1

More information

A Propagating Wave Packet Group Velocity Dispersion

A Propagating Wave Packet Group Velocity Dispersion Lctur 8 Phys 375 A Propagating Wav Packt Group Vlocity Disprsion Ovrviw and Motivation: In th last lctur w lookd at a localizd solution t) to th 1D fr-particl Schrödingr quation (SE) that corrsponds to

More information

DISTRIBUTION OF DIFFERENCE BETWEEN INVERSES OF CONSECUTIVE INTEGERS MODULO P

DISTRIBUTION OF DIFFERENCE BETWEEN INVERSES OF CONSECUTIVE INTEGERS MODULO P DISTRIBUTION OF DIFFERENCE BETWEEN INVERSES OF CONSECUTIVE INTEGERS MODULO P Tsz Ho Chan Dartmnt of Mathmatics, Cas Wstrn Rsrv Univrsity, Clvland, OH 4406, USA txc50@cwru.du Rcivd: /9/03, Rvisd: /9/04,

More information

Basic Polyhedral theory

Basic Polyhedral theory Basic Polyhdral thory Th st P = { A b} is calld a polyhdron. Lmma 1. Eithr th systm A = b, b 0, 0 has a solution or thr is a vctorπ such that π A 0, πb < 0 Thr cass, if solution in top row dos not ist

More information

The van der Waals interaction 1 D. E. Soper 2 University of Oregon 20 April 2012

The van der Waals interaction 1 D. E. Soper 2 University of Oregon 20 April 2012 Th van dr Waals intraction D. E. Sopr 2 Univrsity of Orgon 20 pril 202 Th van dr Waals intraction is discussd in Chaptr 5 of J. J. Sakurai, Modrn Quantum Mchanics. Hr I tak a look at it in a littl mor

More information

10. The Discrete-Time Fourier Transform (DTFT)

10. The Discrete-Time Fourier Transform (DTFT) Th Discrt-Tim Fourir Transform (DTFT Dfinition of th discrt-tim Fourir transform Th Fourir rprsntation of signals plays an important rol in both continuous and discrt signal procssing In this sction w

More information

LINEAR DELAY DIFFERENTIAL EQUATION WITH A POSITIVE AND A NEGATIVE TERM

LINEAR DELAY DIFFERENTIAL EQUATION WITH A POSITIVE AND A NEGATIVE TERM Elctronic Journal of Diffrntial Equations, Vol. 2003(2003), No. 92, pp. 1 6. ISSN: 1072-6691. URL: http://jd.math.swt.du or http://jd.math.unt.du ftp jd.math.swt.du (login: ftp) LINEAR DELAY DIFFERENTIAL

More information

Week 3: Connected Subgraphs

Week 3: Connected Subgraphs Wk 3: Connctd Subgraphs Sptmbr 19, 2016 1 Connctd Graphs Path, Distanc: A path from a vrtx x to a vrtx y in a graph G is rfrrd to an xy-path. Lt X, Y V (G). An (X, Y )-path is an xy-path with x X and y

More information

Problem Set 6 Solutions

Problem Set 6 Solutions 6.04/18.06J Mathmatics for Computr Scinc March 15, 005 Srini Dvadas and Eric Lhman Problm St 6 Solutions Du: Monday, March 8 at 9 PM in Room 3-044 Problm 1. Sammy th Shark is a financial srvic providr

More information

10. Limits involving infinity

10. Limits involving infinity . Limits involving infinity It is known from th it ruls for fundamntal arithmtic oprations (+,-,, ) that if two functions hav finit its at a (finit or infinit) point, that is, thy ar convrgnt, th it of

More information

1 N N(θ;d 1...d l ;N) 1 q l = o(1)

1 N N(θ;d 1...d l ;N) 1 q l = o(1) NORMALITY OF NUMBERS GENERATED BY THE VALUES OF ENTIRE FUNCTIONS MANFRED G. MADRITSCH, JÖRG M. THUSWALDNER, AND ROBERT F. TICHY Abstract. W show that th numbr gnratd by th q-ary intgr part of an ntir function

