International Journal of Advanced and Applied Sciences
|
|
- Ross Kenneth Cannon
- 5 years ago
- Views:
Transcription
1 Itratioal Joural of Advacd ad Applid Scics x(x) xxxx Pags: xx xx Cotts lists availabl at Scic Gat Itratioal Joural of Advacd ad Applid Scics Joural hompag: gatcom/ijaashtml Symmtric Fuctios of th -Fiboacci ad -Lucas umbrs Ali Boussayoud M Krada Sra Araci 3* Mhmt Acigo 4 LMAM Laboratory ad Dpartmt of Mathmatics Mohamd Sddi B Yahia Uivrsity Jijl Algria; aboussayoud@yahoofr mrada@yahoofr 3 Dpartmt of Ecoomics Faculty of Ecoomics Admiistrativ ad Social Scics Hasa Kalyocu Uivrsity TR-740 Gaiatp Tury; mtsr@hotmailcom 4 Dpartmt of Mathmatics Faculty of Arts ad ScicUivrsity of Gaiatp TR-730 Gaiatp Tury; acigo@gatpdutr A R T I C L E I N F O Articl history: Rcivd April 5 07 Rcivd i rvisd form Ju 0 07 Accptd Ju 07 Availabl oli Kywords: -Fiboacci umbrs; -Lucas umbrs; Gratig fuctios; Fiboacci polyomials A B S T R A C T I this papr w itroduc a w oprator i ordr to driv som w symmtric proprtis of -Fiboacci ad -Lucas umbrs ad Fiboacci polyomials By maig us of th w oprator dfid i this papr w giv som w gratig fuctios for -Fiboacci ad Pll umbrs ad Fiboacci polyomials 04 IASE Publishr All rights rsrvd Itroductio ad Notatios Fiboacci umbrs ad thir graliatios hav may itrstig proprtis ad applicatios to almost vry fild of scic ad art (g s [7]) Fiboacci umbrs F ar dfid by th rcurrc rlatio * Corrspodig Author Addrss: mtsr@hotmailcom (S Araci) F0 F F F F Thr xist a lot of proprtis about Fiboacci umbrs I particular thr is a bautiful combiatorial idtity to Fiboacci umbrs [7] F i i 0 i () From () Filippoi [Phlypoi] itroducd th icomplt Fiboacci umbrs F ( s ) ad th icomplt Lucas umbrs L ( s ) Thy ar dfid by s j F ( s ) 0 s ; 0 j 0 j s j L ( s ) 0 s ; 0 j 0 j I [7] Djordjvic gav th icomplt gralid Fiboacci ad Lucas umbrs I [8] Djordjvic ad Srivastava dfid icomplt gralid Jacobsthal ad Jacobsthal-Lucas umbrs I [6] th authors dfi th icomplt Fiboacci ad Lucas umbrs O th othr had may ids of graliatios of Fiboacci umbrs hav b prstd i th litratur I particular a graliatio is th - Fiboacci Numbrs For ay positiv ral umbr th -Fiboacci squc say ( F ) is dfid rcurrtly by F 0 F F F F I [4] -Fiboacci umbrs wr foud by studyig th rcursiv applicatio of two gomtrical trasformatios usd i th four-triagl logst-dg (4TLE) partitio Ths umbrs hav b studid i svral paprs; s [4 5] For ay positiv ral umbr th -Lucas
2 ( ) L squc say is dfid rcurrtly by L 0 L L L L If w hav th classical Pll-Lucas umbrs appars: Q0 Q ad Q Q Q for [3] I this cotributio w shall dfi a w usful oprator dotd by for which w ca formulat xtd ad prov w rsults basd o our prvious os [3 5] I ordr to dtrmi w gratig fuctios of th products of som ow umbrs ad polyomails w combi btw our idicatd past tchiqus ad ths prstd polishig approachs Lt ad b two positiv itgr ad { a a a } ar st of giv variabls rcall [0] that th -th lmtary symmtric fuctio ( a a a ) ad th -th complt homogous symmtric fuctio h( a a a ) ar dfid rspctivly by i i i ( a a a ) a a a 0 ii i with i i i 0 or h ( a a a ) a a a 0 i i i ii i with i i i 0 First w st 0( a a a ) ad h ( a a a ) (by covtio) For or 0 0 w st ( a a a) 0 ad h ( a a a ) 0 Dfiitio Lt E { } a alphabt w dfi th symmtric fuctio S associatd with th alphabt E by with S ( ) h( ) S ( ) h ( ) 0 0 S ( ) h ( ) S ( ) h ( ) Dfiitio Lt A ad B b ay two alphabts th w giv S ( A B ) by th followig form: b B( b ) S ( A B ) () ( a ) aa 0 with th coditio S ( A B ) 0 for 0 (s []) Corollary3 Taig A 0 i () givs ( b ) S ( B ) bb 0 Furthr i th cas A 0 or 0 B w hav S A B S A S B ( ) ( ) ( ) Dfiitio4 Lt g b ay fuctio o th w cosidr th dividd diffrc oprator as th followig form g ( x x x x ) g ( x x x x ) i i i i x ( ) ix g i x i x i (s [9] ) Dfiitio5 [9] Th symmtriig oprator dfid by f ( ) f ( ) is ( f ) for all N Rmar6 Lt E { } a alphabt w hav h ( ) S ( ) ( ) Th Fiboacci Polyomials Not that if is a ral variabl x th F F ad thy corrspod to th Fiboacci x polyomials dfid by [4] if 0 F ( x ) x if xf( x ) F ( x ) if from whr th first Fiboacci polyomials ar F x F x x F 3 x x F 4 x x 3 x F 5 x x 4 3x F 6 x x 5 4x 3 3x F 7 x x 6 5x 4 6x F 8 x x 7 6x 5 0x 3 4x Ad from ths xprssio as for th umbrs w ca writ [4]:?? i i ( ) for 0 i 0 i F x x () -Fiboacci
