Objectives. Configuration. Epoxidation. Epoxidation. Epoxidation. Epoxidation. trans-alkene. trans-ring. O (S)-1,2-epoxypropane

Size: px
Start display at page:

Download "Objectives. Configuration. Epoxidation. Epoxidation. Epoxidation. Epoxidation. trans-alkene. trans-ring. O (S)-1,2-epoxypropane"

Transcription

1 0% % 0% 77% % (2,3)- 3. (2,3)-. (2,3)- 5. (2,3) onfigurtion?-2-hloro-3-fluoroutne LQ #) A) Drw enntiomer. B) Drw distereomer. Lst leture hirlity enntiomers distereomers / nottion This leture hirlity jetives optil & iologil properties hirl moleules ontining more thn one hirlity enter. retions tht rete stereoisomers 2 Dr. Ky nderg xns tht rete stereogeni enter Epoxidtion Epoxidtion 3 2 (peroxyeti id) tereospeifi 3 2 (peroxyeti id) etion 7.9 enntiomers ()-,2-epoxypropne Epoxidtion hirl prohirl enntiotopi (peroxyeti fes id) 3 3 ()-,2-epoxypropne trns-lkene trns-ring 3 etion 7.9 xns tht rete stereogeni enter 3 hirl ptilly intive ()-,2-epoxypropne (±)-,2-epoxypropne hirl Epoxidtion (peroxyeti id) emi mixture ptilly intive 3 ()-,2-epoxypropne 3 ow do we show remi mixture in Mrvin? tik onds imply this is remi mixture. Epoxidtion 3 2 (peroxyeti id) ()-,2-epoxypropne ()-,2-epoxypropne (±)-,2-epoxypropne = etion 7.9 xns tht rete stereogeni enter 5 etion 7.9 6

2 remte () 3 3 hirl () Eletrophili ddition etion hirl optilly intive 3 3 prohirl 2 2 h omintion enntiomers 3 etion 7.9 hirl optilly intive remi mixture 8 ( +_ ) Alohol to lkyl romide hirl ut remi optilly intive 2 2 Fumri id hirl optilly intive etion 7.9 ( +_ ) fumrse (hirl enzyme) hirl ut remi optilly intive ()-( )-mli id hirl optilly tive 9 emi produt mixture? Viinl hlohydrin E-lkene 3 etion nti-ddition Doule whmmy 3 0 xns tht rete stereoisomers Viinl hlohydrin (2,3) onfigurtion Z-lkene (2,3) onfigurtion etion 7.3 Enntiomers remte xns tht rete stereoisomers E-lkene E vs Z lkene etion tereospeifi rxn: nti ddition 2 enntiomers Z-lkene

3 2% 3% 93% % 2% 0% xns tht rete stereoisomers E-lkene 3 3 epresenttion 3 Implies the presene of ll onfigurtions (2,3) (2,3) 3 3 etion enntiomers Z-lkene xns tht rete stereoisomers E-lkene produt etion 7.3 It doesn t mtter whih onds re wedge nd whih re dshed s long s one is nd the other. xns tht rete stereoisomers 3 Z-lkene produts etion 7.3 It doesn t mtter whih onds re wedge nd whih re dshed s long s oth re or oth re. 5. A,B 2. A, 3. A,D. B, 5. B,D 6.,D Whih two from Z-lkene? (Z)- Whih re formed? A 2 3 I like to keep the onds of the reting sites in the plne of the pper (E)- 3 nti-ddition etion B Whih two hve the methyl & ethyl on the sme side of the nti onformtionl plne? D We re going to fous on how the ethyl groups orienttions hnge s the retion progresses l 2 pprohes top fe of doule ond Viinl rotte hlohydrin 90 o l l ethyls: ngled down, wedge l l 2, 2 Both pprohes hve the sme proility of ourring, therefore produt mixture will e from eh pproh. ethyls: stright out, wedge ethyls: ngled up, wedge l l l l 2 pprohes ottom fe of doule ond l l prtil rotionmore plnr onds L ethyl: stright wter pprohes from ottom wter pushes ethyl nd methyl up L ethyl: ngled up T (proton trnsfer) Viinl hlohydrin L ethyl: ngled down wedge l Doule whmmy ethyl: ngled down wedge lens prtilly reking ond l nti-ddition l ethyl: ngled down LQ2 elet the onfigurtion ethyl: ngled down l 3

