Conventional Phrase. Conventional phrase ending in an authentic cadence. Die Zauberflöte K. 620, Act II, no. 21. mein Mäd # # # # # # # # # ( & # ( &

Size: px
Start display at page:

Download "Conventional Phrase. Conventional phrase ending in an authentic cadence. Die Zauberflöte K. 620, Act II, no. 21. mein Mäd # # # # # # # # # ( & # ( &"

Transcription

1 The Phrase

2 Covetioal Phrase Four measures log, termiated by a cadece, usually authetic or half. The idea of the covetioal 4-bar phrase is a coveiet startig poit for study, ad is ot to be take as some kid of holy writ that all phrases are four measures log.

3 Covetioal Phrase Covetioal phrase edig i a authetic cadece Die Zauberflöte K. 620, Act II, o. 21 Allegro (Papageo) " W. A. Mozart Kli - get, Glöck - che, kli - get schafft ' mei Mäd - che her, " ( ( ( ( 5 " ) kli - get, Glöck - che, kli - get brigt mei Mäd - che her. ' " ( * ( ( ( * * *

4 Covetioal Phrase Covetioal phrase edig i a half cadece (first four measures) followed by a secod phrase edig o a authetic cadece. " p " ( 5 Allegretto vivace sf Soata, Op. 31 No. 3, II sf Ludwig va Beethove ' ' ' ' ' ' ' ' ' ' ' ( ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' sf ' ' ' ' ' ' ' ' ( sf ) ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' * ) * *

5 Covetioal Phrase Foud i early music: hyms, troubadour sogs, etc. Estampie (13th cetury) " ' ' " ( ) " " (

6 Covetioal Phrase Notatio may mask four measures sometimes. Fast tempo: two measures = oe measure Slow tempo: oe measure = two measures Lädler Fraz Schubert " mp " 5 3 ' ( (

7 Covetioal Phrase Typically divided ito semiphrases, usually two measures log. These ca i tur divide ito motives ad their maipulatios. Mozart Symphoy No. 40, mai theme "

8 Phrase as Uit Termiated by Cadece Semiphrase is distict from the phrase A 2+2 phrase may ofte seem to have a cadece at midpoit, but it s a cadetial iflectio: Now is the witer of our discotet Made glorious summer by this so of York.

9 Phrase as Uit Termiated by Cadece Chopi Mazurka: oe could opie a plagal cadece i measure 2, but 3-4 provide a solid V7-I cadece. Not every V-I motio is a cadece " f + ", Vivo e risoluto fz Mazurka, Op. 17, No. 1 ' ( ) ), Fréderic Chopi * 5 fz + - ) - * ' ) - * *, ) - -

10 Phrase as Uit Termiated by Cadece Chopi Mazurka: Measures 1-3 form V7-I-V7-I; but the melodic otatio chages our hearig of measure 2. ", " - 7 Moderato, co aima ) ' Mazurka, Op. 24, No. 3 " ( ' ( * ) ) ( p * ),.. / - ) ).. / - ) ) 3 ) * + ) fz dolce 3 Fréderic Chopi * + ) fz * (

11 Phrase as Compoet of a Larger Patter Setece (period) of two phrases Symphoy No. 35, K. 385 Meuetto (Trio) W. A. Mozart " " ' ' ( p ) " ' ' ' ' " * * * + + ) ' * * ' * +

12 Phrase as Compoet of a Larger Patter Double period cosistig of four phrases " p * " Adagio molto Piao Soata Op. 10, No. 1: II Ludwig va Beethove ( ' ) " ( ( ' ) ' ' cresc. fp + +, 3 ( " ' - " ( p *. ) ' - ) ) * ( / / ) sf ( ) '. p / ( / ( ) '.

13 Phrase as Compoet of a Larger Patter Phrase group cosistig of two or more phrases

14 Phrase as a Idepedet Uit A phrase ca be completely isolated. It ca act as prelude, postlude, coda, iterlude, or trasitioal passage. Always thik carefully before idetifyig a phrase as idepedet

15 Repetitios of Phrases Idetical Bergamasca " Allegretto Scheidt ' " '

16 Repetitios of Phrases Idetical " ' ( " Faschigsschwak aus Wie, Op. 26 No. 3: Scherzio ' ' ' ' ' ' R. Schuma

17 Repetitios of Phrases Embellished

18 Repetitios of Phrases Harmoic (ad accompaimet) chages " Allegro co brio Soata i D Major, H XVI: 37 ' Fraz Joseph Hayd ( ' ) * ) " ( ) ( + " ' ) ' + " ) * ' +

19 Repetitios of Phrases Harmoic (ad accompaimet) chages Dace i Hugaria Rhythm "" "" p Béla Bartók ' ' ' ' ' ' ' ( ) ) " ' ' ' ' ' f mf ) * + ' ' * )) ) ) ) '

20 Repetitios of Phrases Register Chage " p ' " Lädler 3 3 Schubert " mp ' 3 3 (

21 Repetitios of Phrases Register Chage " Allegretto p ) " + * * Impromptu, Op. 90 No. 4, D * * Fraz Schubert ' ' ' ( ( ' + * ( ' ) * + * ( ' ( ' + + * * ( ' +