3 Not that F (0) 0 ad x 0 is th oly ral root whil (0) with o ral roots Also for x F w obtai th lmts of th - Fiboacci umbrs By itratig rcurrc rlatio of formula () th followig proprty is straightforwardly dducd Propositio [6] For r holds: F ( x ) F ( x ) F ( x ) F ( x ) F ( x ) r ( r ) r ( r ) Propositio (Bit's formula) Th th Fiboacci polyomial may b writt as ( ) x x 4 F ( x) big Proof Not that th charactristic quatio for - Fiboacci polyomials is r x roots r ad x x 4 r 0 with r from whr Formula () is dducd Propositio3 [5] (Asymptotic bhaviour of th quotit of coscutiv trms) If x x 4 F th lim ( x ) F ( x ) As a cosquc th quotit btw two coscutiv trms of th -Fiboacci umbrs 3 { F; } {0 } tds to th positiv charactristic root For ach itgr is calld th th mtallic ratio [8]: Gold Ratio for ad Bro Ratio for 3 Propositio4 [4] (Hosbrgr's formula) for m itgrs it holds: F ( x ) F ( x ) F ( x ) F ( x ) F ( x ) m m m 3 O th Symmtric Fuctios of Som - Fiboacci Numbrs ad Fiboacci Polyomials I this part w ar ow i a positio to provid Thorm 33 Also w driv th w gratig fuctios of th products of som ow umbrs ad polyomails Dfiitio3 Th symmtriig oprator is dfid by [7] f ( ) f ( ) ( f ) for all ( ) ( ) Lmma3 [7] Lt E { } w dfi th oprator as follows: h ( ) f ( ) f ( ) f ( ) Thorm33 Lt E ad A b two alphabts rspctivly ad a a th w hav 0 aa a a a a aa h a a ( ) 0 for all Proof By applyig th oprator a f ( ) w hav aa 0 h a a ( ) f a a a ( ) 0 O th othr had (3) to th sris a a a a a a 0 0 a a a a 0 0 a a aa aa a a a 0 a a aa aa ( f ) h a a a h a a h a a h a a a 0 h a a 0 h a a h a a a ( ) 0 h a a a ( ) 0 This complts th proof W ow driv th w gratig fuctios of th products of som ow polyomials Idd w cosidr Thorm 33 i ordr to driv -Fiboacci umbrs ad Fiboacci polyomials if Thorm34 Lt E ad A b two alphabts
4 rspctivly 0 ad a a th w hav a a a a h ( ) h ( a a ) ( a a) aa h ( a a ) (3) I th cas aa a a aa A a basd o Thorm 34 w dduc th followig Lmmas Lmma35 Giv two alphabts E A a w hav ad a 0 a ( ) a (33) Lmma36 Giv two alphabts E A a w hav ad a 0 a ( ) a (34) Assumig that ad a i (33) ad (34) w obtai th gratig fuctios giv by Boussayoud t al [ 5] wich rprst: ) Th gratig fuctio of th Fiboacci umbrs F ) Th gratig fuctio of th Lucas umbrs L Choosig ad such that ad substitutig i (33) ad (34) w d up with [3] (35) 0 (36) 0 w dduc th followig thorm Thorm37 [] For th gratig fuctio of th -Fiboacci umbrs is giv by F 0 Multiplyig th quatio (35) by ( ) ad subtract it from (36) by ( ) w obtai 0 ad w hav th followig thorm Thorm38 [] For th gratig fuctio of th -Lucas umbrs is giv by L 0 Put i th rlatioship (37) w hav Q 0 which rprsts a gratig fuctio for Pll-Lucas umbrs [5] Choosig ad such that substitutig i (33) w d up with 0 x ad x x 4 with x Thus w gt th followig thorm Thorm39 W hav th followig a w gratig fuctio of th Fiboacci polyomials as F ( x ) x 0 For th cas E ad A a a with rplacig by a by a i (3) w hav h ( a [ a ]) h ( [ ]) 0 a a a a a a aa h( ) h( a a ) ( a a) aa (38) This cas cosists of thr rlatd parts Firstly th substitutios i (38) giv a a x ad aa x x ( x ) x h( a a ) h( ) F 0 F ( x ) From which w hav th followig thorm Thorm30 W hav th followig a w gratig fuctio of th product of -Fiboacci umbrs ad Fiboacci polyomials as x F F ( x ) (39) 3 4 x ( x ) x i th rlatioship (39) w hav x F F( x ) x ( x 3) x 0 Put which rprsts a w gratigs fuctios of th product of Fiboacci umbrs ad Fiboacci polyomials Put i th rlatioship (39) w hav 5x P F( x ) 3 4 x ( x 6) x 0 which rprsts a w gratigs fuctiosof th product of Pll umbrs ad Fiboacci polyomials Scodly by maig th followig rstrictios:
5 i (38) giv 0 0 a a ad aa h ( a a ) h ( ) ( ) 3 4 F F (30) rprstig a w gratig fuctio of - Fiboacci umbrs F O th othr had w cosidr ( ) 0 0 Sic 0 F F F F F 0 0 F F F F ( ) 3 4 w hav ( ) ( ) F F ( ) 3 4 (s 3 ) from thos applicatios w dduc th followig thorm Thorm3 W hav th followig a w gratig fuctio of th product of two coscutiv -Fiboacci umbrs as ( ) F F (3) ( ) Put i th rlatioship (30) ad (3) w obtai th followig rsults Corollary3 W hav th followig a w gratig fuctio of th product of Fiboacci umbrs as F F Corollary33 For th gratig fuctio of th product of two coscutiv Fiboacci umbrs is giv by F F 0 3 Put i th rlatioship (30) ad (3) w obtai th followig rsults Corollary34 W hav th followig a w gratig fuctio of th product of Pll umbrs as 54 P P 3 4 Corollary35 W hav th followig a w gratig fuctio of th product of two coscutiv Pll umbrs as 4 P P Coclusios I this papr w hav drivd w thorms i ordr to dtrmi gratig fuctios of -Fiboacci umbrs - Lucas umbrs ad Fiboacci polyomials