A Collection of Three Pieces for Solo Instrument and Electronics

Size: px
Start display at page:

Download "A Collection of Three Pieces for Solo Instrument and Electronics"

Transcription

1 Weste Michiga Uivesity ScholaWoks at WMU Maste's Theses Gaduate College -208 A Collectio o Thee ieces o Solo Istumet ad Electoics Caoly Bochedig Weste Michiga Uivesity, cgbochedig@gmail.com ollo this ad additioal oks at: htts://scholaoks.mich.edu/mastes_theses at o the Comositio Commos Recommeded Citatio Bochedig, Caoly, "A Collectio o Thee ieces o Solo Istumet ad Electoics" (208). Maste's Theses. 29. htts://scholaoks.mich.edu/mastes_theses/29 This Mastes Thesis-Oe Access is bought to you o ee ad oe access by the Gaduate College at ScholaWoks at WMU. It has bee acceted o iclusio i Maste's Theses by a authoized admiistato o ScholaWoks at WMU. o moe iomatio, lease cotact maia.budza@mich.edu.

2 A COLLECTION O THREE IECES OR SOLO INSTRUMENT AND ELECTRONICS by Caoly Bochedig A thesis submitted to the Gaduate College i atial ulillmet o the equiemets o the degee o Maste o Music School o Music Weste Michiga Uivesity Ail 208 Thesis Committee: Chistohe Biggs, D.M.A., Chai Lisa Coos, h.d. Richad Johso, D.M.A.

3 Coyight by Caoly Bochedig 208

4 A COLLECTION O THREE IECES OR SOLO INSTRUMENT AND ELECTRONICS Caoly Bochedig, M.M. Weste Michiga Uivesity, 208 This collectio o thee musical comositios o solo istumets ad electoics exloes the iteactio betee acoustic istumets ad electoic media ith a seciic ocus o musical om. The musical om is aaet due to clea musical segmetatios, ad the coheece o each ok is maitaied by the ite-cotextual elatioshis evidet i the musical mateial. Each iece is divided ito thee ats: begiig, middle, ad ed. Each begiig sets u a stog elatioshi betee the to media by usig closely timed actio-eactio gestues. The middle sectio diveges aay om these eactioay gestues, hile the edig the bigs about a shot etu o ealie mateial. Both the electoic ad acoustic ats adhee to this stuctue. Each iece combies live ad ixed electoics ith a dieet istumetal tye i ode to exloe the otetial elatioshis betee both media. Live electoics ocess soud i eal time heeas ixed media is a combiatio o set audio iles that soud the exact same ay each time they ae layed. Tagled Illusios is itte o koto, a lucked stiged istumet, ad tiggeed ixed media. Emyea Tides is itte o alto lute, a id istumet, ad a combiatio o edomiatly tiggeed ixed media ad live electoic eects. Leaig to Seak is itte o oma s voice, a vocal istumet, ad a combiatio o edomiatly live electoic eects ad tiggeed ixed media.

5 TABLE O CONTENTS TANGLED ILLUSIONS Scoe... EMYREAN TIDES Scoe...2 LEARNING TO SEAK Scoe...22 ii

6 Tagled Illusios i a Tisted Deam Caoly Bochedig q = 20 oshide ~~~~ b ~~~~~~~ ight o bidges ~~~~~~~~ Koto? let o bidges Electoics 2 0:09 0:00 ~~~~~~~~~~ 7 b ~~~ ~~~~ ~~ b b ~~~~ ~~ b ~~~ b! :0 0:08 J oshide j b J J vib. 5 b ~ oshide j b > j J 5 6 0:05 5 0:7

7 2 6 vib. 8 0: :07 oshide vib. b 2 vib. 7 0:08 2 b g 2 oshide gadually :05 5 g j... vib.. 0:0 2 0:05!2

8 8 6 oshide gadually 2 6 0:07. b 2 5 0:0 6 0:06 b. 2 0:0 oshide o b. 2 b 5 oshide o b. j b b ~ ~~~~~~~ ~~~~~ 55 b b b b :08 b oshide b b 6 8 ~ ~ ~~~ ~~!

9 b b b b b b add itches doad as aidity iceases b 8 2 ƒ b scae alog stig : 2 oshide j b 8 oshide gadually :2 2 b b 2 ~~~~~~~!

10 :2? 2 0:06 8 b? :02 b b > 9 > J 0 2 0:06!5

11 87? 9 9 Mute b Mute b Mute b b Mute subito b b Mute b b b Mute b b b b b 8 9 ~~~~~~~~~!6

12 97 Mute 5 ~~~~~ 00 Mute j 5 Mute Mute 5 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~ ~~~ ~~~~~~~ 02 5 ƒ 2 0:28 2 subito b b b 20 2 uiously ~~~~~~~~~~~~ ~~~ ~~~ ~~ ~~ ~~ ~~~~~ 25 0: : : ƒ!7

13 26? b j j j b Ï ~~~~~~~~~~~~~~ ~~~~ ~~~ ~~~ ~~~~~~~ ~~~~~~~ ~~~~~~~~~~~~~~ ~~~~ ~~~~~~ ~~~~~~~ ~~~~~~~ ~~~~~~ ~~~~~~~~ ~~~~~~ ~~~~~~~~ 29 5 ȯ ȯ 5 7 meditatively 28 :0 o 5 π o 5. b o ȯ 9 8 b 6 ȯ b o :20 ȯ 8 oshide vib. ȯ!8

14 5 vib.. oshide vib. vib. 0 0:7 ~~~~~~~~~ 59 68? b > 0 alm slas movig uad i itch o both sides o the bidges b > 0!9

15 7? π ~~~~~~~~~~~~~~ ige luck b oshide vib. 0:0 b :50 2 0:06 oshide vib. b 6 5 b 6 5 0:06 6 0:0 7 0:07 b oshide j b 5 6 7!0

16 j 7 7 0:06 8 oshide vib. b J 9 oshide j N b oshide vib. 9 9 ige luck oshide vib. b 6 8 0:09 b b. J oshide vib.. J j o. j b J b 9 6 b 9 0:2 7 j oshide vib. ȯ π ~~ 22 j j o o b o 2 o π j ou. J 0 0:7 0!

17 Emyea Tides Caoly Bochedig Alto lute Electoics q = 0 b > b [] 0: > 2 2 b > b > b [2] 0:0" > " b [] 0:0" > b J π [] 0:0" > 5 b b b. b b b b > b > [5]. 0:05" > [6] 0:" > # 2 ƒ b b ẇ ȯ [7] 0:8" >!2

18 > b. b π b [8] 0:05" > b b. lz. π ȯ [9] 0:5" >.. o. [ssssshhhhhhaaaaaaa] π b [0] 0:06" > 5" o b [ssssshhh - [] 0:09" > hhhhha - 5 aaaaaaa] lz.. [2] 0:0" >.. b [] 0:0" > b. 50. j lz.. o b R. b b. b [] 0:5" >!

19 55 b J > b >. b > b b J. b 58 o b b > b. b. [5] 0:5" > b b - " b b o Sh... o o 7 6 ƒ [6] 0:2" > b b. π sh R b.... b b π sh sh sh!