22 Repetitios of Phrases Color (orchestratio) Chage " ' ( ) mf p * " ' (, - p mf

23 Voice " Die ihr ei - em eu - e Gra - ' de der Er - ket - iss - u euch " ' ' ( ' acht, wa - dert fest auf eu - rem - Orgel ) * * * ( ' ( ' ( ' * 6 " Pfa ' ( + u * Ma - - de, wisst, es " ' ) + ist der * * Weis - heit Pfad. Nur der - ver - dross e + ' ( + mag dem 11 " Quell des ' ( sich ah'; ( ( e * (, + - Lichts ur der u - ver dross - Ma " ' ' ' ' ' ( ( ( ' ) ( ' ( ' ( ' ( ' ' ( ( ' ' + + ( mag dem Quell des Lichts sich ah'. + - * + -

24 Aalyzig Phrases Measure Letter Cadece Notes 1-4 a HC Delayed 5-8 b IAC Exteded

25

ARRANGEMENTS IN A CIRCLE

ARRANGEMENTS IN A CIRCLE ARRANGEMENTS IN A CIRCLE Whe objects are arraged i a circle, the total umber of arragemets is reduced. The arragemet of (say) four people i a lie is easy ad o problem (if they liste of course!!). With

More information

Ray Jackendoff. Sonata. For basset horn and piano SCORE

Ray Jackendoff. Sonata. For basset horn and piano SCORE Ray ackedo Soata For asset hor ad iao SCORE Comoser s ote Ray ackedo Soata or asset hor ad iao Gregory Tucker a isirig iaist comoser ad teacher served o the aculty o MIT or may years Whe I was a graduate

More information

Sigma notation. 2.1 Introduction

Sigma notation. 2.1 Introduction Sigma otatio. Itroductio We use sigma otatio to idicate the summatio process whe we have several (or ifiitely may) terms to add up. You may have see sigma otatio i earlier courses. It is used to idicate

More information

Preview Only W PREVIEW PREVIEW PREVIEW PREVIEW EW PREVIEW PREVIEW PREVIE REVIEW PREVIEW EVIEW PREVIEW PREVIEW PREV ALREADY HOME.

Preview Only W PREVIEW PREVIEW PREVIEW PREVIEW EW PREVIEW PREVIEW PREVIE REVIEW PREVIEW EVIEW PREVIEW PREVIEW PREV ALREADY HOME. 2 Arraged y JAY ALTHOUSE SOPRANO I SOPRANO II C C ALTO PIANO 5 ALREADY HOME for S.S.A. voices ad piao ith optioal SoudTrax CD* Music y ANDREW LLOYD WEBBER Lyrics y TIM RICE Moderately slo (h = ca.72),

More information

Preview Only W PREVIEW PREVIEW PREVIEW PREVIEW EW PREVIEW PREVIEW PREVIE REVIEW PREVIEW EVIEW PREVIEW PREVIEW PREV ALREADY HOME. &b b b b C.

Preview Only W PREVIEW PREVIEW PREVIEW PREVIEW EW PREVIEW PREVIEW PREVIE REVIEW PREVIEW EVIEW PREVIEW PREVIEW PREV ALREADY HOME. &b b b b C. 2 Arraged y JAY ALTHOUSE SOPRANO ALTO TENOR BASS PIANO 5 C C ALREADY HOME for S.A.T.B. voices ad piao ith optioal SoudTrax CD* Music y ANDREW LLOYD WEBBER Lyrics y TIM RICE Moderately slo (h = ca.72),

More information

Ray Jackendoff. Sonata. For bassoon and piano SCORE

Ray Jackendoff. Sonata. For bassoon and piano SCORE ! Ray ackedo! Soata!! For assoo ad iao!!!! SCORE ! Comoser s ote! Ray ackedo Soata or assoo ad Piao Gregory Tucker, a isirig iaist, comoser, ad teacher, served o the aculty o MIT or may years Whe I was

More information

1. Hydrogen Atom: 3p State

1. Hydrogen Atom: 3p State 7633A QUANTUM MECHANICS I - solutio set - autum. Hydroge Atom: 3p State Let us assume that a hydroge atom is i a 3p state. Show that the radial part of its wave fuctio is r u 3(r) = 4 8 6 e r 3 r(6 r).

More information

5.76 Lecture #33 5/08/91 Page 1 of 10 pages. Lecture #33: Vibronic Coupling

5.76 Lecture #33 5/08/91 Page 1 of 10 pages. Lecture #33: Vibronic Coupling 5.76 Lecture #33 5/8/9 Page of pages Lecture #33: Vibroic Couplig Last time: H CO A A X A Electroically forbidde if A -state is plaar vibroically allowed to alterate v if A -state is plaar iertial defect

More information

A UDITION V IOLIN 1 & VIOLIN 2 TUTTI

A UDITION V IOLIN 1 & VIOLIN 2 TUTTI T HE ROYAL SWEDISH OPERA A UDITION V IOLIN 1 & VIOLIN 2 TUTTI 6 TH AND 7 TH OF F EBRUARY 2019 ORCHESTRAL EXCERPTS VIOLIN 1 1. Strauss: from Der Rosenkavalier, act 3, Einleitung 2. Tjajkovskij: from Nutcracker,

More information

SPEC/4/PHYSI/SPM/ENG/TZ0/XX PHYSICS PAPER 1 SPECIMEN PAPER. 45 minutes INSTRUCTIONS TO CANDIDATES

SPEC/4/PHYSI/SPM/ENG/TZ0/XX PHYSICS PAPER 1 SPECIMEN PAPER. 45 minutes INSTRUCTIONS TO CANDIDATES SPEC/4/PHYSI/SPM/ENG/TZ0/XX PHYSICS STANDARD LEVEL PAPER 1 SPECIMEN PAPER 45 miutes INSTRUCTIONS TO CANDIDATES Do ot ope this examiatio paper util istructed to do so. Aswer all the questios. For each questio,

More information

PROPERTIES OF SQUARES

PROPERTIES OF SQUARES PROPERTIES OF SQUARES The square is oe of the simplest two-dimesioal geometric figures. It is recogized by most pre-idergarters through programs such as Sesame Street, the comic strip Spoge Bob Square

More information

Physics 324, Fall Dirac Notation. These notes were produced by David Kaplan for Phys. 324 in Autumn 2001.