More information

The graph of y = x (or y = ) consists of two branches, As x 0, y + ; as x 0, y +. x = 0 is the

The graph of y = x (or y = ) consists of two branches, As x 0, y + ; as x 0, y +. x = 0 is the Copyright itutcom 005 Fr download & print from wwwitutcom Do not rproduc by othr mans Functions and graphs Powr functions Th graph of n y, for n Q (st of rational numbrs) y is a straight lin through th

More information

u x v x dx u x v x v x u x dx d u x v x u x v x dx u x v x dx Integration by Parts Formula

u x v x dx u x v x v x u x dx d u x v x u x v x dx u x v x dx Integration by Parts Formula 7. Intgration by Parts Each drivativ formula givs ris to a corrsponding intgral formula, as w v sn many tims. Th drivativ product rul yilds a vry usful intgration tchniqu calld intgration by parts. Starting

More information

On spanning trees and cycles of multicolored point sets with few intersections

On spanning trees and cycles of multicolored point sets with few intersections On spanning trs and cycls of multicolord point sts with fw intrsctions M. Kano, C. Mrino, and J. Urrutia April, 00 Abstract Lt P 1,..., P k b a collction of disjoint point sts in R in gnral position. W

More information

CPSC 665 : An Algorithmist s Toolkit Lecture 4 : 21 Jan Linear Programming

CPSC 665 : An Algorithmist s Toolkit Lecture 4 : 21 Jan Linear Programming CPSC 665 : An Algorithmist s Toolkit Lctur 4 : 21 Jan 2015 Lcturr: Sushant Sachdva Linar Programming Scrib: Rasmus Kyng 1. Introduction An optimization problm rquirs us to find th minimum or maximum) of

More information

First order differential equation Linear equation; Method of integrating factors

First order differential equation Linear equation; Method of integrating factors First orr iffrntial quation Linar quation; Mtho of intgrating factors Exampl 1: Rwrit th lft han si as th rivativ of th prouct of y an som function by prouct rul irctly. Solving th first orr iffrntial

More information

First derivative analysis

First derivative analysis Robrto s Nots on Dirntial Calculus Chaptr 8: Graphical analysis Sction First drivativ analysis What you nd to know alrady: How to us drivativs to idntiy th critical valus o a unction and its trm points

More information

International Journal of Scientific & Engineering Research, Volume 6, Issue 7, July ISSN

International Journal of Scientific & Engineering Research, Volume 6, Issue 7, July ISSN Intrnational Journal of Scintific & Enginring Rsarch, Volum 6, Issu 7, July-25 64 ISSN 2229-558 HARATERISTIS OF EDGE UTSET MATRIX OF PETERSON GRAPH WITH ALGEBRAI GRAPH THEORY Dr. G. Nirmala M. Murugan

More information

That is, we start with a general matrix: And end with a simpler matrix:

That is, we start with a general matrix: And end with a simpler matrix: DIAGON ALIZATION OF THE STR ESS TEN SOR INTRO DUCTIO N By th us of Cauchy s thorm w ar abl to rduc th numbr of strss componnts in th strss tnsor to only nin valus. An additional simplification of th strss

More information

Lecture 6.4: Galois groups

Lecture 6.4: Galois groups Lctur 6.4: Galois groups Matthw Macauly Dpartmnt of Mathmatical Scincs Clmson Univrsity http://www.math.clmson.du/~macaul/ Math 4120, Modrn Algbra M. Macauly (Clmson) Lctur 6.4: Galois groups Math 4120,

More information

ON A SECOND ORDER RATIONAL DIFFERENCE EQUATION

ON A SECOND ORDER RATIONAL DIFFERENCE EQUATION Hacttp Journal of Mathmatics and Statistics Volum 41(6) (2012), 867 874 ON A SECOND ORDER RATIONAL DIFFERENCE EQUATION Nourssadat Touafk Rcivd 06:07:2011 : Accptd 26:12:2011 Abstract In this papr, w invstigat