Th drivd thorms ad Lmmas ar basd o symmtric fuctios ad products of ths umbrs ad polyomials Rfrcs AAbdrra Gééralisatio d la trasformatio d'eulr d'u séri formll Adv Math (994) A Boussayoud M Boulyr M Krada O Som Idtitis ad Symmtric Fuctios for Lucas ad Pll Numbrs ElctroJMathAalysisAppl (07) A Boussayoud N Harrouch Complt Symmtric Fuctios ad - Fiboacci Numbrs Commu Appl Aal (06) A Abdrra M Krada A Boussayoud Graliatio of Som Hadamard Product Commu Appl Aal (06) A Boussayoud M Krada M Boulyr A simpl ad accurat mthod for dtrmiatio of som gralid squc of umbrs It J Pur Appl Math (06) A Boussayoud A Abdrra M Krada Som applicatios of symmtric fuctios Itgrs 5 A#48-7 (05) A Boussayoud R Sahali Th applicatio of th oprator L j j bb i th sris j 0a jb J Adv Rs Appl Math (05) A Boussayoud M Krada R Sahali W Rouibah Som Applicatios o Gratig Fuctios J Cocr Appl Math (04) A Boussayoud M Krada A Abdrra A Graliatio of Som Orthogoal Polyomials Sprigr Proc Math Stat (03) AT Bjami JJ Qui Proofs That Rally Cout: Th Art of Combiatorial Proof Mathmatical Associatio of Amrica Washigto DC 003 AF Horadam Basic proprtis of a crtai gralid squc of umbrsfiboacci Q (965) AF Horadam Gratig fuctios for powrs of a crtai gralid squc of umbrs Du Math J (965) AF Horadam JM Maho Pll ad Pll-Lucas Polyomials Fiboacci Q (985) A Lascoux AM Fua Partitio aalysis ad symmtriig oprators J Comb Thory A 09
6 (005) C Bolat H Kos O th Proprtis of -Fiboacci Numbrs It J Cotmp Math Scics (00) D Tasci M Cti Firgi Icomplt Fiboacci ad Lucas p-umbrs Math Comput Modllig (00) G B Djordjvic Gratig fuctios of th icomplt gralid Fiboacci ad gralid Lucas umbrs Fiboacci Q (004) G B Djordjvic H M Srivastava Icomplt gralid Jacobsthal ad Jacobsthal-Lucas umbrs Math Comput Modllig (005) IG Macdoald Symmtric fuctios ad Hall polyomias Oxford Uivrsity Prss 979 M Mrca A Graliatio of th symmtry btw complt ad lmtary symmtric fuctios Idia J Pur Appl Math (04) P A Pao O th gratig fuctios of Mrs ad Frmat prims Collct Math (0) P Filippoi Icomplt Fiboacci ad Lucas umbrs Rd Circ Mat Palrmo (996) S Araci Novl idtitis ivolvig Gocchi umbrs ad polyomials arisig from applicatios of umbral calculus Appl Math Comput (04) S Falco A Plaa O th Fiboacci - umbrs Chaos Sulutios & Fractals (007) S Falco A Plaa Th - Fiboacci squc ad th Pascal -triagl Chaos Sulutios & Fractals (008) S Falco A Plaa O - Fiboacci squcs ad Polyomials ad thir drivativs Chaos Sulutios & Fractals (009) T Koshy Fiboacci ad Lucas Numbrs with Applicatios Wily-Itrscic 00 VW Spiadl Thmtallic mas family ad forbidd symmtris It Math J (00)
SOME IDENTITIES FOR THE GENERALIZED POLY-GENOCCHI POLYNOMIALS WITH THE PARAMETERS A, B AND C
Joural of Mathatical Aalysis ISSN: 2217-3412, URL: www.ilirias.co/ja Volu 8 Issu 1 2017, Pags 156-163 SOME IDENTITIES FOR THE GENERALIZED POLY-GENOCCHI POLYNOMIALS WITH THE PARAMETERS A, B AND C BURAK
More informationOn the approximation of the constant of Napier
Stud. Uiv. Babş-Bolyai Math. 560, No., 609 64 O th approximatio of th costat of Napir Adri Vrscu Abstract. Startig from som oldr idas of [] ad [6], w show w facts cocrig th approximatio of th costat of
More informationRestricted Factorial And A Remark On The Reduced Residue Classes
Applid Mathmatics E-Nots, 162016, 244-250 c ISSN 1607-2510 Availabl fr at mirror sits of http://www.math.thu.du.tw/ am/ Rstrictd Factorial Ad A Rmark O Th Rducd Rsidu Classs Mhdi Hassai Rcivd 26 March
More informationPURE MATHEMATICS A-LEVEL PAPER 1
-AL P MATH PAPER HONG KONG EXAMINATIONS AUTHORITY HONG KONG ADVANCED LEVEL EXAMINATION PURE MATHEMATICS A-LEVEL PAPER 8 am am ( hours) This papr must b aswrd i Eglish This papr cosists of Sctio A ad Sctio
More informationHadamard Exponential Hankel Matrix, Its Eigenvalues and Some Norms
Math Sci Ltt Vol No 8-87 (0) adamard Exotial al Matrix, Its Eigvalus ad Som Norms İ ad M bula Mathmatical Scics Lttrs Itratioal Joural @ 0 NSP Natural Scics Publishig Cor Dartmt of Mathmatics, aculty of
More informationz 1+ 3 z = Π n =1 z f() z = n e - z = ( 1-z) e z e n z z 1- n = ( 1-z/2) 1+ 2n z e 2n e n -1 ( 1-z )/2 e 2n-1 1-2n -1 1 () z
Sris Expasio of Rciprocal of Gamma Fuctio. Fuctios with Itgrs as Roots Fuctio f with gativ itgrs as roots ca b dscribd as follows. f() Howvr, this ifiit product divrgs. That is, such a fuctio caot xist
More informationFolding of Hyperbolic Manifolds
