Chapter 17 Study Guide

Size: px
Start display at page:

Download "Chapter 17 Study Guide"

Transcription

1 Chptr 7 Stuy Gui Som smpl pk.8 N α-hyrogns(hyrogns on rons nxt to ronyls) r firly ii. This is u to th rsonn-stiliztion of th rsulting rnion. - Bs rnion nolt KET form + ENL form ) Know Kto-Enol Tutomrism spil typ of onstitutionl isomrs, quikly intrhng in prsn of i

2 ) Know formtion of Enolt/Crnion y s - N ) Know rmiztion/pimriztion of α-ron if th α-ron is strontr, rmiztion n our quit simply with th ition of s. C N Bss turn α-rons into rnions whih thn n onvrt into nolts. Th nolt ron is sp hyriiz, whih rsults in loss of strohmistry. ) Know hli ition nolts n pik up groups othr thn hyrogn ) N ) Br Br ) Know hloform rtion mthyl ktons will thr hlogns to form trihli, whih is susptil to hyrolysis to n i n hloform(ron with thr hlogns tth, suh s hloroform). ) Br / N + CBr )

3 A lortory tst for mthyl ktons or sonry lohols(nxt to mthyl groups) is to I. If yllow pripitt of iooform(ci ) forms, this onfirms th prsn of mthyl ktons or mthyl sonry lohols. ) Know lol ition/onnstion rtions sin α-hyrogns r so sily rmov to form nolts/rnions, this hmistry is of grt importn. n of th most importnt rtions is th Alol rtion. In this rtion rnion is form y ing s to rmov n α- hyrogn. Thn rtiv lhy is. Th rnion ttks th lhy forming nw ron-ron on. Th oxygn of th lhy forms n lohol tht n rmov with ht to form n lkn. ) N ) pntnl ) + ) N ) pntnl ) + /ht Th lol rtions shown ov r whn th rnion tht forms is rt with sprt lhy. This is spifilly ll ross-lol sin you us two iffrnt moluls. Alols n lso form whn th originl molul us to form th rnion is lso us s th lhy tht gts ttk in th son stp. IMPRTANT. It is lwys th CARBANIN rsonn form tht is rtiv. Evn though th nolt is mor stl, it n not rt to form th lol prout.

4 ) N ) N nothr molul of strting mtril - 7) Know mhnism for lol rtion - - +

5 As sn oth on prvious xmpl n low, th rnion os th ttking not th nol. Bs KET rtiv form ENL nonrtiv form + /ht + lol onnstion lol ition ow woul you gnrt ths lol prouts? I II III IV Br

6 TE ALDL IS A REACTIN WERE A CARBNYL (KETNE R ALDEYDE) IS DEPRTNATED WIT A BASE (N most sustitut or LDA lst sustitut ). most sustitut ( o ) N f g f g LDA f g lst sustitut ( o ) TE RESULTIN CARBANIN WILL TEN ATTACK AN CARBNYL. TE CARBNYL WEN ATTACKED BY A NUCLEPILE WILL TURN INT AN ALCL (). g f ) N ) Pntnl ) + g f g f ) LDA ) Pntnl ) + f g

7 IF EAT IS APPLIED, TE DEYDRATES T AN ALKENE. g f ) N ) Pntnl ) + /ht g f g f ) LDA ) Pntnl ) + /ht g f IF A CMPUND CNTAINS CARBNYLS SEPARATED BY R CARBNS, AN INTRAMLECULAR ALDL WILL YIELD A R MEMBERED RING. f N f f N f 7

8 8 EXAMPLES ) N ) Pntnl ) + ) N ) Butnl ) + ) N ) ) + ) LDA ) Butnl ) + ) LDA ) Propnl ) + ) LDA ) ) +

9 9 ) N ) ) + ) LDA ) Butnl ) N NTE: NLY or mmr rings will form, not, 7, Wht is th prout(s) of th following? ) N ) Propnl ) + /ht ) LDA ) Propnl ) + N

10 0 ) N ) Propnl ) + /ht ) LDA ) Propnl ) N N 7 8

11 8) Know rvrs lol s you noti in th rtions, quilirium rrows r rwn. This mns th rtion n rvrs itslf k to th originl ronyls. - + B - 9) Know Clisn-Shmit Rtion sin most synthsis n to mploy th ross lol s oppos to th slf-onnsing lol, r ns to tkn in th sltion of rgnts. If you us somthing lik pntnl n try n mk n nolt of it n ross it with nzlhy, you n gt th prout you r looking for. But you n lso gt slf-onnstion of pntnl with itslf. ftn ktons r us sin thy o not unrgo slf-onnstion. Whn ktons r us, this is ll th Clisn-Shmit Rtions. Quit oftn hyrtion ours in ths rtions vn without th ition of ht. 0) Know nitril/nitrolkn onnstion inst of strting with ronyl, nitrils n nitro n us. Th -rons r still quit ii. ) N CN ) Propnl CN N ) N ) Propnl N

12 ) Know intrmolulr lol yliztion rtions if ikton or ilhy is us, slf-onnstion ling to ring formtion is oftn sn spilly if n mmr rings r form. If multipl rtions r possil, n mmr rings r prfrr. N N ) Thrmoynmi vs. kinti nolt us of this phnomn, you n sily hoos whih si of th ronyl to hv th lol/ition rtion tk pl y hnging th s. Th us of lithium form nolts is vry prvlnt in ross lol(clisn-shmit) rtions. TD nolt mor sustitut ron, wk s(n), proti solvnt(mthnol,thnol ) Kinti nolt lss sustitut ron, strong s(lda), proti solvnt(tf, DMS, DME ) N - LDA -

13 ) Alkyltion y lithium nolt inst of prforming n lol rtion to hin of rons, irt lkyltion n tk pl. In this rtion th nolt tht forms rts with lkyl hli in n S N typ rtion. Primry lkyl hlis, primry nzyli hlis n primry llyli hlis must us(sonry n trtiry l to limintions). ) LDA ) mthyl ioi ) Silyltion of nolt in vry rtion so fr th rnion hs n th rtiv spis. Th lkoxi nion n rt in spil ss, spifilly with th ition of TMS-Cl, TBDMS-Cl or TBDPS-Cl. Ths silyl groups hv trmnous ttrtion for oxygn. Thrfor SN typ of rtion ours twn th lkoxi nion n th silyl group. This trps th nolt in non-rtiv form. thr hmistry n tk pl(for xmpl if you wnt to rt on ronyl inst of nothr, you n trp th first ronyl s silyl nol thr n thn rt th othr). As for TBAF(or mthyl lithium n us to rgnrt th nolt. ) LDA ) TMS-Cl TMS T M S-C l C l C S i C T B A F or C L i C C C T B D M S-C l C l Si C C - C C C T B D PS -C l C l S i C C C

14 ) α-slntion/ oul on α,β to th ronyl n y ing slni to th -ron n thn rmoving th slni unr mil, nutrl onitions. This ls to α, β-unsturt ronyls. ) LDA ) C SBr SC S - C ), vs, ition to α, β-unsturt ronyls just lik w sw with ins in hptr, th prsn of two unsturt groups llows us th possiility of hving, vs, ition. In ins,, ition l to movmnt of th oul on. Although th oul on is mov in th mhnism, it ultimtly is rmov., ition ours t th ronyl. r th ronyl is turn into n lohol n th nulophil s t th ronyl ron whil th oul on is unhng. This is known s simpl ition. In, ition, th nulophil s on th outsi of th oul on using n nolt to form(from th movmnt of th oul on). Th nolt is tutomriz to th kto form in th finl prout. Th offshoot is th nulophil s to th outsi of th oul on, with th oul on isppring. This is ll onjugt ition. Most ss involv omintion of oth simpl n onjugt ition. Simpl ition is fvor y strong nulophils(orgnomtllis, grignrs) whil onjugt ition is fvor y wk nulophils(uprts, mins n CN)., strong nulophils, wk nulophils, uprts to ylis giv trns prouts if vill