More information

Equidistribution and Weyl s criterion

Equidistribution and Weyl s criterion Euidistribution and Wyl s critrion by Brad Hannigan-Daly W introduc th ida of a sunc of numbrs bing uidistributd (mod ), and w stat and prov a thorm of Hrmann Wyl which charactrizs such suncs. W also discuss

More information

1 Isoparametric Concept

1 Isoparametric Concept UNIVERSITY OF CALIFORNIA BERKELEY Dpartmnt of Civil Enginring Spring 06 Structural Enginring, Mchanics and Matrials Profssor: S. Govindj Nots on D isoparamtric lmnts Isoparamtric Concpt Th isoparamtric

More information

Superposition. Thinning

Superposition. Thinning Suprposition STAT253/317 Wintr 213 Lctur 11 Yibi Huang Fbruary 1, 213 5.3 Th Poisson Procsss 5.4 Gnralizations of th Poisson Procsss Th sum of two indpndnt Poisson procsss with rspctiv rats λ 1 and λ 2,

More information

Homotopy perturbation technique

Homotopy perturbation technique Comput. Mthods Appl. Mch. Engrg. 178 (1999) 257±262 www.lsvir.com/locat/cma Homotopy prturbation tchniqu Ji-Huan H 1 Shanghai Univrsity, Shanghai Institut of Applid Mathmatics and Mchanics, Shanghai 272,

More information

Mutually Independent Hamiltonian Cycles of Pancake Networks

Mutually Independent Hamiltonian Cycles of Pancake Networks Mutually Indpndnt Hamiltonian Cycls of Pancak Ntworks Chng-Kuan Lin Dpartmnt of Mathmatics National Cntral Univrsity, Chung-Li, Taiwan 00, R O C discipl@ms0urlcomtw Hua-Min Huang Dpartmnt of Mathmatics

More information

Homework #3. 1 x. dx. It therefore follows that a sum of the

Homework #3. 1 x. dx. It therefore follows that a sum of the Danil Cannon CS 62 / Luan March 5, 2009 Homwork # 1. Th natural logarithm is dfind by ln n = n 1 dx. It thrfor follows that a sum of th 1 x sam addnd ovr th sam intrval should b both asymptotically uppr-

More information

An arithmetic formula of Liouville

An arithmetic formula of Liouville Journal d Théori ds Nombrs d Bordaux (006), 3 39 An arithmtic formula of Liouvill par Erin McAFEE t Knnth S. WILLIAMS Résumé. Un pruv élémntair st donné d un formul arithmétiqu qui fut présnté mais non

More information

Continuous probability distributions

Continuous probability distributions Continuous probability distributions Many continuous probability distributions, including: Uniform Normal Gamma Eponntial Chi-Squard Lognormal Wibull EGR 5 Ch. 6 Uniform distribution Simplst charactrizd

More information

UNTYPED LAMBDA CALCULUS (II)

UNTYPED LAMBDA CALCULUS (II) 1 UNTYPED LAMBDA CALCULUS (II) RECALL: CALL-BY-VALUE O.S. Basic rul Sarch ruls: (\x.) v [v/x] 1 1 1 1 v v CALL-BY-VALUE EVALUATION EXAMPLE (\x. x x) (\y. y) x x [\y. y / x] = (\y. y) (\y. y) y [\y. y /

More information

1 Minimum Cut Problem

1 Minimum Cut Problem CS 6 Lctur 6 Min Cut and argr s Algorithm Scribs: Png Hui How (05), Virginia Dat: May 4, 06 Minimum Cut Problm Today, w introduc th minimum cut problm. This problm has many motivations, on of which coms

More information

ON A CONJECTURE OF RYSElt AND MINC

ON A CONJECTURE OF RYSElt AND MINC MA THEMATICS ON A CONJECTURE OF RYSElt AND MINC BY ALBERT NIJE~HUIS*) AND HERBERT S. WILF *) (Communicatd at th mting of January 31, 1970) 1. Introduction Lt A b an n x n matrix of zros and ons, and suppos

More information

Thus, because if either [G : H] or [H : K] is infinite, then [G : K] is infinite, then [G : K] = [G : H][H : K] for all infinite cases.