20 . " bbbbbbbb 7 75 o [ssshhh - hhaaaa] o b j j b> b bb b j > b [7] 0:08" > ch ch ch ch. [8] 0:0" > 78. b> ch ch ch b t t t t t t t t lz. T.R.. t k t t k t [9] 0:0" > [20] 0:0" > begi delay + eveb [2] 0:0" 82. b b b t k t t k t ch ch ch [22] 0:09" > t k t k t k t k t k t k t k 85 b # b b t k t k t k t T.R.. j # > b > > t T.R. b b t k t ch t k t k [2] 0:05" >!5

21 89 9 b b j > T.R. b > b ch ch ch t k b > [2] 0:02" > b. ch t ch b ch ch ch b> > T.R. b J T.R. [25] 0:09" > ed delay + eveb. [sshha] b o lz.. t k t k [26] 0:08" > [27] 0:05" > 99 b b > j > j > j T.R. b b ch k ch k t k ch k ch k [28] 0:0" > [29] 0:02" > [0] 0:0" > 0 08 b >..... t k ch k t k t [shhaa]. [] 0:05" >. b. R ch b.. R R [5] 0:0" >.. K b... b. [2] o lz. b..... J. 0:0" > > ch. b R. R... R t k ch [] [] 0:0" > 0:0" > T.R. T.R. k t t t [6] 0:0" >!6

22 5.. b. R > b R J ch 9 b o.. b t ch o b ch t [shhhhhh] ch [7] 0:0" > R >. R > #.. k t. [9] 0:02" > R... b R. b j R.. > b T.R. [] # # # # # [2] 0:22" > o J 0:0" > " b. J. [0] 0:06" >. b b. # b > b J ch t k t k.. #. # b.. [8] T.R. π 0:06" >. R R... T.R.. # > # # o ch [shhh]!7

23 29.. b. R b R b. ch t ch ch k b t k t. T.R... b. b. b. b. b >. R. [] 0:06" > t 2 5 b b ch k t k b b j > shule b t, k, ch b b. R o k t k ch [sssha] shule t, k, ch b b b b j. oj [sha] j K b K b T.R. [] 0:02" > > J [5] 0:02" > [6] 0:0" > [7] 0:05" > 8 # > b J b b b b t k t k t k t k t k t k t k. # #. ch t k t k ch k ch R. j. b ch k ch k [8] 0:56" > b b J. b b b. b t k t k!8

24 5 b b k t k [shh] o b # # b. R t k b. J.. 8 b. b.... b b b. b R. b b ƒ 5 b 5 R ch b # b " b o b [shha], b Ï ch 56 ẇ o [9] 0:" >!9

25 65 j b> S π 2 2 b b.. [50] 0:07" > o [52] 0:0" > [5] 0:0" > b b b b. o. π lz. [5] 0:02" > [5]. J > 0:0" > begi delay + eveb b T.R. j [55] 0:0" > T.R. T.R. j b j. [56] 0:05" > 8 #. #. #.. #. #. #.. b. [57] 0:0" > [58] 0:0" > 87 J j # # #. π. b R R.. b.. b -, - π [59] [60] 0:0" > 0:02" >!20

26 , -. π 92. R > t. R J k ch # π # # #. b [6] 0:0" > [62] 0:0" > [6] 0:06" > 97 ẇ o [6] 0:0" > ed delay + eveb 20 # π # #. # # # U [65] 0:0" > [66]!2

27 Leaig to Seak Caoly Bochedig Voice To Mid Bottom Elec. q = 60 meek, ith evous eegy π π π π quick, eely Hm Hm Hm Hm. U.. Hm Hm Hm N Nah ah [] [2] sectal delay: o sectal delay: o. R classically sug π.. [i] [e] [a] [i] I I? I? I? I Yuh! Yuh Yuh Yuh Yuh Yuh [] [] time delay: o time delay: o soud: "yuh gas" classically sug.... π U. j j [e] [a] [i] [o] [u] I? I? I Yuh! Hm [e] [a] [o] [u] [5] [6] [7] [8] lage: o time delay: o lage: o time delay: o 2 /2 /2 5 6 /2 /2!22

28 J > [i] [e] [a] [o] π. J. [a] mm [e] [u] [a] I am m m m m [9] time delay: o π - -. j. >. U R [e] [u] [i] [u] you [u] [o] m m [a] [e] [i] I I. RÔ. K RÔ delicately π almost to gol > [0] [] soud "you" soud "heave gas" K.. J... K K K.. RÔ.. j.. [e][u] [e] [u] I Am I Am mm [e][u] [e][u][e][u][e][u] [i] [e] [i] [e] [i] [e] ah ustated almost to gol > [2] [] soud "you" time delay shotes soud "e-u" gol - - >.. Ah [u] I [e] Am N N [a] Nah [o] Yuh Ah [e] I [u] You [] [5] [6] sectal delay: o time delay legthes ƒ time delay: o sectal delay: o /2 /2!2

29 hiseed beathy. j > j >. j... 2 /2 2 /2. π R You Y(ou) N(eed) Nee -? Nee? Nee? Nee? d d d d Nee d [] U [7] [8] time delay: o time delay: o π j. d d d d d d d do d d d [9] [20] lage: o lage: o... K > j.. sibilat shule icoheet mumble N ah - t sho sho sho sho sho sho d sh sh sh sh I I shou d [2] [22] sibilat shule soud "sh" sectal delay: o. icoheet mumble Mm Mm do [o] [2] [2] [u] I [e] time delay: o soud "gas" icoheet mumble sectal delay: o time delay: o lage: o /2 2 /2 /2 /2 6!2

30 K R J j I should ot do't icoheet d d d d d d d(o't) [o] [u] mumble [25] lage: o soud "you" π subito R classically sug classically hiseed sug beathy J >. R (eed) d(o't) ha ha ha ha ha you sh(ould) [u] You! Y(ou). Hm [26] [27] [28] soud "you" soud "ha" soud "sh" R..... I I I [e] [a] [o] [u] K... RÔ â.... R You You You ha ha ha ha ha ha ha I I should eed do't I should ot do't should. R R R Ô Yuh! do't am ot I do't eed eed you you you I should eed do't do't do't am N N I [0] [] time delay: o soud "you" soud "gas" classically sug [29] soud "sh" soud "ha" icoheet mumble!25

31 seak-sug seak oceully do't eed do't eed do't eed do't eed icoheet I Am Not! mumble ƒ > R suddely act evous agai. [2] Wait o electoics to ade out comletely. time delay: o evely, obotic j j j t t t t t t t t t t t t t t t t t t t t t t t t t t â t [] gol - - > laboed sibiliat shule soud "hiss" soud "hises" >. icoheet mumble. RÔ >. > > > â J duh duh ah! I tuh tuh tuh tuh ha(t) π j. J j. gol - - > gol ha(t) o(t) t ah ha ha ha ah I d d d do t sibilat Am You Mm shule sibilat shule [] lage: o /2 6!26