15 N (wk nulophil) C MgBr N C C Cu (C C )CuLi 7) Mihl ition/roinson nnultion if n nolt is us to ttk n α,β-unsturt ronyl it will o so through, ition. This is ll Mihl ition(ny, ition is ll Mihl ition). If th originl nolt is ikton whih n follow th Mihl ition with slf-lol onnstion ling to yliztion, tht is ll Roinson Annultion. Mihl ition, ition y lithium nolt Roinson nnultion Mihl ition follow y lol yliztion N +

16 REVERSE ALDL I + II I + II I + II

17 7 REVERSE ALDL Br Br I II - N - - mmr ring mmr ring mmr ring

18 - N - - f f f mmr ring 7 mmr ring 8

19 N G F B C D E A - - G F E D C B A + G F B C D E A + B 9

20 i g f h j k i g f h j k R f j k h i g i h f g j k + 0

21 Complt th following rtion shms A. ) LDA ) hxnl ) + /ht ) LDA ) Br ) N ) TMS-Cl TBDMS ) N/ Br ) ) LDA ) Propyl ioi ) TBAF ) PCC ) LDA ) N ) hxnl ) +

22 B. C Li N (C C ) CuLi + N Giv th mhnism for th following ) N ) Pntnl ) +

23 KEY Complt th following rtion shms A. TBDMS TBDMS ) LDA ) hxnl ) + /ht ) LDA ) Br ) N ) TMS-Cl TBDMS Br TMS TBDMS ) N/ Br ) TBDMS ) LDA ) Propyl ioi ) TBAF ) PCC ) LDA ) N ) hxnl ) + TBDMS TBDMS

24 B. points h N C Li N (C C ) CuLi C C + N Giv th mhnism for th following ) N ) Pntnl )

25 Nm: ) LDA ) Ethyl Ioi ) LDA ) TBDMS-Cl ) N ) Br ) LDA ) Pntnl ) + ) LDA ) Pntnl ) + /ht ) LDA ) PhSBr ) ) LDA ) Butnl ) + ) N ) Butnl ) + NC C MgBr TBDMS (C ) CuLi

26 Show th prout n mhnism for th following rtions: ) LDA )

27 7 N

28 8 Roinson nnultion

29 9 KEY TBDMS ) LDA ) Ethyl Ioi ) LDA ) TBDMS-Cl Br ) N ) Br ) LDA ) Pntnl ) + ) LDA ) Pntnl ) + /ht ) LDA ) PhSBr )

30 0 ) LDA ) Butnl ) + ) N ) Butnl ) + A B A B C NC C C MgBr TBDMS (C ) CuLi C

31 Show th prout n mhnism for th following rtions: ) LDA ) - LDA N 7 7 8

32 Bs Bs

33 Prt I: Giv th prout for th following. PCC Cr ) DIBAL- ) C N ) N ) Pntnl ) + (C C C C ) CuLi Cl -romonzn, AlCl

34 + N NC N(C ) ) N ) Pntyl ioi (Ph) PCC C mcpba

35 ) LDA ) PhSBr ) (C ) CuLi ) C MgBr ) + N

36 Prt II: Giv th rgnts n for th following. C This isomr is th only on prou.

37 7 Giv th hmils n rgnts n to prou th following lol prouts.

38 8 Prt I: Giv th prout for th following. PCC Cr ) DIBAL-h= ) C N ) N ) Pntnl ) + C N Cl (C C C C ) CuLi Br -romonzn, AlCl

39 9 + N NC NN C N N(C ) ) N ) Pntyl ioi C C C (Ph) PCC C mcpba

40 0 ) LDA ) PhSBr ) C B A (C ) CuLi ) C MgBr ) + N A B C A A B B C C

41 Prt II: Giv th rgnts n for th following. ) + ) CN ) Cl,, t C ) KMn, - ht ) + R ) ) Zn, A ) DIBAL- ) + gs, S Et P CC C Et This isomr is th only on prou. ) BF SS ) Rny Ni

42 Giv th hmils n rgnts n to prou th following lol prouts. ) N ) Butnl ) + LDA ) N ) Propnl ) +, t ) N ) +

43 MEWRK # Nm: Giv th prouts of th following. CN ) N ) Pntnl ) LDA ) PhSBr ) ) N ) utyl hlori ) PCC ) N ) Pntnl ) = /ht TBDMS ) TBAF ) Jons ) N

44 TBDMS ) TBAF ) PCC ) N ) C C Li ) + NCN (C C ) CuLi N Giv th mhnisms for th following ) N ) Propnl ) +

45 MEWRK # KEY Giv th prouts of th following. CN CN ) N ) Pntnl ) LDA ) PhSBr ) ) N ) utyl hlori TBDMS ) PCC ) N ) Pntnl ) = /ht TBDMS ) TBAF ) Jons ) N

46 TBDMS ) TBAF ) PCC ) N CN ) C C Li ) + NCN (C C ) CuLi N Giv th mhnisms for th following ) N ) Propnl )

CSE 373: More on graphs; DFS and BFS. Michael Lee Wednesday, Feb 14, 2018

CSE 373: More on graphs; DFS and BFS. Michael Lee Wednesday, Feb 14, 2018 CSE 373: Mor on grphs; DFS n BFS Mihl L Wnsy, F 14, 2018 1 Wrmup Wrmup: Disuss with your nighor: Rmin your nighor: wht is simpl grph? Suppos w hv simpl, irt grph with x nos. Wht is th mximum numr of gs

More information

COMP108 Algorithmic Foundations

COMP108 Algorithmic Foundations Grdy mthods Prudn Wong http://www.s.liv..uk/~pwong/thing/omp108/01617 Coin Chng Prolm Suppos w hv 3 typs of oins 10p 0p 50p Minimum numr of oins to mk 0.8, 1.0, 1.? Grdy mthod Lrning outoms Undrstnd wht

More information

Math 61 : Discrete Structures Final Exam Instructor: Ciprian Manolescu. You have 180 minutes.

Math 61 : Discrete Structures Final Exam Instructor: Ciprian Manolescu. You have 180 minutes. Nm: UCA ID Numr: Stion lttr: th 61 : Disrt Struturs Finl Exm Instrutor: Ciprin nolsu You hv 180 minuts. No ooks, nots or lultors r llow. Do not us your own srth ppr. 1. (2 points h) Tru/Fls: Cirl th right

More information

(2) If we multiplied a row of B by λ, then the value is also multiplied by λ(here lambda could be 0). namely

(2) If we multiplied a row of B by λ, then the value is also multiplied by λ(here lambda could be 0). namely . DETERMINANT.. Dtrminnt. Introution:I you think row vtor o mtrix s oorint o vtors in sp, thn th gomtri mning o th rnk o th mtrix is th imnsion o th prlllppi spnn y thm. But w r not only r out th imnsion,

More information

# 1 ' 10 ' 100. Decimal point = 4 hundred. = 6 tens (or sixty) = 5 ones (or five) = 2 tenths. = 7 hundredths.

# 1 ' 10 ' 100. Decimal point = 4 hundred. = 6 tens (or sixty) = 5 ones (or five) = 2 tenths. = 7 hundredths. How os it work? Pl vlu o imls rprsnt prts o whol numr or ojt # 0 000 Tns o thousns # 000 # 00 Thousns Hunrs Tns Ons # 0 Diml point st iml pl: ' 0 # 0 on tnth n iml pl: ' 0 # 00 on hunrth r iml pl: ' 0