Thus, because if either [G : H] or [H : K] is infinite, then [G : K] is infinite, then [G : K] = [G : H][H : K] for all infinite cases. Homwork 5 M 373K Solutions Mark Lindbrg and Travis Schdlr 1. Prov that th ring Z/mZ (for m 0) is a fild if and only if m is prim. ( ) Proof by Contrapositiv: Hr, thr ar thr cass for m not prim. m 0: Whn

More information

2.3 Matrix Formulation

2.3 Matrix Formulation 23 Matrix Formulation 43 A mor complicatd xampl ariss for a nonlinar systm of diffrntial quations Considr th following xampl Exampl 23 x y + x( x 2 y 2 y x + y( x 2 y 2 (233 Transforming to polar coordinats,

More information

6.1 Integration by Parts and Present Value. Copyright Cengage Learning. All rights reserved.

6.1 Integration by Parts and Present Value. Copyright Cengage Learning. All rights reserved. 6.1 Intgration by Parts and Prsnt Valu Copyright Cngag Larning. All rights rsrvd. Warm-Up: Find f () 1. F() = ln(+1). F() = 3 3. F() =. F() = ln ( 1) 5. F() = 6. F() = - Objctivs, Day #1 Studnts will b

More information

CLASSIFYING EXTENSIONS OF THE FIELD OF FORMAL LAURENT SERIES OVER F p

CLASSIFYING EXTENSIONS OF THE FIELD OF FORMAL LAURENT SERIES OVER F p CLASSIFYING EXTENSIONS OF THE FIELD OF FORMAL LAURENT SERIES OVER F p JIM BROWN, ALFEEN HASMANI, LINDSEY HILTNER, ANGELA RAFT, DANIEL SCOFIELD, AND IRSTI WASH Abstract. In prvious works, Jons-Robrts and

More information

ABEL TYPE THEOREMS FOR THE WAVELET TRANSFORM THROUGH THE QUASIASYMPTOTIC BOUNDEDNESS

ABEL TYPE THEOREMS FOR THE WAVELET TRANSFORM THROUGH THE QUASIASYMPTOTIC BOUNDEDNESS Novi Sad J. Math. Vol. 45, No. 1, 2015, 201-206 ABEL TYPE THEOREMS FOR THE WAVELET TRANSFORM THROUGH THE QUASIASYMPTOTIC BOUNDEDNESS Mirjana Vuković 1 and Ivana Zubac 2 Ddicatd to Acadmician Bogoljub Stanković

More information

MATH 319, WEEK 15: The Fundamental Matrix, Non-Homogeneous Systems of Differential Equations

MATH 319, WEEK 15: The Fundamental Matrix, Non-Homogeneous Systems of Differential Equations MATH 39, WEEK 5: Th Fundamntal Matrix, Non-Homognous Systms of Diffrntial Equations Fundamntal Matrics Considr th problm of dtrmining th particular solution for an nsmbl of initial conditions For instanc,

More information

ME 321 Kinematics and Dynamics of Machines S. Lambert Winter 2002

ME 321 Kinematics and Dynamics of Machines S. Lambert Winter 2002 3.4 Forc Analysis of Linkas An undrstandin of forc analysis of linkas is rquird to: Dtrmin th raction forcs on pins, tc. as a consqunc of a spcifid motion (don t undrstimat th sinificanc of dynamic or

More information

The second condition says that a node α of the tree has exactly n children if the arity of its label is n.