32 U. hiseed beathy K t t t t t t t t t t t h(o) I ha t t t t t t ha ha ha [5] lage: o soud "beath" [6] π gol.. J >. ha Hm Mm Yuh Yuh Yuh Yuh You Mm Mm Mm gol. [7] /2 /2 [8] distotio: o distotio: o π J Who a(m) a a a a a a a a ha ha ha ha ha ha ha ha [a] [i] [0] [9] a coidet laugh to yousel time delay: o time delay: o soud "laughte" hiseed beathy icoheet mumble. [a] Am Wha soke. J I [] electoics cut o. t t t t t t t t t Am Mm Mm [2] time delay: o sectal delay: o gol R [] soud "cuchy" /2 6 /2!27

33 . hiseed beathy soke. delicately, lightly. K. K Mm Mm ha(t) a(m) ha ha ha ha ha ha ha I a - m I am I am [a] - mm [5] [] [6] time delay: o soud "laughte" soud "hise" soud "i-u" gol π π. j. j. K..... K. K K K soke I Am Mm N Ah t t t t t t t t t t t s s s do't ho ha ha ha(t) do't eed eed h ha ha ha ha [e][u] [e][u] [e][u] ha icoheet You mumble [8] [9] [7] [50] hiseed beathy sectal delay: o time delay: o soud "e-u" soud "laughte" time delay: o j K R. j.. R soke ha(t) t t t t ha t t t s s s t t t t t t t t Yuh! Hm You You [52] [5] ha ha ha ha ha ha ha ha [5] hiseed beathy j soud "beath" soud "beath" soud "cuchy" >..... J movig om hiseed to sug od. time delay: o lage: o >.. J >. K > gol - - > R > U ha(t) t ah I I ah I am ot sib. I I? I duh duh ha ha ha ah I I I ah I am ot ot you you I Am Not Am Not ho ot you ot ha t t s say! [55] [5] agily ƒ [56] soud "ha" soud "hiss" lage: o soud "gas" /2 /2!28

34 . K. K. K K.. gol... gol - - > you I Am I Am I Am I I I I I I I I Am. Am Hm Hm yuh yuh [58] [59] [57] distotio: o distotio: o time delay: o J.. j... J. [60] distotio: o distotio: o J.. J. You yuh yuh S Say Say Say Say s s s s s s s s Say Say Say Say Say [62] [6] soud "you" soud "i-u".... R Ô. [6] soud "beath" time delay: o. RÔ. gol I say I am I am am am mm ha ha ho K R. hat am I I I say say hat Mm Hm [65] [66] [6] lage: o soud "hiss". j. gol soud "you" distotio: o distotio: o icoheet mumble ƒ... [67] gol - - > am ho say you I ho hat ho hat hu hu hu hu hu hu hu hu [o] Yuh You ha ha ha ha ha ha ha ha hat ho ho N Ah I Am [69] [70] [68] time delay: o soud "gas" soud "laughte" time delay: o [7] distotio: o time delay: o 8 7 8!29

35 aste ezied ah. t say ho ot > > j tuhtuh tuh tuh Ah! > I am ot I am ot I am ot Ah [7] [7] [72] sibilat shule sectal delay: o lage: o distotio iceases ith some golig J > I Am Not Who Not What Who icoheet hat hat hat hat hat hat hat hat hat hat mumble soud "ah!" time delay: o sectal delay: o [77] [76] soud "delay laugh" time delay iceases distotio: o shule ods: am, I, ot, say, hat, ho, you [75] soud "you" soud "beath" [78] distotio: o soud "gas" gol to sceam > R ƒ uiously, shoutig > > > > ä ä Ah I am ot hat you say! [80] [8] [79] time delay: o distotio: o cut o electoics distotio: o distotio iceases electoics o Ï [82]!0

干珠満珠. The Tide Jewels

干珠満珠. The Tide Jewels 干珠満珠 The Tide eels o Ochesta [201] Chistohe Laosa chislaosacom 干珠満珠 The Tide eels o Ochesta [201] Duatio: ' Chistohe Laosa chislaosacom 2 Istumetatio lutes ( d doules icc) 2 Ooes Eglish Ho 2 Claiets i

More information

High Voice Two Violins Viola & Accordion

High Voice Two Violins Viola & Accordion CHISTOHE WILLIAM IECE ON A OEM O ADELAIE e t'adoe à l'égal de la voûte noctune om Les leus du Mal High Two Violins Viola Accodion 00 Chistohe William iece ASCA, All ights eseved o Katy e t'adoe à l'égal

More information

Henrik Denerin. seals II. for cello and soprano. full score

Henrik Denerin. seals II. for cello and soprano. full score Henik Denein seals II o and ull scoe Henik Denein seals II (2014-15) o and duation: 5 seconds Notation: GENERA (all instuments) Scoe witten at laying itch All instuments ae tansosed and thei ats ae identical

More information

Fortgeschrittene Datenstrukturen Vorlesung 11

Fortgeschrittene Datenstrukturen Vorlesung 11 Fortgeschrittee Datestruture Vorlesug 11 Schriftführer: Marti Weider 19.01.2012 1 Succict Data Structures (ctd.) 1.1 Select-Queries A slightly differet approach, compared to ra, is used for select. B represets

More information

ON EUCLID S AND EULER S PROOF THAT THE NUMBER OF PRIMES IS INFINITE AND SOME APPLICATIONS

ON EUCLID S AND EULER S PROOF THAT THE NUMBER OF PRIMES IS INFINITE AND SOME APPLICATIONS Joual of Pue ad Alied Mathematics: Advaces ad Alicatios Volume 0 Numbe 03 Pages 5-58 ON EUCLID S AND EULER S PROOF THAT THE NUMBER OF PRIMES IS INFINITE AND SOME APPLICATIONS ALI H HAKAMI Deatmet of Mathematics

More information

Math 7409 Homework 2 Fall from which we can calculate the cycle index of the action of S 5 on pairs of vertices as

Math 7409 Homework 2 Fall from which we can calculate the cycle index of the action of S 5 on pairs of vertices as Math 7409 Hoewok 2 Fall 2010 1. Eueate the equivalece classes of siple gaphs o 5 vetices by usig the patte ivetoy as a guide. The cycle idex of S 5 actig o 5 vetices is 1 x 5 120 1 10 x 3 1 x 2 15 x 1

More information

Preview Only. Legal Use Requires Purchase. Too Close for Comfort JAZZ

Preview Only. Legal Use Requires Purchase. Too Close for Comfort JAZZ Too Close or Comort Words and Music y JERRY BOC, LARRY HOLOCENER and GEORGE DAVID WEISS Arranged y GORDON GOODWIN Conductor Solo 1st E Alto Saxohone 2nd E Alto Saxohone 1st B Tenor Saxohone 2nd B Tenor

More information

BINOMIAL THEOREM An expression consisting of two terms, connected by + or sign is called a

BINOMIAL THEOREM An expression consisting of two terms, connected by + or sign is called a BINOMIAL THEOREM hapte 8 8. Oveview: 8.. A epessio cosistig of two tems, coected by + o sig is called a biomial epessio. Fo eample, + a, y,,7 4 5y, etc., ae all biomial epessios. 8.. Biomial theoem If

More information

By the end of this section you will be able to prove the Chinese Remainder Theorem apply this theorem to solve simultaneous linear congruences