More information

DUET WITH DIAMONDS COLOR SHIFTING BRACELET By Leslie Rogalski

DUET WITH DIAMONDS COLOR SHIFTING BRACELET By Leslie Rogalski Dut with Dimons Brlt DUET WITH DIAMONDS COLOR SHIFTING BRACELET By Lsli Roglski Photo y Anrw Wirth Supruo DUETS TM from BSmith rt olor shifting fft tht mks your work tk on lif of its own s you mov! This

More information

CSE 373: AVL trees. Warmup: Warmup. Interlude: Exploring the balance invariant. AVL Trees: Invariants. AVL tree invariants review

CSE 373: AVL trees. Warmup: Warmup. Interlude: Exploring the balance invariant. AVL Trees: Invariants. AVL tree invariants review rmup CSE 7: AVL trs rmup: ht is n invrint? Mihl L Friy, Jn 9, 0 ht r th AVL tr invrints, xtly? Disuss with your nighor. AVL Trs: Invrints Intrlu: Exploring th ln invrint Cor i: xtr invrint to BSTs tht

More information

EE1000 Project 4 Digital Volt Meter

EE1000 Project 4 Digital Volt Meter Ovrviw EE1000 Projt 4 Diitl Volt Mtr In this projt, w mk vi tht n msur volts in th rn o 0 to 4 Volts with on iit o ury. Th input is n nlo volt n th output is sinl 7-smnt iit tht tlls us wht tht input s

More information

Cycles and Simple Cycles. Paths and Simple Paths. Trees. Problem: There is No Completely Standard Terminology!

Cycles and Simple Cycles. Paths and Simple Paths. Trees. Problem: There is No Completely Standard Terminology! Outlin Computr Sin 331, Spnnin, n Surphs Mik Joson Dprtmnt o Computr Sin Univrsity o Clry Ltur #30 1 Introution 2 3 Dinition 4 Spnnin 5 6 Mik Joson (Univrsity o Clry) Computr Sin 331 Ltur #30 1 / 20 Mik

More information

S i m p l i f y i n g A l g e b r a SIMPLIFYING ALGEBRA.

S i m p l i f y i n g A l g e b r a SIMPLIFYING ALGEBRA. S i m p l i y i n g A l g r SIMPLIFYING ALGEBRA www.mthltis.o.nz Simpliying SIMPLIFYING Algr ALGEBRA Algr is mthmtis with mor thn just numrs. Numrs hv ix vlu, ut lgr introus vrils whos vlus n hng. Ths

More information

Graph Isomorphism. Graphs - II. Cayley s Formula. Planar Graphs. Outline. Is K 5 planar? The number of labeled trees on n nodes is n n-2

Graph Isomorphism. Graphs - II. Cayley s Formula. Planar Graphs. Outline. Is K 5 planar? The number of labeled trees on n nodes is n n-2 Grt Thortil Is In Computr Sin Vitor Amhik CS 15-251 Ltur 9 Grphs - II Crngi Mllon Univrsity Grph Isomorphism finition. Two simpl grphs G n H r isomorphi G H if thr is vrtx ijtion V H ->V G tht prsrvs jny

More information

Module graph.py. 1 Introduction. 2 Graph basics. 3 Module graph.py. 3.1 Objects. CS 231 Naomi Nishimura

Module graph.py. 1 Introduction. 2 Graph basics. 3 Module graph.py. 3.1 Objects. CS 231 Naomi Nishimura Moul grph.py CS 231 Nomi Nishimur 1 Introution Just lik th Python list n th Python itionry provi wys of storing, ssing, n moifying t, grph n viw s wy of storing, ssing, n moifying t. Bus Python os not

More information

Why the Junction Tree Algorithm? The Junction Tree Algorithm. Clique Potential Representation. Overview. Chris Williams 1.

Why the Junction Tree Algorithm? The Junction Tree Algorithm. Clique Potential Representation. Overview. Chris Williams 1. Why th Juntion Tr lgorithm? Th Juntion Tr lgorithm hris Willims 1 Shool of Informtis, Univrsity of Einurgh Otor 2009 Th JT is gnrl-purpos lgorithm for omputing (onitionl) mrginls on grphs. It os this y

More information

Paths. Connectivity. Euler and Hamilton Paths. Planar graphs.

Paths. Connectivity. Euler and Hamilton Paths. Planar graphs. Pths.. Eulr n Hmilton Pths.. Pth D. A pth rom s to t is squn o gs {x 0, x 1 }, {x 1, x 2 },... {x n 1, x n }, whr x 0 = s, n x n = t. D. Th lngth o pth is th numr o gs in it. {, } {, } {, } {, } {, } {,

More information

Present state Next state Q + M N

Present state Next state Q + M N Qustion 1. An M-N lip-lop works s ollows: I MN=00, th nxt stt o th lip lop is 0. I MN=01, th nxt stt o th lip-lop is th sm s th prsnt stt I MN=10, th nxt stt o th lip-lop is th omplmnt o th prsnt stt I

More information

ECE COMBINATIONAL BUILDING BLOCKS - INVEST 13 DECODERS AND ENCODERS

ECE COMBINATIONAL BUILDING BLOCKS - INVEST 13 DECODERS AND ENCODERS C 24 - COMBINATIONAL BUILDING BLOCKS - INVST 3 DCODS AND NCODS FALL 23 AP FLZ To o "wll" on this invstition you must not only t th riht nswrs ut must lso o nt, omplt n onis writups tht mk ovious wht h

More information

1 Introduction to Modulo 7 Arithmetic

1 Introduction to Modulo 7 Arithmetic 1 Introution to Moulo 7 Arithmti Bor w try our hn t solvin som hr Moulr KnKns, lt s tk los look t on moulr rithmti, mo 7 rithmti. You ll s in this sminr tht rithmti moulo prim is quit irnt rom th ons w

More information

CSE 373. Graphs 1: Concepts, Depth/Breadth-First Search reading: Weiss Ch. 9. slides created by Marty Stepp

CSE 373. Graphs 1: Concepts, Depth/Breadth-First Search reading: Weiss Ch. 9. slides created by Marty Stepp CSE 373 Grphs 1: Conpts, Dpth/Brth-First Srh ring: Wiss Ch. 9 slis rt y Mrty Stpp http://www.s.wshington.u/373/ Univrsity o Wshington, ll rights rsrv. 1 Wht is grph? 56 Tokyo Sttl Soul 128 16 30 181 140

More information

12/3/12. Outline. Part 10. Graphs. Circuits. Euler paths/circuits. Euler s bridge problem (Bridges of Konigsberg Problem)

12/3/12. Outline. Part 10. Graphs. Circuits. Euler paths/circuits. Euler s bridge problem (Bridges of Konigsberg Problem) 12/3/12 Outlin Prt 10. Grphs CS 200 Algorithms n Dt Struturs Introution Trminology Implmnting Grphs Grph Trvrsls Topologil Sorting Shortst Pths Spnning Trs Minimum Spnning Trs Ciruits 1 Ciruits Cyl 2 Eulr

More information

5/9/13. Part 10. Graphs. Outline. Circuits. Introduction Terminology Implementing Graphs

5/9/13. Part 10. Graphs. Outline. Circuits. Introduction Terminology Implementing Graphs Prt 10. Grphs CS 200 Algorithms n Dt Struturs 1 Introution Trminology Implmnting Grphs Outlin Grph Trvrsls Topologil Sorting Shortst Pths Spnning Trs Minimum Spnning Trs Ciruits 2 Ciruits Cyl A spil yl

More information

Page 1. Question 19.1b Electric Charge II Question 19.2a Conductors I. ConcepTest Clicker Questions Chapter 19. Physics, 4 th Edition James S.