It. J. Cotmp. Math. Scics, Vol. 7, 0, o. 6, 79-799 Foldig of Hyprbolic Maifolds H. I. Attiya Basic Scic Dpartmt, Collg of Idustrial Educatio BANE - SUEF Uivrsity, Egypt hala_attiya005@yahoo.com Abstract
More informationDTFT Properties. Example - Determine the DTFT Y ( e ) of n. Let. We can therefore write. From Table 3.1, the DTFT of x[n] is given by 1
DTFT Proprtis Exampl - Dtrmi th DTFT Y of y α µ, α < Lt x α µ, α < W ca thrfor writ y x x From Tabl 3., th DTFT of x is giv by ω X ω α ω Copyright, S. K. Mitra Copyright, S. K. Mitra DTFT Proprtis DTFT
More informationNEW VERSION OF SZEGED INDEX AND ITS COMPUTATION FOR SOME NANOTUBES
Digst Joural of Naomatrials ad Biostructurs Vol 4, No, March 009, p 67-76 NEW VERSION OF SZEGED INDEX AND ITS COMPUTATION FOR SOME NANOTUBES A IRANMANESH a*, O KHORMALI b, I NAJAFI KHALILSARAEE c, B SOLEIMANI
More information07 - SEQUENCES AND SERIES Page 1 ( Answers at he end of all questions ) b, z = n
07 - SEQUENCES AND SERIES Pag ( Aswrs at h d of all qustios ) ( ) If = a, y = b, z = c, whr a, b, c ar i A.P. ad = 0 = 0 = 0 l a l
More informationDiscrete Fourier Transform (DFT)
Discrt Fourir Trasorm DFT Major: All Egirig Majors Authors: Duc guy http://umricalmthods.g.us.du umrical Mthods or STEM udrgraduats 8/3/29 http://umricalmthods.g.us.du Discrt Fourir Trasorm Rcalld th xpotial
More informationThe Interplay between l-max, l-min, p-max and p-min Stable Distributions
DOI: 0.545/mjis.05.4006 Th Itrplay btw lma lmi pma ad pmi Stabl Distributios S Ravi ad TS Mavitha Dpartmt of Studis i Statistics Uivrsity of Mysor Maasagagotri Mysuru 570006 Idia. Email:ravi@statistics.uimysor.ac.i
More informationCharacter sums over generalized Lehmer numbers
Ma t al. Joural of Iualitis ad Applicatios 206 206:270 DOI 0.86/s3660-06-23-y R E S E A R C H Op Accss Charactr sums ovr gralizd Lhmr umbrs Yuakui Ma, Hui Ch 2, Zhzh Qi 2 ad Tiapig Zhag 2* * Corrspodc:
More information1985 AP Calculus BC: Section I
985 AP Calculus BC: Sctio I 9 Miuts No Calculator Nots: () I this amiatio, l dots th atural logarithm of (that is, logarithm to th bas ). () Ulss othrwis spcifid, th domai of a fuctio f is assumd to b
More informationFurther Results on Pair Sum Graphs
Applid Mathmatis, 0,, 67-75 http://dx.doi.org/0.46/am.0.04 Publishd Oli Marh 0 (http://www.sirp.org/joural/am) Furthr Rsults o Pair Sum Graphs Raja Poraj, Jyaraj Vijaya Xavir Parthipa, Rukhmoi Kala Dpartmt
More informationA Simple Proof that e is Irrational
Two of th most bautiful ad sigificat umbrs i mathmatics ar π ad. π (approximatly qual to 3.459) rprsts th ratio of th circumfrc of a circl to its diamtr. (approximatly qual to.788) is th bas of th atural
More informationJournal of Modern Applied Statistical Methods
Joural of Modr Applid Statistical Mthods Volum Issu Articl 6 --03 O Som Proprtis of a Htrogous Trasfr Fuctio Ivolvig Symmtric Saturatd Liar (SATLINS) with Hyprbolic Tagt (TANH) Trasfr Fuctios Christophr
More informationH2 Mathematics Arithmetic & Geometric Series ( )
H Mathmatics Arithmtic & Gomtric Sris (08 09) Basic Mastry Qustios Arithmtic Progrssio ad Sris. Th rth trm of a squc is 4r 7. (i) Stat th first four trms ad th 0th trm. (ii) Show that th squc is a arithmtic
More informationSTIRLING'S 1 FORMULA AND ITS APPLICATION
MAT-KOL (Baja Luka) XXIV ()(08) 57-64 http://wwwimviblorg/dmbl/dmblhtm DOI: 075/МК80057A ISSN 0354-6969 (o) ISSN 986-588 (o) STIRLING'S FORMULA AND ITS APPLICATION Šfkt Arslaagić Sarajvo B&H Abstract:
More informationOn Some Identities and Generating Functions for Mersenne Numbers and Polynomials
Turish Joural of Aalysis ad Number Theory, 8, Vol 6, No, 9-97 Available olie at htt://ubsscieubcom/tjat/6//5 Sciece ad Educatio Publishig DOI:69/tjat-6--5 O Some Idetities ad Geeratig Fuctios for Mersee
More informationChapter Five. More Dimensions. is simply the set of all ordered n-tuples of real numbers x = ( x 1
Chatr Fiv Mor Dimsios 51 Th Sac R W ar ow rard to mov o to sacs of dimsio gratr tha thr Ths sacs ar a straightforward gralizatio of our Euclida sac of thr dimsios Lt b a ositiv itgr Th -dimsioal Euclida
More informationTriple Play: From De Morgan to Stirling To Euler to Maclaurin to Stirling
Tripl Play: From D Morga to Stirlig To Eulr to Maclauri to Stirlig Augustus D Morga (186-1871) was a sigificat Victoria Mathmaticia who mad cotributios to Mathmatics History, Mathmatical Rcratios, Mathmatical
More informationLinear Algebra Existence of the determinant. Expansion according to a row.