The second condition says that a node α of the tree has exactly n children if the arity of its label is n. CS 6110 S14 Hanout 2 Proof of Conflunc 27 January 2014 In this supplmntary lctur w prov that th λ-calculus is conflunt. This is rsult is u to lonzo Church (1903 1995) an J. arkly Rossr (1907 1989) an is

More information

Recounting the Rationals

Recounting the Rationals Rconting th Rationals Nil Calkin and Hrbrt S. Wilf pril, 000 It is wll known (indd, as Pal Erd}os might hav said, vry child knows) that th rationals ar contabl. Howvr, th standard prsntations of this fact

More information

A. Limits and Horizontal Asymptotes ( ) f x f x. f x. x "±# ( ).

A. Limits and Horizontal Asymptotes ( ) f x f x. f x. x ±# ( ). A. Limits and Horizontal Asymptots What you ar finding: You can b askd to find lim x "a H.A.) problm is asking you find lim x "# and lim x "$#. or lim x "±#. Typically, a horizontal asymptot algbraically,

More information

ON THE DISTRIBUTION OF THE ELLIPTIC SUBSET SUM GENERATOR OF PSEUDORANDOM NUMBERS

ON THE DISTRIBUTION OF THE ELLIPTIC SUBSET SUM GENERATOR OF PSEUDORANDOM NUMBERS INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 2008, #A3 ON THE DISTRIBUTION OF THE ELLIPTIC SUBSET SUM GENERATOR OF PSEUDORANDOM NUMBERS Edwin D. El-Mahassni Dpartmnt of Computing, Macquari

More information

Finding low cost TSP and 2-matching solutions using certain half integer subtour vertices

Finding low cost TSP and 2-matching solutions using certain half integer subtour vertices Finding low cost TSP and 2-matching solutions using crtain half intgr subtour vrtics Sylvia Boyd and Robrt Carr Novmbr 996 Introduction Givn th complt graph K n = (V, E) on n nods with dg costs c R E,

More information

COMPUTER GENERATED HOLOGRAMS Optical Sciences 627 W.J. Dallas (Monday, April 04, 2005, 8:35 AM) PART I: CHAPTER TWO COMB MATH.

COMPUTER GENERATED HOLOGRAMS Optical Sciences 627 W.J. Dallas (Monday, April 04, 2005, 8:35 AM) PART I: CHAPTER TWO COMB MATH. C:\Dallas\0_Courss\03A_OpSci_67\0 Cgh_Book\0_athmaticalPrliminaris\0_0 Combath.doc of 8 COPUTER GENERATED HOLOGRAS Optical Scincs 67 W.J. Dallas (onday, April 04, 005, 8:35 A) PART I: CHAPTER TWO COB ATH

More information

Some Results on E - Cordial Graphs

Some Results on E - Cordial Graphs Intrnational Journal of Mathmatics Trnds and Tchnology Volum 7 Numbr 2 March 24 Som Rsults on E - Cordial Graphs S.Vnkatsh, Jamal Salah 2, G.Sthuraman 3 Corrsponding author, Dpartmnt of Basic Scincs, Collg

More information

5.80 Small-Molecule Spectroscopy and Dynamics

5.80 Small-Molecule Spectroscopy and Dynamics MIT OpnCoursWar http://ocw.mit.du 5.80 Small-Molcul Spctroscopy and Dynamics Fall 008 For information about citing ths matrials or our Trms of Us, visit: http://ocw.mit.du/trms. Lctur # 3 Supplmnt Contnts

More information

BSc Engineering Sciences A. Y. 2017/18 Written exam of the course Mathematical Analysis 2 August 30, x n, ) n 2

BSc Engineering Sciences A. Y. 2017/18 Written exam of the course Mathematical Analysis 2 August 30, x n, ) n 2 BSc Enginring Scincs A. Y. 27/8 Writtn xam of th cours Mathmatical Analysis 2 August, 28. Givn th powr sris + n + n 2 x n, n n dtrmin its radius of convrgnc r, and study th convrgnc for x ±r. By th root

More information

Properties of Phase Space Wavefunctions and Eigenvalue Equation of Momentum Dispersion Operator

Properties of Phase Space Wavefunctions and Eigenvalue Equation of Momentum Dispersion Operator Proprtis of Phas Spac Wavfunctions and Eignvalu Equation of Momntum Disprsion Oprator Ravo Tokiniaina Ranaivoson 1, Raolina Andriambololona 2, Hanitriarivo Rakotoson 3 raolinasp@yahoo.fr 1 ;jacqulinraolina@hotmail.com