4 3% 6% 8% 83% l l Don t forget understood Viinl hlohydrin l l L ethyl: ngled down, wedge wter pushes ethyl nd methyl down Viinl hlohydrin wter pprohes from top l ethyl: ngled up, wedge T (proton trnsfer) ethyl: ngled up, wedge ethyls: ngled up, wedge l l L ethyl: flttens, wedge I like to keep the onds of the reting speies in the plne of the pper. ow for the pproh from the ottom fe. l l l lenge l 2, 2 l l l l X l l X A B D 2-hloro--methylylohexnol In WeAssign nd erlier in the text, we llowed you to show just one of the enntiomers, ut e wre tht the produt mixture tully ontins oth. y ttention to instrutions involving stereospeifi retions. 2 xns tht rete distereomers tereospeifi rxn: syn ddition t ydrogention 3 2 dded from ottom fe etion dded from top fe Distereomers elet the true sttement regrding the produts. t. oth re hirl 2. oth re hirl 3. only is is hirl. only trns is hirl hirl? is 3 2 dded from ottom fe trns dded from top fe xns tht rete distereomers 3 2 M of hydrogention syn ddition 2 t Less hindered fe tereoseletive rxn: preferene for one stereoisomer tereospeifi rxn: 3 2 etion 7.3,2-dimethylylohexne 3 2 LQ #3 A) M B) mop ) stereoisomer not formed D) desription of M & mop. 2

5 mino group etion Distereomers (2,3)-3-minoutn-2-ol ( 2,3 )-3-minoutn-2-ol liquid olid, mp 9 o esolution of enntiomers eprtion of remi mixture into its enntiomeri omponents esolution of enntiomers etion 7. We n use the different melting points to seprte them But how do we seprte enntiomers whih hve the sme melting points, oiling points, et.? 25, slt distereomers, slt 26 Mximum # of stereoisomers = 2 n When stereogeni enters re equivlently sustituted (possile ugrs forms), the # is less thn 2 n. ldohexose (sugr) llose (2,3,,5)- (2,3,,5)- (2,3,,5)- (2,3,,5)- (2,3,,5)- ltrose (2,3,,5)- (2,3,,5)- (2,3,,5)- et. gluose etion totl. 27 2,3,,5,6-penthydroxyhexnl 3 holi id 3 holi 3 id # of hirlity enters? (ile) etion 7.2 LQ A) ow mny enntiomers of holi id re there? B) ow mny distereomers of holi id re there? # of stereoisomers? 2 = 208 Any possiility of -forms? 28 The mrine orgnism new highly potent nlogue utilizes hirl enzymes of ryosttin to tlyze only one of the stereoisomers yosttin yosttin synthesis Bugul neritin, orl-like ritter lled ryozon tht is the soure of the promising ner drug ryosttin. The urrent reipe lls for tons of hrvested Bugul to produe out hlf n oune of ryosttin. ientists hve sueeded t mking ext replis of ryosttin from hemils tht n e ought in ny l tlog, ut their proesses n tke up to 70 expensive steps, ompred to the 0 to 5 used to mke most ommerilly vile drugs steps nd 2% yield 29 ws produed in >3% yield nd only 7 steps

6 . 2. yosttin synthesis symmetri enter is synonymous with hirl enter Dniel omins: ew syntheti methodologies, strtegies for symmetri synthesis, direted metlltion retions, stereoseletive redutions, tlyti symmetri synthesis, new hirl uxiliries, nd the totl synthesis of nturl produts nd iologilly tive ompounds Dniel L. omins, jn. Joseph, o ong, im. Al-wr, hristopher J. Foti, Yue-mei Zhng, Xinghi hen, Donld. LMunyon nd Mri Guerr-Weltzien, "Asymmetri ynthesis nd yntheti Utility of 2,3- Dihydro--pyridones," ure nd Applied hem., 997, 69, 77. Dn omins work Dniel L. omins, resh M. Thkker, Mtthew F. Bevsky, nd Mohmmed M. Bdwi, "hirl Auxiliry Medited itet-pengler etions: Asymmetri yntheses of (-)-Ludnosine, (+)-Gluine nd (-)-Xylopinine," Tetrhedron 997, 53, Dniel L. omins, Xinghi hen nd Lwrene A. Morgn, "Enntiopure - Ayldihydropyridones s yntheti Intermedites: Asymmetri ynthesis of (-)-eptiine nd (-)-Tylophorine," J. rg. hem., 997, 62, 735. Jeffrey T. Kuethe nd Dniel L. omins, "Asymmetri Totl ynthesis of (+)-nnistivine," J. rg. hem., 200, 69, tereogeni enters other thn ron i: four tetrhedrlly distriuted sustituents i & etion 7.6 sp 3 hyridized toms with 3 different sustituents nd lone pir 33 tereogeni enters other thn ron very fst o resolution euse of immedite remiztion due to rpid inversion (smll energy rrier) etion tereogeni enters other thn ron slow etion 7.6 sp 3 hyridized toms with 3 different sustituents nd lone pir ine the inversion is slow, enntiomers n e resolved (seprted) (lrger energy rrier for inversion from to.) 35 tereogeni enters other thn ron onfigurtion + etion 7.6 sp 3 hyridized toms with 3 different sustituents nd lone pir Given tht the lone pir is the lowest rnking, selet the onfigurtion. 2 Inversion from to does not our redily euse of the lrge energy rrier of inversion. 3 (_)-utyl methyl sulfoxide 0 % 90 % 36 6