Physics 324, Fall Dirac Notation. These notes were produced by David Kaplan for Phys. 324 in Autumn 2001. Physics 324, Fall 2002 Dirac Notatio These otes were produced by David Kapla for Phys. 324 i Autum 2001. 1 Vectors 1.1 Ier product Recall from liear algebra: we ca represet a vector V as a colum vector;

More information

6.1. Sequences as Discrete Functions. Investigate

6.1. Sequences as Discrete Functions. Investigate 6.1 Sequeces as Discrete Fuctios The word sequece is used i everyday laguage. I a sequece, the order i which evets occur is importat. For example, builders must complete work i the proper sequece to costruct

More information

MATHEMATICS. The assessment objectives of the Compulsory Part are to test the candidates :

MATHEMATICS. The assessment objectives of the Compulsory Part are to test the candidates : MATHEMATICS INTRODUCTION The public assessmet of this subject is based o the Curriculum ad Assessmet Guide (Secodary 4 6) Mathematics joitly prepared by the Curriculum Developmet Coucil ad the Hog Kog

More information

Orthogonal transformations

Orthogonal transformations Orthogoal trasformatios October 12, 2014 1 Defiig property The squared legth of a vector is give by takig the dot product of a vector with itself, v 2 v v g ij v i v j A orthogoal trasformatio is a liear

More information

MATHEMATICS. The assessment objectives of the Compulsory Part are to test the candidates :

MATHEMATICS. The assessment objectives of the Compulsory Part are to test the candidates : MATHEMATICS INTRODUCTION The public assessmet of this subject is based o the Curriculum ad Assessmet Guide (Secodary 4 6) Mathematics joitly prepared by the Curriculum Developmet Coucil ad the Hog Kog

More information

(3) If you replace row i of A by its sum with a multiple of another row, then the determinant is unchanged! Expand across the i th row:

(3) If you replace row i of A by its sum with a multiple of another row, then the determinant is unchanged! Expand across the i th row: Math 5-4 Tue Feb 4 Cotiue with sectio 36 Determiats The effective way to compute determiats for larger-sized matrices without lots of zeroes is to ot use the defiitio, but rather to use the followig facts,

More information

September 2012 C1 Note. C1 Notes (Edexcel) Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 1

September 2012 C1 Note. C1 Notes (Edexcel) Copyright   - For AS, A2 notes and IGCSE / GCSE worksheets 1 September 0 s (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright

More information

Chemical Kinetics CHAPTER 14. Chemistry: The Molecular Nature of Matter, 6 th edition By Jesperson, Brady, & Hyslop. CHAPTER 14 Chemical Kinetics

Chemical Kinetics CHAPTER 14. Chemistry: The Molecular Nature of Matter, 6 th edition By Jesperson, Brady, & Hyslop. CHAPTER 14 Chemical Kinetics Chemical Kietics CHAPTER 14 Chemistry: The Molecular Nature of Matter, 6 th editio By Jesperso, Brady, & Hyslop CHAPTER 14 Chemical Kietics Learig Objectives: Factors Affectig Reactio Rate: o Cocetratio

More information

Mathematical Notation Math Finite Mathematics

Mathematical Notation Math Finite Mathematics Mathematical Notatio Math 60 - Fiite Mathematics Use Word or WordPerfect to recreate the followig documets. Each article is worth 0 poits ad should be emailed to the istructor at james@richlad.edu. If

More information

REDUCING THE POSSIBILITY OF SUBJECTIVE ERROR IN THE DETERMINATION OF THE STRUCTURE-FUNCTION-BASED EFFECTIVE THERMAL CONDUCTIVITY OF BOARDS

REDUCING THE POSSIBILITY OF SUBJECTIVE ERROR IN THE DETERMINATION OF THE STRUCTURE-FUNCTION-BASED EFFECTIVE THERMAL CONDUCTIVITY OF BOARDS Nice, Côte d Azur, Frace, 27-29 Septeber 2006 REDUCING THE POSSIBILITY OF SUBJECTIVE ERROR IN THE DETERMINATION OF THE STRUCTURE-FUNCTION-BASED EFFECTIVE THERMAL CONDUCTIVITY OF BOARDS Erő Kollár, Vladiír

More information

(3) If you replace row i of A by its sum with a multiple of another row, then the determinant is unchanged! Expand across the i th row:

(3) If you replace row i of A by its sum with a multiple of another row, then the determinant is unchanged! Expand across the i th row: Math 50-004 Tue Feb 4 Cotiue with sectio 36 Determiats The effective way to compute determiats for larger-sized matrices without lots of zeroes is to ot use the defiitio, but rather to use the followig

More information

MA131 - Analysis 1. Workbook 3 Sequences II

MA131 - Analysis 1. Workbook 3 Sequences II MA3 - Aalysis Workbook 3 Sequeces II Autum 2004 Cotets 2.8 Coverget Sequeces........................ 2.9 Algebra of Limits......................... 2 2.0 Further Useful Results........................

