Reconstruction of mass spectra on the basis of partial pressures

Size: px
Start display at page:

Download "Reconstruction of mass spectra on the basis of partial pressures"

Transcription

1 Tutoril: Residul Gs Alysis Reostrutio of mss spetr o the sis of prtil pressures olo Chiggito & Berthold Jeiger 06-16/06/017 CER Aelertor Shool 017 1

2 Tutoril: Residul Gs Alysis Reostrutio of mss spetr o the sis of prtil pressures Why reostrutig? - Visul: Alyser resolutio oise offset d rtefts e iluded - RGA re OT idel istrumets: High uertities i gs the speifi sesitivities d frgmettio. Reostrutio y ssimiltio to mesured spetrum is ltertive to solvig (y deovolutio) for prtil pressures. - Simultio llows triig of persol o my ses without eed for mesuremets. - Chek for suspeted otmits y ddig frgmettio ptter of suspeted ompoet to lirry - Coviig if residul gs ompositio is well idetified. Disussios o epte e more speifi /06/017 CER Aelertor Shool 017

3 Tutoril: Residul Gs Alysis Reostrutio of mss spetr o the sis of prtil pressures Some defiitios used here Sesitivity io guge: e roportiolity ftor of the mesured hde i olletor urret s retio to hge i the prtil pressure of gs speies divided y the eletro emissio urret e. Sesitivity residul gs lyser RGA: M roportiolity ftor of the mesured detetor urret t mss M s retio to hge i the prtil pressure of gs speies. f M is ot idited it refers ot mss with mximum itesity i frgmettio ptter. ormlised or reltive sesitivity: Sesitivity of o guge to gs speies with respet to the sesitivity for itroge (here) Sesitivity of RGA to gs speies with respet to the sesitivity for itroge (here) 06-16/06/017 CER Aelertor Shool 017 3

4 Tutoril: Residul Gs Alysis Reostrutio of mss spetr o the sis of prtil pressures Mss spetrum: Mesuremet of o detetor urret s futi of m/ o (detetor) urrets t mss m [ A ] Cotriutio of ompoet to io urret sigl t mss m Sesitivity of lyser to referee pek [ A / mr ] rtil pressure [ mr ] m m m m... m C 8 = 1 C 8 CO = 0.1 C 8 CH4 = 0 CO CH 4 m/ m/ m/ /06/017 CER Aelertor Shool 017 4

5 Compositio of gses Mss spetrum Frgmettio ptter Reltive height to referee pek Sesitivity to referee pek rtil pressure Equtios for ll msses 1... Reostrutio of mss spetr o the sis of prtil pressures Tutoril: Residul Gs Alysis 06-16/06/017 CER Aelertor Shool 017 5

6 Frgmettio ptter mtrix C Sesitivity mtrix α rtil pressur vetor o urret vetor Ԧ C Whole spetrum expressed i simple mthemtil form usig mtries d vetors. Equtio for spetr simultio with C s frgmettio ptter lirry d prtil pressures i vuum system. Reostrutio of mss spetr o the sis of prtil pressures Tutoril: Residul Gs Alysis 06-16/06/017 CER Aelertor Shool 017 6

7 Tutoril: Residul Gs Alysis Reostrutio of mss spetr o the sis of prtil pressures Br grph spetrum Alogue spetrum o urret rtil pressures Gs speifi sesitivities Referee spetr lirry (frgmettio ptters) FWMH = 0.5 Assimiltio of mesured logue spetrum with reostruted simulted oe d ross-hek sum of prtil pressures with io guge redig 06-16/06/017 CER Aelertor Shool 017 7

8 Tutoril: Residul Gs Alysis Reostrutio of mss spetr o the sis of prtil pressures Cross-hek: Compriso with ioitio guge Colletor urret is sum of the otriutio of ll prtil pressures multiplied y their respetive sesitivity e e equivlet pressure iditio... 1 eq tot 1... _ RGA Cross-hek RGA e rtil pressure of speies BA-guge olletor urret BA-guge emissio urret Sesitivity of BA guge to speies [mr -1 ] Reltive sesitivity of BA guge to with respet to 06-16/06/017 CER Aelertor Shool 017 8

9 Tutoril: Residul Gs Alysis Reostrutio of mss spetr o the sis of prtil pressures Rel. sesitivity to QMS for UHV pplitios QMA15 (QMG4) QMA15 (HiqhQud) rism VGQ Hide H He CH4 Ar Kr ormlised sesitivity for differet qudrupole mss spetrometers (QMS) t ftory settigs -> High dispersio t low msses Oly iditive Comprig reltive sesitivities with mss Opposite to io guges oistio ross-setio + extrtio + trsmissio proility RGA: Sesitivity to referee peks of frgmettio ptters o guges ollets ll frgmets Whe omprig with io guges sum up ll frgmets of residul gs ompoet o guges Gs f He 0.18 e 0.3 Ar 1.9 Kr 1.94 Xe.87 H O 1.01 Air 1 CO 1.05 H O 1.1 CO 1.4 CH C H 6.6 C 3 H 8 4. Uertity < 0 % Rtio mily determied y ioistio ross-setio 06-16/06/017 CER Aelertor Shool 017 9

10 Tutoril: Residul Gs Alysis Reostrutio of mss spetr o the sis of prtil pressures Exmple of reostrutio of mesured with simulted spetrum F-Cotmitio with CHF3 (Fluoroform) Cofirmed ESD peks O d F 06-16/06/017 CER Aelertor Shool

11 Tutoril: Residul Gs Alysis Reostrutio of mss spetr o the sis of prtil pressures terprettio of the mss spetrum 06-16/06/017 CER Aelertor Shool