1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. 1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the

More information

CHEM1611 Answers to Problem Sheet 9

CHEM1611 Answers to Problem Sheet 9 CEM1611 Answers to Prolem Sheet 9 1. Tutomers re struturl isomers whih re relte y migrtion of hyrogen tom n the exhnge of single on n jent oule on. Compoun Tutomer 2 2 2 2 2 2 2 2 2 2 2 2. () Whih pir

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

Project 6: Minigoals Towards Simplifying and Rewriting Expressions

Project 6: Minigoals Towards Simplifying and Rewriting Expressions MAT 51 Wldis Projet 6: Minigols Towrds Simplifying nd Rewriting Expressions The distriutive property nd like terms You hve proly lerned in previous lsses out dding like terms ut one prolem with the wy

More information

3.15 NMR spectroscopy Different types of NMR There are two main types of NMR 1. C 13 NMR 2. H (proton) NMR

3.15 NMR spectroscopy Different types of NMR There are two main types of NMR 1. C 13 NMR 2. H (proton) NMR .5 NMR spetrosopy Different types of NMR There re two min types of NMR. NMR. (proton) NMR There is only round % in orgni moleules ut modern NMR mhines re sensitive enough to give full spetr for The spetr

More information

6.3.2 Spectroscopy. N Goalby chemrevise.org 1 NO 2 CH 3. CH 3 C a. NMR spectroscopy. Different types of NMR

6.3.2 Spectroscopy. N Goalby chemrevise.org 1 NO 2 CH 3. CH 3 C a. NMR spectroscopy. Different types of NMR 6.. Spetrosopy NMR spetrosopy Different types of NMR NMR spetrosopy involves intertion of mterils with the lowenergy rdiowve region of the eletromgneti spetrum NMR spetrosopy is the sme tehnology s tht

More information

General Organic Chemistry 1033

General Organic Chemistry 1033 Generl rgni hemistry 0 or Exmple, is, Dihloropentene (igh dipolemoment) Similr is the se with hexene-. isexene (more polr) trns,dihloropentene ( Less dipolemoment ) trns exene (Less polr) (iv) By studying

More information

CHAPTER 3. base O N R R = C 5 H 11 OTHP. OBn. OBn. Ti(OiPr) 4, D-( )-DET t-buooh, MS 4Å 20 C, 4 d MOMO. OBn + MOMO OMOM MOMO OH

CHAPTER 3. base O N R R = C 5 H 11 OTHP. OBn. OBn. Ti(OiPr) 4, D-( )-DET t-buooh, MS 4Å 20 C, 4 d MOMO. OBn + MOMO OMOM MOMO OH Chpter 3 1 CPTER 3 1. The produt is the ldehyde, nd the mehnism is nlogous to the DMS-sed oxidtions disussed in Setion 3.2.C. resonle mehnism is shown. Pyridine -oxide ttks the romomethyl moiety vi n S

More information

Intermediate Math Circles Wednesday 17 October 2012 Geometry II: Side Lengths

Intermediate Math Circles Wednesday 17 October 2012 Geometry II: Side Lengths Intermedite Mth Cirles Wednesdy 17 Otoer 01 Geometry II: Side Lengths Lst week we disussed vrious ngle properties. As we progressed through the evening, we proved mny results. This week, we will look t

More information

BOND ORDER (BO): Single bond Þ BO = 1; Double bond Þ BO = 2; Triple bond Þ BO = 3 Bond Order Þ bond strength and bond length

BOND ORDER (BO): Single bond Þ BO = 1; Double bond Þ BO = 2; Triple bond Þ BO = 3 Bond Order Þ bond strength and bond length EMISTRY 104 elp Sheet #1 hem 103 Review (Text: h 6, h 7) Do topics pproprite for your lecture Prepred y Dr. Tony Jco http://www.chem.wisc.edu/res/clc (Resource pge) Nuggets: Electronegtivity (6.7), Bond

More information

anti-addition acidic conditions S N basic conditions CH 18: Ethers and Epoxides

anti-addition acidic conditions S N basic conditions CH 18: Ethers and Epoxides rgnic Chemistry Notes y Jim Mxk C 18: Ethers nd Epoxides jim.mxk@nu.edu Topics: Nming ethers nd understnding ethers inertness Prepring ethers vi Willimson ether synthesis: S N 2 vs E 2 ection of ethers

More information

Spacetime and the Quantum World Questions Fall 2010

Spacetime and the Quantum World Questions Fall 2010 Spetime nd the Quntum World Questions Fll 2010 1. Cliker Questions from Clss: (1) In toss of two die, wht is the proility tht the sum of the outomes is 6? () P (x 1 + x 2 = 6) = 1 36 - out 3% () P (x 1