More information

ii. O = {x x = 2k + 1 for some integer k} (This set could have been listed O = { -3, -1, 1, 3, 5 }.)

ii. O = {x x = 2k + 1 for some integer k} (This set could have been listed O = { -3, -1, 1, 3, 5 }.) Sets 1 Math 3312 Set Theory Sprig 2008 Itroductio Set theory is a brach of mathematics that deals with the properties of welldefied collectios of objects, which may or may ot be of a mathematical ature,

More information

Permutations, Combinations, and the Binomial Theorem

Permutations, Combinations, and the Binomial Theorem Permutatios, ombiatios, ad the Biomial Theorem Sectio Permutatios outig methods are used to determie the umber of members of a specific set as well as outcomes of a evet. There are may differet ways to

More information

6.3 Testing Series With Positive Terms

6.3 Testing Series With Positive Terms 6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial

More information

Polynomial Equations and Tangents

Polynomial Equations and Tangents Polyomial Equatios ad Tagets Jim lowers presetatio to the Mathematical ssociatio of merica MD-DC-V Sectio Meetig 07 pril 9 9:5 am rilliat.org Puzzle Problem appeared i a Facebook post this past witer What

More information

Math 105 TOPICS IN MATHEMATICS REVIEW OF LECTURES VII. 7. Binomial formula. Three lectures ago ( in Review of Lectuires IV ), we have covered

Math 105 TOPICS IN MATHEMATICS REVIEW OF LECTURES VII. 7. Binomial formula. Three lectures ago ( in Review of Lectuires IV ), we have covered Math 5 TOPICS IN MATHEMATICS REVIEW OF LECTURES VII Istructor: Lie #: 59 Yasuyuki Kachi 7 Biomial formula February 4 Wed) 5 Three lectures ago i Review of Lectuires IV ) we have covered / \ / \ / \ / \

More information

PROPERTIES OF AN EULER SQUARE

PROPERTIES OF AN EULER SQUARE PROPERTIES OF N EULER SQURE bout 0 the mathematicia Leoard Euler discussed the properties a x array of letters or itegers ow kow as a Euler or Graeco-Lati Square Such squares have the property that every

More information

( ) GENERATING FUNCTIONS

( ) GENERATING FUNCTIONS GENERATING FUNCTIONS Solve a ifiite umber of related problems i oe swoop. *Code the problems, maipulate the code, the decode the aswer! Really a algebraic cocept but ca be eteded to aalytic basis for iterestig

More information

Definitions and Theorems. where x are the decision variables. c, b, and a are constant coefficients.

Definitions and Theorems. where x are the decision variables. c, b, and a are constant coefficients. Defiitios ad Theorems Remember the scalar form of the liear programmig problem, Miimize, Subject to, f(x) = c i x i a 1i x i = b 1 a mi x i = b m x i 0 i = 1,2,, where x are the decisio variables. c, b,

More information

Median and IQR The median is the value which divides the ordered data values in half.

Median and IQR The median is the value which divides the ordered data values in half. STA 666 Fall 2007 Web-based Course Notes 4: Describig Distributios Numerically Numerical summaries for quatitative variables media ad iterquartile rage (IQR) 5-umber summary mea ad stadard deviatio Media

More information

Enumerative & Asymptotic Combinatorics

Enumerative & Asymptotic Combinatorics C50 Eumerative & Asymptotic Combiatorics Notes 4 Sprig 2003 Much of the eumerative combiatorics of sets ad fuctios ca be geeralised i a maer which, at first sight, seems a bit umotivated I this chapter,

More information

LECTURE 11 LINEAR PROCESSES III: ASYMPTOTIC RESULTS

LECTURE 11 LINEAR PROCESSES III: ASYMPTOTIC RESULTS PRIL 7, 9 where LECTURE LINER PROCESSES III: SYMPTOTIC RESULTS (Phillips ad Solo (99) ad Phillips Lecture Notes o Statioary ad Nostatioary Time Series) I this lecture, we discuss the LLN ad CLT for a liear

More information

CSE 21 Mathematics for

CSE 21 Mathematics for CSE 2 Mathematics for Algorithm ad System Aalysis Summer, 2005 Outlie What a geeratig fuctio is How to create a geeratig fuctio to model a problem Fidig the desired coefficiet Partitios Expoetial geeratig

More information

Algebra II Notes Unit Seven: Powers, Roots, and Radicals

Algebra II Notes Unit Seven: Powers, Roots, and Radicals Syllabus Objectives: 7. The studets will use properties of ratioal epoets to simplify ad evaluate epressios. 7.8 The studet will solve equatios cotaiig radicals or ratioal epoets. b a, the b is the radical.