12 Tutoril: Residul Gs Alysis Reostrutio of mss spetr o the sis of prtil pressures Mesured / reostruted mss spetrum F-Cotmitio ESD peks O d F. Remiig tres of Fe d Cr just fter degssig d depositio o hot filmet. Cr d Fe dispperd fter severl hours of opertio. Doule-ioistio COF 47 -> 3.5 (ot simulted) sotope 13 COF (ot simulted) (B. Jeiger CER TE-VSC) SEM_Multipl Sesitivity Ar (FAR) [A/mr] 1.00E-04 Bkgroud dist 4.00E-14 FWHM 0.5 Bkgroud offs 1.0E-13 i Gs S [A/mr] [mr] Hydroge H 1.00E E-11 Helium He 1.00E+00.00E-13 Methe CH4 1.00E E-1 Wter HO 1.00E E-1 Cromooxide CO 1.00E E-11 Ethe CH6 1.00E E-13 Crodioxide CO 1.00E+00.00E-1 HydrogeFluoride HF 1.00E+00.00E-1 Ferrum Fe 1.00E E-13 Chromium Cr 1.00E+00.00E-13 Oxyge (tomi) O 1.00E E-1 Fluorie(tomi) F 1.00E E-11 TrifluoroAhydridAetiAid C4O3F6 1.00E E-13 Croi Difluoride COF 1.00E E-1 erfluororope C3F6 1.00E E-1 _tot = 1.7E-10 Spetrum e fully explied with presee of Fluorie + kow ompositio i hot io soure (C O H Fe Cr) 06-16/06/017 CER Aelertor Shool 017 1

13 Tutoril: Residul Gs Alysis Reostrutio of mss spetr o the sis of prtil pressures SO 1491: Vuum guges Defiitios d speifitios for qudrupole mss spetrometers SO/ TS 0175 (i preprtio): Vuum tehology -- Vuum guges - Clirtio of qudrupole mss spetrometers for prtil pressure mesuremet SO/ TS 0177 (i preprtio): Vuum tehology - Vuum guges -- roedures to mesure d report outgssig rtes /06/017 CER Aelertor Shool

14 Tutoril: Residul Gs Alysis Reostrutio of mss spetr o the sis of prtil pressures OW Exerise: Reostrutig mesured mss spetrum 06-16/06/017 CER Aelertor Shool

Project 3: Using Identities to Rewrite Expressions

Project 3: Using Identities to Rewrite Expressions MAT 5 Projet 3: Usig Idetities to Rewrite Expressios Wldis I lger, equtios tht desrie properties or ptters re ofte lled idetities. Idetities desrie expressio e repled with equl or equivlet expressio tht

More information

Riemann Integral Oct 31, such that

Riemann Integral Oct 31, such that Riem Itegrl Ot 31, 2007 Itegrtio of Step Futios A prtitio P of [, ] is olletio {x k } k=0 suh tht = x 0 < x 1 < < x 1 < x =. More suitly, prtitio is fiite suset of [, ] otiig d. It is helpful to thik of

More information

Section 11.5 Notes Page Partial Fraction Decomposition. . You will get: +. Therefore we come to the following: x x

Section 11.5 Notes Page Partial Fraction Decomposition. . You will get: +. Therefore we come to the following: x x Setio Notes Pge Prtil Frtio Deompositio Suppose we were sked to write the followig s sigle frtio: We would eed to get ommo deomitors: You will get: Distributig o top will give you: 8 This simplifies to:

More information

Section 6.3: Geometric Sequences

Section 6.3: Geometric Sequences 40 Chpter 6 Sectio 6.: Geometric Sequeces My jobs offer ul cost-of-livig icrese to keep slries cosistet with ifltio. Suppose, for exmple, recet college grdute fids positio s sles mger erig ul slry of $6,000.

More information

EXPONENTS AND LOGARITHMS

EXPONENTS AND LOGARITHMS 978--07-6- Mthemtis Stdrd Level for IB Diplom Eerpt EXPONENTS AND LOGARITHMS WHAT YOU NEED TO KNOW The rules of epoets: m = m+ m = m ( m ) = m m m = = () = The reltioship etwee epoets d rithms: = g where

More information

Handout #2. Introduction to Matrix: Matrix operations & Geometric meaning

Handout #2. Introduction to Matrix: Matrix operations & Geometric meaning Hdout # Title: FAE Course: Eco 8/ Sprig/5 Istructor: Dr I-Mig Chiu Itroductio to Mtrix: Mtrix opertios & Geometric meig Mtrix: rectgulr rry of umers eclosed i pretheses or squre rckets It is covetiolly

More information

Fig. 1. I a. V ag I c. I n. V cg. Z n Z Y. I b. V bg

Fig. 1. I a. V ag I c. I n. V cg. Z n Z Y. I b. V bg ymmetricl Compoets equece impedces Although the followig focuses o lods, the results pply eqully well to lies, or lies d lods. Red these otes together with sectios.6 d.9 of text. Cosider the -coected lced

More information

Lens Design II. Lecture 7: Chromatical correction II Herbert Gross. Winter term

Lens Design II. Lecture 7: Chromatical correction II Herbert Gross. Winter term Les Desig II Leture 7: Chromtil orretio II 05--0 Herbert Gross Witer term 05 www.ip.ui-e.de relimiry Shedule 0.0. Aberrtios d optimiztio Repetitio 7.0. Struturl modifitios Zero operds, les splittig, les

More information

Section 7.3, Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors (the variable vector of the system) and

Section 7.3, Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors (the variable vector of the system) and Sec. 7., Boyce & DiPrim, p. Sectio 7., Systems of Lier Algeric Equtios; Lier Idepedece, Eigevlues, Eigevectors I. Systems of Lier Algeric Equtios.. We c represet the system...... usig mtrices d vectors

More information

Section 2.2. Matrix Multiplication

Section 2.2. Matrix Multiplication Mtri Alger Mtri Multiplitio Setio.. Mtri Multiplitio Mtri multiplitio is little more omplite th mtri itio or slr multiplitio. If A is the prout A of A is the ompute s follow: m mtri, the is k mtri, 9 m