More information

6.3.2 Spectroscopy. N Goalby chemrevise.org 1 NO 2 H 3 CH3 C. NMR spectroscopy. Different types of NMR

6.3.2 Spectroscopy. N Goalby chemrevise.org 1 NO 2 H 3 CH3 C. NMR spectroscopy. Different types of NMR 6.. Spetrosopy NMR spetrosopy Different types of NMR NMR spetrosopy involves intertion of mterils with the lowenergy rdiowve region of the eletromgneti spetrum NMR spetrosopy is the sme tehnology s tht

More information

Arrow s Impossibility Theorem

Arrow s Impossibility Theorem Rep Fun Gme Properties Arrow s Theorem Arrow s Impossiility Theorem Leture 12 Arrow s Impossiility Theorem Leture 12, Slide 1 Rep Fun Gme Properties Arrow s Theorem Leture Overview 1 Rep 2 Fun Gme 3 Properties

More information

2.4 Linear Inequalities and Interval Notation

2.4 Linear Inequalities and Interval Notation .4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or

More information

Unit 4. Combinational Circuits

Unit 4. Combinational Circuits Unit 4. Comintionl Ciruits Digitl Eletroni Ciruits (Ciruitos Eletrónios Digitles) E.T.S.I. Informáti Universidd de Sevill 5/10/2012 Jorge Jun 2010, 2011, 2012 You re free to opy, distriute

More information

AP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals

AP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals AP Clulus BC Chpter 8: Integrtion Tehniques, L Hopitl s Rule nd Improper Integrls 8. Bsi Integrtion Rules In this setion we will review vrious integrtion strtegies. Strtegies: I. Seprte the integrnd into

More information

Trigonometry Revision Sheet Q5 of Paper 2

Trigonometry Revision Sheet Q5 of Paper 2 Trigonometry Revision Sheet Q of Pper The Bsis - The Trigonometry setion is ll out tringles. We will normlly e given some of the sides or ngles of tringle nd we use formule nd rules to find the others.

More information

Lecture 27: Diffusion of Ions: Part 2: coupled diffusion of cations and

Lecture 27: Diffusion of Ions: Part 2: coupled diffusion of cations and Leture 7: iffusion of Ions: Prt : oupled diffusion of tions nd nions s desried y Nernst-Plnk Eqution Tody s topis Continue to understnd the fundmentl kinetis prmeters of diffusion of ions within n eletrilly

More information

6.5 Improper integrals

6.5 Improper integrals Eerpt from "Clulus" 3 AoPS In. www.rtofprolemsolving.om 6.5. IMPROPER INTEGRALS 6.5 Improper integrls As we ve seen, we use the definite integrl R f to ompute the re of the region under the grph of y =

More information

Parse trees, ambiguity, and Chomsky normal form

Parse trees, ambiguity, and Chomsky normal form Prse trees, miguity, nd Chomsky norml form In this lecture we will discuss few importnt notions connected with contextfree grmmrs, including prse trees, miguity, nd specil form for context-free grmmrs

More information

CALCULATING REACTING QUANTITIES

CALCULATING REACTING QUANTITIES MODULE 2 14 WORKSHEET WORKSHEET For multiple-hoie questions 1 5 irle the letter orresponding to the most orret nswer. 1 The lned eqution for the urning of utnol (C 4 H 9 OH) is given elow: C 4 H 9 OH(l)

More information

12.4 Similarity in Right Triangles

12.4 Similarity in Right Triangles Nme lss Dte 12.4 Similrit in Right Tringles Essentil Question: How does the ltitude to the hpotenuse of right tringle help ou use similr right tringles to solve prolems? Eplore Identifing Similrit in Right

More information

Introduction to Olympiad Inequalities

Introduction to Olympiad Inequalities Introdution to Olympid Inequlities Edutionl Studies Progrm HSSP Msshusetts Institute of Tehnology Snj Simonovikj Spring 207 Contents Wrm up nd Am-Gm inequlity 2. Elementry inequlities......................

More information

Algorithm Design and Analysis

Algorithm Design and Analysis Algorithm Design nd Anlysis LECTURE 5 Supplement Greedy Algorithms Cont d Minimizing lteness Ching (NOT overed in leture) Adm Smith 9/8/10 A. Smith; sed on slides y E. Demine, C. Leiserson, S. Rskhodnikov,

More information

April 8, 2017 Math 9. Geometry. Solving vector problems. Problem. Prove that if vectors and satisfy, then.