More information

Complex Analysis Spring 2001 Homework I Solution

Complex Analysis Spring 2001 Homework I Solution Complex Aalysis Sprig 2001 Homework I Solutio 1. Coway, Chapter 1, sectio 3, problem 3. Describe the set of poits satisfyig the equatio z a z + a = 2c, where c > 0 ad a R. To begi, we see from the triagle

More information

BIOSTATS 640 Intermediate Biostatistics Frequently Asked Questions Topic 1 FAQ 1 Review of BIOSTATS 540 Introductory Biostatistics

BIOSTATS 640 Intermediate Biostatistics Frequently Asked Questions Topic 1 FAQ 1 Review of BIOSTATS 540 Introductory Biostatistics BIOTAT 640 Itermediate Biostatistics Frequetly Asked Questios Topic FAQ Review of BIOTAT 540 Itroductory Biostatistics. I m cofused about the jargo ad otatio, especially populatio versus sample. Could

More information

Sequences, Sums, and Products

Sequences, Sums, and Products CSCE 222 Discrete Structures for Computig Sequeces, Sums, ad Products Dr. Philip C. Ritchey Sequeces A sequece is a fuctio from a subset of the itegers to a set S. A discrete structure used to represet

More information

mx bx kx F t. dt IR I LI V t, Q LQ RQ V t,

mx bx kx F t. dt IR I LI V t, Q LQ RQ V t, Lecture 5 omplex Variables II (Applicatios i Physics) (See hapter i Boas) To see why complex variables are so useful cosider first the (liear) mechaics of a sigle particle described by Newto s equatio

More information

CS161 Handout 05 Summer 2013 July 10, 2013 Mathematical Terms and Identities

CS161 Handout 05 Summer 2013 July 10, 2013 Mathematical Terms and Identities CS161 Hadout 05 Summer 2013 July 10, 2013 Mathematical Terms ad Idetities Thaks to Ady Nguye ad Julie Tibshirai for their advice o this hadout. This hadout covers mathematical otatio ad idetities that

More information

Areas and Distances. We can easily find areas of certain geometric figures using well-known formulas:

Areas and Distances. We can easily find areas of certain geometric figures using well-known formulas: Areas ad Distaces We ca easily fid areas of certai geometric figures usig well-kow formulas: However, it is t easy to fid the area of a regio with curved sides: METHOD: To evaluate the area of the regio

More information

C/CS/Phys C191 Deutsch and Deutsch-Josza algorithms 10/20/07 Fall 2007 Lecture 17

C/CS/Phys C191 Deutsch and Deutsch-Josza algorithms 10/20/07 Fall 2007 Lecture 17 C/CS/Phs C9 Deutsch ad Deutsch-Josza algorithms 0/0/07 Fall 007 Lecture 7 Readigs Beeti et al., Ch. 3.9-3.9. Stolze ad Suter, Quatum Computig, Ch. 8. - 8..5) Nielse ad Chuag, Quatum Computatio ad Quatum

More information

Linear Regression Demystified

Linear Regression Demystified Liear Regressio Demystified Liear regressio is a importat subject i statistics. I elemetary statistics courses, formulae related to liear regressio are ofte stated without derivatio. This ote iteds to

More information

Know your major scales

Know your major scales page 8 Know your major scales Major and minor scales use every note name (A B C D E F G), but start on different notes. The notes of a scale move stepwise by intervals of a tone or semitone. The pattern

More information

MEI Conference 2009 Stretching students: A2 Core

MEI Conference 2009 Stretching students: A2 Core MEI Coferece 009 Stretchig studets: A Core Preseter: Berard Murph berard.murph@mei.org.uk Workshop G How ca ou prove that these si right-agled triagles fit together eactl to make a 3-4-5 triagle? What

More information

Conditional Probability. Given an event M with non zero probability and the condition P( M ) > 0, Independent Events P A P (AB) B P (B)

Conditional Probability. Given an event M with non zero probability and the condition P( M ) > 0, Independent Events P A P (AB) B P (B) Coditioal robability Give a evet M with o zero probability ad the coditio ( M ) > 0, ( / M ) ( M ) ( M ) Idepedet Evets () ( ) ( ) () ( ) ( ) ( ) Examples ) Let items be chose at radom from a lot cotaiig

More information

Introduction To Discrete Mathematics

Introduction To Discrete Mathematics Itroductio To Discrete Mathematics Review If you put + pigeos i pigeoholes the at least oe hole would have more tha oe pigeo. If (r + objects are put ito boxes, the at least oe of the boxes cotais r or

More information

CHAPTER 5. Theory and Solution Using Matrix Techniques

CHAPTER 5. Theory and Solution Using Matrix Techniques A SERIES OF CLASS NOTES FOR 2005-2006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 3 A COLLECTION OF HANDOUTS ON SYSTEMS OF ORDINARY DIFFERENTIAL

More information

Symphonic Literature Excerpts

Symphonic Literature Excerpts Symhonic Literature Excerts CBW Symhonic Band Audition 017018 Perorm the ollowing excerts rom Symhonic Band literature in any order and include them in your recording The temos listed should be ollowed

More information

Butterworth LC Filter Designer

Butterworth LC Filter Designer Butterworth LC Filter Desiger R S = g g 4 g - V S g g 3 g R L = Fig. : LC filter used for odd-order aalysis g R S = g 4 g V S g g 3 g - R L = useful fuctios ad idetities Uits Costats Table of Cotets I.

More information

Goodness-of-Fit Tests and Categorical Data Analysis (Devore Chapter Fourteen)

Goodness-of-Fit Tests and Categorical Data Analysis (Devore Chapter Fourteen) Goodess-of-Fit Tests ad Categorical Data Aalysis (Devore Chapter Fourtee) MATH-252-01: Probability ad Statistics II Sprig 2019 Cotets 1 Chi-Squared Tests with Kow Probabilities 1 1.1 Chi-Squared Testig................