More information

Lecture 4 Recursive Algorithm Analysis. Merge Sort Solving Recurrences The Master Theorem

Lecture 4 Recursive Algorithm Analysis. Merge Sort Solving Recurrences The Master Theorem Lecture 4 Recursive Algorithm Alysis Merge Sort Solvig Recurreces The Mster Theorem Merge Sort MergeSortA, left, right) { if left < right) { mid = floorleft + right) / 2); MergeSortA, left, mid); MergeSortA,

More information

MAS221 Analysis, Semester 2 Exercises

MAS221 Analysis, Semester 2 Exercises MAS22 Alysis, Semester 2 Exercises Srh Whitehouse (Exercises lbelled * my be more demdig.) Chpter Problems: Revisio Questio () Stte the defiitio of covergece of sequece of rel umbers, ( ), to limit. (b)

More information

Surds, Indices, and Logarithms Radical

Surds, Indices, and Logarithms Radical MAT 6 Surds, Idices, d Logrithms Rdicl Defiitio of the Rdicl For ll rel, y > 0, d ll itegers > 0, y if d oly if y where is the ide is the rdicl is the rdicd. Surds A umber which c be epressed s frctio

More information

1/16/2013. Overview. 05-Three Phase Analysis Text: Three Phase. Three-Phase Voltages. Benefits of Three-Phase Systems.

1/16/2013. Overview. 05-Three Phase Analysis Text: Three Phase. Three-Phase Voltages. Benefits of Three-Phase Systems. oltge () 1/16/21 Overview 5Three Phse Alysis Text: 2.4 2.7 ECEGR 451 Power Systems ThreePhse Soures Delt d Y Coetios ThreePhse Lods ThreePhse Power ThreePhse Alysis PerPhse Alysis Dr. Louie 2 ThreePhse

More information

z line a) Draw the single phase equivalent circuit. b) Calculate I BC.

z line a) Draw the single phase equivalent circuit. b) Calculate I BC. ECE 2260 F 08 HW 7 prob 4 solutio EX: V gyb' b' b B V gyc' c' c C = 101 0 V = 1 + j0.2 Ω V gyb' = 101 120 V = 6 + j0. Ω V gyc' = 101 +120 V z LΔ = 9 j1.5 Ω ) Drw the sigle phse equivlet circuit. b) Clculte

More information

Module B3 3.1 Sinusoidal steady-state analysis (single-phase), a review 3.2 Three-phase analysis. Kirtley

Module B3 3.1 Sinusoidal steady-state analysis (single-phase), a review 3.2 Three-phase analysis. Kirtley Module B.1 Siusoidl stedy-stte lysis (sigle-phse), review.2 Three-phse lysis Kirtley Chpter 2: AC Voltge, Curret d Power 2.1 Soures d Power 2.2 Resistors, Idutors, d Cpitors Chpter 4: Polyphse systems

More information

= 47.5 ;! R. = 34.0 ; n air =

= 47.5 ;! R. = 34.0 ; n air = Setio 9: Refratio ad Total Iteral Refletio Tutorial Pratie, page 449 The agle of iidee is 65 The fat that the experimet takes plae i water does ot hage the agle of iidee Give:! i = 475 ;! R = 340 ; air

More information

The Atmosphere. Atmospheric composition Measures of concentration Atmospheric pressure Barometric law

The Atmosphere. Atmospheric composition Measures of concentration Atmospheric pressure Barometric law The Atmosphere Atmospheric compositio Mesures of cocetrtio Atmospheric pressure Brometric lw Literture coected with tody s lecture: Jcob, chpter 1-2 Exercises: 1:1 1:6; 2:1 2:4 The Atmosphere The tmosphere

More information

SPH3UW Unit 7.5 Snell s Law Page 1 of Total Internal Reflection occurs when the incoming refraction angle is

SPH3UW Unit 7.5 Snell s Law Page 1 of Total Internal Reflection occurs when the incoming refraction angle is SPH3UW Uit 7.5 Sell s Lw Pge 1 of 7 Notes Physis Tool ox Refrtio is the hge i diretio of wve due to hge i its speed. This is most ommoly see whe wve psses from oe medium to other. Idex of refrtio lso lled

More information

Chapter System of Equations

Chapter System of Equations hpter 4.5 System of Equtios After redig th chpter, you should be ble to:. setup simulteous lier equtios i mtrix form d vice-vers,. uderstd the cocept of the iverse of mtrix, 3. kow the differece betwee

More information

0 otherwise. sin( nx)sin( kx) 0 otherwise. cos( nx) sin( kx) dx 0 for all integers n, k.

0 otherwise. sin( nx)sin( kx) 0 otherwise. cos( nx) sin( kx) dx 0 for all integers n, k. . Computtio of Fourier Series I this sectio, we compute the Fourier coefficiets, f ( x) cos( x) b si( x) d b, i the Fourier series To do this, we eed the followig result o the orthogolity of the trigoometric

More information

Test Info. Test may change slightly.

Test Info. Test may change slightly. 9. 9.6 Test Ifo Test my chge slightly. Short swer (0 questios 6 poits ech) o Must choose your ow test o Tests my oly be used oce o Tests/types you re resposible for: Geometric (kow sum) Telescopig (kow

More information

AP Calculus BC Formulas, Definitions, Concepts & Theorems to Know

AP Calculus BC Formulas, Definitions, Concepts & Theorems to Know P Clls BC Formls, Deiitios, Coepts & Theorems to Kow Deiitio o e : solte Vle: i 0 i 0 e lim Deiitio o Derivtive: h lim h0 h ltertive orm o De o Derivtive: lim Deiitio o Cotiity: is otios t i oly i lim

More information

LESSON 11: TRANSFORMER NAME PLATE DATA AND CONNECTIONS

LESSON 11: TRANSFORMER NAME PLATE DATA AND CONNECTIONS ET 332 Motors, Geertors d Power Systems LESSON 11: TRNSFORMER NME PLTE DT ND ONNETIONS 1 LERNING OJETIVES fter this presettio you will e le to: Idetify trsformer polrity usig dot d ovetiol lelig. Expli

