On a problem of J. de Graaf connected with algebras of unbounded operators de Bruijn, N.G.

Size: px
Start display at page:

Download "On a problem of J. de Graaf connected with algebras of unbounded operators de Bruijn, N.G."

Transcription

1 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 th documt vrsio of this publicatio: A submittd mauscript is th author's vrsio of th articl upo submissio ad bfor pr-rviw. Thr ca b importat diffrcs btw th submittd vrsio ad th official publishd vrsio of rcord. Popl itrstd i th rsarch ar advisd to cotact th author for th fial vrsio of th publicatio, or visit th DOI to th publishr's wbsit. Th fial author vrsio ad th gally proof ar vrsios of th publicatio aftr pr rviw. Th fial publishd vrsio faturs th fial layout of th papr icludig th volum, issu ad pag umbrs. Lik to publicatio Citatio for publishd vrsio (APA): Bruij, d, N. G. (1984). O a problm of J. d Graaf coctd with algbras of uboudd oprators. (Eidhov Uivrsity of Tchology : Dpt of Mathmatics : mmoradum; Vol. 8402). Eidhov: Tchisch Hogschool Eidhov. Gral rights Copyright ad moral rights for th publicatios mad accssibl i th public portal ar rtaid by th authors ad/or othr copyright owrs ad it is a coditio of accssig publicatios that usrs rcogis ad abid by th lgal rquirmts associatd with ths rights. Usrs may dowload ad prit o copy of ay publicatio from th public portal for th purpos of privat study or rsarch. You may ot furthr distribut th matrial or us it for ay profit-makig activity or commrcial gai You may frly distribut th URL idtifyig th publicatio i th public portal? Tak dow policy If you bliv that this documt brachs copyright plas cotact us providig dtails, ad w will rmov accss to th work immdiatly ad ivstigat your claim. Dowload dat: 28. Nov. 2017

2 N 62 TECHNISCHE HOGESCHOOL EINDHOVEN O a problm of J. d Graaf coctd with algbras of uboudd oprators. by N.G. d Bruij Mmoradum Fbruary 1984 Dpartmt of Mathmatics ad Computig Scic, Eidhov Uivrsity of Tchology, PO BOX 513, 5600 MB Eidhov, Th Nthrlads.

3 O a problm of J. d Graaf, coctd with algbras of uboudd oprators by N.G. d Bruij I coctio with his work o uboudd oprators my collagu J. d Graaf raisd th followig qustio. Tak N = {I,2,.. }, ad assum x ~ 0 (m, N), V E > o "10 > 0 E m, N x -E-om < 00 (1) x mil +E-om = co (2) Do thr xist ral umbrs S (m, E N) such that S ~ 0 (m, N) ; "IE> > 0 ( m, E N -E+om < (0) (3) L m, N 00? (4) Th aswr 1S gativ. W ca giv th followig coutrxampl: 2 x = xp (- (-) ). +m (5) W shall show (1) ad (2), ad that for vry doubl squc {S} with S ~ 0 such that (3) is satisfid, w also hav

4 2. Sic a ~ 1~ (1) is trivial. W xt show (2). Tak ay E > O. If w succd i fidig 0 such that 0 > 0 ad /(+m) + E - om > 0 (6 ) for ifiitly may pairs (m,), th (2) is obvious. 1 3/2 W tak =(~) Thr ar ifiitly may pairs (m,) with m, ~ ad OE}-! -1 < ~ < (!E)-! (7) (it is irrlvat whthr th lft-had sid is or is ot positiv). If m ad satisfy (7) w hav (~)2 <.f. +m 2 ad (6) follows. W ow tak ay doubl squc {13 } of o-gativ umbrs such that (3) holds. W prov (4) by showig that both /. ad m, E E al3 < 00 ~, / (+m) < P E a 13 < 00, IIlIl m, E ~, / (m+) 2: P (8) (9) whr p has b chos as follows. By (3), with ; Y > 0 such that B 'to.t m, E ~~ -+ym < 00, = 1, thr xists ( 10) W fix p by y p =! l+y ( 11) I th squl w shall us th abbrviatio +m =, so m = 1. For th pairs (m,) occurrig l. (8) w hav 0 < <! II y, ad thrfor 1 y(g - 1),

5 3. whc! y (+m) < - + ym. Sic Cl S 1 ' w ow hav by (10) : m, h(+mj a < 00. E: N, l (+m) < E W ow gt (8), sic r!y(+m) covrgs. W fially prov (9). W tak ay E with 0 < E < p2, ad w put p(p2_e)=q. By (3) w hav m, E::N!1I1'l - < 00 For th pairs (m,) occurrig ~ (9 ) w hav p S 0 S 1, ad thrfor S _ 0 2 S S = p - E - 0 S - E. 0 P It follows that q(m+) 2 (+m)!il - E', whc m, E: q(m+) -E Cl i3 S m < 00. m N, l (+m) ~ p m, E: N Sic -q(m+) r covrgs, this provs (9).