More information

Is mathematics discovered or

Is mathematics discovered or 996 Chapter 1 Sequeces, Iductio, ad Probability Sectio 1. Objectives Evaluate a biomial coefficiet. Expad a biomial raised to a power. Fid a particular term i a biomial expasio. The Biomial Theorem Galaxies

More information

18th Bay Area Mathematical Olympiad. Problems and Solutions. February 23, 2016

18th Bay Area Mathematical Olympiad. Problems and Solutions. February 23, 2016 18th Bay Area Mathematical Olympiad February 3, 016 Problems ad Solutios BAMO-8 ad BAMO-1 are each 5-questio essay-proof exams, for middle- ad high-school studets, respectively. The problems i each exam

More information

Chapter 0. Review of set theory. 0.1 Sets

Chapter 0. Review of set theory. 0.1 Sets Chapter 0 Review of set theory Set theory plays a cetral role i the theory of probability. Thus, we will ope this course with a quick review of those otios of set theory which will be used repeatedly.

More information

CALCULATING FIBONACCI VECTORS

CALCULATING FIBONACCI VECTORS THE GENERALIZED BINET FORMULA FOR CALCULATING FIBONACCI VECTORS Stuart D Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithacaedu ad Dai Novak Departmet

More information

The Binomial Theorem

The Binomial Theorem The Biomial Theorem Robert Marti Itroductio The Biomial Theorem is used to expad biomials, that is, brackets cosistig of two distict terms The formula for the Biomial Theorem is as follows: (a + b ( k

More information

Find a formula for the exponential function whose graph is given , 1 2,16 1, 6

Find a formula for the exponential function whose graph is given , 1 2,16 1, 6 Math 4 Activity (Due by EOC Apr. ) Graph the followig epoetial fuctios by modifyig the graph of f. Fid the rage of each fuctio.. g. g. g 4. g. g 6. g Fid a formula for the epoetial fuctio whose graph is

More information

Compositions of Random Functions on a Finite Set

Compositions of Random Functions on a Finite Set Compositios of Radom Fuctios o a Fiite Set Aviash Dalal MCS Departmet, Drexel Uiversity Philadelphia, Pa. 904 ADalal@drexel.edu Eric Schmutz Drexel Uiversity ad Swarthmore College Philadelphia, Pa., 904

More information

0, otherwise. EX = E(X 1 + X n ) = EX j = np and. Var(X j ) = np(1 p). Var(X) = Var(X X n ) =

0, otherwise. EX = E(X 1 + X n ) = EX j = np and. Var(X j ) = np(1 p). Var(X) = Var(X X n ) = PROBABILITY MODELS 35 10. Discrete probability distributios I this sectio, we discuss several well-ow discrete probability distributios ad study some of their properties. Some of these distributios, lie

More information

Image Spaces. What might an image space be

Image Spaces. What might an image space be Image Spaces What might a image space be Map each image to a poit i a space Defie a distace betwee two poits i that space Mabe also a shortest path (morph) We have alread see a simple versio of this, i

More information

Shannon s noiseless coding theorem

Shannon s noiseless coding theorem 18.310 lecture otes May 4, 2015 Shao s oiseless codig theorem Lecturer: Michel Goemas I these otes we discuss Shao s oiseless codig theorem, which is oe of the foudig results of the field of iformatio

More information

8. СОВЕТУВАЊЕ. Охрид, септември ANALYSIS OF NO LOAD APPARENT POWER AND FREQUENCY SPECTRUM OF MAGNETIZING CURRENT FOR DIFFERENT CORE TYPES

8. СОВЕТУВАЊЕ. Охрид, септември ANALYSIS OF NO LOAD APPARENT POWER AND FREQUENCY SPECTRUM OF MAGNETIZING CURRENT FOR DIFFERENT CORE TYPES 8. СОВЕТУВАЊЕ Охрид, 22 24 септември Leoardo Štrac Frajo Keleme Kočar Power Trasformers Ltd. ANALYSS OF NO LOAD APPARENT POWER AND FREQENCY SPECTRM OF MAGNETZNG CRRENT FOR DFFERENT CORE TYPES ABSTRACT

More information

The Pendulum. Purpose

The Pendulum. Purpose The Pedulum Purpose To carry out a example illustratig how physics approaches ad solves problems. The example used here is to explore the differet factors that determie the period of motio of a pedulum.

More information

Lecture 25 (Dec. 6, 2017)

Lecture 25 (Dec. 6, 2017) Lecture 5 8.31 Quatum Theory I, Fall 017 106 Lecture 5 (Dec. 6, 017) 5.1 Degeerate Perturbatio Theory Previously, whe discussig perturbatio theory, we restricted ourselves to the case where the uperturbed

More information

Student s Printed Name:

Student s Printed Name: Studet s Prited Name: Istructor: XID: C Sectio: No questios will be aswered durig this eam. If you cosider a questio to be ambiguous, state your assumptios i the margi ad do the best you ca to provide

More information

The Fizeau Experiment with Moving Water. Sokolov Gennadiy, Sokolov Vitali

The Fizeau Experiment with Moving Water. Sokolov Gennadiy, Sokolov Vitali The Fizeau Experimet with Movig Water. Sokolov Geadiy, Sokolov itali geadiy@vtmedicalstaffig.com I all papers o the Fizeau experimet with movig water, a aalysis cotais the statemet: "The beams travel relative

More information

Erratum to: An empirical central limit theorem for intermittent maps

Erratum to: An empirical central limit theorem for intermittent maps Probab. Theory Relat. Fields (2013) 155:487 491 DOI 10.1007/s00440-011-0393-0 ERRATUM Erratum to: A empirical cetral limit theorem for itermittet maps J. Dedecker Published olie: 25 October 2011 Spriger-Verlag