Page 1. Question 19.1b Electric Charge II Question 19.2a Conductors I. ConcepTest Clicker Questions Chapter 19. Physics, 4 th Edition James S. ConTst Clikr ustions Chtr 19 Physis, 4 th Eition Jms S. Wlkr ustion 19.1 Two hrg blls r rlling h othr s thy hng from th iling. Wht n you sy bout thir hrgs? Eltri Chrg I on is ositiv, th othr is ngtiv both

More information

Outline. Circuits. Euler paths/circuits 4/25/12. Part 10. Graphs. Euler s bridge problem (Bridges of Konigsberg Problem)

Outline. Circuits. Euler paths/circuits 4/25/12. Part 10. Graphs. Euler s bridge problem (Bridges of Konigsberg Problem) 4/25/12 Outlin Prt 10. Grphs CS 200 Algorithms n Dt Struturs Introution Trminology Implmnting Grphs Grph Trvrsls Topologil Sorting Shortst Pths Spnning Trs Minimum Spnning Trs Ciruits 1 2 Eulr s rig prolm

More information

, each of which is a tree, and whose roots r 1. , respectively, are children of r. Data Structures & File Management

, each of which is a tree, and whose roots r 1. , respectively, are children of r. Data Structures & File Management nrl tr T is init st o on or mor nos suh tht thr is on sint no r, ll th root o T, n th rminin nos r prtition into n isjoint susts T, T,, T n, h o whih is tr, n whos roots r, r,, r n, rsptivly, r hilrn o

More information

a b c cat CAT A B C Aa Bb Cc cat cat Lesson 1 (Part 1) Verbal lesson: Capital Letters Make The Same Sound Lesson 1 (Part 1) continued...

a b c cat CAT A B C Aa Bb Cc cat cat Lesson 1 (Part 1) Verbal lesson: Capital Letters Make The Same Sound Lesson 1 (Part 1) continued... Progrssiv Printing T.M. CPITLS g 4½+ Th sy, fun (n FR!) wy to tch cpitl lttrs. ook : C o - For Kinrgrtn or First Gr (not for pr-school). - Tchs tht cpitl lttrs mk th sm souns s th littl lttrs. - Tchs th

More information

V={A,B,C,D,E} E={ (A,D),(A,E),(B,D), (B,E),(C,D),(C,E)}

V={A,B,C,D,E} E={ (A,D),(A,E),(B,D), (B,E),(C,D),(C,E)} Introution Computr Sin & Enginring 423/823 Dsign n Anlysis of Algorithms Ltur 03 Elmntry Grph Algorithms (Chptr 22) Stphn Sott (Apt from Vinohnrn N. Vriym) I Grphs r strt t typs tht r pplil to numrous

More information

Decimals DECIMALS.

Decimals DECIMALS. Dimls DECIMALS www.mthltis.o.uk ow os it work? Solutions Dimls P qustions Pl vlu o imls 0 000 00 000 0 000 00 0 000 00 0 000 00 0 000 tnths or 0 thousnths or 000 hunrths or 00 hunrths or 00 0 tn thousnths

More information

V={A,B,C,D,E} E={ (A,D),(A,E),(B,D), (B,E),(C,D),(C,E)}

V={A,B,C,D,E} E={ (A,D),(A,E),(B,D), (B,E),(C,D),(C,E)} s s of s Computr Sin & Enginring 423/823 Dsign n Anlysis of Ltur 03 (Chptr 22) Stphn Sott (Apt from Vinohnrn N. Vriym) s of s s r strt t typs tht r pplil to numrous prolms Cn ptur ntitis, rltionships twn

More information

CSC Design and Analysis of Algorithms. Example: Change-Making Problem

CSC Design and Analysis of Algorithms. Example: Change-Making Problem CSC 801- Dsign n Anlysis of Algorithms Ltur 11 Gry Thniqu Exmpl: Chng-Mking Prolm Givn unlimit mounts of oins of nomintions 1 > > m, giv hng for mount n with th lst numr of oins Exmpl: 1 = 25, 2 =10, =

More information

Garnir Polynomial and their Properties

Garnir Polynomial and their Properties Univrsity of Cliforni, Dvis Dprtmnt of Mthmtis Grnir Polynomil n thir Proprtis Author: Yu Wng Suprvisor: Prof. Gorsky Eugny My 8, 07 Grnir Polynomil n thir Proprtis Yu Wng mil: uywng@uvis.u. In this ppr,

More information

Planar Upward Drawings

Planar Upward Drawings C.S. 252 Pro. Rorto Tmssi Computtionl Gomtry Sm. II, 1992 1993 Dt: My 3, 1993 Sri: Shmsi Moussvi Plnr Upwr Drwings 1 Thorm: G is yli i n only i it hs upwr rwing. Proo: 1. An upwr rwing is yli. Follow th

More information

QUESTIONS BEGIN HERE!

QUESTIONS BEGIN HERE! Points miss: Stunt's Nm: Totl sor: /100 points Est Tnnss Stt Univrsity Dprtmnt of Computr n Informtion Sins CSCI 710 (Trnoff) Disrt Struturs TEST for Fll Smstr, 00 R this for strtin! This tst is los ook

More information

Chem 104A, Fall 2016, Midterm 1 Key

Chem 104A, Fall 2016, Midterm 1 Key hm 104A, ll 2016, Mitrm 1 Ky 1) onstruct microstt tl for p 4 configurtion. Pls numrt th ms n ml for ch lctron in ch microstt in th tl. (Us th formt ml m s. Tht is spin -½ lctron in n s oritl woul writtn

More information

Self-Adjusting Top Trees

Self-Adjusting Top Trees Th Polm Sl-jsting Top Ts ynmi ts: ol: mintin n n-tx ost tht hngs o tim. link(,w): ts n g twn tis n w. t(,w): lts g (,w). pplition-spii t ssoit with gs n/o tis. ont xmpls: in minimm-wight g in th pth twn

More information

Graphs. CSC 1300 Discrete Structures Villanova University. Villanova CSC Dr Papalaskari

Graphs. CSC 1300 Discrete Structures Villanova University. Villanova CSC Dr Papalaskari Grphs CSC 1300 Disrt Struturs Villnov Univrsity Grphs Grphs r isrt struturs onsis?ng of vr?s n gs tht onnt ths vr?s. Grphs n us to mol: omputr systms/ntworks mthm?l rl?ons logi iruit lyout jos/prosss f

More information

Last time: introduced our first computational model the DFA.