Chapter 2 Infinite Series Page 1 of 11. Chapter 2 : Infinite Series

Chapter 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 information

A Simple Proof that e is Irrational

A 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 information

1985 AP Calculus BC: Section I

1985 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 information

PURE MATHEMATICS A-LEVEL PAPER 1

PURE 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 information

z 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

z 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 information

STIRLING'S 1 FORMULA AND ITS APPLICATION

STIRLING'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 information

H2 Mathematics Arithmetic & Geometric Series ( )

H2 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 information

Triple Play: From De Morgan to Stirling To Euler to Maclaurin to Stirling

Triple 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 information

NEW VERSION OF SZEGED INDEX AND ITS COMPUTATION FOR SOME NANOTUBES

NEW 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 information

A note on non-periodic tilings of the plane

A note on non-periodic tilings of the plane A note on non-periodic tilings of the plane de Bruijn, N.G. Published: 01/01/1985 Document Version Publisher s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please

More information

07 - SEQUENCES AND SERIES Page 1 ( Answers at he end of all questions ) b, z = n

07 - 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 information

The influence of the carriage speed speed on the compliance of the tool-holder Kals, H.J.J.; Hoogenboom, A.J.

The influence of the carriage speed speed on the compliance of the tool-holder Kals, H.J.J.; Hoogenboom, A.J. The ifluece of the carriage speed speed o the compliace of the tool-holder Kals, HJJ; Hoogeboom, AJ Published: 111969 Documet Versio Publisher s PDF, also kow as Versio of Record (icludes fial page, issue

More information

On the approximation of the constant of Napier

On 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 information

Law of large numbers

Law 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 information

Washington State University

Washington State University he 3 Ktics ad Ractor Dsig Sprg, 00 Washgto Stat Uivrsity Dpartmt of hmical Egrg Richard L. Zollars Exam # You will hav o hour (60 muts) to complt this xam which cosists of four (4) problms. You may us

More information

Asymptotics of a nonlinear delay differential equation

Asymptotics of a nonlinear delay differential equation Asymptotics of a oliear delay differetial equatio Brads, J.J.A.M. Published: 01/01/1979 Documet Versio Publisher s PDF, also kow as Versio of Record (icludes fial page, issue ad volume umbers) Please check

More information

Solution to 1223 The Evil Warden.

Solution to 1223 The Evil Warden. Solutio to 1 Th Evil Ward. This is o of thos vry rar PoWs (I caot thik of aothr cas) that o o solvd. About 10 of you submittd th basic approach, which givs a probability of 47%. I was shockd wh I foud

More information

Worksheet: Taylor Series, Lagrange Error Bound ilearnmath.net

Worksheet: 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 information

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

Narayana IIT Academy

Narayana IIT Academy INDIA Sc: LT-IIT-SPARK Dat: 9--8 6_P Max.Mars: 86 KEY SHEET PHYSIS A 5 D 6 7 A,B 8 B,D 9 A,B A,,D A,B, A,B B, A,B 5 A 6 D 7 8 A HEMISTRY 9 A B D B B 5 A,B,,D 6 A,,D 7 B,,D 8 A,B,,D 9 A,B, A,B, A,B,,D A,B,

More information

Lectures 9 IIR Systems: First Order System

Lectures 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 information

Fooling Newton s Method a) Find a formula for the Newton sequence, and verify that it converges to a nonzero of f. A Stirling-like Inequality

Fooling Newton s Method a) Find a formula for the Newton sequence, and verify that it converges to a nonzero of f. A Stirling-like Inequality Foolig Nwto s Mthod a Fid a formla for th Nwto sqc, ad vrify that it covrgs to a ozro of f. ( si si + cos 4 4 3 4 8 8 bt f. b Fid a formla for f ( ad dtrmi its bhavior as. f ( cos si + as A Stirlig-li

More information

Review Exercises. 1. Evaluate using the definition of the definite integral as a Riemann Sum. Does the answer represent an area? 2