More information

Section 3.6: Rational Exponents

Section 3.6: Rational Exponents CHAPTER Sectio.6: Rtiol Epoets Sectio.6: Rtiol Epoets Objectives: Covert betwee rdicl ottio d epoetil ottio. Siplif epressios with rtiol epoets usig the properties of epoets. Multipl d divide rdicl epressios

More information

Lens Design II. Lecture 7: Chromatical correction II Herbert Gross. Winter term

Lens Design II. Lecture 7: Chromatical correction II Herbert Gross. Winter term Les Desig II Leture 7: Chromtil orretio II 07--7 Herert Gross Witer term 07 www.ip.ui-e.de relimiry Shedule Les Desig II 07 6.0. Aerrtios d optimiztio Repetitio 3.0. Struturl modiitios Zero operds, les

More information

Chapter 7 Infinite Series

Chapter 7 Infinite Series MA Ifiite Series Asst.Prof.Dr.Supree Liswdi Chpter 7 Ifiite Series Sectio 7. Sequece A sequece c be thought of s list of umbers writte i defiite order:,,...,,... 2 The umber is clled the first term, 2

More information

The limit comparison test

The limit comparison test Roerto s Notes o Ifiite Series Chpter : Covergece tests Sectio 4 The limit compriso test Wht you eed to kow lredy: Bsics of series d direct compriso test. Wht you c ler here: Aother compriso test tht does

More information

FREE Download Study Package from website: &

FREE Download Study Package from website:  & FREE Dolod Study Pkge from esite:.tekolsses.om &.MthsBySuhg.om Get Solutio of These Pkges & Ler y Video Tutorils o.mthsbysuhg.om SHORT REVISION. Defiitio : Retgulr rry of m umers. Ulike determits it hs

More information

5.6 Absolute Convergence and The Ratio and Root Tests

5.6 Absolute Convergence and The Ratio and Root Tests 5.6 Absolute Covergece ad The Ratio ad Root Tests Bria E. Veitch 5.6 Absolute Covergece ad The Ratio ad Root Tests Recall from our previous sectio that diverged but ( ) coverged. Both of these sequeces

More information

10.5 Test Info. Test may change slightly.

10.5 Test Info. Test may change slightly. 0.5 Test Ifo Test my chge slightly. Short swer (0 questios 6 poits ech) o Must choose your ow test o Tests my oly be used oce o Tests/types you re resposible for: Geometric (kow sum) Telescopig (kow sum)

More information

Lens Design II. Lecture 7: Chromatical correction II Herbert Gross. Winter term

Lens Design II. Lecture 7: Chromatical correction II Herbert Gross. Winter term Les Desig II Leture 7: Chromtil orretio II 06--30 Herert Gross Witer term 06 www.ip.ui-e.de relimiry Shedule 9.0. Aerrtios d optimiztio Repetitio 6.0. Struturl modiitios Zero operds, les splittig, les

More information

Exponents and Radical

Exponents and Radical Expoets d Rdil Rule : If the root is eve d iside the rdil is egtive, the the swer is o rel umber, meig tht If is eve d is egtive, the Beuse rel umber multiplied eve times by itself will be lwys positive.

More information

0 x < 5 PIECEWISE FUNCTIONS DAY1 4.7

0 x < 5 PIECEWISE FUNCTIONS DAY1 4.7 PIECEWISE FUNCTIONS DAY 7 GOAL Red fuctios of grphs represeted by more th oe equtio fuctios of grphs represeted by more th oe equtio Grph piecewise fuctios PIECEWISE FUNCTION A fuctio defied by two or

More information

5. Solving recurrences

5. Solving recurrences 5. Solvig recurreces Time Complexity Alysis of Merge Sort T( ) 0 if 1 2T ( / 2) otherwise sortig oth hlves mergig Q. How to prove tht the ru-time of merge sort is O( )? A. 2 Time Complexity Alysis of Merge

More information

Mathematics 116 HWK 21 Solutions 8.2 p580

Mathematics 116 HWK 21 Solutions 8.2 p580 Mathematics 6 HWK Solutios 8. p580 A abbreviatio: iff is a abbreviatio for if ad oly if. Geometric Series: Several of these problems use what we worked out i class cocerig the geometric series, which I

More information

Review of Sections

Review of Sections Review of Sectios.-.6 Mrch 24, 204 Abstrct This is the set of otes tht reviews the mi ides from Chpter coverig sequeces d series. The specific sectios tht we covered re s follows:.: Sequces..2: Series,

More information

Infinite Series Sequences: terms nth term Listing Terms of a Sequence 2 n recursively defined n+1 Pattern Recognition for Sequences Ex:

Infinite Series Sequences: terms nth term Listing Terms of a Sequence 2 n recursively defined n+1 Pattern Recognition for Sequences Ex: Ifiite Series Sequeces: A sequece i defied s fuctio whose domi is the set of positive itegers. Usully it s esier to deote sequece i subscript form rther th fuctio ottio.,, 3, re the terms of the sequece

More information

Name: A2RCC Midterm Review Unit 1: Functions and Relations Know your parent functions!