Last time: introduced our first computational model the DFA. Lctur 7 Homwork #7: 2.2.1, 2.2.2, 2.2.3 (hnd in c nd d), Misc: Givn: M, NFA Prov: (q,xy) * (p,y) iff (q,x) * (p,) (follow proof don in clss tody) Lst tim: introducd our first computtionl modl th DFA. Tody

More information

Using the Printable Sticker Function. Using the Edit Screen. Computer. Tablet. ScanNCutCanvas

Using the Printable Sticker Function. Using the Edit Screen. Computer. Tablet. ScanNCutCanvas SnNCutCnvs Using th Printl Stikr Funtion On-o--kin stikrs n sily rt y using your inkjt printr n th Dirt Cut untion o th SnNCut mhin. For inormtion on si oprtions o th SnNCutCnvs, rr to th Hlp. To viw th

More information

Nefertiti. Echoes of. Regal components evoke visions of the past MULTIPLE STITCHES. designed by Helena Tang-Lim

Nefertiti. Echoes of. Regal components evoke visions of the past MULTIPLE STITCHES. designed by Helena Tang-Lim MULTIPLE STITCHES Nrtiti Ehos o Rgl omponnts vok visions o th pst sign y Hln Tng-Lim Us vrity o stiths to rt this rgl yt wrl sign. Prt sping llows squr s to mk roun omponnts tht rp utiully. FCT-SC-030617-07

More information

A Simple Code Generator. Code generation Algorithm. Register and Address Descriptors. Example 3/31/2008. Code Generation

A Simple Code Generator. Code generation Algorithm. Register and Address Descriptors. Example 3/31/2008. Code Generation A Simpl Co Gnrtor Co Gnrtion Chptr 8 II Gnrt o for singl si lok How to us rgistrs? In most mhin rhitturs, som or ll of th oprnsmust in rgistrs Rgistrs mk goo tmporris Hol vlus tht r omput in on si lok

More information

CS61B Lecture #33. Administrivia: Autograder will run this evening. Today s Readings: Graph Structures: DSIJ, Chapter 12

CS61B Lecture #33. Administrivia: Autograder will run this evening. Today s Readings: Graph Structures: DSIJ, Chapter 12 Aministrivi: CS61B Ltur #33 Autogrr will run this vning. Toy s Rings: Grph Struturs: DSIJ, Chptr 12 Lst moifi: W Nov 8 00:39:28 2017 CS61B: Ltur #33 1 Why Grphs? For xprssing non-hirrhilly rlt itms Exmpls:

More information

Solutions for HW11. Exercise 34. (a) Use the recurrence relation t(g) = t(g e) + t(g/e) to count the number of spanning trees of v 1

Solutions for HW11. Exercise 34. (a) Use the recurrence relation t(g) = t(g e) + t(g/e) to count the number of spanning trees of v 1 Solutions for HW Exris. () Us th rurrn rltion t(g) = t(g ) + t(g/) to ount th numr of spnning trs of v v v u u u Rmmr to kp multipl gs!! First rrw G so tht non of th gs ross: v u v Rursing on = (v, u ):

More information

Register Allocation. Register Allocation. Principle Phases. Principle Phases. Example: Build. Spills 11/14/2012

Register Allocation. Register Allocation. Principle Phases. Principle Phases. Example: Build. Spills 11/14/2012 Rgistr Allotion W now r l to o rgistr llotion on our intrfrn grph. W wnt to l with two typs of onstrints: 1. Two vlus r liv t ovrlpping points (intrfrn grph) 2. A vlu must or must not in prtiulr rhitturl

More information

FSA. CmSc 365 Theory of Computation. Finite State Automata and Regular Expressions (Chapter 2, Section 2.3) ALPHABET operations: U, concatenation, *

FSA. CmSc 365 Theory of Computation. Finite State Automata and Regular Expressions (Chapter 2, Section 2.3) ALPHABET operations: U, concatenation, * CmSc 365 Thory of Computtion Finit Stt Automt nd Rgulr Exprssions (Chptr 2, Sction 2.3) ALPHABET oprtions: U, conctntion, * otin otin Strings Form Rgulr xprssions dscri Closd undr U, conctntion nd * (if

More information

SOLVED EXAMPLES. be the foci of an ellipse with eccentricity e. For any point P on the ellipse, prove that. tan

SOLVED EXAMPLES. be the foci of an ellipse with eccentricity e. For any point P on the ellipse, prove that. tan LOCUS 58 SOLVED EXAMPLES Empl Lt F n F th foci of n llips with ccntricit. For n point P on th llips, prov tht tn PF F tn PF F Assum th llips to, n lt P th point (, sin ). P(, sin ) F F F = (-, 0) F = (,

More information

Aquauno Video 6 Plus Page 1

Aquauno Video 6 Plus Page 1 Connt th timr to th tp. Aquuno Vio 6 Plus Pg 1 Usr mnul 3 lik! For Aquuno Vio 6 (p/n): 8456 For Aquuno Vio 6 Plus (p/n): 8413 Opn th timr unit y prssing th two uttons on th sis, n fit 9V lklin ttry. Whn

More information

In which direction do compass needles always align? Why?

In which direction do compass needles always align? Why? AQA Trloy Unt 6.7 Mntsm n Eltromntsm - Hr 1 Complt t p ll: Mnt or s typ o or n t s stronst t t o t mnt. Tr r two typs o mnt pol: n. Wrt wt woul ppn twn t pols n o t mnt ntrtons low: Drw t mnt l lns on

More information

Seven-Segment Display Driver

Seven-Segment Display Driver 7-Smnt Disply Drivr, Ron s in 7-Smnt Disply Drivr, Ron s in Prolm 62. 00 0 0 00 0000 000 00 000 0 000 00 0 00 00 0 0 0 000 00 0 00 BCD Diits in inry Dsin Drivr Loi 4 inputs, 7 outputs 7 mps, h with 6 on

More information

Outline. 1 Introduction. 2 Min-Cost Spanning Trees. 4 Example

Outline. 1 Introduction. 2 Min-Cost Spanning Trees. 4 Example Outlin Computr Sin 33 Computtion o Minimum-Cost Spnnin Trs Prim's Alorithm Introution Mik Joson Dprtmnt o Computr Sin Univrsity o Clry Ltur #33 3 Alorithm Gnrl Constrution Mik Joson (Univrsity o Clry)

More information

More Foundations. Undirected Graphs. Degree. A Theorem. Graphs, Products, & Relations

More Foundations. Undirected Graphs. Degree. A Theorem. Graphs, Products, & Relations Mr Funtins Grphs, Pruts, & Rltins Unirt Grphs An unirt grph is pir f 1. A st f ns 2. A st f gs (whr n g is st f tw ns*) Friy, Sptmr 2, 2011 Ring: Sipsr 0.2 ginning f 0.4; Stughtn 1.1.5 ({,,,,}, {{,}, {,},

More information

This chapter covers special properties of planar graphs.

This chapter covers special properties of planar graphs. Chptr 21 Plnr Grphs This hptr ovrs spil proprtis of plnr grphs. 21.1 Plnr grphs A plnr grph is grph whih n b rwn in th pln without ny gs rossing. Som piturs of plnr grph might hv rossing gs, but it s possibl

More information

The University of Sydney MATH2969/2069. Graph Theory Tutorial 5 (Week 12) Solutions 2008

The University of Sydney MATH2969/2069. Graph Theory Tutorial 5 (Week 12) Solutions 2008 Th Univrsity o Syny MATH2969/2069 Grph Thory Tutoril 5 (Wk 12) Solutions 2008 1. (i) Lt G th isonnt plnr grph shown. Drw its ul G, n th ul o th ul (G ). (ii) Show tht i G is isonnt plnr grph, thn G is

More information

Compression. Compression. Compression. This part of the course... Ifi, UiO Norsk Regnesentral Vårsemester 2005 Wolfgang Leister

Compression. Compression. Compression. This part of the course... Ifi, UiO Norsk Regnesentral Vårsemester 2005 Wolfgang Leister Kurs INF5080 Ifi, UiO Norsk Rgnsntrl Vårsmstr 2005 Wolfgng Listr This prt of th ours...... is hl t Ifi, UiO... (Wolfgng Listr) n t ontins mtril from Univrsity Collg Krlsruh (Ptr Ol, Clmns Knorzr) Informtion

More information

Practice Test I Bonding and Geometry Name Per

Practice Test I Bonding and Geometry Name Per Prti Tst Boning n Gomtry Nm Pr This is prti - Do NOT ht yourslf of fining out wht you r pl of oing. B sur you follow th tsting onitions outlin low. DO NOT USE A CALCULATOR. You my us ONLY th lu prioi tl.