Review 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 information

Chapter (8) Estimation and Confedence Intervals Examples

Chapter (8) Estimation and Confedence Intervals Examples Chaptr (8) Estimatio ad Cofdc Itrvals Exampls Typs of stimatio: i. Poit stimatio: Exampl (1): Cosidr th sampl obsrvatios, 17,3,5,1,18,6,16,10 8 X i i1 17 3 5 118 6 16 10 116 X 14.5 8 8 8 14.5 is a poit

More information

Chapter Taylor Theorem Revisited

Chapter 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 information

Differentiation of Exponential Functions

Differentiation of Exponential Functions Calculus Modul C Diffrntiation of Eponntial Functions Copyright This publication Th Northrn Albrta Institut of Tchnology 007. All Rights Rsrvd. LAST REVISED March, 009 Introduction to Diffrntiation of

More information

Brief Introduction to Statistical Mechanics

Brief Introduction to Statistical Mechanics Brif Introduction to Statistical Mchanics. Purpos: Ths nots ar intndd to provid a vry quick introduction to Statistical Mchanics. Th fild is of cours far mor vast than could b containd in ths fw pags.

More information

Restricted Factorial And A Remark On The Reduced Residue Classes

Restricted 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 information

Chapter Five. More Dimensions. is simply the set of all ordered n-tuples of real numbers x = ( x 1

Chapter 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 information

On a set of diophantine equations

On a set of diophantine equations On a set of diophantine equations Citation for published version (APA): van Lint, J. H. (1968). On a set of diophantine equations. (EUT report. WSK, Dept. of Mathematics and Computing Science; Vol. 68-WSK-03).

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

State-space behaviours 2 using eigenvalues

State-space behaviours 2 using eigenvalues 1 Stat-spac bhaviours 2 using ignvalus J A Rossitr Slids by Anthony Rossitr Introduction Th first vido dmonstratd that on can solv 2 x x( ( x(0) Th stat transition matrix Φ( can b computd using Laplac

More information

1973 AP Calculus BC: Section I

1973 AP Calculus BC: Section I 97 AP Calculus BC: Scio I 9 Mius No Calculaor No: I his amiaio, l dos h aural logarihm of (ha is, logarihm o h bas ).. If f ( ) =, h f ( ) = ( ). ( ) + d = 7 6. If f( ) = +, h h s of valus for which f

More information

A GENERALIZED RAMANUJAN-NAGELL EQUATION RELATED TO CERTAIN STRONGLY REGULAR GRAPHS

A 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 information

Hadamard Exponential Hankel Matrix, Its Eigenvalues and Some Norms

Hadamard 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 information

CDS 101: Lecture 5.1 Reachability and State Space Feedback

CDS 101: Lecture 5.1 Reachability and State Space Feedback CDS, Lctur 5. CDS : Lctur 5. Rachability ad Stat Spac Fdback Richard M. Murray 7 Octobr 3 Goals: Di rachability o a cotrol systm Giv tsts or rachability o liar systms ad apply to ampls Dscrib th dsig o

More information

Option 3. b) xe dx = and therefore the series is convergent. 12 a) Divergent b) Convergent Proof 15 For. p = 1 1so the series diverges.

Option 3. b) xe dx = and therefore the series is convergent. 12 a) Divergent b) Convergent Proof 15 For. p = 1 1so the series diverges. Optio Chaptr Ercis. Covrgs to Covrgs to Covrgs to Divrgs Covrgs to Covrgs to Divrgs 8 Divrgs Covrgs to Covrgs to Divrgs Covrgs to Covrgs to Covrgs to Covrgs to 8 Proof Covrgs to π l 8 l a b Divrgt π Divrgt

More information

Discrete Fourier Transform (DFT)

Discrete 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 information

The M/G/1 FIFO queue with several customer classes

The M/G/1 FIFO queue with several customer classes The M/G/1 FIFO queue with several customer classes Boxma, O.J.; Takine, T. Published: 01/01/2003 Document Version Publisher s PDF, also known as Version of Record (includes final page, issue and volume

More information

Nyquist pulse shaping criterion for two-dimensional systems

Nyquist pulse shaping criterion for two-dimensional systems Nyquist pulse shapig criterio for two-dimesioal systems Citatio for published versio (APA): Yag, H., Padharipade, A., & Bergmas, J. W. M. (8). Nyquist pulse shapig criterio for two-dimesioal systems. I