More information

EEO 401 Digital Signal Processing Prof. Mark Fowler

EEO 401 Digital Signal Processing Prof. Mark Fowler EEO 401 Digital Signal Procssing Prof. Mark Fowlr ot St #18 Introduction to DFT (via th DTFT) Rading Assignmnt: Sct. 7.1 of Proakis & Manolakis 1/24 Discrt Fourir Transform (DFT) W v sn that th DTFT is

More information

Exponential inequalities and the law of the iterated logarithm in the unbounded forecasting game

Exponential inequalities and the law of the iterated logarithm in the unbounded forecasting game Ann Inst Stat Math (01 64:615 63 DOI 101007/s10463-010-03-5 Exponntial inqualitis and th law of th itratd logarithm in th unboundd forcasting gam Shin-ichiro Takazawa Rcivd: 14 Dcmbr 009 / Rvisd: 5 Octobr

More information

sets and continuity in fuzzy topological spaces

sets and continuity in fuzzy topological spaces Journal of Linar and Topological Algbra Vol. 06, No. 02, 2017, 125-134 Fuzzy sts and continuity in fuzzy topological spacs A. Vadivl a, B. Vijayalakshmi b a Dpartmnt of Mathmatics, Annamalai Univrsity,

More information

September 23, Honors Chem Atomic structure.notebook. Atomic Structure

September 23, Honors Chem Atomic structure.notebook. Atomic Structure Atomic Structur Topics covrd Atomic structur Subatomic particls Atomic numbr Mass numbr Charg Cations Anions Isotops Avrag atomic mass Practic qustions atomic structur Sp 27 8:16 PM 1 Powr Standards/ Larning

More information

Mapping properties of the elliptic maximal function

Mapping properties of the elliptic maximal function Rv. Mat. Ibroamricana 19 (2003), 221 234 Mapping proprtis of th lliptic maximal function M. Burak Erdoğan Abstract W prov that th lliptic maximal function maps th Sobolv spac W 4,η (R 2 )intol 4 (R 2 )

More information

Application of Vague Soft Sets in students evaluation

Application of Vague Soft Sets in students evaluation Availabl onlin at www.plagiarsarchlibrary.com Advancs in Applid Scinc Rsarch, 0, (6):48-43 ISSN: 0976-860 CODEN (USA): AASRFC Application of Vagu Soft Sts in studnts valuation B. Chtia*and P. K. Das Dpartmnt

More information

arxiv: v1 [math.nt] 9 Jan 2014

arxiv: v1 [math.nt] 9 Jan 2014 ON A FORM OF DEGREE d IN 2d+ VARIABLES (d 4 MANOJ VERMA arxiv:4.2366v [math.nt] 9 Jan 24 Abstract. For k 2, w driv an asymptotic formula for th numbr of zros of th forms k (x2 2i +x2 2i + k (x2 2k+2i +x2

More information

Complex Powers and Logs (5A) Young Won Lim 10/17/13

Complex Powers and Logs (5A) Young Won Lim 10/17/13 Complx Powrs and Logs (5A) Copyright (c) 202, 203 Young W. Lim. Prmission is grantd to copy, distribut and/or modify this documnt undr th trms of th GNU Fr Documntation Licns, Vrsion.2 or any latr vrsion

More information

Quasi-Classical States of the Simple Harmonic Oscillator

Quasi-Classical States of the Simple Harmonic Oscillator Quasi-Classical Stats of th Simpl Harmonic Oscillator (Draft Vrsion) Introduction: Why Look for Eignstats of th Annihilation Oprator? Excpt for th ground stat, th corrspondnc btwn th quantum nrgy ignstats

More information

CS 361 Meeting 12 10/3/18

CS 361 Meeting 12 10/3/18 CS 36 Mting 2 /3/8 Announcmnts. Homwork 4 is du Friday. If Friday is Mountain Day, homwork should b turnd in at my offic or th dpartmnt offic bfor 4. 2. Homwork 5 will b availabl ovr th wknd. 3. Our midtrm

More information

GEOMETRICAL PHENOMENA IN THE PHYSICS OF SUBATOMIC PARTICLES. Eduard N. Klenov* Rostov-on-Don, Russia