More information

CALCULATION OF FIBONACCI VECTORS

CALCULATION OF FIBONACCI VECTORS CALCULATION OF FIBONACCI VECTORS Stuart D. Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithaca.edu ad Dai Novak Departmet of Mathematics, Ithaca College

More information

Chapter 1 : Combinatorial Analysis

Chapter 1 : Combinatorial Analysis STAT/MATH 394 A - PROBABILITY I UW Autum Quarter 205 Néhémy Lim Chapter : Combiatorial Aalysis A major brach of combiatorial aalysis called eumerative combiatorics cosists of studyig methods for coutig

More information

CS276A Practice Problem Set 1 Solutions

CS276A Practice Problem Set 1 Solutions CS76A Practice Problem Set Solutios Problem. (i) (ii) 8 (iii) 6 Compute the gamma-codes for the followig itegers: (i) (ii) 8 (iii) 6 Problem. For this problem, we will be dealig with a collectio of millio

More information

ON THE RELIABILITY OF AN n-component SYSTEM 1. INTRODUCTION

ON THE RELIABILITY OF AN n-component SYSTEM 1. INTRODUCTION ON THE RELIABILITY OF AN -COMPONENT SYSTEM Do Rawligs ad Lawrece Sze Uder assumptios compatible with the theory of Markov chais, we use a property of Vadermode matrices to examie the reliability of a -compoet

More information

Unit 6: Sequences and Series

Unit 6: Sequences and Series AMHS Hoors Algebra 2 - Uit 6 Uit 6: Sequeces ad Series 26 Sequeces Defiitio: A sequece is a ordered list of umbers ad is formally defied as a fuctio whose domai is the set of positive itegers. It is commo

More information

Maximum and Minimum Values

Maximum and Minimum Values Sec 4.1 Maimum ad Miimum Values A. Absolute Maimum or Miimum / Etreme Values A fuctio Similarly, f has a Absolute Maimum at c if c f f has a Absolute Miimum at c if c f f for every poit i the domai. f

More information

Lecture 7: Density Estimation: k-nearest Neighbor and Basis Approach

Lecture 7: Density Estimation: k-nearest Neighbor and Basis Approach STAT 425: Itroductio to Noparametric Statistics Witer 28 Lecture 7: Desity Estimatio: k-nearest Neighbor ad Basis Approach Istructor: Ye-Chi Che Referece: Sectio 8.4 of All of Noparametric Statistics.

More information

Solutions to Final Exam

Solutions to Final Exam Solutios to Fial Exam 1. Three married couples are seated together at the couter at Moty s Blue Plate Dier, occupyig six cosecutive seats. How may arragemets are there with o wife sittig ext to her ow

More information

Probability theory and mathematical statistics:

Probability theory and mathematical statistics: N.I. Lobachevsky State Uiversity of Nizhi Novgorod Probability theory ad mathematical statistics: Law of Total Probability. Associate Professor A.V. Zorie Law of Total Probability. 1 / 14 Theorem Let H

More information

COMPUTING SUMS AND THE AVERAGE VALUE OF THE DIVISOR FUNCTION (x 1) + x = n = n.

COMPUTING SUMS AND THE AVERAGE VALUE OF THE DIVISOR FUNCTION (x 1) + x = n = n. COMPUTING SUMS AND THE AVERAGE VALUE OF THE DIVISOR FUNCTION Abstract. We itroduce a method for computig sums of the form f( where f( is ice. We apply this method to study the average value of d(, where

More information

Representations of State Vectors and Operators

Representations of State Vectors and Operators Chapter 10 Represetatios of State Vectors ad Operators I the precedig Chapters, the mathematical ideas uderpiig the quatum theory have bee developed i a fairly geeral (though, admittedly, ot a mathematically

More information

MISCELLANEOUS SEQUENCES & SERIES QUESTIONS

MISCELLANEOUS SEQUENCES & SERIES QUESTIONS MISCELLANEOUS SEQUENCES & SERIES QUESTIONS Questio (***+) Evaluate the followig sum 30 r ( 2) 4r 78. 3 MP2-V, 75,822,200 Questio 2 (***+) Three umbers, A, B, C i that order, are i geometric progressio

More information

The picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled

The picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled 1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how

More information

In the preceding Chapters, the mathematical ideas underpinning the quantum theory have been

In the preceding Chapters, the mathematical ideas underpinning the quantum theory have been Chapter Matrix Represetatios of State Vectors ad Operators I the precedig Chapters, the mathematical ideas uderpiig the quatum theory have bee developed i a fairly geeral (though, admittedly, ot a mathematically

More information

*X203/701* X203/701. APPLIED MATHEMATICS ADVANCED HIGHER Numerical Analysis. Read carefully

*X203/701* X203/701. APPLIED MATHEMATICS ADVANCED HIGHER Numerical Analysis. Read carefully X0/70 NATIONAL QUALIFICATIONS 006 MONDAY, MAY.00 PM.00 PM APPLIED MATHEMATICS ADVANCED HIGHER Numerical Aalysis Read carefully. Calculators may be used i this paper.. Cadidates should aswer all questios.