Lir Algbr 2270 1 Existc of th dtrmit. Expsio ccordig to row. W dfi th dtrmit for 1 1 mtrics s dt([]) = (1) It is sy chck tht it stisfis D1)-D3). For y othr w dfi th dtrmit s follows. Assumig th dtrmit
More informationAn Introduction to Asymptotic Expansions
A Itroductio to Asmptotic Expasios R. Shaar Subramaia Asmptotic xpasios ar usd i aalsis to dscrib th bhavior of a fuctio i a limitig situatio. Wh a fuctio ( x, dpds o a small paramtr, ad th solutio of
More informationDigital Signal Processing, Fall 2006
Digital Sigal Procssig, Fall 6 Lctur 9: Th Discrt Fourir Trasfor Zhg-Hua Ta Dpartt of Elctroic Systs Aalborg Uivrsity, Dar zt@o.aau.d Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Cours at a glac MM Discrt-ti
More informationA GENERALIZED RAMANUJAN-NAGELL EQUATION RELATED TO CERTAIN STRONGLY REGULAR GRAPHS
#A35 INTEGERS 4 (204) A GENERALIZED RAMANUJAN-NAGELL EQUATION RELATED TO CERTAIN STRONGLY REGULAR GRAPHS B d Wgr Faculty of Mathmatics ad Computr Scic, Eidhov Uivrsity of Tchology, Eidhov, Th Nthrlads
More informationChapter 2 Infinite Series Page 1 of 11. Chapter 2 : Infinite Series
Chatr Ifiit Sris Pag of Sctio F Itgral Tst Chatr : Ifiit Sris By th d of this sctio you will b abl to valuat imror itgrals tst a sris for covrgc by alyig th itgral tst aly th itgral tst to rov th -sris
More informationUNIT 2: MATHEMATICAL ENVIRONMENT
UNIT : MATHEMATICAL ENVIRONMENT. Itroductio This uit itroducs som basic mathmatical cocpts ad rlats thm to th otatio usd i th cours. Wh ou hav workd through this uit ou should: apprciat that a mathmatical
More informationDiscrete Fourier Transform. Discrete Fourier Transform. Discrete Fourier Transform. Discrete Fourier Transform. Discrete Fourier Transform
Discrt Fourir Trasform Dfiitio - T simplst rlatio btw a lt- squc x dfid for ω ad its DTFT X ( ) is ω obtaid by uiformly sampli X ( ) o t ω-axis btw ω < at ω From t dfiitio of t DTFT w tus av X X( ω ) ω
More informationMONTGOMERY COLLEGE Department of Mathematics Rockville Campus. 6x dx a. b. cos 2x dx ( ) 7. arctan x dx e. cos 2x dx. 2 cos3x dx
MONTGOMERY COLLEGE Dpartmt of Mathmatics Rockvill Campus MATH 8 - REVIEW PROBLEMS. Stat whthr ach of th followig ca b itgratd by partial fractios (PF), itgratio by parts (PI), u-substitutio (U), or o of
More informationLaw of large numbers
Law of larg umbrs Saya Mukhrj W rvisit th law of larg umbrs ad study i som dtail two typs of law of larg umbrs ( 0 = lim S ) p ε ε > 0, Wak law of larrg umbrs [ ] S = ω : lim = p, Strog law of larg umbrs
More informationWorksheet: Taylor Series, Lagrange Error Bound ilearnmath.net
Taylor s Thorm & Lagrag Error Bouds Actual Error This is th ral amout o rror, ot th rror boud (worst cas scario). It is th dirc btw th actual () ad th polyomial. Stps:. Plug -valu ito () to gt a valu.
More informationTime : 1 hr. Test Paper 08 Date 04/01/15 Batch - R Marks : 120
Tim : hr. Tst Papr 8 D 4//5 Bch - R Marks : SINGLE CORRECT CHOICE TYPE [4, ]. If th compl umbr z sisfis th coditio z 3, th th last valu of z is qual to : z (A) 5/3 (B) 8/3 (C) /3 (D) o of ths 5 4. Th itgral,
More informationHardy-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 informationLimiting 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 informationOn a problem of J. de Graaf connected with algebras of unbounded operators de Bruijn, N.G.