Name: A2RCC Midterm Review Unit 1: Functions and Relations Know your parent functions! Nme: ARCC Midterm Review Uit 1: Fuctios d Reltios Kow your pret fuctios! 1. The ccompyig grph shows the mout of rdio-ctivity over time. Defiitio of fuctio. Defiitio of 1-1. Which digrm represets oe-to-oe

More information

National Conference on Advances in Mechanical Engineering Science (NCAMES-2016)

National Conference on Advances in Mechanical Engineering Science (NCAMES-2016) Ntiol Coferee o Adves i Mehil Egieerig Siee (NCAMES-2016) Modelig Stepper Motor Cotrol System with Miro-Steppig Exittio Mode Nvy Thirumleshwr, Hegde 1 Dr. S. Meethi Sudrm 2,, Aldri Vz 3 1 Assistt Professor,

More information

Introduction to Algorithms 6.046J/18.401J

Introduction to Algorithms 6.046J/18.401J Itrodutio to Algorithms.04J/8.40J The divide-d-oquer desig prdigm. Divide the problem (iste) ito subproblems.. Coquer the subproblems by solvig them reursively. 3. Combie subproblem solutios. Leture 3

More information

8.3 Sequences & Series: Convergence & Divergence

8.3 Sequences & Series: Convergence & Divergence 8.3 Sequeces & Series: Covergece & Divergece A sequece is simply list of thigs geerted by rule More formlly, sequece is fuctio whose domi is the set of positive itegers, or turl umbers,,,3,. The rge of

More information

Review Topics on Stoichiometry

Review Topics on Stoichiometry Itroductory Chemistry Review Topics o Stoichiometry Ucertity i Mesuremets Ucertity i Derived Qutity 5.55 ± 0.05 +.55 ± 0.05 8.1 ± 0.1 derived result mesuremets 0. + 0.1 10.0 + 0.1 X 0. + 0.5% X = + = 10.0

More information

Similar idea to multiplication in N, C. Divide and conquer approach provides unexpected improvements. Naïve matrix multiplication

Similar idea to multiplication in N, C. Divide and conquer approach provides unexpected improvements. Naïve matrix multiplication Next. Covered bsics of simple desig techique (Divided-coquer) Ch. of the text.. Next, Strsse s lgorithm. Lter: more desig d coquer lgorithms: MergeSort. Solvig recurreces d the Mster Theorem. Similr ide

More information

lecture 16: Introduction to Least Squares Approximation

lecture 16: Introduction to Least Squares Approximation 97 lecture 16: Itroductio to Lest Squres Approximtio.4 Lest squres pproximtio The miimx criterio is ituitive objective for pproximtig fuctio. However, i my cses it is more ppelig (for both computtio d

More information

( a n ) converges or diverges.

( a n ) converges or diverges. Chpter Ifiite Series Pge of Sectio E Rtio Test Chpter : Ifiite Series By the ed of this sectio you will be ble to uderstd the proof of the rtio test test series for covergece by pplyig the rtio test pprecite

More information

POWER SERIES R. E. SHOWALTER

POWER SERIES R. E. SHOWALTER POWER SERIES R. E. SHOWALTER. sequeces We deote by lim = tht the limit of the sequece { } is the umber. By this we me tht for y ε > 0 there is iteger N such tht < ε for ll itegers N. This mkes precise

More information

CS 331 Design and Analysis of Algorithms. -- Divide and Conquer. Dr. Daisy Tang

CS 331 Design and Analysis of Algorithms. -- Divide and Conquer. Dr. Daisy Tang CS 33 Desig d Alysis of Algorithms -- Divide d Coquer Dr. Disy Tg Divide-Ad-Coquer Geerl ide: Divide problem ito subproblems of the sme id; solve subproblems usig the sme pproh, d ombie prtil solutios,

More information

Chapter Real Numbers

Chapter Real Numbers Chpter. - Rel Numbers Itegers: coutig umbers, zero, d the egtive of the coutig umbers. ex: {,-3, -, -,,,, 3, } Rtiol Numbers: quotiets of two itegers with ozero deomitor; termitig or repetig decimls. ex:

More information

BC Calculus Review Sheet

BC Calculus Review Sheet BC Clculus Review Sheet Whe you see the words. 1. Fid the re of the ubouded regio represeted by the itegrl (sometimes 1 f ( ) clled horizotl improper itegrl). This is wht you thik of doig.... Fid the re

More information

( x) [ ] ( ) ( ) ( ) ( )( ) ACID-BASE EQUILIBRIUM PROBLEMS. Because HCO / K = 2.8 < 100 the xin the denominator is not going to be negligible.

( x) [ ] ( ) ( ) ( ) ( )( ) ACID-BASE EQUILIBRIUM PROBLEMS. Because HCO / K = 2.8 < 100 the xin the denominator is not going to be negligible. ACID-BASE EQUILIBRIUM PROBLEMS 1. Clculte the ph of solutio tht cotis 0.165 M olic cid. Clculte the cocetrtio of the olte io i this solutio. 1 H C O 4 + H O Ý H 3O + + HC O 1 5.9 HC O + H O Ý H 3O + +

More information

UNIT #5 SEQUENCES AND SERIES COMMON CORE ALGEBRA II

UNIT #5 SEQUENCES AND SERIES COMMON CORE ALGEBRA II Awer Key Nme: Dte: UNIT # SEQUENCES AND SERIES COMMON CORE ALGEBRA II Prt I Quetio. For equece defied by f? () () 08 6 6 f d f f, which of the followig i the vlue of f f f f f f 0 6 6 08 (). I the viul

More information

ECE 102 Engineering Computation

ECE 102 Engineering Computation ECE Egieerig Computtio Phillip Wog Mth Review Vetor Bsis Mtri Bsis System of Lier Equtios Summtio Symol is the symol for summtio. Emple: N k N... 9 k k k k k the, If e e e f e f k Vetor Bsis A vetor is

More information

Presentation for use with the textbook, Algorithm Design and Applications, by M. T. Goodrich and R. Tamassia, Wiley, Divide-and-Conquer

Presentation for use with the textbook, Algorithm Design and Applications, by M. T. Goodrich and R. Tamassia, Wiley, Divide-and-Conquer Presettio for use with the textook, Algorithm Desig d Applictios, y M. T. Goodrich d R. Tmssi, Wiley, 25 Divide-d-Coquer Divide-d-Coquer Divide-d coquer is geerl lgorithm desig prdigm: Divide: divide the

More information

THE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK SUMMER EXAMINATION 2005 FIRST ENGINEERING

THE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK SUMMER EXAMINATION 2005 FIRST ENGINEERING OLLSCOIL NA héireann, CORCAIGH THE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK SUMMER EXAMINATION 2005 FIRST ENGINEERING MATHEMATICS MA008 Clculus d Lier

More information

is completely general whenever you have waves from two sources interfering. 2

is completely general whenever you have waves from two sources interfering. 2 MAKNG SENSE OF THE EQUATON SHEET terferece & Diffrctio NTERFERENCE r1 r d si. Equtio for pth legth differece. r1 r is completely geerl. Use si oly whe the two sources re fr wy from the observtio poit.