April 8, 2017 Math 9. Geometry. Solving vector problems. Problem. Prove that if vectors and satisfy, then. pril 8, 2017 Mth 9 Geometry Solving vetor prolems Prolem Prove tht if vetors nd stisfy, then Solution 1 onsider the vetor ddition prllelogrm shown in the Figure Sine its digonls hve equl length,, the prllelogrm

More information

Comparing the Pre-image and Image of a Dilation

Comparing the Pre-image and Image of a Dilation hpter Summry Key Terms Postultes nd Theorems similr tringles (.1) inluded ngle (.2) inluded side (.2) geometri men (.) indiret mesurement (.6) ngle-ngle Similrity Theorem (.2) Side-Side-Side Similrity

More information

H 4 H 8 N 2. Example 1 A compound is found to have an accurate relative formula mass of It is thought to be either CH 3.

H 4 H 8 N 2. Example 1 A compound is found to have an accurate relative formula mass of It is thought to be either CH 3. . Spetrosopy Mss spetrosopy igh resolution mss spetrometry n e used to determine the moleulr formul of ompound from the urte mss of the moleulr ion For exmple, the following moleulr formuls ll hve rough

More information

ENERGY AND PACKING. Outline: MATERIALS AND PACKING. Crystal Structure

ENERGY AND PACKING. Outline: MATERIALS AND PACKING. Crystal Structure EERGY AD PACKIG Outline: Crstlline versus morphous strutures Crstl struture - Unit ell - Coordintion numer - Atomi pking ftor Crstl sstems on dense, rndom pking Dense, regulr pking tpil neighor ond energ

More information

Matrices SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics (c) 1. Definition of a Matrix

Matrices SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics (c) 1. Definition of a Matrix tries Definition of tri mtri is regulr rry of numers enlosed inside rkets SCHOOL OF ENGINEERING & UIL ENVIRONEN Emple he following re ll mtries: ), ) 9, themtis ), d) tries Definition of tri Size of tri

More information

Arrow s Impossibility Theorem

Arrow s Impossibility Theorem Rep Voting Prdoxes Properties Arrow s Theorem Arrow s Impossiility Theorem Leture 12 Arrow s Impossiility Theorem Leture 12, Slide 1 Rep Voting Prdoxes Properties Arrow s Theorem Leture Overview 1 Rep

More information

In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.

In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle. Mth 3329-Uniform Geometries Leture 06 1. Review of trigonometry While we re looking t Eulid s Elements, I d like to look t some si trigonometry. Figure 1. The Pythgoren theorem sttes tht if = 90, then

More information

Section 1.3 Triangles

Section 1.3 Triangles Se 1.3 Tringles 21 Setion 1.3 Tringles LELING TRINGLE The line segments tht form tringle re lled the sides of the tringle. Eh pir of sides forms n ngle, lled n interior ngle, nd eh tringle hs three interior

More information

CS 573 Automata Theory and Formal Languages

CS 573 Automata Theory and Formal Languages Non-determinism Automt Theory nd Forml Lnguges Professor Leslie Lnder Leture # 3 Septemer 6, 2 To hieve our gol, we need the onept of Non-deterministi Finite Automton with -moves (NFA) An NFA is tuple

More information

ANSWERS TO REVIEW PROBLEM SET CH F

ANSWERS TO REVIEW PROBLEM SET CH F ANSWERS T REVIEW PRBLEM SET C41-018F [1] () The chirl isomer is stilized y intrmoleculr hydrogen onding etween the two guche groups. Such n rrngement lso keeps the ulky tert-utyl groups frther prt s they

More information

CS311 Computational Structures Regular Languages and Regular Grammars. Lecture 6

CS311 Computational Structures Regular Languages and Regular Grammars. Lecture 6 CS311 Computtionl Strutures Regulr Lnguges nd Regulr Grmmrs Leture 6 1 Wht we know so fr: RLs re losed under produt, union nd * Every RL n e written s RE, nd every RE represents RL Every RL n e reognized

More information

a) Read over steps (1)- (4) below and sketch the path of the cycle on a P V plot on the graph below. Label all appropriate points.

a) Read over steps (1)- (4) below and sketch the path of the cycle on a P V plot on the graph below. Label all appropriate points. Prole 3: Crnot Cyle of n Idel Gs In this prole, the strting pressure P nd volue of n idel gs in stte, re given he rtio R = / > of the volues of the sttes nd is given Finlly onstnt γ = 5/3 is given You

More information

CS 491G Combinatorial Optimization Lecture Notes

CS 491G Combinatorial Optimization Lecture Notes CS 491G Comintoril Optimiztion Leture Notes Dvi Owen July 30, August 1 1 Mthings Figure 1: two possile mthings in simple grph. Definition 1 Given grph G = V, E, mthing is olletion of eges M suh tht e i,

More information

Chem Homework 11 due Monday, Apr. 28, 2014, 2 PM

Chem Homework 11 due Monday, Apr. 28, 2014, 2 PM Chem 44 - Homework due ondy, pr. 8, 4, P.. . Put this in eq 8.4 terms: E m = m h /m e L for L=d The degenery in the ring system nd the inresed sping per level (4x bigger) mkes the sping between the HOO