By the end of this section you will be able to prove the Chinese Remainder Theorem apply this theorem to solve simultaneous linear congruences Chapte : Theoy of Modula Aithmetic 8 Sectio D Chiese Remaide Theoem By the ed of this sectio you will be able to pove the Chiese Remaide Theoem apply this theoem to solve simultaeous liea cogueces The

More information

BINOMIAL THEOREM NCERT An expression consisting of two terms, connected by + or sign is called a

BINOMIAL THEOREM NCERT An expression consisting of two terms, connected by + or sign is called a 8. Oveview: 8.. A epessio cosistig of two tems, coected by + o sig is called a biomial epessio. Fo eample, + a, y,,7 4, etc., ae all biomial 5y epessios. 8.. Biomial theoem BINOMIAL THEOREM If a ad b ae

More information

Auchmuty High School Mathematics Department Sequences & Series Notes Teacher Version

Auchmuty High School Mathematics Department Sequences & Series Notes Teacher Version equeces ad eies Auchmuty High chool Mathematics Depatmet equeces & eies Notes Teache Vesio A sequece takes the fom,,7,0,, while 7 0 is a seies. Thee ae two types of sequece/seies aithmetic ad geometic.

More information

Five Loons LLC No Erik Ferguson PERSPECTIVES OF A MAN SONG CYCLE FOR TENOR & FLUTE. Text by Stephen Crane. Complete Score.

Five Loons LLC No Erik Ferguson PERSPECTIVES OF A MAN SONG CYCLE FOR TENOR & FLUTE. Text by Stephen Crane. Complete Score. ive Loons LLC No. 102 Erik erguson ERSECTIVES O A MAN SONG CYCLE OR TENOR LUTE Text by Stehen Crane Comlete Score Comosed 2016 Erik erguson ERSECTIVES O A MAN SONG CYCLE OR TENOR LUTE Comosed 2015-2016

More information

Continuity, Derivatives, and All That

Continuity, Derivatives, and All That Chater Seve Cotiuity, Derivatives, ad All That 7 imits ad Cotiuity et x R ad r > The set B( a; r) { x R : x a < r} is called the oe ball of radius r cetered at x The closed ball of radius r cetered at

More information

BOUND FOR SOUTH AUSTRALIA

BOUND FOR SOUTH AUSTRALIA FULL SCOE $3.50 BUNDLED VESION BOUND FO SOUTH AUSTALIA or 3-5 oct. hadbells Traditioal sea shaty arr. Alex Guebert (2015) 2015 Alex Guebert. Distributed by The Golde Dace..HadbellMusic.com This ull score

More information

Disjoint Sets { 9} { 1} { 11} Disjoint Sets (cont) Operations. Disjoint Sets (cont) Disjoint Sets (cont) n elements

Disjoint Sets { 9} { 1} { 11} Disjoint Sets (cont) Operations. Disjoint Sets (cont) Disjoint Sets (cont) n elements Disjoit Sets elemets { x, x, } X =, K Opeatios x Patitioed ito k sets (disjoit sets S, S,, K Fid-Set(x - etu set cotaiig x Uio(x,y - make a ew set by combiig the sets cotaiig x ad y (destoyig them S k

More information

The Discrete Fourier Transform

The Discrete Fourier Transform (7) The Discete Fouie Tasfom The Discete Fouie Tasfom hat is Discete Fouie Tasfom (DFT)? (ote: It s ot DTFT discete-time Fouie tasfom) A liea tasfomatio (mati) Samples of the Fouie tasfom (DTFT) of a apeiodic

More information

Spectral colours. for Ensemble and tape. Javier Alejandro Garavaglia (1997)

Spectral colours. for Ensemble and tape. Javier Alejandro Garavaglia (1997) Sectal clus Ensemble and tae avie Aleand Gaavaglia (997) Ntes the emance The iece can be layed ith ithut the tae In the secnd case thee must be a ntice n the gam that claiies it ( examle: " Vesin ithut

More information

The Pigeonhole Principle 3.4 Binomial Coefficients

The Pigeonhole Principle 3.4 Binomial Coefficients Discete M athematic Chapte 3: Coutig 3. The Pigeohole Piciple 3.4 Biomial Coefficiets D Patic Cha School of Compute Sciece ad Egieeig South Chia Uivesity of Techology Ageda Ch 3. The Pigeohole Piciple

More information

Ch 3.4 Binomial Coefficients. Pascal's Identit y and Triangle. Chapter 3.2 & 3.4. South China University of Technology

Ch 3.4 Binomial Coefficients. Pascal's Identit y and Triangle. Chapter 3.2 & 3.4. South China University of Technology Disc ete Mathem atic Chapte 3: Coutig 3. The Pigeohole Piciple 3.4 Biomial Coefficiets D Patic Cha School of Compute Sciece ad Egieeig South Chia Uivesity of Techology Pigeohole Piciple Suppose that a

More information

Balázs HORVÁTH. kábé 10

Balázs HORVÁTH. kábé 10 Balázs HOVÁTH káé 10 in eoia E o Alto Saxohone in E-lat and Ha 016 to the Duo Sea AYING SCOE Balázs Hováth, 016 káé 10 (cica 10) was coosed in the eoy o the Hungaian wite éte Esteházy in 016 It is dedicated

More information

A note on random minimum length spanning trees

A note on random minimum length spanning trees A ote o adom miimum legth spaig tees Ala Fieze Miklós Ruszikó Lubos Thoma Depatmet of Mathematical Scieces Caegie Mello Uivesity Pittsbugh PA15213, USA ala@adom.math.cmu.edu, usziko@luta.sztaki.hu, thoma@qwes.math.cmu.edu

More information

ANSWERS, HINTS & SOLUTIONS HALF COURSE TEST VII (Main)

ANSWERS, HINTS & SOLUTIONS HALF COURSE TEST VII (Main) AIITS-HT-VII-PM-JEE(Mai)-Sol./7 I JEE Advaced 06, FIITJEE Studets bag 6 i Top 00 AIR, 7 i Top 00 AIR, 8 i Top 00 AIR. Studets fom Log Tem lassoom/ Itegated School Pogam & Studets fom All Pogams have qualified

More information

Preview Only. Legal Use Requires Purchase. Recorda me. JOE HENDERSON Arranged by KRIS BERG INSTRUMENTATION

Preview Only. Legal Use Requires Purchase. Recorda me. JOE HENDERSON Arranged by KRIS BERG INSTRUMENTATION Recorda me 1st E Alto Saxohone 2nd E Alto Saxohone 1st B Tenor Saxohone 2nd B Tenor Saxohone E Baritone Saxohone 1st B Trumet 2nd B Trumet rd B Trumet th B Trumet JOE HENDERSON Arranged y RIS BERG INSTRUMENTATION

More information

4.3 Growth Rates of Solutions to Recurrences

4.3 Growth Rates of Solutions to Recurrences 4.3. GROWTH RATES OF SOLUTIONS TO RECURRENCES 81 4.3 Growth Rates of Solutios to Recurreces 4.3.1 Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer.