More information

Chapter Real Numbers

Chapter Real Numbers Chpter. - Rel Numbers Itegers: coutig umbers, zero, d the egtive of the coutig umbers. ex: {,-3, -, -, 0,,, 3, } Rtiol Numbers: quotiets of two itegers with ozero deomitor; termitig or repetig decimls.

More information

Discrete Mathematics I Tutorial 12

Discrete Mathematics I Tutorial 12 Discrete Mthemtics I Tutoril Refer to Chpter 4., 4., 4.4. For ech of these sequeces fid recurrece reltio stisfied by this sequece. (The swers re ot uique becuse there re ifiitely my differet recurrece

More information

4. Optical Resonators

4. Optical Resonators S. Blair September 3, 2003 47 4. Optial Resoators Optial resoators are used to build up large itesities with moderate iput. Iput Iteral Resoators are typially haraterized by their quality fator: Q w stored

More information

In an algebraic expression of the form (1), like terms are terms with the same power of the variables (in this case

In an algebraic expression of the form (1), like terms are terms with the same power of the variables (in this case Chpter : Algebr: A. Bckgroud lgebr: A. Like ters: I lgebric expressio of the for: () x b y c z x y o z d x... p x.. we cosider x, y, z to be vribles d, b, c, d,,, o,.. to be costts. I lgebric expressio

More information

All the Laplace Transform you will encounter has the following form: Rational function X(s)

All the Laplace Transform you will encounter has the following form: Rational function X(s) EE G Note: Chpter Itructor: Cheug Pge - - Iverio of Rtiol Fuctio All the Lplce Trform you will ecouter h the followig form: m m m m e τ 0...... Rtiol fuctio Dely Why? Rtiol fuctio come out turlly from

More information

Grammar-based Compression for Directed and Undirected Generalized Series-parallel Graphs using Integer Linear Programming

Grammar-based Compression for Directed and Undirected Generalized Series-parallel Graphs using Integer Linear Programming Grmmr-bsed ompressio for Direted d Udireted Geerlized eries-prllel Grphs usig Iteger Lier Progrmmig Morihiro Hyshid, Hitoshi Koyo 2 d Ttsuy kutsu 3 tiol Istitute of Tehology, Mtsue ollege, 4-4, ishiikumho,

More information

Material Balances on Reactive Processes F&R

Material Balances on Reactive Processes F&R Material Balaces o Reactive Processes F&R 4.6-4.8 What does a reactio do to the geeral balace equatio? Accumulatio = I Out + Geeratio Cosumptio For a reactive process at steady-state, the geeral balace

More information

Section 11.6 Absolute and Conditional Convergence, Root and Ratio Tests

Section 11.6 Absolute and Conditional Convergence, Root and Ratio Tests Sectio.6 Absolute ad Coditioal Covergece, Root ad Ratio Tests I this chapter we have see several examples of covergece tests that oly apply to series whose terms are oegative. I this sectio, we will lear

More information

(Figure 2.9), we observe x. and we write. (b) as x, x 1. and we write. We say that the line y 0 is a horizontal asymptote of the graph of f.

(Figure 2.9), we observe x. and we write. (b) as x, x 1. and we write. We say that the line y 0 is a horizontal asymptote of the graph of f. The symbol for ifiity ( ) does ot represet a real umber. We use to describe the behavior of a fuctio whe the values i its domai or rage outgrow all fiite bouds. For eample, whe we say the limit of f as

More information

Remarks: (a) The Dirac delta is the function zero on the domain R {0}.

Remarks: (a) The Dirac delta is the function zero on the domain R {0}. Sectio Objective(s): The Dirc s Delt. Mi Properties. Applictios. The Impulse Respose Fuctio. 4.4.. The Dirc Delt. 4.4. Geerlized Sources Defiitio 4.4.. The Dirc delt geerlized fuctio is the limit δ(t)

More information

MAT 271 Project: Partial Fractions for certain rational functions

MAT 271 Project: Partial Fractions for certain rational functions MAT 7 Project: Partial Fractios for certai ratioal fuctios Prerequisite kowledge: partial fractios from MAT 7, a very good commad of factorig ad complex umbers from Precalculus. To complete this project,

More information

Chapter 2. LOGARITHMS

Chapter 2. LOGARITHMS Chpter. LOGARITHMS Dte: - 009 A. INTRODUCTION At the lst hpter, you hve studied bout Idies d Surds. Now you re omig to Logrithms. Logrithm is ivers of idies form. So Logrithms, Idies, d Surds hve strog

More information

Phys 102 Lecture 25 The quantum mechanical model of light

Phys 102 Lecture 25 The quantum mechanical model of light Phys 102 Lecture 25 The quatum mechaical model of light 1 Recall last time Problems with classical physics Stability of atoms Atomic spectra Photoelectric effect Quatum model of the atom Bohr model oly

More information

Dynamics of Structures

Dynamics of Structures UNION Dymis of Strutures Prt Zbigiew Wójii Je Grosel Projet o-fied by Europe Uio withi Europe Soil Fud UNION Mtries Defiitio of mtri mtri is set of umbers or lgebri epressios rrged i retgulr form with

More information

Content. Languages, Alphabets and Strings. Operations on Strings. a ab abba baba. aaabbbaaba b 5. Languages. A language is a set of strings