O a problm of J. d Graaf coctd with algbras of uboudd oprators d Bruij, N.G. Publishd: 01/01/1984 Documt Vrsio Publishr s PDF, also kow as Vrsio of Rcord (icluds fial pag, issu ad volum umbrs) Plas chck
More informationDerangements 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 informationIterative Methods of Order Four for Solving Nonlinear Equations
Itrativ Mods of Ordr Four for Solvig Noliar Equatios V.B. Kumar,Vatti, Shouri Domii ad Mouia,V Dpartmt of Egirig Mamatis, Formr Studt of Chmial Egirig Adhra Uivrsity Collg of Egirig A, Adhra Uivrsity Visakhapatam
More informationNET/JRF, GATE, IIT JAM, JEST, TIFR
Istitut for NET/JRF, GATE, IIT JAM, JEST, TIFR ad GRE i PHYSICAL SCIENCES Mathmatical Physics JEST-6 Q. Giv th coditio φ, th solutio of th quatio ψ φ φ is giv by k. kφ kφ lφ kφ lφ (a) ψ (b) ψ kφ (c) ψ
More informationThomas J. Osler. 1. INTRODUCTION. This paper gives another proof for the remarkable simple
5/24/5 A PROOF OF THE CONTINUED FRACTION EXPANSION OF / Thomas J Oslr INTRODUCTION This ar givs aothr roof for th rmarkabl siml cotiud fractio = 3 5 / Hr is ay ositiv umbr W us th otatio x= [ a; a, a2,
More informationA Note on Quantile Coupling Inequalities and Their Applications
A Not o Quatil Couplig Iqualitis ad Thir Applicatios Harriso H. Zhou Dpartmt of Statistics, Yal Uivrsity, Nw Hav, CT 06520, USA. E-mail:huibi.zhou@yal.du Ju 2, 2006 Abstract A rlatioship btw th larg dviatio
More information10. Joint Moments and Joint Characteristic Functions
0 Joit Momts ad Joit Charactristic Fctios Followig sctio 6 i this sctio w shall itrodc varios paramtrs to compactly rprst th iformatio cotaid i th joit pdf of two rvs Giv two rvs ad ad a fctio g x y dfi
More informationThe 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 informationOn the Reformulated Zagreb Indices of Certain Nanostructures
lobal Joural o Pur ad Applid Mathmatics. ISSN 097-768 Volum, Numbr 07, pp. 87-87 Rsarch Idia Publicatios http://www.ripublicatio.com O th Rormulatd Zagrb Idics o Crtai Naostructurs Krthi. Mirajar ad Priyaa
More informationGENERATING FUNCTIONS K-FIBONACCI AND K-JACOBSTHAL NUMBERS AT NEGATIVE INDICES
Electronic Journal of Mathematical Analysis and Applications Vol. 6(2) July 2018, pp. 195-202. ISSN: 2090-729X(online) http://fcag-egypt.com/journals/ejmaa/ GENERATING FUNCTIONS K-FIBONACCI AND K-JACOBSTHAL
More informationConstruction 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 informationA Review of Complex Arithmetic
/0/005 Rviw of omplx Arithmti.do /9 A Rviw of omplx Arithmti A omplx valu has both a ral ad imagiary ompot: { } ad Im{ } a R b so that w a xprss this omplx valu as: whr. a + b Just as a ral valu a b xprssd
More informationBINOMIAL 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 informationAustralian Journal of Basic and Applied Sciences, 4(9): , 2010 ISSN
Australia Joural of Basic ad Applid Scics, 4(9): 4-43, ISSN 99-878 Th Caoical Product of th Diffrtial Equatio with O Turig Poit ad Sigular Poit A Dabbaghia, R Darzi, 3 ANaty ad 4 A Jodayr Akbarfa, Islaic
More informationRecall 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 informationThe 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 informationDiscrete Fourier Transform. Nuno Vasconcelos UCSD
Discrt Fourir Trasform uo Vascoclos UCSD Liar Shift Ivariat (LSI) systms o of th most importat cocpts i liar systms thory is that of a LSI systm Dfiitio: a systm T that maps [ ito y[ is LSI if ad oly if
More informationBINOMIAL 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 informationEinstein 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 informationChapter 3 Fourier Series Representation of Periodic Signals
Chptr Fourir Sris Rprsttio of Priodic Sigls If ritrry sigl x(t or x[] is xprssd s lir comitio of som sic sigls th rspos of LI systm coms th sum of th idividul rsposs of thos sic sigls Such sic sigl must:
More informationChapter 11.00C Physical Problem for Fast Fourier Transform Civil Engineering
haptr. Physical Problm for Fast Fourir Trasform ivil Egirig Itroductio I this chaptr, applicatios of FFT algorithms [-5] for solvig ral-lif problms such as computig th dyamical (displacmt rspos [6-7] of
More informationTwo Products Manufacturer s Production Decisions with Carbon Constraint
Managmnt Scinc and Enginring Vol 7 No 3 pp 3-34 DOI:3968/jms9335X374 ISSN 93-34 [Print] ISSN 93-35X [Onlin] wwwcscanadant wwwcscanadaorg Two Products Manufacturr s Production Dcisions with Carbon Constraint
More informationNEW 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 information2.29 Numerical Fluid Mechanics Spring 2015 Lecture 12
REVIEW Lctur 11: Numrical Fluid Mchaics Sprig 2015 Lctur 12 Fiit Diffrcs basd Polyomial approximatios Obtai polyomial (i gral u-qually spacd), th diffrtiat as dd Nwto s itrpolatig polyomial formulas Triagular
More informationCOLLECTION OF SUPPLEMENTARY PROBLEMS CALCULUS II
COLLECTION OF SUPPLEMENTARY PROBLEMS I. CHAPTER 6 --- Trscdtl Fuctios CALCULUS II A. FROM CALCULUS BY J. STEWART:. ( How is th umbr dfid? ( Wht is pproimt vlu for? (c ) Sktch th grph of th turl potil fuctios.