More information

Discussion 02 Solutions

Discussion 02 Solutions STAT 400 Discussio 0 Solutios Spig 08. ~.5 ~.6 At the begiig of a cetai study of a goup of pesos, 5% wee classified as heavy smoes, 30% as light smoes, ad 55% as osmoes. I the fiveyea study, it was detemied

More information

MMEA Central District - Jr. Jazz Audition Music. Quabbin Blues

MMEA Central District - Jr. Jazz Audition Music. Quabbin Blues r. azz Audition Music This iece has been commissioned by the or the exclusive use o district schools articiating in the 202 junior jazz auditions. Teachers o auditioning schools may rint and coy these

More information

Recursion. Algorithm : Design & Analysis [3]

Recursion. Algorithm : Design & Analysis [3] Recusio Algoithm : Desig & Aalysis [] I the last class Asymptotic gowth ate he Sets Ο, Ω ad Θ Complexity Class A Example: Maximum Susequece Sum Impovemet of Algoithm Compaiso of Asymptotic Behavio Aothe

More information

= 5! 3! 2! = 5! 3! (5 3)!. In general, the number of different groups of r items out of n items (when the order is ignored) is given by n!

= 5! 3! 2! = 5! 3! (5 3)!. In general, the number of different groups of r items out of n items (when the order is ignored) is given by n! 0 Combiatoial Aalysis Copyight by Deiz Kalı 4 Combiatios Questio 4 What is the diffeece betwee the followig questio i How may 3-lette wods ca you wite usig the lettes A, B, C, D, E ii How may 3-elemet

More information

When dealing with series, n is always a positive integer. Remember at every, sine has a value of zero, which means

When dealing with series, n is always a positive integer. Remember at every, sine has a value of zero, which means Fourier Series Some Prelimiar Ideas: Odd/Eve Fuctios: Sie is odd, which meas si ( ) si Cosie is eve, which meas cos ( ) cos Secial values of siie a cosie at Whe dealig with series, is alwas a ositive iteger.

More information

(b) What is the probability that a particle reaches the upper boundary n before the lower boundary m?

(b) What is the probability that a particle reaches the upper boundary n before the lower boundary m? MATH 529 The Boudary Problem The drukard s walk (or boudary problem) is oe of the most famous problems i the theory of radom walks. Oe versio of the problem is described as follows: Suppose a particle

More information

Nanomaterials for Photovoltaics (v11) 10. Bulk-Heterojunction Solar Cells

Nanomaterials for Photovoltaics (v11) 10. Bulk-Heterojunction Solar Cells 1 10. Bulk-Heteojuctio Sola Cells Naostuctued sola cells We ca imagie a spectum of aostuctue sola cells agig fom a ogaic, bulk heteojuctio type just discussed to a iogaic, 3-D ogaized type. Example: CuSCN/TiO

More information

THE ANALYTIC LARGE SIEVE

THE ANALYTIC LARGE SIEVE THE ANALYTIC LAGE SIEVE 1. The aalytic lage sieve I the last lectue we saw how to apply the aalytic lage sieve to deive a aithmetic fomulatio of the lage sieve, which we applied to the poblem of boudig

More information

GRADUATE ENTERING PROFICIENCY EXAMINATION MUSIC THEORY: WRITTEN NAME (PRINT) SIGNATURE STUDENT NUMBER

GRADUATE ENTERING PROFICIENCY EXAMINATION MUSIC THEORY: WRITTEN NAME (PRINT) SIGNATURE STUDENT NUMBER GRADUATE ENTERING PROFICIENCY EXAMINATION MUSIC THEORY: WRITTEN NAME (PRINT) SIGNATURE STUDENT NUMBER Is this a REPEAT examiatio (hek oe) Yes No Please omplete the folloig iformatio: Iteded degree pla

More information

Born-Oppenheimer Approximation and Nonadiabatic Effects. Hans Lischka University of Vienna

Born-Oppenheimer Approximation and Nonadiabatic Effects. Hans Lischka University of Vienna Bo-Oppeheie Appoxiatio ad Noadiabatic Effects Has Lischa Uivesity of Viea Typical situatio. Fac-Codo excitatio fo the iiu of the goud state. Covetioal dyaics possibly M* ad TS 3. Coical itesectio fuel

More information

Review of the H-O model. Problem 1. Assume that the production functions in the standard H-O model are the following:

Review of the H-O model. Problem 1. Assume that the production functions in the standard H-O model are the following: Revie of the H-O model Poblem 1 Assume that the poduction functions in the standad H-O model ae the folloing: f 1 L 1 1 ) L 1/ 1 1/ 1 f L ) L 1/3 /3 In addition e assume that the consume pefeences ae given

More information

Masses and orbits of minor planets with the GAIA mission

Masses and orbits of minor planets with the GAIA mission asses ad obits of io laets with the GAIA issio Sege ouet Suevisos : F.igad D.Hestoffe PLAN Itoductio Puose of the PhD Iotace of asses The diffeet ethods to estiate these asses Descitio of close aoach Diffeet

More information

Preview Only. Legal Use Requires Purchase. Thomas Fats WAller Arranged by Eric Richards. Conductor 2nd E% Alto Saxophone

Preview Only. Legal Use Requires Purchase. Thomas Fats WAller Arranged by Eric Richards. Conductor 2nd E% Alto Saxophone The Jitterug Waltz Thomas ats WAller Arranged y Eric Richards instrumentation Conductor 1st E% Alto Saxohone 2nd E% Alto Saxohone 1st B% Tenor Saxohone 2nd B% Tenor Saxohone E% Baritone Saxohone 1st B%

More information

5.1 Review of Singular Value Decomposition (SVD)

5.1 Review of Singular Value Decomposition (SVD) MGMT 69000: Topics i High-dimesioal Data Aalysis Falll 06 Lecture 5: Spectral Clusterig: Overview (cotd) ad Aalysis Lecturer: Jiamig Xu Scribe: Adarsh Barik, Taotao He, September 3, 06 Outlie Review of

More information

Curve Sketching Handout #5 Topic Interpretation Rational Functions

Curve Sketching Handout #5 Topic Interpretation Rational Functions Curve Sketchig Hadout #5 Topic Iterpretatio Ratioal Fuctios A ratioal fuctio is a fuctio f that is a quotiet of two polyomials. I other words, p ( ) ( ) f is a ratioal fuctio if p ( ) ad q ( ) are polyomials

More information

Week 03 Discussion. 30% are serious, and 50% are stable. Of the critical ones, 30% die; of the serious, 10% die; and of the stable, 2% die.

Week 03 Discussion. 30% are serious, and 50% are stable. Of the critical ones, 30% die; of the serious, 10% die; and of the stable, 2% die. STAT 400 Wee 03 Discussio Fall 07. ~.5- ~.6- At the begiig of a cetai study of a gou of esos, 5% wee classified as heavy smoes, 30% as light smoes, ad 55% as osmoes. I the five-yea study, it was detemied

More information

Homework 3. = k 1. Let S be a set of n elements, and let a, b, c be distinct elements of S. The number of k-subsets of S is

Homework 3. = k 1. Let S be a set of n elements, and let a, b, c be distinct elements of S. The number of k-subsets of S is Homewor 3 Chapter 5 pp53: 3 40 45 Chapter 6 p85: 4 6 4 30 Use combiatorial reasoig to prove the idetity 3 3 Proof Let S be a set of elemets ad let a b c be distict elemets of S The umber of -subsets of

More information

Sound II. Sound intensity level. Question. Energy and Intensity of sound waves

Sound II. Sound intensity level. Question. Energy and Intensity of sound waves Soud. Eergy ad tesity terferece of soud waes Stadig waes Complex soud waes Eergy ad tesity of soud waes power tesity eergy P time power P area A area A (uits W/m ) Soud itesity leel β 0log o o 0 - W/m

More information

K owi g yourself is the begi i g of all wisdo.