More information

Exam 1 Solution. CS 542 Advanced Data Structures and Algorithms 2/14/2013

Exam 1 Solution. CS 542 Advanced Data Structures and Algorithms 2/14/2013 CS Avn Dt Struturs n Algorithms Exm Solution Jon Turnr //. ( points) Suppos you r givn grph G=(V,E) with g wights w() n minimum spnning tr T o G. Now, suppos nw g {u,v} is to G. Dsri (in wors) mtho or

More information

ph controlled assembly of a polybutadiene poly(methacrylic acid) copolymer in water: packing considerations and kinetic limitations

ph controlled assembly of a polybutadiene poly(methacrylic acid) copolymer in water: packing considerations and kinetic limitations Supplmtry Mtril (ESI) for Soft Mttr This jourl is Th Royl Soity of hmistry 2009 p otroll ssmly of polyuti poly(mthryli i) opolymr i wtr pkig osirtios kiti limittios hristi Fryhough, Athoy J. Ry, Giuspp

More information

Graphs. Graphs. Graphs: Basic Terminology. Directed Graphs. Dr Papalaskari 1

Graphs. Graphs. Graphs: Basic Terminology. Directed Graphs. Dr Papalaskari 1 CSC 00 Disrt Struturs : Introuon to Grph Thory Grphs Grphs CSC 00 Disrt Struturs Villnov Univrsity Grphs r isrt struturs onsisng o vrs n gs tht onnt ths vrs. Grphs n us to mol: omputr systms/ntworks mthml

More information

Section 3: Antiderivatives of Formulas

Section 3: Antiderivatives of Formulas Chptr Th Intgrl Appli Clculus 96 Sction : Antirivtivs of Formuls Now w cn put th is of rs n ntirivtivs togthr to gt wy of vluting finit intgrls tht is ct n oftn sy. To vlut finit intgrl f(t) t, w cn fin

More information

12 - M G P L Z - M9BW. Port type. Bore size ø12, ø16 20/25/32/40/50/ MPa 10 C to 60 C (With no condensation) 50 to 400 mm/s +1.

12 - M G P L Z - M9BW. Port type. Bore size ø12, ø16 20/25/32/40/50/ MPa 10 C to 60 C (With no condensation) 50 to 400 mm/s +1. ris - MP - Compt gui ylinr ø, ø, ø, ø, ø, ø, ø, ø ow to Orr Cln sris lif typ (with spilly trt sliing prts) Vuum sution typ (with spilly trt sliing prts) ir ylinr otry tutor - M P - - MW ll ushing ring

More information

Section 10.4 Connectivity (up to paths and isomorphism, not including)

Section 10.4 Connectivity (up to paths and isomorphism, not including) Toy w will isuss two stions: Stion 10.3 Rprsnting Grphs n Grph Isomorphism Stion 10.4 Conntivity (up to pths n isomorphism, not inluing) 1 10.3 Rprsnting Grphs n Grph Isomorphism Whn w r working on n lgorithm

More information

Fundamental Algorithms for System Modeling, Analysis, and Optimization

Fundamental Algorithms for System Modeling, Analysis, and Optimization Fundmntl Algorithms for Sstm Modling, Anlsis, nd Optimiztion Edwrd A. L, Jijt Rohowdhur, Snjit A. Sshi UC Brkl EECS 144/244 Fll 2011 Copright 2010-11, E. A. L, J. Rohowdhur, S. A. Sshi, All rights rsrvd

More information

Binomials and Pascal s Triangle

Binomials and Pascal s Triangle Binomils n Psl s Tringl Binomils n Psl s Tringl Curriulum R AC: 0, 0, 08 ACS: 00 www.mthltis.om Binomils n Psl s Tringl Bsis 0. Intif th prts of th polnomil: 8. (i) Th gr. Th gr is. (Sin is th highst

More information

Tangram Fractions Overview: Students will analyze standard and nonstandard

Tangram Fractions Overview: Students will analyze standard and nonstandard ACTIVITY 1 Mtrils: Stunt opis o tnrm mstrs trnsprnis o tnrm mstrs sissors PROCEDURE Skills: Dsriin n nmin polyons Stuyin onrun Comprin rtions Tnrm Frtions Ovrviw: Stunts will nlyz stnr n nonstnr tnrms

More information

0.1. Exercise 1: the distances between four points in a graph

0.1. Exercise 1: the distances between four points in a graph Mth 707 Spring 2017 (Drij Grinrg): mitrm 3 pg 1 Mth 707 Spring 2017 (Drij Grinrg): mitrm 3 u: W, 3 My 2017, in lss or y mil (grinr@umn.u) or lss S th wsit or rlvnt mtril. Rsults provn in th nots, or in

More information

CEM143 MWF 8:00 8:50 am. October 5, 2018

CEM143 MWF 8:00 8:50 am. October 5, 2018 CEM43, Fll 208 st Miterm CEM43 MWF 8:00 8:50 m st Miterm toer 5, 208 Nme: Setion: PID: TA: This is lose ook n note exmintion. This exm hs 35 questions. Answer ll questions on the seprte nswer sheet (ule

More information

CEM143 MWF 8:00 8:50 am. October 5, 2018

CEM143 MWF 8:00 8:50 am. October 5, 2018 CEM43, Fll 208 st Miterm CEM43 MWF 8:00 8:50 m st Miterm toer 5, 208 Nme: Setion: PID: TA: This is lose ook n note exmintion. This exm hs 35 questions. Answer ll questions on the seprte nswer sheet (ule

More information

Constructive Geometric Constraint Solving

Constructive Geometric Constraint Solving Construtiv Gomtri Constrint Solving Antoni Soto i Rir Dprtmnt Llngutgs i Sistms Inormàtis Univrsitt Politèni Ctluny Brlon, Sptmr 2002 CGCS p.1/37 Prliminris CGCS p.2/37 Gomtri onstrint prolm C 2 D L BC

More information

Chem 107: Inorganic Chemistry (40720)

Chem 107: Inorganic Chemistry (40720) Chm 107: Inorgni Chmistry (40720) Prossor Mtt Lw -mil: lwm@ui.u Oi Hours: W 3:00-4:00p n Thurs 11-noon in NS2 2127 TAs Julit Khosrowi -mil: jkhosrow@ui.u Oi Hours: Tus 2:00-3:00p, 3 r loor tls, Rins Hll

More information

The Plan. Honey, I Shrunk the Data. Why Compress. Data Compression Concepts. Braille Example. Braille. x y xˆ

The Plan. Honey, I Shrunk the Data. Why Compress. Data Compression Concepts. Braille Example. Braille. x y xˆ h ln ony, hrunk th t ihr nr omputr in n nginring nivrsity of shington t omprssion onpts ossy t omprssion osslss t omprssion rfix os uffmn os th y 24 2 t omprssion onpts originl omprss o x y xˆ nor or omprss

More information

d e c b a d c b a d e c b a a c a d c c e b

d e c b a d c b a d e c b a a c a d c c e b FLAT PEYOTE STITCH Bin y mkin stoppr -- sw trou n pull it lon t tr until it is out 6 rom t n. Sw trou t in witout splittin t tr. You soul l to sli it up n own t tr ut it will sty in pl wn lt lon. Evn-Count