More informationFORBIDDING RAINBOW-COLORED STARS
FORBIDDING RAINBOW-COLORED STARS CARLOS HOPPEN, HANNO LEFMANN, KNUT ODERMANN, AND JULIANA SANCHES Abstract. W cosidr a xtrmal problm motivatd by a papr of Balogh [J. Balogh, A rmark o th umbr of dg colorigs
More informationReview Article Incomplete Bivariate Fibonacci and Lucas p-polynomials
Discrete Dyamics i Nature ad Society Volume 2012, Article ID 840345, 11 pages doi:10.1155/2012/840345 Review Article Icomplete Bivariate Fiboacci ad Lucas p-polyomials Dursu Tasci, 1 Mirac Ceti Firegiz,
More informationCramér-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 informationMATH 681 Notes Combinatorics and Graph Theory I. ( 4) n. This will actually turn out to be marvelously simplifiable: C n = 2 ( 4) n n + 1. ) (n + 1)!
MATH 681 Nots Combiatorics ad Graph Thory I 1 Catala umbrs Prviously, w usd gratig fuctios to discovr th closd form C = ( 1/ +1) ( 4). This will actually tur out to b marvlously simplifiabl: ( ) 1/ C =
More informationChapter 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 informationCalculus & analytic geometry
Calculus & aalytic gomtry B Sc MATHEMATICS Admissio owards IV SEMESTER CORE COURSE UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION CALICUT UNIVERSITYPO, MALAPPURAM, KERALA, INDIA 67 65 5 School of Distac
More informationFigure 2-18 Thevenin Equivalent Circuit of a Noisy Resistor
.8 NOISE.8. Th Nyquist Nois Thorm W ow wat to tur our atttio to ois. W will start with th basic dfiitio of ois as usd i radar thory ad th discuss ois figur. Th typ of ois of itrst i radar thory is trmd
More informationPier Franz Roggero, Michele Nardelli, Francesco Di Noto
Vrsio.0 9/06/04 Pagia di 7 O SOME EQUATIOS COCEIG THE IEMA S PIME UMBE FOMULA AD O A SECUE AD EFFICIET PIMALITY TEST. MATHEMATICAL COECTIOS WITH SOME SECTOS OF STIG THEOY Pir Fra oggro, Michl ardlli, Fracsco
More informationLINEAR 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 informationAPPENDIX: STATISTICAL TOOLS
I. Nots o radom samplig Why do you d to sampl radomly? APPENDI: STATISTICAL TOOLS I ordr to masur som valu o a populatio of orgaisms, you usually caot masur all orgaisms, so you sampl a subst of th populatio.
More informationIntroduction to Quantum Information Processing. Overview. A classical randomised algorithm. q 3,3 00 0,0. p 0,0. Lecture 10.
Itroductio to Quatum Iformatio Procssig Lctur Michl Mosca Ovrviw! Classical Radomizd vs. Quatum Computig! Dutsch-Jozsa ad Brsti- Vazirai algorithms! Th quatum Fourir trasform ad phas stimatio A classical
More informationOn 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 informationTechnical Support Document Bias of the Minimum Statistic
Tchical Support Documt Bias o th Miimum Stattic Itroductio Th papr pla how to driv th bias o th miimum stattic i a radom sampl o siz rom dtributios with a shit paramtr (also kow as thrshold paramtr. Ths
More informationStatistics 3858 : Likelihood Ratio for Exponential Distribution
Statistics 3858 : Liklihood Ratio for Expotial Distributio I ths two xampl th rjctio rjctio rgio is of th form {x : 2 log (Λ(x)) > c} for a appropriat costat c. For a siz α tst, usig Thorm 9.5A w obtai
More informationTraveling Salesperson Problem and Neural Networks. A Complete Algorithm in Matrix Form
Procdigs of th th WSEAS Itratioal Cofrc o COMPUTERS, Agios Nikolaos, Crt Islad, Grc, July 6-8, 7 47 Travlig Salsprso Problm ad Nural Ntworks A Complt Algorithm i Matrix Form NICOLAE POPOVICIU Faculty of
More informationEmpirical Study in Finite Correlation Coefficient in Two Phase Estimation
M. Khoshvisa Griffith Uivrsity Griffith Busiss School Australia F. Kaymarm Massachustts Istitut of Tchology Dpartmt of Mchaical girig USA H. P. Sigh R. Sigh Vikram Uivrsity Dpartmt of Mathmatics ad Statistics
More informationChapter Taylor Theorem Revisited
Captr 0.07 Taylor Torm Rvisitd Atr radig tis captr, you sould b abl to. udrstad t basics o Taylor s torm,. writ trascdtal ad trigoomtric uctios as Taylor s polyomial,. us Taylor s torm to id t valus o
More informationReview Exercises. 1. Evaluate using the definition of the definite integral as a Riemann Sum. Does the answer represent an area? 2
MATHEMATIS --RE Itgral alculus Marti Huard Witr 9 Rviw Erciss. Evaluat usig th dfiitio of th dfiit itgral as a Rima Sum. Dos th aswr rprst a ara? a ( d b ( d c ( ( d d ( d. Fid f ( usig th Fudamtal Thorm
More informationON 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 informationGaps in samples of geometric random variables
Discrt Mathmatics 37 7 871 89 Not Gaps i sampls of gomtric radom variabls William M.Y. Goh a, Pawl Hitczko b,1 a Dpartmt of Mathmatics, Drxl Uivrsity, Philadlphia, PA 1914, USA b Dpartmts of Mathmatics
More informationSome remarks on Kurepa s left factorial
Som rmarks on Kurpa s lft factorial arxiv:math/0410477v1 [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).