K owi g yourself is the begi i g of all wisdo. I t odu tio K owi g yourself is the begi i g of all wisdo. A istotle Why You Need Insight Whe is the last ti e ou a e e e taki g ti e to thi k a out ou life, ou alues, ou d ea s o ou pu pose i ei g o this

More information

HOMEWORK 2 SOLUTIONS

HOMEWORK 2 SOLUTIONS HOMEWORK SOLUTIONS CSE 55 RANDOMIZED AND APPROXIMATION ALGORITHMS 1. Questio 1. a) The larger the value of k is, the smaller the expected umber of days util we get all the coupos we eed. I fact if = k

More information

Progression. CATsyllabus.com. CATsyllabus.com. Sequence & Series. Arithmetic Progression (A.P.) n th term of an A.P.

Progression. CATsyllabus.com. CATsyllabus.com. Sequence & Series. Arithmetic Progression (A.P.) n th term of an A.P. Pogessio Sequece & Seies A set of umbes whose domai is a eal umbe is called a SEQUENCE ad sum of the sequece is called a SERIES. If a, a, a, a 4,., a, is a sequece, the the expessio a + a + a + a 4 + a

More information

Chapter 7. Support Vector Machine

Chapter 7. Support Vector Machine Chapter 7 Support Vector Machie able of Cotet Margi ad support vectors SVM formulatio Slack variables ad hige loss SVM for multiple class SVM ith Kerels Relevace Vector Machie Support Vector Machie (SVM)

More information

Discrete-Time Systems, LTI Systems, and Discrete-Time Convolution

Discrete-Time Systems, LTI Systems, and Discrete-Time Convolution EEL5: Discrete-Time Sigals ad Systems. Itroductio I this set of otes, we begi our mathematical treatmet of discrete-time s. As show i Figure, a discrete-time operates or trasforms some iput sequece x [

More information

Dream Theater. Paul Richards

Dream Theater. Paul Richards Commissioned y The niesity o loida Symhony Ochesta, Raymond Choaz, conducto, in celeation o thei 00 th anniesay season Deam Theate aul Richads Deam Theate aul Richads I Theshold II lace o Seeal Mysteies

More information

Semiconductor Optical Communication Components and Devices Lecture 15: Light Emitting Diode (LED)

Semiconductor Optical Communication Components and Devices Lecture 15: Light Emitting Diode (LED) Semicoducto Otical Commuicatio Comoets ad Devices Lectue 15: Light mittig Diode (LD) Pof. Utal Das Pofesso, Deatmet of lectical gieeig, Lase Techology Pogam, Idia Istitute of Techology, Kau htt://www.iitk.ac.i/ee/faculty/det_esume/utal.html

More information

Chapter 11 Output Analysis for a Single Model. Banks, Carson, Nelson & Nicol Discrete-Event System Simulation

Chapter 11 Output Analysis for a Single Model. Banks, Carson, Nelson & Nicol Discrete-Event System Simulation Chapter Output Aalysis for a Sigle Model Baks, Carso, Nelso & Nicol Discrete-Evet System Simulatio Error Estimatio If {,, } are ot statistically idepedet, the S / is a biased estimator of the true variace.

More information

Technical Report: Bessel Filter Analysis

Technical Report: Bessel Filter Analysis Sasa Mahmoodi 1 Techical Repot: Bessel Filte Aalysis 1 School of Electoics ad Compute Sciece, Buildig 1, Southampto Uivesity, Southampto, S17 1BJ, UK, Email: sm3@ecs.soto.ac.uk I this techical epot, we

More information

Sebastian Hilli. Paraphrase III. Soave dolore for Flute (+Alto Flute), Clarinet in Bb (+Bass Clarinet), Oboe (+English Horn), Horn and Bassoon

Sebastian Hilli. Paraphrase III. Soave dolore for Flute (+Alto Flute), Clarinet in Bb (+Bass Clarinet), Oboe (+English Horn), Horn and Bassoon 1 84 Sebastian Hilli Parahrase III Soave dolore or Flute (+Alto Flute), Clarinet in Bb (+Bass Clarinet), Oboe (+English Horn), Horn and Bassoon 2016 Coyright by the Comoser All Rights Reserved No art o

More information

TO PAINT THEIR MADNESS

TO PAINT THEIR MADNESS IECE GADNE T AINT THEI MADNESS o telve musicias Coyight 01 amescest Music (ASCA) All ights eseved Istumetatio 1 lute (doulig iccolo) 1 oe (doulig Eglish Ho i ) 1 Claiet i -lat 1 Alto Saxohoe i E-lat (doulig

More information

Optimally Sparse SVMs

Optimally Sparse SVMs A. Proof of Lemma 3. We here prove a lower boud o the umber of support vectors to achieve geeralizatio bouds of the form which we cosider. Importatly, this result holds ot oly for liear classifiers, but

More information

CHAPTER I: Vector Spaces

CHAPTER I: Vector Spaces CHAPTER I: Vector Spaces Sectio 1: Itroductio ad Examples This first chapter is largely a review of topics you probably saw i your liear algebra course. So why cover it? (1) Not everyoe remembers everythig

More information

GRAVITATIONAL FORCE IN HYDROGEN ATOM

GRAVITATIONAL FORCE IN HYDROGEN ATOM Fudametal Joual of Mode Physics Vol. 8, Issue, 015, Pages 141-145 Published olie at http://www.fdit.com/ GRAVITATIONAL FORCE IN HYDROGEN ATOM Uiesitas Pedidika Idoesia Jl DR Setyabudhi No. 9 Badug Idoesia

More information

Then the number of elements of S of weight n is exactly the number of compositions of n into k parts.