More information

Lecture 3: Equivalence Relations

Lecture 3: Equivalence Relations Mthcmp Crsh Course Instructor: Pdric Brtlett Lecture 3: Equivlence Reltions Week 1 Mthcmp 2014 In our lst three tlks of this clss, we shift the focus of our tlks from proof techniques to proof concepts

More information

Lesson 2.1 Inductive Reasoning

Lesson 2.1 Inductive Reasoning Lesson 2.1 Inutive Resoning Nme Perio Dte For Eerises 1 7, use inutive resoning to fin the net two terms in eh sequene. 1. 4, 8, 12, 16,, 2. 400, 200, 100, 50, 25,, 3. 1 8, 2 7, 1 2, 4, 5, 4. 5, 3, 2,

More information

Technische Universität München Winter term 2009/10 I7 Prof. J. Esparza / J. Křetínský / M. Luttenberger 11. Februar Solution

Technische Universität München Winter term 2009/10 I7 Prof. J. Esparza / J. Křetínský / M. Luttenberger 11. Februar Solution Tehnishe Universität Münhen Winter term 29/ I7 Prof. J. Esprz / J. Křetínský / M. Luttenerger. Ferur 2 Solution Automt nd Forml Lnguges Homework 2 Due 5..29. Exerise 2. Let A e the following finite utomton:

More information

Lecture 6: Coding theory

Lecture 6: Coding theory Leture 6: Coing theory Biology 429 Crl Bergstrom Ferury 4, 2008 Soures: This leture loosely follows Cover n Thoms Chpter 5 n Yeung Chpter 3. As usul, some of the text n equtions re tken iretly from those

More information

Chemical Equilibrium

Chemical Equilibrium Chpter 16 Questions 5, 7, 31, 33, 35, 43, 71 Chemil Equilibrium Exmples of Equilibrium Wter n exist simultneously in the gs nd liquid phse. The vpor pressure of H O t given temperture is property ssoited

More information

Physics 9 Fall 2011 Homework 2 - Solutions Friday September 2, 2011

Physics 9 Fall 2011 Homework 2 - Solutions Friday September 2, 2011 Physics 9 Fll 0 Homework - s Fridy September, 0 Mke sure your nme is on your homework, nd plese box your finl nswer. Becuse we will be giving prtil credit, be sure to ttempt ll the problems, even if you

More information

Selectivities, including stereoselectivity, would be affected by the relative energy of each. possible product. k 1 K k 2 A (1)

Selectivities, including stereoselectivity, would be affected by the relative energy of each. possible product. k 1 K k 2 A (1) . tereoontrol in rgni etions () Priniples in seletive orgni retions () Curtin mmett priniple eletivities re ontrolle y the reltive energies of two trnsition sttes. The ifferene of retnts or prouts oes

More information

1.3 SCALARS AND VECTORS

1.3 SCALARS AND VECTORS Bridge Course Phy I PUC 24 1.3 SCLRS ND VECTORS Introdution: Physis is the study of nturl phenomen. The study of ny nturl phenomenon involves mesurements. For exmple, the distne etween the plnet erth nd

More information

Numbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point

Numbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point GCSE C Emple 7 Work out 9 Give your nswer in its simplest form Numers n inies Reiprote mens invert or turn upsie own The reiprol of is 9 9 Mke sure you only invert the frtion you re iviing y 7 You multiply

More information

Solutions for HW9. Bipartite: put the red vertices in V 1 and the black in V 2. Not bipartite!

Solutions for HW9. Bipartite: put the red vertices in V 1 and the black in V 2. Not bipartite! Solutions for HW9 Exerise 28. () Drw C 6, W 6 K 6, n K 5,3. C 6 : W 6 : K 6 : K 5,3 : () Whih of the following re iprtite? Justify your nswer. Biprtite: put the re verties in V 1 n the lk in V 2. Biprtite:

More information

Surface maps into free groups

Surface maps into free groups Surfce mps into free groups lden Wlker Novemer 10, 2014 Free groups wedge X of two circles: Set F = π 1 (X ) =,. We write cpitl letters for inverse, so = 1. e.g. () 1 = Commuttors Let x nd y e loops. The

More information

Probability. b a b. a b 32.

Probability. b a b. a b 32. Proility If n event n hppen in '' wys nd fil in '' wys, nd eh of these wys is eqully likely, then proility or the hne, or its hppening is, nd tht of its filing is eg, If in lottery there re prizes nd lnks,

More information

H (2a, a) (u 2a) 2 (E) Show that u v 4a. Explain why this implies that u v 4a, with equality if and only u a if u v 2a.