GEOMETRICAL PHENOMENA IN THE PHYSICS OF SUBATOMIC PARTICLES. Eduard N. Klenov* Rostov-on-Don, Russia GEOMETRICAL PHENOMENA IN THE PHYSICS OF SUBATOMIC PARTICLES Eduard N. Klnov* Rostov-on-Don, Russia Th articl considrs phnomnal gomtry figurs bing th carrirs of valu spctra for th pairs of th rmaining additiv

More information

COHORT MBA. Exponential function. MATH review (part2) by Lucian Mitroiu. The LOG and EXP functions. Properties: e e. lim.

COHORT MBA. Exponential function. MATH review (part2) by Lucian Mitroiu. The LOG and EXP functions. Properties: e e. lim. MTH rviw part b Lucian Mitroiu Th LOG and EXP functions Th ponntial function p : R, dfind as Proprtis: lim > lim p Eponntial function Y 8 6 - -8-6 - - X Th natural logarithm function ln in US- log: function

More information

A Prey-Predator Model with an Alternative Food for the Predator, Harvesting of Both the Species and with A Gestation Period for Interaction

A Prey-Predator Model with an Alternative Food for the Predator, Harvesting of Both the Species and with A Gestation Period for Interaction Int. J. Opn Problms Compt. Math., Vol., o., Jun 008 A Pry-Prdator Modl with an Altrnativ Food for th Prdator, Harvsting of Both th Spcis and with A Gstation Priod for Intraction K. L. arayan and. CH. P.

More information

ECE602 Exam 1 April 5, You must show ALL of your work for full credit.

ECE602 Exam 1 April 5, You must show ALL of your work for full credit. ECE62 Exam April 5, 27 Nam: Solution Scor: / This xam is closd-book. You must show ALL of your work for full crdit. Plas rad th qustions carfully. Plas chck your answrs carfully. Calculators may NOT b

More information

Ewald s Method Revisited: Rapidly Convergent Series Representations of Certain Green s Functions. Vassilis G. Papanicolaou 1

Ewald s Method Revisited: Rapidly Convergent Series Representations of Certain Green s Functions. Vassilis G. Papanicolaou 1 wald s Mthod Rvisitd: Rapidly Convrgnt Sris Rprsntations of Crtain Grn s Functions Vassilis G. Papanicolaou 1 Suggstd Running Had: wald s Mthod Rvisitd Complt Mailing Addrss of Contact Author for offic

More information

Sundials and Linear Algebra

Sundials and Linear Algebra Sundials and Linar Algbra M. Scot Swan July 2, 25 Most txts on crating sundials ar dirctd towards thos who ar solly intrstd in making and using sundials and usually assums minimal mathmatical background.

More information

WHAT LIES BETWEEN + AND (and beyond)? H.P.Williams

WHAT LIES BETWEEN + AND (and beyond)? H.P.Williams Working Par LSEOR 10-119 ISSN 2041-4668 (Onlin) WHAT LIES BETWEEN + AND (and byond)? HPWilliams London School of Economics hwilliams@lsacuk First ublishd in Grat Britain in 2010 by th Orational Rsarch

More information

Square of Hamilton cycle in a random graph

Square of Hamilton cycle in a random graph Squar of Hamilton cycl in a random graph Andrzj Dudk Alan Friz Jun 28, 2016 Abstract W show that p = n is a sharp thrshold for th random graph G n,p to contain th squar of a Hamilton cycl. This improvs

More information

perm4 A cnt 0 for for if A i 1 A i cnt cnt 1 cnt i j. j k. k l. i k. j l. i l

perm4 A cnt 0 for for if A i 1 A i cnt cnt 1 cnt i j. j k. k l. i k. j l. i l h 4D, 4th Rank, Antisytric nsor and th 4D Equivalnt to th Cross Product or Mor Fun with nsors!!! Richard R Shiffan Digital Graphics Assoc 8 Dunkirk Av LA, Ca 95 rrs@isidu his docunt dscribs th four dinsional

More information