More information

NATIONAL SENIOR CERTIFICATE GRADE 12

NATIONAL SENIOR CERTIFICATE GRADE 12 NAIONAL SENI CERIFICAE GRADE MAHEMAICS P NOVEMBER 008 MEMANDUM MARKS: 0 his memoradum cosists of pages. Copright reserved Please tur over Mathematics/P DoE/November 008 NSC Memoradum Cotiued Accurac will

More information

Summary: Congruences. j=1. 1 Here we use the Mathematica syntax for the function. In Maple worksheets, the function

Summary: Congruences. j=1. 1 Here we use the Mathematica syntax for the function. In Maple worksheets, the function Summary: Cogrueces j whe divided by, ad determiig the additive order of a iteger mod. As described i the Prelab sectio, cogrueces ca be thought of i terms of coutig with rows, ad for some questios this

More information

(c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is

(c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is Calculus BC Fial Review Name: Revised 7 EXAM Date: Tuesday, May 9 Remiders:. Put ew batteries i your calculator. Make sure your calculator is i RADIAN mode.. Get a good ight s sleep. Eat breakfast. Brig:

More information

Chapter 2 Feedback Control Theory Continued

Chapter 2 Feedback Control Theory Continued Chapter Feedback Cotrol Theor Cotiued. Itroductio I the previous chapter, the respose characteristic of simple first ad secod order trasfer fuctios were studied. It was show that first order trasfer fuctio,

More information

Chapter 12 Sound Waves

Chapter 12 Sound Waves Chapter 2 Soud Waves We study the properties ad detectio o a particular type o wave soud waves. A speaker geerates soud. The desity o the air chages as the wave propagates. The rage o requecies that ca

More information

Different kinds of Mathematical Induction

Different kinds of Mathematical Induction Differet ids of Mathematical Iductio () Mathematical Iductio Give A N, [ A (a A a A)] A N () (First) Priciple of Mathematical Iductio Let P() be a propositio (ope setece), if we put A { : N p() is true}

More information

Mathematical Methods for Physics and Engineering

Mathematical Methods for Physics and Engineering Mathematical Methods for Physics ad Egieerig Lecture otes Sergei V. Shabaov Departmet of Mathematics, Uiversity of Florida, Gaiesville, FL 326 USA CHAPTER The theory of covergece. Numerical sequeces..

More information

Solutions 3.2-Page 215

Solutions 3.2-Page 215 Solutios.-Page Problem Fid the geeral solutios i powers of of the differetial equatios. State the reurree relatios ad the guarateed radius of overgee i eah ase. ) Substitutig,, ad ito the differetial equatio

More information

The Maximum-Likelihood Decoding Performance of Error-Correcting Codes

The Maximum-Likelihood Decoding Performance of Error-Correcting Codes The Maximum-Lielihood Decodig Performace of Error-Correctig Codes Hery D. Pfister ECE Departmet Texas A&M Uiversity August 27th, 2007 (rev. 0) November 2st, 203 (rev. ) Performace of Codes. Notatio X,

More information

VICTORIA JUNIOR COLLEGE Preliminary Examination. Paper 1 September 2015

VICTORIA JUNIOR COLLEGE Preliminary Examination. Paper 1 September 2015 VICTORIA JUNIOR COLLEGE Prelimiary Eamiatio MATHEMATICS (Higher ) 70/0 Paper September 05 Additioal Materials: Aswer Paper Graph Paper List of Formulae (MF5) 3 hours READ THESE INSTRUCTIONS FIRST Write

More information

Notes for Lecture 11

Notes for Lecture 11 U.C. Berkeley CS78: Computatioal Complexity Hadout N Professor Luca Trevisa 3/4/008 Notes for Lecture Eigevalues, Expasio, ad Radom Walks As usual by ow, let G = (V, E) be a udirected d-regular graph with

More information

Introductions to PartitionsP

Introductions to PartitionsP Itroductios to PartitiosP Itroductio to partitios Geeral Iterest i partitios appeared i the 7th cetury whe G. W. Leibiz (669) ivestigated the umber of ways a give positive iteger ca be decomposed ito a

More information

Sound event detection and rhythmic parsing of music signals

Sound event detection and rhythmic parsing of music signals Soud evet detectio ad rhythmic parsig of music sigals ISMIR Graduate School, October 4th-9th, 2004 Cotets: Itroductio Measurig the degree of chage i music sigals Oset detectio Rhythmic pulse estimatio

More information

Exponential Rules and How to Use Them Together

Exponential Rules and How to Use Them Together Epoetial Rules ad How to Use Them Together Welcome back! Before we look at problems that ivolve the use of all of our epoetial rules, let s review the epoetial rules ad get a strateg for attackig these

More information

Signal Processing in Mechatronics. Lecture 3, Convolution, Fourier Series and Fourier Transform

Signal Processing in Mechatronics. Lecture 3, Convolution, Fourier Series and Fourier Transform Sigal Processig i Mechatroics Summer semester, 1 Lecture 3, Covolutio, Fourier Series ad Fourier rasform Dr. Zhu K.P. AIS, UM 1 1. Covolutio Covolutio Descriptio of LI Systems he mai premise is that the

More information

Statistics 511 Additional Materials

Statistics 511 Additional Materials Cofidece Itervals o mu Statistics 511 Additioal Materials This topic officially moves us from probability to statistics. We begi to discuss makig ifereces about the populatio. Oe way to differetiate probability

More information

Physics 232 Gauge invariance of the magnetic susceptibilty

Physics 232 Gauge invariance of the magnetic susceptibilty Physics 232 Gauge ivariace of the magetic susceptibilty Peter Youg (Dated: Jauary 16, 2006) I. INTRODUCTION We have see i class that the followig additioal terms appear i the Hamiltoia o addig a magetic

More information