H (2a, a) (u 2a) 2 (E) Show that u v 4a. Explain why this implies that u v 4a, with equality if and only u a if u v 2a. Chpter Review 89 IGURE ol hord GH of the prol 4. G u v H (, ) (A) Use the distne formul to show tht u. (B) Show tht G nd H lie on the line m, where m ( )/( ). (C) Solve m for nd sustitute in 4, otining

More information

3.2.2 Kinetics. Maxwell Boltzmann distribution. 128 minutes. 128 marks. Page 1 of 12

3.2.2 Kinetics. Maxwell Boltzmann distribution. 128 minutes. 128 marks. Page 1 of 12 3.. Kinetics Mxwell Boltzmnn distribution 8 minutes 8 mrks Pge of M. () M On the energy xis E mp t the mximum of the originl pek M The limits for the horizontl position of E mp re defined s bove the word

More information

] dx (3) = [15x] 2 0

] dx (3) = [15x] 2 0 Leture 6. Double Integrls nd Volume on etngle Welome to Cl IV!!!! These notes re designed to be redble nd desribe the w I will eplin the mteril in lss. Hopefull the re thorough, but it s good ide to hve

More information

Area and Perimeter. Area and Perimeter. Solutions. Curriculum Ready.

Area and Perimeter. Area and Perimeter. Solutions. Curriculum Ready. Are n Perimeter Are n Perimeter Solutions Curriulum Rey www.mthletis.om How oes it work? Solutions Are n Perimeter Pge questions Are using unit squres Are = whole squres Are = 6 whole squres = units =

More information

22: Union Find. CS 473u - Algorithms - Spring April 14, We want to maintain a collection of sets, under the operations of:

22: Union Find. CS 473u - Algorithms - Spring April 14, We want to maintain a collection of sets, under the operations of: 22: Union Fin CS 473u - Algorithms - Spring 2005 April 14, 2005 1 Union-Fin We wnt to mintin olletion of sets, uner the opertions of: 1. MkeSet(x) - rete set tht ontins the single element x. 2. Fin(x)

More information

Lecture 6. Notes. Notes. Notes. Representations Z A B and A B R. BTE Electronics Fundamentals August Bern University of Applied Sciences

Lecture 6. Notes. Notes. Notes. Representations Z A B and A B R. BTE Electronics Fundamentals August Bern University of Applied Sciences Lecture 6 epresenttions epresenttions TE52 - Electronics Fundmentls ugust 24 ern University of pplied ciences ev. c2d5c88 6. Integers () sign-nd-mgnitude representtion The set of integers contins the Nturl

More information

Algorithm Design and Analysis

Algorithm Design and Analysis Algorithm Design nd Anlysis LECTURE 8 Mx. lteness ont d Optiml Ching Adm Smith 9/12/2008 A. Smith; sed on slides y E. Demine, C. Leiserson, S. Rskhodnikov, K. Wyne Sheduling to Minimizing Lteness Minimizing

More information

Exercise 3 Logic Control

Exercise 3 Logic Control Exerise 3 Logi Control OBJECTIVE The ojetive of this exerise is giving n introdution to pplition of Logi Control System (LCS). Tody, LCS is implemented through Progrmmle Logi Controller (PLC) whih is lled

More information

Thermodynamics. Question 1. Question 2. Question 3 3/10/2010. Practice Questions PV TR PV T R

Thermodynamics. Question 1. Question 2. Question 3 3/10/2010. Practice Questions PV TR PV T R /10/010 Question 1 1 mole of idel gs is rought to finl stte F y one of three proesses tht hve different initil sttes s shown in the figure. Wht is true for the temperture hnge etween initil nd finl sttes?

More information

ERT 316: REACTION ENGINEERING CHAPTER 3 RATE LAWS & STOICHIOMETRY

ERT 316: REACTION ENGINEERING CHAPTER 3 RATE LAWS & STOICHIOMETRY ER 316: REIO EGIEERIG HER 3 RE LWS & SOIHIOMERY 1 OULIE R 1: Rte Lws Reltive Rtes of Retion Retion Orer & Rte Lw Retion Rte onstnt, k R 2: Stoihiometry th System Stoihiometri le low System Stoihiometri

More information

SECTION A STUDENT MATERIAL. Part 1. What and Why.?

SECTION A STUDENT MATERIAL. Part 1. What and Why.? SECTION A STUDENT MATERIAL Prt Wht nd Wh.? Student Mteril Prt Prolem n > 0 n > 0 Is the onverse true? Prolem If n is even then n is even. If n is even then n is even. Wht nd Wh? Eploring Pure Mths Are

More information

Chapter 4 State-Space Planning

Chapter 4 State-Space Planning Leture slides for Automted Plnning: Theory nd Prtie Chpter 4 Stte-Spe Plnning Dn S. Nu CMSC 722, AI Plnning University of Mrylnd, Spring 2008 1 Motivtion Nerly ll plnning proedures re serh proedures Different

More information

Welcome. Balanced search trees. Balanced Search Trees. Inge Li Gørtz

Welcome. Balanced search trees. Balanced Search Trees. Inge Li Gørtz Welome nge Li Gørt. everse tehing n isussion of exerises: 02110 nge Li Gørt 3 tehing ssistnts 8.00-9.15 Group work 9.15-9.45 isussions of your solutions in lss 10.00-11.15 Leture 11.15-11.45 Work on exerises

More information

1 This diagram represents the energy change that occurs when a d electron in a transition metal ion is excited by visible light.