Content. Languages, Alphabets and Strings. Operations on Strings. a ab abba baba. aaabbbaaba b 5. Languages. A language is a set of strings CD5560 FABER Forml guges, Automt d Models of Computtio ecture Mälrdle Uiversity 006 Cotet guges, Alphets d Strigs Strigs & Strig Opertios guges & guge Opertios Regulr Expressios Fiite Automt, FA Determiistic

More information

Autar Kaw Benjamin Rigsby. Transforming Numerical Methods Education for STEM Undergraduates

Autar Kaw Benjamin Rigsby.   Transforming Numerical Methods Education for STEM Undergraduates Autr Kw Bejmi Rigsby http://m.mthforcollege.com Trsformig Numericl Methods Eductio for STEM Udergrdutes http://m.mthforcollege.com . solve set of simulteous lier equtios usig Nïve Guss elimitio,. ler the

More information

G8-11 Congruence Rules

G8-11 Congruence Rules G8-11 ogruee Rules If two polgos re ogruet, ou ple the oe o top of the other so tht the th etl. The verties tht th re lled orrespodig verties. The gles tht th re lled orrespodig gles. The sides tht th

More information

n 2 + 3n + 1 4n = n2 + 3n + 1 n n 2 = n + 1

n 2 + 3n + 1 4n = n2 + 3n + 1 n n 2 = n + 1 Ifiite Series Some Tests for Divergece d Covergece Divergece Test: If lim u or if the limit does ot exist, the series diverget. + 3 + 4 + 3 EXAMPLE: Show tht the series diverges. = u = + 3 + 4 + 3 + 3

More information

A GENERAL METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS: THE FROBENIUS (OR SERIES) METHOD

A GENERAL METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS: THE FROBENIUS (OR SERIES) METHOD Diol Bgoo () A GENERAL METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS: THE FROBENIUS (OR SERIES) METHOD I. Itroductio The first seprtio of vribles (see pplictios to Newto s equtios) is ver useful method

More information

What Is Required? You need to determine the hydronium ion concentration in an aqueous solution. K w = [H 3 O + ][OH ] =

What Is Required? You need to determine the hydronium ion concentration in an aqueous solution. K w = [H 3 O + ][OH ] = Calculatig the [H3O + ] or [OH ] i Aqueous Solutio (Studet textbook page 500) 11. The cocetratio of hydroxide ios, OH (aq), i a solutio at 5C is 0.150 /. Determie the cocetratio of hydroium ios, H 3 O

More information

General properties of definite integrals

General properties of definite integrals Roerto s Notes o Itegrl Clculus Chpter 4: Defiite itegrls d the FTC Sectio Geerl properties of defiite itegrls Wht you eed to kow lredy: Wht defiite Riem itegrl is. Wht you c ler here: Some key properties

More information

The Structure of Z p when p is Prime

The Structure of Z p when p is Prime LECTURE 13 The Structure of Z p whe p is Prime Theorem 131 If p > 1 is a iteger, the the followig properties are equivalet (1) p is prime (2) For ay [0] p i Z p, the equatio X = [1] p has a solutio i Z

More information

Math 3B Midterm Review

Math 3B Midterm Review Mth 3B Midterm Review Writte by Victori Kl vtkl@mth.ucsb.edu SH 643u Office Hours: R 11:00 m - 1:00 pm Lst updted /15/015 Here re some short otes o Sectios 7.1-7.8 i your ebook. The best idictio of wht

More information

HE ATOM & APPROXIMATION METHODS MORE GENERAL VARIATIONAL TREATMENT. Examples:

HE ATOM & APPROXIMATION METHODS MORE GENERAL VARIATIONAL TREATMENT. Examples: 5.6 4 Lecture #3-4 page HE ATOM & APPROXIMATION METHODS MORE GENERAL VARIATIONAL TREATMENT Do t restrict the wavefuctio to a sigle term! Could be a liear combiatio of several wavefuctios e.g. two terms:

More information

Frequency-domain Characteristics of Discrete-time LTI Systems

Frequency-domain Characteristics of Discrete-time LTI Systems requecy-domi Chrcteristics of Discrete-time LTI Systems Prof. Siripog Potisuk LTI System descriptio Previous bsis fuctio: uit smple or DT impulse The iput sequece is represeted s lier combitio of shifted

More information

CH 20 SOLVING FORMULAS

CH 20 SOLVING FORMULAS CH 20 SOLVING FORMULAS 179 Itrodutio S olvig equtios suh s 2 + 7 20 is oviousl the orerstoe of lger. But i siee, usiess, d omputers it is lso eessr to solve equtios tht might hve vriet of letters i them.

More information

AP Calculus AB AP Review

AP Calculus AB AP Review AP Clulus AB Chpters. Re limit vlues from grphsleft-h Limits Right H Limits Uerst tht f() vlues eist ut tht the limit t oes ot hve to.. Be le to ietify lel isotiuities from grphs. Do t forget out the 3-step

More information

Sect 5.3 Proportions

Sect 5.3 Proportions Sect 5. Proportios Objective a: Defiitio of a Proportio. A proportio is a statemet that two ratios or two rates are equal. A proportio takes a form that is similar to a aalogy i Eglish class. For example,

More information

MATRIX ALGEBRA, Systems Linear Equations

MATRIX ALGEBRA, Systems Linear Equations MATRIX ALGEBRA, Systes Lier Equtios Now we chge to the LINEAR ALGEBRA perspective o vectors d trices to reforulte systes of lier equtios. If you fid the discussio i ters of geerl d gets lost i geerlity,

More information

DETERMINANT. = 0. The expression a 1. is called a determinant of the second order, and is denoted by : y + c 1

DETERMINANT. = 0. The expression a 1. is called a determinant of the second order, and is denoted by : y + c 1 NOD6 (\Dt\04\Kot\J-Advced\SMP\Mths\Uit#0\NG\Prt-\0.Determits\0.Theory.p65. INTRODUCTION : If the equtios x + b 0, x + b 0 re stisfied by the sme vlue of x, the b b 0. The expressio b b is clled determit

More information

Unit 5. Gases (Answers)

Unit 5. Gases (Answers) Uit 5. Gases (Aswers) Upo successful completio of this uit, the studets should be able to: 5. Describe what is meat by gas pressure.. The ca had a small amout of water o the bottom to begi with. Upo heatig

More information

CH 19 SOLVING FORMULAS

CH 19 SOLVING FORMULAS 1 CH 19 SOLVING FORMULAS INTRODUCTION S olvig equtios suh s 2 + 7 20 is oviousl the orerstoe of lger. But i siee, usiess, d omputers it is lso eessr to solve equtios tht might hve vriet of letters i them.