Then the number of elements of S of weight n is exactly the number of compositions of n into k parts. Geneating Function In a geneal combinatoial poblem, we have a univee S of object, and we want to count the numbe of object with a cetain popety. Fo example, if S i the et of all gaph, we might want to

More information

Yamaha Virago V-twin. Instruction manual with visual guide for Yamaha XV

Yamaha Virago V-twin. Instruction manual with visual guide for Yamaha XV Yamaha Virago V-twin Instruction manual with visual guide for Yamaha XV700-1100 PHOTO HOWN FOR ILLU TRATION PURPO E ONLY We o use a o e pie e housi g a d s all si gle to e oils fo i p o ed ope aio. If

More information

NICK DUFRESNE. 1 1 p(x). To determine some formulas for the generating function of the Schröder numbers, r(x) = a(x) =

NICK DUFRESNE. 1 1 p(x). To determine some formulas for the generating function of the Schröder numbers, r(x) = a(x) = AN INTRODUCTION TO SCHRÖDER AND UNKNOWN NUMBERS NICK DUFRESNE Abstract. I this article we will itroduce two types of lattice paths, Schröder paths ad Ukow paths. We will examie differet properties of each,

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

Solution to Problem First, the firm minimizes the cost of the inputs: min wl + rk + sf

Solution to Problem First, the firm minimizes the cost of the inputs: min wl + rk + sf Econ 0A Poblem Set 4 Solutions ue in class on Tu 4 Novembe. No late Poblem Sets accepted, so! This Poblem set tests the knoledge that ou accumulated mainl in lectues 5 to 9. Some of the mateial ill onl

More information

OH BOY! Story. N a r r a t iv e a n d o bj e c t s th ea t e r Fo r a l l a g e s, fr o m th e a ge of 9

OH BOY! Story. N a r r a t iv e a n d o bj e c t s th ea t e r Fo r a l l a g e s, fr o m th e a ge of 9 OH BOY! O h Boy!, was or igin a lly cr eat ed in F r en ch an d was a m a jor s u cc ess on t h e Fr en ch st a ge f or young au di enc es. It h a s b een s een by ap pr ox i ma t ely 175,000 sp ect at

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

International Journal of Mathematics Trends and Technology (IJMTT) Volume 47 Number 1 July 2017

International Journal of Mathematics Trends and Technology (IJMTT) Volume 47 Number 1 July 2017 Iteatioal Joual of Matheatics Teds ad Techology (IJMTT) Volue 47 Nube July 07 Coe Metic Saces, Coe Rectagula Metic Saces ad Coo Fixed Poit Theoes M. Sivastava; S.C. Ghosh Deatet of Matheatics, D.A.V. College

More information

The Basic Space Model

The Basic Space Model The Basic Space Model Let x i be the ith idividual s (i=,, ) reported positio o the th issue ( =,, m) ad let X 0 be the by m matrix of observed data here the 0 subscript idicates that elemets are missig

More information

ELOQUENTIA espacio para flauta y orquesta

ELOQUENTIA espacio para flauta y orquesta Violin I 1 Manuel Sosa ELOQENTIA esacio aa lauta y oquesta MusikaSilentis New Yok 2010 The imay uose o the ba-lines is to seve as a device o synchonization. Ba-lines do not indicate accentuation o the

More information

Beechwood Music Department Staff

Beechwood Music Department Staff Beechwood Music Department Staff MRS SARAH KERSHAW - HEAD OF MUSIC S a ra h K e rs h a w t r a i n e d a t t h e R oy a l We ls h C o l le g e of M u s i c a n d D ra m a w h e re s h e ob t a i n e d

More information

= y and Normed Linear Spaces

= y and Normed Linear Spaces 304-50 LINER SYSTEMS Lectue 8: Solutos to = ad Nomed Lea Spaces 73 Fdg N To fd N, we eed to chaacteze all solutos to = 0 Recall that ow opeatos peseve N, so that = 0 = 0 We ca solve = 0 ecusvel backwads

More information

(ii) Two-permutations of {a, b, c}. Answer. (B) P (3, 3) = 3! (C) 3! = 6, and there are 6 items in (A). ... Answer.

(ii) Two-permutations of {a, b, c}. Answer. (B) P (3, 3) = 3! (C) 3! = 6, and there are 6 items in (A). ... Answer. SOLUTIONS Homewor 5 Due /6/19 Exercise. (a Cosider the set {a, b, c}. For each of the followig, (A list the objects described, (B give a formula that tells you how may you should have listed, ad (C verify

More information

a) The average (mean) of the two fractions is halfway between them: b) The answer is yes. Assume without loss of generality that p < r.

a) The average (mean) of the two fractions is halfway between them: b) The answer is yes. Assume without loss of generality that p < r. Solutios to MAML Olympiad Level 00. Factioated a) The aveage (mea) of the two factios is halfway betwee them: p ps+ q ps+ q + q s qs qs b) The aswe is yes. Assume without loss of geeality that p

More information

Recursive Algorithm for Generating Partitions of an Integer. 1 Preliminary

Recursive Algorithm for Generating Partitions of an Integer. 1 Preliminary Recursive Algorithm for Geeratig Partitios of a Iteger Sug-Hyuk Cha Computer Sciece Departmet, Pace Uiversity 1 Pace Plaza, New York, NY 10038 USA scha@pace.edu Abstract. This article first reviews the

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

A representation approach to the tower of Hanoi problem

A representation approach to the tower of Hanoi problem Uiversity of Wollogog Research Olie Departmet of Computig Sciece Workig Paper Series Faculty of Egieerig ad Iformatio Scieces 98 A represetatio approach to the tower of Haoi problem M. C. Er Uiversity

More information

Robbery > J. œ œ. œ b œ. œ œ œ œ. œ œ œ œ œ œ œ. œ J. œ œ œ. b œ œ œ # œ ? 4. œ J. b œ œ œ # œ Ó. # b œœ. œ œ. œ J. œ Œ Ó & œ - œ - œ - œ - Kbds, Str

Robbery > J. œ œ. œ b œ. œ œ œ œ. œ œ œ œ œ œ œ. œ J. œ œ œ. b œ œ œ # œ ? 4. œ J. b œ œ œ # œ Ó. # b œœ. œ œ. œ J. œ Œ Ó & œ - œ - œ - œ - Kbds, Str IANO CONDCTOR 171 LES MISÉRABLES 10 Robbery e = q Allegro A b b Kbds, Str 8 e = e C 1 b b Ba - bet, b Claque-sous b b b pon - ine, Take care! b Andante {q = 8} Kbd1, Xylo 5 Ev - 'ry - one here b sim b

More information

Slide 1. Slide 2. Slide 3. Solids of Rotation:

Slide 1. Slide 2. Slide 3. Solids of Rotation: Slide 1 Solids of Rotatio: The Eggplat Experiece Suz Atik Palo Alto High School Palo Alto, Ca EdD; NBCT, AYA Math satik@pausd.org May thaks to my colleague, Kathy Weiss, NBCT, AYA Math, who origially desiged

More information

Shedding light on atomic energy levels (segment of Hydrogen spectrum)

Shedding light on atomic energy levels (segment of Hydrogen spectrum) 3.0 ev.85 ev.55 ev.69 ev Fri. 8.4-.7 More Eergy Quatizatio RE 8.b Mo. Tues. Wed. Lab Fri. 9.-., (.8) Mometum ad Eergy i Multiparticle Systems 9.3 Rotatioal Eergy Quiz 8 Review Exam (Ch 5-8) Exam (Ch 5-8)

More information

Do k is lo ated at the East e d of The La di g. You ill see a i k oad ight efo e gett g o to the Mai

Do k is lo ated at the East e d of The La di g. You ill see a i k oad ight efo e gett g o to the Mai Ma keti g & E e ts Coo di ato : E i A de se The Ja kso ille La di g is e ited to host the Ea th Da Cele atio fo the th ea. This e e t ill i lude a ele atio of ou ho e, Pla et Ea th, a d featu e ea th-f