More information

Designing A Concrete Arch Bridge

Designing A Concrete Arch Bridge This is th mous Shwnh ri in Switzrln, sin y Rort Millrt in 1933. It spns 37.4 mtrs (122 t) n ws sin usin th sm rphil mths tht will monstrt in this lsson. To pro with this lsson, lik on th Nxt utton hr

More information

An undirected graph G = (V, E) V a set of vertices E a set of unordered edges (v,w) where v, w in V

An undirected graph G = (V, E) V a set of vertices E a set of unordered edges (v,w) where v, w in V Unirt Grphs An unirt grph G = (V, E) V st o vrtis E st o unorr gs (v,w) whr v, w in V USE: to mol symmtri rltionships twn ntitis vrtis v n w r jnt i thr is n g (v,w) [or (w,v)] th g (v,w) is inint upon

More information

CS 461, Lecture 17. Today s Outline. Example Run

CS 461, Lecture 17. Today s Outline. Example Run Prim s Algorithm CS 461, Ltur 17 Jr Si Univrsity o Nw Mxio In Prim s lgorithm, th st A mintin y th lgorithm orms singl tr. Th tr strts rom n ritrry root vrtx n grows until it spns ll th vrtis in V At h

More information

Walk Like a Mathematician Learning Task:

Walk Like a Mathematician Learning Task: Gori Dprtmnt of Euction Wlk Lik Mthmticin Lrnin Tsk: Mtrics llow us to prform mny usful mthmticl tsks which orinrily rquir lr numbr of computtions. Som typs of problms which cn b on fficintly with mtrics

More information

CSE303 - Introduction to the Theory of Computing Sample Solutions for Exercises on Finite Automata

CSE303 - Introduction to the Theory of Computing Sample Solutions for Exercises on Finite Automata CSE303 - Introduction to th Thory of Computing Smpl Solutions for Exrciss on Finit Automt Exrcis 2.1.1 A dtrministic finit utomton M ccpts th mpty string (i.., L(M)) if nd only if its initil stt is finl

More information

Numbering Boundary Nodes

Numbering Boundary Nodes Numring Bounry Nos Lh MBri Empori Stt Univrsity August 10, 2001 1 Introution Th purpos of this ppr is to xplor how numring ltril rsistor ntworks ffts thir rspons mtrix, Λ. Morovr, wht n lrn from Λ out

More information

Algorithmic and NP-Completeness Aspects of a Total Lict Domination Number of a Graph

Algorithmic and NP-Completeness Aspects of a Total Lict Domination Number of a Graph Intrntionl J.Mth. Comin. Vol.1(2014), 80-86 Algorithmi n NP-Compltnss Aspts of Totl Lit Domintion Numr of Grph Girish.V.R. (PES Institut of Thnology(South Cmpus), Bnglor, Krntk Stt, Ini) P.Ush (Dprtmnt

More information

Case Study VI Answers PHA 5127 Fall 2006

Case Study VI Answers PHA 5127 Fall 2006 Qustion. A ptint is givn 250 mg immit-rls thophyllin tblt (Tblt A). A wk ltr, th sm ptint is givn 250 mg sustin-rls thophyllin tblt (Tblt B). Th tblts follow on-comprtmntl mol n hv first-orr bsorption

More information

(PP/2-1)x S(PP) SIGNAL CONTACTS 0.50 FRONT POLARIZATION KEY SEE TABLE DIM A ( 10.00) D D

(PP/2-1)x S(PP) SIGNAL CONTACTS 0.50 FRONT POLARIZATION KEY SEE TABLE DIM A ( 10.00) D D IM #0. P[(+NN)/+] P(+NN) 0. PITH.0 PITH (/)x (NN/)x.. PITH (PP/)x (.) S[(PP/)+]... P POWR ONTTS P[(+NN)/] 0.0 RONT POLRIZTION KY S TL S SIGNL ONTTS S(PP/) IM.0 IM (.9) ROSS PS G R POSITION WHN ULL MT NOT...

More information

ALDEHYDES AND KETONES

ALDEHYDES AND KETONES CE 325 ALDEYDES AND KETNES CAP 17 ASSGN NUCLEPLC ADDTN T TE CARBNYL GRUP 1. What is the correct UPAC name for the following compound A. 2-Methyl-5-heptanone B. 7-Methyl-4-octanone C. 6-sopropyl-4-octanone

More information

CS September 2018

CS September 2018 Loil los Distriut Systms 06. Loil los Assin squn numrs to msss All ooprtin prosss n r on orr o vnts vs. physil los: rport tim o y Assum no ntrl tim sour Eh systm mintins its own lol lo No totl orrin o

More information

Chapter 7 1 CHAPTER 7

Chapter 7 1 CHAPTER 7 Chpter 7 1 CAPTER 7 1. () Reuing n lehye in the presene of ketone is very iffiult. Although the lehye is more retive, the ifferene in retivity is so smll tht fining regent to seletively reue one in the

More information

Outline. Computer Science 331. Computation of Min-Cost Spanning Trees. Costs of Spanning Trees in Weighted Graphs

Outline. Computer Science 331. Computation of Min-Cost Spanning Trees. Costs of Spanning Trees in Weighted Graphs Outlin Computr Sin 33 Computtion o Minimum-Cost Spnnin Trs Prim s Mik Joson Dprtmnt o Computr Sin Univrsity o Clry Ltur #34 Introution Min-Cost Spnnin Trs 3 Gnrl Constrution 4 5 Trmintion n Eiiny 6 Aitionl

More information

ME 522 PRINCIPLES OF ROBOTICS. FIRST MIDTERM EXAMINATION April 19, M. Kemal Özgören

ME 522 PRINCIPLES OF ROBOTICS. FIRST MIDTERM EXAMINATION April 19, M. Kemal Özgören ME 522 PINCIPLES OF OBOTICS FIST MIDTEM EXAMINATION April 9, 202 Nm Lst Nm M. Kml Özgörn 2 4 60 40 40 0 80 250 USEFUL FOMULAS cos( ) cos cos sin sin sin( ) sin cos cos sin sin y/ r, cos x/ r, r 0 tn 2(

More information

INTEGRALS. Chapter 7. d dx. 7.1 Overview Let d dx F (x) = f (x). Then, we write f ( x)

INTEGRALS. Chapter 7. d dx. 7.1 Overview Let d dx F (x) = f (x). Then, we write f ( x) Chptr 7 INTEGRALS 7. Ovrviw 7.. Lt d d F () f (). Thn, w writ f ( ) d F () + C. Ths intgrls r clld indfinit intgrls or gnrl intgrls, C is clld constnt of intgrtion. All ths intgrls diffr y constnt. 7..