More information

MTH 146 Class 16 Notes

MTH 146 Class 16 Notes MTH 46 Clss 6 Notes 0.4- Cotiued Motivtio: We ow cosider the rc legth of polr curve. Suppose we wish to fid the legth of polr curve curve i terms of prmetric equtios s: r f where b. We c view the cos si

More information

Repeated Root and Common Root

Repeated Root and Common Root Repeted Root d Commo Root 1 (Method 1) Let α, β, γ e the roots of p(x) x + x + 0 (1) The α + β + γ 0, αβ + βγ + γα, αβγ - () (α - β) (α + β) - αβ (α + β) [ (βγ + γα)] + [(α + β) + γ (α + β)] +γ (α + β)

More information

y udv uv y v du 7.1 INTEGRATION BY PARTS

y udv uv y v du 7.1 INTEGRATION BY PARTS 7. INTEGRATION BY PARTS Ever differetitio rule hs correspodig itegrtio rule. For istce, the Substitutio Rule for itegrtio correspods to the Chi Rule for differetitio. The rule tht correspods to the Product

More information

Numerical Solutions of Fredholm Integral Equations Using Bernstein Polynomials

Numerical Solutions of Fredholm Integral Equations Using Bernstein Polynomials Numericl Solutios of Fredholm Itegrl Equtios Usig erstei Polyomils A. Shiri, M. S. Islm Istitute of Nturl Scieces, Uited Itertiol Uiversity, Dhk-, gldesh Deprtmet of Mthemtics, Uiversity of Dhk, Dhk-,

More information

10.5 Power Series. In this section, we are going to start talking about power series. A power series is a series of the form

10.5 Power Series. In this section, we are going to start talking about power series. A power series is a series of the form 0.5 Power Series I the lst three sectios, we ve spet most of tht time tlkig bout how to determie if series is coverget or ot. Now it is time to strt lookig t some specific kids of series d we will evetully

More information

Closed Newton-Cotes Integration

Closed Newton-Cotes Integration Closed Newto-Cotes Itegrtio Jmes Keeslig This documet will discuss Newto-Cotes Itegrtio. Other methods of umericl itegrtio will be discussed i other posts. The other methods will iclude the Trpezoidl Rule,

More information

Schrödinger Equation Via Laplace-Beltrami Operator

Schrödinger Equation Via Laplace-Beltrami Operator IOSR Jourl of Mthemtics (IOSR-JM) e-issn: 78-578, p-issn: 39-765X. Volume 3, Issue 6 Ver. III (Nov. - Dec. 7), PP 9-95 www.iosrjourls.org Schrödiger Equtio Vi Lplce-Beltrmi Opertor Esi İ Eskitşçioğlu,

More information

Department of Mechanical Engineering ME 322 Mechanical Engineering Thermodynamics. Lecture 33. Psychrometric Properties of Moist Air

Department of Mechanical Engineering ME 322 Mechanical Engineering Thermodynamics. Lecture 33. Psychrometric Properties of Moist Air Deprtment of Mechnicl Engineering ME 3 Mechnicl Engineering hermodynmics Lecture 33 sychrometric roperties of Moist Air Air-Wter Vpor Mixtures Atmospheric ir A binry mixture of dry ir () + ter vpor ()

More information

Vectors. Vectors in Plane ( 2

Vectors. Vectors in Plane ( 2 Vectors Vectors i Ple ( ) The ide bout vector is to represet directiol force Tht mes tht every vector should hve two compoets directio (directiol slope) d mgitude (the legth) I the ple we preset vector

More information

a f(x)dx is divergent.

a f(x)dx is divergent. Mth 250 Exm 2 review. Thursdy Mrh 5. Brig TI 30 lultor but NO NOTES. Emphsis o setios 5.5, 6., 6.2, 6.3, 3.7, 6.6, 8., 8.2, 8.3, prt of 8.4; HW- 2; Q-. Kow for trig futios tht 0.707 2/2 d 0.866 3/2. From

More information

Merge Sort. Outline and Reading. Divide-and-Conquer. Divide-and-conquer paradigm ( 4.1.1) Merge-sort ( 4.1.1)

Merge Sort. Outline and Reading. Divide-and-Conquer. Divide-and-conquer paradigm ( 4.1.1) Merge-sort ( 4.1.1) Merge Sort 7 2 9 4 2 4 7 9 7 2 2 7 9 4 4 9 7 7 2 2 9 9 4 4 Merge Sort versio 1.3 1 Outlie d Redig Divide-d-coquer prdigm ( 4.1.1 Merge-sort ( 4.1.1 Algorithm Mergig two sorted sequeces Merge-sort tree

More information

GRADE 12 SEPTEMBER 2016 MATHEMATICS P1

GRADE 12 SEPTEMBER 2016 MATHEMATICS P1 NATIONAL SENIOR CERTIFICATE GRADE SEPTEMBER 06 MATHEMATICS P MARKS: 50 TIME: 3 hours *MATHE* This questio pper cosists of pges icludig iformtio sheet MATHEMATICS P (EC/SEPTEMBER 06 INSTRUCTIONS AND INFORMATION

More information