More information2617 Mark Scheme June 2005 Mark Scheme 2617 June 2005
Mark Schm 67 Ju 5 GENERAL INSTRUCTIONS Marks i th mark schm ar plicitly dsigatd as M, A, B, E or G. M marks ("mthod" ar for a attmpt to us a corrct mthod (ot mrly for statig th mthod. A marks ("accuracy"
More informationLectures 9 IIR Systems: First Order System
EE3054 Sigals ad Systms Lcturs 9 IIR Systms: First Ordr Systm Yao Wag Polytchic Uivrsity Som slids icludd ar xtractd from lctur prstatios prpard by McCllla ad Schafr Lics Ifo for SPFirst Slids This work
More informationAn Introduction to Asymptotic Expansions
A Itroductio to Asmptotic Expasios R. Shaar Subramaia Dpartmt o Chmical ad Biomolcular Egirig Clarso Uivrsit Asmptotic xpasios ar usd i aalsis to dscrib th bhavior o a uctio i a limitig situatio. Wh a
More informationUNIVERSALITY LIMITS INVOLVING ORTHOGONAL POLYNOMIALS ON AN ARC OF THE UNIT CIRCLE
UNIVERSALITY LIMITS INVOLVING ORTHOGONAL POLYNOMIALS ON AN ARC OF THE UNIT CIRCLE DORON S. LUBINSKY AND VY NGUYEN A. W stablish uivrsality limits for masurs o a subarc of th uit circl. Assum that µ is
More informationDISTRIBUTION 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 informationOn Deterministic Finite Automata and Syntactic Monoid Size, Continued
O Dtrmiistic Fiit Automata ad Sytactic Mooid Siz, Cotiud Markus Holzr ad Barbara Köig Istitut für Iformatik, Tchisch Uivrsität Müch, Boltzmastraß 3, D-85748 Garchig bi Müch, Grmay mail: {holzr,koigb}@iformatik.tu-much.d
More informationBayesian Estimations in Insurance Theory and Practice
Advacs i Mathmatical ad Computatioal Mthods Baysia Estimatios i Isurac Thory ad Practic VIERA PACÁKOVÁ Dpartmt o Mathmatics ad Quatitativ Mthods Uivrsity o Pardubic Studtská 95, 53 0 Pardubic CZECH REPUBLIC
More informationNormal Form for Systems with Linear Part N 3(n)
Applid Mathmatics 64-647 http://dxdoiorg/46/am7 Publishd Oli ovmbr (http://wwwscirporg/joural/am) ormal Form or Systms with Liar Part () Grac Gachigua * David Maloza Johaa Sigy Dpartmt o Mathmatics Collg
More informationOrdinary Differential Equations
Ordiary Diffrtial Equatio Aftr radig thi chaptr, you hould b abl to:. dfi a ordiary diffrtial quatio,. diffrtiat btw a ordiary ad partial diffrtial quatio, ad. Solv liar ordiary diffrtial quatio with fid
More informationAn 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 informationEuler s Method for Solving Initial Value Problems in Ordinary Differential Equations.
Eulr s Mthod for Solvig Iitial Valu Problms i Ordiar Diffrtial Equatios. Suda Fadugba, M.Sc. * ; Bosd Ogurid, Ph.D. ; ad Tao Okulola, M.Sc. 3 Dpartmt of Mathmatical ad Phsical Scics, Af Babalola Uivrsit,
More informationЭлектронный архив УГЛТУ ЭКО-ПОТЕНЦИАЛ 2 (14),
УДК 004.93'; 004.93 Электронный архив УГЛТУ ЭКО-ПОТЕНЦИАЛ (4), 06 V. Labuts, I. Artmov, S. Martyugi & E. Osthimr Ural dral Uivrsity, Ykatriburg, Russia Capricat LLC, USA AST RACTIOAL OURIER TRASORMS BASED
More informationFirst 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 informationASSERTION AND REASON
ASSERTION AND REASON Som qustios (Assrtio Rso typ) r giv low. Ech qustio cotis Sttmt (Assrtio) d Sttmt (Rso). Ech qustio hs choics (A), (B), (C) d (D) out of which ONLY ONE is corrct. So slct th corrct
More informationBayesian Test for Lifetime Performance Index of Exponential Distribution under Symmetric Entropy Loss Function
Mathmatics ttrs 08; 4(): 0-4 http://www.scicpublishiggroup.com/j/ml doi: 0.648/j.ml.08040.5 ISSN: 575-503X (Prit); ISSN: 575-5056 (Oli) aysia Tst for iftim Prformac Idx of Expotial Distributio udr Symmtric
More informationCORRECTIONS TO THE WU-SPRUNG POTENTIAL FOR THE RIEMANN ZEROS AND A NEW HAMILTONIAN WHOSE ENERGIES ARE THE PRIME NUMBERS
CORRECTIONS TO THE WU-SPRUNG POTENTIAL FOR THE RIEMANN ZEROS AND A NEW HAMILTONIAN WHOSE ENERGIES ARE THE PRIME NUMBERS Jos Javir Garcia Morta Graduat studt of Physics at th UPV/EHU (Uivrsity of Basqu
More informationBipolar Junction Transistors
ipolar Juctio Trasistors ipolar juctio trasistors (JT) ar activ 3-trmial dvics with aras of applicatios: amplifirs, switch tc. high-powr circuits high-spd logic circuits for high-spd computrs. JT structur:
More informationDerivation of a Predictor of Combination #1 and the MSE for a Predictor of a Position in Two Stage Sampling with Response Error.
Drivatio of a Prdictor of Cobiatio # ad th SE for a Prdictor of a Positio i Two Stag Saplig with Rspos Error troductio Ed Stak W driv th prdictor ad its SE of a prdictor for a rado fuctio corrspodig to
More informationarxiv: v1 [math.fa] 18 Feb 2016
SPECTRAL PROPERTIES OF WEIGHTE COMPOSITION OPERATORS ON THE BLOCH AN IRICHLET SPACES arxiv:60.05805v [math.fa] 8 Fb 06 TE EKLUN, MIKAEL LINSTRÖM, AN PAWE L MLECZKO Abstract. Th spctra of ivrtibl wightd
More information