More information

APPENDIX: STATISTICAL TOOLS

APPENDIX: 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 information

CDS 101: Lecture 5.1 Reachability and State Space Feedback

CDS 101: Lecture 5.1 Reachability and State Space Feedback CDS, Lctur 5. CDS : Lctur 5. Rachability ad Stat Spac Fdback Richard M. Murray ad Hido Mabuchi 5 Octobr 4 Goals: Di rachability o a cotrol systm Giv tsts or rachability o liar systms ad apply to ampls

More information

Ordinary Differential Equations

Ordinary 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 information

Class #24 Monday, April 16, φ φ φ

Class #24 Monday, April 16, φ φ φ lass #4 Moday, April 6, 08 haptr 3: Partial Diffrtial Equatios (PDE s First of all, this sctio is vry, vry difficult. But it s also supr cool. PDE s thr is mor tha o idpdt variabl. Exampl: φ φ φ φ = 0

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

Thomas J. Osler. 1. INTRODUCTION. This paper gives another proof for the remarkable simple

Thomas 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 information

Equation Sheet Please tear off this page and keep it with you

Equation Sheet Please tear off this page and keep it with you ECE 30L, Exam Fall 05 Equatio Sht Plas tar off this ag ad k it with you Gral Smicoductor: 0 i ( EF EFi ) kt 0 i ( EFi EF ) kt Eg i N C NV kt 0 0 V IR L, D, τ, d d τ c g L D kt I J diff D D µ * m σ (µ +

More information

Differential Equations

Differential Equations Prfac Hr ar m onlin nots for m diffrntial quations cours that I tach hr at Lamar Univrsit. Dspit th fact that ths ar m class nots, th should b accssibl to anon wanting to larn how to solv diffrntial quations

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

MONTGOMERY 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 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 information

A Review of Complex Arithmetic

A 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 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

Probability & Statistics,

Probability & Statistics, Probability & Statistics, BITS Pilai K K Birla Goa Campus Dr. Jajati Kshari Sahoo Dpartmt of Mathmatics BITS Pilai, K K Birla Goa Campus Poisso Distributio Poisso Distributio: A radom variabl X is said

More information

(1) Then we could wave our hands over this and it would become:

(1) Then we could wave our hands over this and it would become: MAT* K285 Spring 28 Anthony Bnoit 4/17/28 Wk 12: Laplac Tranform Rading: Kohlr & Johnon, Chaptr 5 to p. 35 HW: 5.1: 3, 7, 1*, 19 5.2: 1, 5*, 13*, 19, 45* 5.3: 1, 11*, 19 * Pla writ-up th problm natly and

More information

Pier Franz Roggero, Michele Nardelli, Francesco Di Noto

Pier 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 information

Time : 1 hr. Test Paper 08 Date 04/01/15 Batch - R Marks : 120

Time : 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 information

Examples and applications on SSSP and MST

Examples and applications on SSSP and MST Exampls an applications on SSSP an MST Dan (Doris) H & Junhao Gan ITEE Univrsity of Qunslan COMP3506/7505, Uni of Qunslan Exampls an applications on SSSP an MST Dijkstra s Algorithm Th algorithm solvs

More information

BOUNDS FOR THE COMPONENTWISE DISTANCE TO THE NEAREST SINGULAR MATRIX

BOUNDS FOR THE COMPONENTWISE DISTANCE TO THE NEAREST SINGULAR MATRIX SIAM J. Matrix Aal. Appl. (SIMAX), 8():83 03, 997 BOUNDS FOR THE COMPONENTWISE DISTANCE TO THE NEAREST SINGULAR MATRIX S. M. RUMP Abstract. Th ormwis distac of a matrix A to th arst sigular matrix is wll

More information

SOME IDENTITIES FOR THE GENERALIZED POLY-GENOCCHI POLYNOMIALS WITH THE PARAMETERS A, B AND C

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 information

Department of Mathematics and Statistics Indian Institute of Technology Kanpur MSO202A/MSO202 Assignment 3 Solutions Introduction To Complex Analysis

Department of Mathematics and Statistics Indian Institute of Technology Kanpur MSO202A/MSO202 Assignment 3 Solutions Introduction To Complex Analysis Dpartmt of Mathmatcs ad Statstcs Ida Isttut of Tchology Kapur MSOA/MSO Assgmt 3 Solutos Itroducto To omplx Aalyss Th problms markd (T) d a xplct dscusso th tutoral class. Othr problms ar for hacd practc..