More information

CSE 191, Class Note 05: Counting Methods Computer Sci & Eng Dept SUNY Buffalo

CSE 191, Class Note 05: Counting Methods Computer Sci & Eng Dept SUNY Buffalo Coutig Methods CSE 191, Class Note 05: Coutig Methods Computer Sci & Eg Dept SUNY Buffalo c Xi He (Uiversity at Buffalo CSE 191 Discrete Structures 1 / 48 Need for Coutig The problem of coutig the umber

More information

Executive Committee and Officers ( )

Executive Committee and Officers ( ) Gifted and Talented International V o l u m e 2 4, N u m b e r 2, D e c e m b e r, 2 0 0 9. G i f t e d a n d T a l e n t e d I n t e r n a t i o n a2 l 4 ( 2), D e c e m b e r, 2 0 0 9. 1 T h e W o r

More information

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,

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

Math 166 Week-in-Review - S. Nite 11/10/2012 Page 1 of 5 WIR #9 = 1+ r eff. , where r. is the effective interest rate, r is the annual

Math 166 Week-in-Review - S. Nite 11/10/2012 Page 1 of 5 WIR #9 = 1+ r eff. , where r. is the effective interest rate, r is the annual Math 66 Week-i-Review - S. Nite // Page of Week i Review #9 (F-F.4, 4.-4.4,.-.) Simple Iteest I = Pt, whee I is the iteest, P is the picipal, is the iteest ate, ad t is the time i yeas. P( + t), whee A

More information

Fall 2018 Exam 3 HAND IN PART 0 10 PIN: 17 INSTRUCTIONS

Fall 2018 Exam 3 HAND IN PART 0 10 PIN: 17 INSTRUCTIONS MARK BOX problem poits HAND IN PART 0 10 1 10 2 5 NAME: Solutios 3 10 PIN: 17 4 16 65=13x5 % 100 INSTRUCTIONS This exam comes i two parts. (1) HAND IN PART. Had i oly this part. (2) STATEMENT OF MULTIPLE

More information

Radioactive Decay and Half Life Simulation 2/17 Integrated Science 2 Redwood High School Name: Period:

Radioactive Decay and Half Life Simulation 2/17 Integrated Science 2 Redwood High School Name: Period: Radioactive Decay and Half Life Simulation 2/17 Integrated Science 2 Redwood High School Name: Period: Background I t was not until the end of the 1800 s that scientists found a method for determining

More information

the silence that reigns... for large ensemble Heather Frasch A dissertation submitted in partial satisfaction of the requirements for the degree of

the silence that reigns... for large ensemble Heather Frasch A dissertation submitted in partial satisfaction of the requirements for the degree of the silence that eigns fo lage ensemble b Heathe Fasch A dissetation submitted in atial satisfaction of the equiements fo the degee of Docto of hilosoh in Music and the Designated Emhasis in Ne Media in

More information

EE692 Applied EM- FDTD Method One-Dimensional Transmission Lines Notes- Lecture 4

EE692 Applied EM- FDTD Method One-Dimensional Transmission Lines Notes- Lecture 4 ee692_fdtd_d_tras_lie_lecture4.doc Page of 6 EE692 Applied EM FDTD Method OeDimesioal Trasmissio ies Notes ecture 4 FDTD Modelig of Voltage ources ad Termiatios with Parallel/eries R oads As the fial step

More information

Classification of DT signals

Classification of DT signals Comlex exoetial A discrete time sigal may be comlex valued I digital commuicatios comlex sigals arise aturally A comlex sigal may be rereseted i two forms: jarg { z( ) } { } z ( ) = Re { z ( )} + jim {

More information

Greatest term (numerically) in the expansion of (1 + x) Method 1 Let T

Greatest term (numerically) in the expansion of (1 + x) Method 1 Let T BINOMIAL THEOREM_SYNOPSIS Geatest tem (umeically) i the epasio of ( + ) Method Let T ( The th tem) be the geatest tem. Fid T, T, T fom the give epasio. Put T T T ad. Th will give a iequality fom whee value

More information

In algebra one spends much time finding common denominators and thus simplifying rational expressions. For example:

In algebra one spends much time finding common denominators and thus simplifying rational expressions. For example: 74 The Method of Partial Fractios I algebra oe speds much time fidig commo deomiators ad thus simplifyig ratioal epressios For eample: + + + 6 5 + = + = = + + + + + ( )( ) 5 It may the seem odd to be watig

More information

A Hallelujah for My Father

A Hallelujah for My Father Univeity o New Mexico UNM Digital Repoitoy New Mexico Compoe' chive Reeach Collection and Data 1312011 Hallelujah o My Fathe lan Stinge Robet Fanci Follow thi and additional wok at: http://digitalepoitoyunmedu/nm_compoe_achive

More information

Disjoint unions of complete graphs characterized by their Laplacian spectrum

Disjoint unions of complete graphs characterized by their Laplacian spectrum Electroic Joural of Liear Algebra Volume 18 Volume 18 (009) Article 56 009 Disjoit uios of complete graphs characterized by their Laplacia spectrum Romai Boulet boulet@uiv-tlse.fr Follow this ad additioal

More information

Northwest High School s Algebra 2/Honors Algebra 2 Summer Review Packet

Northwest High School s Algebra 2/Honors Algebra 2 Summer Review Packet Northwest High School s Algebra /Hoors Algebra Summer Review Packet This packet is optioal! It will NOT be collected for a grade et school year! This packet has bee desiged to help you review various mathematical

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

Polymerization Lab p. 1 Polymerization Lab

Polymerization Lab p. 1 Polymerization Lab Polymerizatio Lab p. 1 Polymerizatio Lab Itroductio: Polymers (Greek-PLY...may ad MES...parts) have existed sice the begiig of life. Both "atural" ad "sythetic" polymers are a itegral part of our life.

More information

RASPBERRY ISLAND DREAMING I. THE RIVER IS LIBBY LARSEN

RASPBERRY ISLAND DREAMING I. THE RIVER IS LIBBY LARSEN OYCE SUTPHE RASPBERRY ISLAD DREAMIG I THE RIVER IS LIBBY LARSE 6 11 sky (drone) 5 5 5 q = 66-72 freely throughout f II ed P f ro-ing it-self a-cross the ed I II ull ack temo rimo Í land f 5 5 5 (drone)

More information

Commutativity in Permutation Groups

Commutativity in Permutation Groups Commutativity i Permutatio Groups Richard Wito, PhD Abstract I the group Sym(S) of permutatios o a oempty set S, fixed poits ad trasiet poits are defied Prelimiary results o fixed ad trasiet poits are

More information

H STO RY OF TH E SA NT

H STO RY OF TH E SA NT O RY OF E N G L R R VER ritten for the entennial of th e Foundin g of t lair oun t y on ay 8 82 Y EEL N E JEN K RP O N! R ENJ F ] jun E 3 1 92! Ph in t ed b y h e t l a i r R ep u b l i c a n O 4 1922

More information

176 5 t h Fl oo r. 337 P o ly me r Ma te ri al s

176 5 t h Fl oo r. 337 P o ly me r Ma te ri al s A g la di ou s F. L. 462 E l ec tr on ic D ev el op me nt A i ng er A.W.S. 371 C. A. M. A l ex an de r 236 A d mi ni st ra ti on R. H. (M rs ) A n dr ew s P. V. 326 O p ti ca l Tr an sm is si on A p ps

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