More information

CS200: Graphs. Graphs. Directed Graphs. Graphs/Networks Around Us. What can this represent? Sometimes we want to represent directionality:

CS200: Graphs. Graphs. Directed Graphs. Graphs/Networks Around Us. What can this represent? Sometimes we want to represent directionality: CS2: Grphs Prihr Ch. 4 Rosn Ch. Grphs A olltion of nos n gs Wht n this rprsnt? n A omputr ntwork n Astrtion of mp n Soil ntwork CS2 - Hsh Tls 2 Dirt Grphs Grphs/Ntworks Aroun Us A olltion of nos n irt

More information

CHE 325 ALDEHYDES AND KETONES I CHAP 18 ASSIGN OCH 3 HCN C 4 H 7 NO. would be: CH 3 B. CH 3CH 2COOCH 3

CHE 325 ALDEHYDES AND KETONES I CHAP 18 ASSIGN OCH 3 HCN C 4 H 7 NO. would be: CH 3 B. CH 3CH 2COOCH 3 CE 325 ALDEYDES AND KETNES I CAP 18 ASSIGN 1. What is the correct IUPAC name for the following compound A. 2-Methyl-5-heptanone B. 7-Methyl-4-octanone C. 6-Isopropyl-4-octanone D. Isobutyl propyl ketone

More information

b. How many ternary words of length 23 with eight 0 s, nine 1 s and six 2 s?

b. How many ternary words of length 23 with eight 0 s, nine 1 s and six 2 s? MATH 3012 Finl Exm, My 4, 2006, WTT Stunt Nm n ID Numr 1. All our prts o this prolm r onrn with trnry strings o lngth n, i.., wors o lngth n with lttrs rom th lpht {0, 1, 2}.. How mny trnry wors o lngth

More information

HIGHER ORDER DIFFERENTIAL EQUATIONS

HIGHER ORDER DIFFERENTIAL EQUATIONS Prof Enriqu Mtus Nivs PhD in Mthmtis Edution IGER ORDER DIFFERENTIAL EQUATIONS omognous linr qutions with onstnt offiints of ordr two highr Appl rdution mthod to dtrmin solution of th nonhomognous qution

More information

Integration Continued. Integration by Parts Solving Definite Integrals: Area Under a Curve Improper Integrals

Integration Continued. Integration by Parts Solving Definite Integrals: Area Under a Curve Improper Integrals Intgrtion Continud Intgrtion y Prts Solving Dinit Intgrls: Ar Undr Curv Impropr Intgrls Intgrtion y Prts Prticulrly usul whn you r trying to tk th intgrl o som unction tht is th product o n lgric prssion

More information

Factorising FACTORISING.

Factorising FACTORISING. Ftorising FACTORISING www.mthletis.om.u Ftorising FACTORISING Ftorising is the opposite of expning. It is the proess of putting expressions into rkets rther thn expning them out. In this setion you will

More information

Multi-Section Coupled Line Couplers

Multi-Section Coupled Line Couplers /0/009 MultiSction Coupld Lin Couplrs /8 Multi-Sction Coupld Lin Couplrs W cn dd multipl coupld lins in sris to incrs couplr ndwidth. Figur 7.5 (p. 6) An N-sction coupld lin l W typiclly dsign th couplr

More information

DEVELOPING COMPUTER PROGRAM FOR COMPUTING EIGENPAIRS OF 2 2 MATRICES AND 3 3 UPPER TRIANGULAR MATRICES USING THE SIMPLE ALGORITHM

DEVELOPING COMPUTER PROGRAM FOR COMPUTING EIGENPAIRS OF 2 2 MATRICES AND 3 3 UPPER TRIANGULAR MATRICES USING THE SIMPLE ALGORITHM Fr Est Journl o Mthtil Sins (FJMS) Volu 6 Nur Pgs 8- Pulish Onlin: Sptr This ppr is vill onlin t http://pphjo/journls/jsht Pushp Pulishing Hous DEVELOPING COMPUTER PROGRAM FOR COMPUTING EIGENPAIRS OF MATRICES

More information

Jonathan Turner Exam 2-10/28/03

Jonathan Turner Exam 2-10/28/03 CS Algorihm n Progrm Prolm Exm Soluion S Soluion Jonhn Turnr Exm //. ( poin) In h Fioni hp ruur, u wn vrx u n i prn v u ing u v i v h lry lo hil in i l m hil o om ohr vrx. Suppo w hng hi, o h ing u i prorm

More information

Problem solving by search

Problem solving by search Prolm solving y srh Tomáš voo Dprtmnt o Cyrntis, Vision or Roots n Autonomous ystms Mrh 5, 208 / 3 Outlin rh prolm. tt sp grphs. rh trs. trtgis, whih tr rnhs to hoos? trtgy/algorithm proprtis? Progrmming

More information

CS553 Lecture Register Allocation I 3

CS553 Lecture Register Allocation I 3 Low-Lvl Issus Last ltur Intrproural analysis Toay Start low-lvl issus Rgistr alloation Latr Mor rgistr alloation Instrution shuling CS553 Ltur Rgistr Alloation I 2 Rgistr Alloation Prolm Assign an unoun

More information

QUESTIONS BEGIN HERE!

QUESTIONS BEGIN HERE! Points miss: Stunt's Nm: Totl sor: /100 points Est Tnnss Stt Univrsity Dprtmnt o Computr n Inormtion Sins CSCI 2710 (Trno) Disrt Struturs TEST or Sprin Smstr, 2005 R this or strtin! This tst is los ook

More information

Intramolecular Quaternization as Folding Strategy for the Synthesis of Catalytically Active Imidazolium-based Single Chain Nanoparticles

Intramolecular Quaternization as Folding Strategy for the Synthesis of Catalytically Active Imidazolium-based Single Chain Nanoparticles Supporting Informtion Intrmolulr Qutrniztion s Foling Strtgy for th Synthsis of Ctlytilly Ativ Imizolium-s Singl Chin Nnoprtils Romin Lmrt,, Ann-Lur Wirotius,, n Dnil Tton, * Lortoir Chimi s Polymèrs Orgniqus

More information

ADORO TE DEVOTE (Godhead Here in Hiding) te, stus bat mas, la te. in so non mor Je nunc. la in. tis. ne, su a. tum. tas: tur: tas: or: ni, ne, o:

ADORO TE DEVOTE (Godhead Here in Hiding) te, stus bat mas, la te. in so non mor Je nunc. la in. tis. ne, su a. tum. tas: tur: tas: or: ni, ne, o: R TE EVTE (dhd H Hdg) L / Mld Kbrd gú s v l m sl c m qu gs v nns V n P P rs l mul m d lud 7 súb Fí cón ví f f dó, cru gs,, j l f c r s m l qum t pr qud ct, us: ns,,,, cs, cut r l sns m / m fí hó sn sí

More information

DETAIL B DETAIL A 7 8 APPLY PRODUCT ID LABEL SB838XXXX ADJ FOUR POST RACK SQUARE HOLE RAIL B REVISION

DETAIL B DETAIL A 7 8 APPLY PRODUCT ID LABEL SB838XXXX ADJ FOUR POST RACK SQUARE HOLE RAIL B REVISION RVISION RV SRIPTION Y T HNG NO NOT OR PROUT LL JJH // LR TIL PPLY PROUT I LL TIL INSI UPPR ROSS MMR ON PR RK IS J OUR POST RK SQUR HOL RIL IS MN MS G NUT, PNL RNG 99 PPLY PROUT I LL INSI UPPR ROSS MMR

More information

Surds and Indices. Surds and Indices. Curriculum Ready ACMNA: 233,

Surds and Indices. Surds and Indices. Curriculum Ready ACMNA: 233, Surs n Inies Surs n Inies Curriulum Rey ACMNA:, 6 www.mthletis.om Surs SURDS & & Inies INDICES Inies n surs re very losely relte. A numer uner (squre root sign) is lle sur if the squre root n t e simplifie.

More information

5/1/2018. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees

5/1/2018. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees /1/018 W usully no strns y ssnn -lnt os to ll rtrs n t lpt (or mpl, 8-t on n ASCII). Howvr, rnt rtrs our wt rnt rquns, w n sv mmory n ru trnsmttl tm y usn vrl-lnt non. T s to ssn sortr os to rtrs tt our

More information