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

Folding of Hyperbolic Manifolds

Folding 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 information

Lecture 37 (Schrödinger Equation) Physics Spring 2018 Douglas Fields

Lecture 37 (Schrödinger Equation) Physics Spring 2018 Douglas Fields Lctur 37 (Schrödingr Equation) Physics 6-01 Spring 018 Douglas Filds Rducd Mass OK, so th Bohr modl of th atom givs nrgy lvls: E n 1 k m n 4 But, this has on problm it was dvlopd assuming th acclration

More information

A Simple Formula for the Hilbert Metric with Respect to a Sub-Gaussian Cone

A Simple Formula for the Hilbert Metric with Respect to a Sub-Gaussian Cone mathmatics Articl A Simpl Formula for th Hilbrt Mtric with Rspct to a Sub-Gaussian Con Stéphan Chrétin 1, * and Juan-Pablo Ortga 2 1 National Physical Laboratory, Hampton Road, Tddinton TW11 0LW, UK 2

More information

Where k is either given or determined from the data and c is an arbitrary constant.

Where k is either given or determined from the data and c is an arbitrary constant. Exponntial growth and dcay applications W wish to solv an quation that has a drivativ. dy ky k > dx This quation says that th rat of chang of th function is proportional to th function. Th solution is

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

International Journal of Advanced and Applied Sciences

International Journal of Advanced and Applied Sciences 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: http://wwwscic gatcom/ijaashtml Symmtric Fuctios

More information

Further Results on Pair Sum Graphs

Further 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 information

10. Joint Moments and Joint Characteristic Functions

10. 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 information

2617 Mark Scheme June 2005 Mark Scheme 2617 June 2005

2617 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 information

Character sums over generalized Lehmer numbers

Character 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 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

Ch. 24 Molecular Reaction Dynamics 1. Collision Theory

Ch. 24 Molecular Reaction Dynamics 1. Collision Theory Ch. 4 Molcular Raction Dynamics 1. Collision Thory Lctur 16. Diffusion-Controlld Raction 3. Th Matrial Balanc Equation 4. Transition Stat Thory: Th Eyring Equation 5. Transition Stat Thory: Thrmodynamic

More information

The Interplay between l-max, l-min, p-max and p-min Stable Distributions

The 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 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

n n ee (for index notation practice, see if you can verify the derivation given in class)

n n ee (for index notation practice, see if you can verify the derivation given in class) EN24: Computatioal mthods i Structural ad Solid Mchaics Homwork 6: Noliar matrials Du Wd Oct 2, 205 School of Egirig Brow Uivrsity I this homwork you will xtd EN24FEA to solv problms for oliar matrials.

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

NET/JRF, GATE, IIT JAM, JEST, TIFR

NET/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 information

Function Spaces. a x 3. (Letting x = 1 =)) a(0) + b + c (1) = 0. Row reducing the matrix. b 1. e 4 3. e 9. >: (x = 1 =)) a(0) + b + c (1) = 0

Function Spaces. a x 3. (Letting x = 1 =)) a(0) + b + c (1) = 0. Row reducing the matrix. b 1. e 4 3. e 9. >: (x = 1 =)) a(0) + b + c (1) = 0 unction Spacs Prrquisit: Sction 4.7, Coordinatization n this sction, w apply th tchniqus of Chaptr 4 to vctor spacs whos lmnts ar functions. Th vctor spacs P n and P ar familiar xampls of such spacs. Othr

More information

Counting the compositions of a positive integer n using Generating Functions Start with, 1. x = 3 ), the number of compositions of 4.

Counting the compositions of a positive integer n using Generating Functions Start with, 1. x = 3 ), the number of compositions of 4. Coutg th compostos of a postv tgr usg Gratg Fuctos Start wth,... - Whr, for ampl, th co-ff of s, for o summad composto of aml,. To obta umbr of compostos of, w d th co-ff of (...) ( ) ( ) Hr for stac w

More information

COLLECTION OF SUPPLEMENTARY PROBLEMS CALCULUS II

COLLECTION 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 information

5.1 The Nuclear Atom

5.1 The Nuclear Atom Sav My Exams! Th Hom of Rvisio For mor awsom GSE ad lvl rsourcs, visit us at www.savmyxams.co.uk/ 5.1 Th Nuclar tom Qustio Papr Lvl IGSE Subjct Physics (0625) Exam oard Topic Sub Topic ooklt ambridg Itratioal

More information

(Reference: sections in Silberberg 5 th ed.)

(Reference: sections in Silberberg 5 th ed.) ALE. Atomic Structur Nam HEM K. Marr Tam No. Sctio What is a atom? What is th structur of a atom? Th Modl th structur of a atom (Rfrc: sctios.4 -. i Silbrbrg 5 th d.) Th subatomic articls that chmists

More information

Calculus concepts derivatives

Calculus concepts derivatives All rasonabl fforts hav bn mad to mak sur th nots ar accurat. Th author cannot b hld rsponsibl for any damags arising from th us of ths nots in any fashion. Calculus concpts drivativs Concpts involving

More information

SOLVED EXAMPLES. Ex.1 If f(x) = , then. is equal to- Ex.5. f(x) equals - (A) 2 (B) 1/2 (C) 0 (D) 1 (A) 1 (B) 2. (D) Does not exist = [2(1 h)+1]= 3

SOLVED EXAMPLES. Ex.1 If f(x) = , then. is equal to- Ex.5. f(x) equals - (A) 2 (B) 1/2 (C) 0 (D) 1 (A) 1 (B) 2. (D) Does not exist = [2(1 h)+1]= 3 SOLVED EXAMPLES E. If f() E.,,, th f() f() h h LHL RHL, so / / Lim f() quls - (D) Dos ot ist [( h)+] [(+h) + ] f(). LHL E. RHL h h h / h / h / h / h / h / h As.[C] (D) Dos ot ist LHL RHL, so giv it dos

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

Document Version Publisher s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Document Version Publisher s PDF, also known as Version of Record (includes final page, issue and volume numbers) Magnetic susceptibility of PbSnMnTe in the transition region between ferromagnetic and spin glass phase Story, T.; Galazka, R.R.; Eggenkamp, Paul; Swagten, H.J.M.; de Jonge, W.J.M. Published in: Acta Physica

More information

Traveling Salesperson Problem and Neural Networks. A Complete Algorithm in Matrix Form

Traveling 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 information

Addition of angular momentum

Addition of angular momentum Addition of angular momntum April, 07 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat

More information

Addition of angular momentum

Addition of angular momentum Addition of angular momntum April, 0 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat th

More information

Computing and Communications -- Network Coding

Computing and Communications -- Network Coding 89 90 98 00 Computing and Communications -- Ntwork Coding Dr. Zhiyong Chn Institut of Wirlss Communications Tchnology Shanghai Jiao Tong Univrsity China Lctur 5- Nov. 05 0 Classical Information Thory Sourc

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

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

3 2x. 3x 2. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

3 2x. 3x 2.   Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Math B Intgration Rviw (Solutions) Do ths intgrals. Solutions ar postd at th wbsit blow. If you hav troubl with thm, sk hlp immdiatly! () 8 d () 5 d () d () sin d (5) d (6) cos d (7) d www.clas.ucsb.du/staff/vinc

More information

The calculation of stable cutting conditions in turning

The calculation of stable cutting conditions in turning The calculation of stable cutting conditions in turning Citation for published version (APA): Kals, H. J. J. (1972). The calculation of stable cutting conditions in turning. (TH Eindhoven. Afd. Werktuigbouwkunde,

More information

Document Version Publisher s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Document Version Publisher s PDF, also known as Version of Record (includes final page, issue and volume numbers) Effect of molar mass ratio of monomers on the mass distribution of chain lengths and compositions in copolymers: extension of the Stockmayer theory Tacx, J.C.J.F.; Linssen, H.N.; German, A.L. Published

More information

Multi (building)physics modeling

Multi (building)physics modeling Multi (building)physics modeling van Schijndel, A.W.M. Published: 01/01/2010 Document Version Publisher s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check

More information

Math 34A. Final Review

Math 34A. Final Review Math A Final Rviw 1) Us th graph of y10 to find approimat valus: a) 50 0. b) y (0.65) solution for part a) first writ an quation: 50 0. now tak th logarithm of both sids: log() log(50 0. ) pand th right

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

Discrete Mathematics and Probability Theory Fall 2014 Anant Sahai Homework 11. This homework is due November 17, 2014, at 12:00 noon.

Discrete Mathematics and Probability Theory Fall 2014 Anant Sahai Homework 11. This homework is due November 17, 2014, at 12:00 noon. EECS 70 Discrt Mathmatics ad Probability Thory Fall 2014 Aat Sahai Homwork 11 This homwork is du Novmbr 17, 2014, at 12:00 oo. 1. Sctio Rollcall! I your slf-gradig for this qustio, giv yourslf a 10, ad

More information