1 This diagram represents the energy change that occurs when a d electron in a transition metal ion is excited by visible light. 1 This igrm represents the energy hnge tht ours when eletron in trnsition metl ion is exite y visile light. Give the eqution tht reltes the energy hnge ΔE to the Plnk onstnt, h, n the frequeny, v, of the

More information

2. Topic: Summation of Series (Mathematical Induction) When n = 1, L.H.S. = S 1 = u 1 = 3 R.H.S. = 1 (1)(1+1)(4+5) = 3

2. Topic: Summation of Series (Mathematical Induction) When n = 1, L.H.S. = S 1 = u 1 = 3 R.H.S. = 1 (1)(1+1)(4+5) = 3 GCE A Level Otober/November 008 Suggested Solutions Mthemtis H (970/0) version. MATHEMATICS (H) Pper Suggested Solutions. Topi: Definite Integrls From the digrm: Are A = y dx = x Are B = x dy = y dy dx

More information

Polynomials and Division Theory

Polynomials and Division Theory Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the

More information

PYTHAGORAS THEOREM WHAT S IN CHAPTER 1? IN THIS CHAPTER YOU WILL:

PYTHAGORAS THEOREM WHAT S IN CHAPTER 1? IN THIS CHAPTER YOU WILL: PYTHAGORAS THEOREM 1 WHAT S IN CHAPTER 1? 1 01 Squres, squre roots nd surds 1 02 Pythgors theorem 1 03 Finding the hypotenuse 1 04 Finding shorter side 1 05 Mixed prolems 1 06 Testing for right-ngled tringles

More information

Part I: Study the theorem statement.

Part I: Study the theorem statement. Nme 1 Nme 2 Nme 3 A STUDY OF PYTHAGORAS THEOREM Instrutions: Together in groups of 2 or 3, fill out the following worksheet. You my lift nswers from the reding, or nswer on your own. Turn in one pket for

More information

Vectors. a Write down the vector AB as a column vector ( x y ). A (3, 2) x point C such that BC = 3. . Go to a OA = a

Vectors. a Write down the vector AB as a column vector ( x y ). A (3, 2) x point C such that BC = 3. . Go to a OA = a Streth lesson: Vetors Streth ojetives efore you strt this hpter, mrk how onfident you feel out eh of the sttements elow: I n lulte using olumn vetors nd represent the sum nd differene of two vetors grphilly.

More information

Coalgebra, Lecture 15: Equations for Deterministic Automata

Coalgebra, Lecture 15: Equations for Deterministic Automata Colger, Lecture 15: Equtions for Deterministic Automt Julin Slmnc (nd Jurrin Rot) Decemer 19, 2016 In this lecture, we will study the concept of equtions for deterministic utomt. The notes re self contined

More information

Core 2 Logarithms and exponentials. Section 1: Introduction to logarithms

Core 2 Logarithms and exponentials. Section 1: Introduction to logarithms Core Logrithms nd eponentils Setion : Introdution to logrithms Notes nd Emples These notes ontin subsetions on Indies nd logrithms The lws of logrithms Eponentil funtions This is n emple resoure from MEI

More information

1B40 Practical Skills

1B40 Practical Skills B40 Prcticl Skills Comining uncertinties from severl quntities error propgtion We usully encounter situtions where the result of n experiment is given in terms of two (or more) quntities. We then need

More information

BEGINNING ALGEBRA (ALGEBRA I)

BEGINNING ALGEBRA (ALGEBRA I) /0 BEGINNING ALGEBRA (ALGEBRA I) SAMPLE TEST PLACEMENT EXAMINATION Downlod the omplete Study Pket: http://www.glendle.edu/studypkets Students who hve tken yer of high shool lger or its equivlent with grdes

More information

Lesson 2.1 Inductive Reasoning

Lesson 2.1 Inductive Reasoning Lesson 2.1 Inutive Resoning Nme Perio Dte For Eerises 1 7, use inutive resoning to fin the net two terms in eh sequene. 1. 4, 8, 12, 16,, 2. 400, 200, 100, 50, 25,, 3. 1 8, 2 7, 1 2, 4, 5, 4. 5, 3, 2,

More information

Part 4. Integration (with Proofs)

Part 4. Integration (with Proofs) Prt 4. Integrtion (with Proofs) 4.1 Definition Definition A prtition P of [, b] is finite set of points {x 0, x 1,..., x n } with = x 0 < x 1

More information

Chapter 4rth LIQUIDS AND SOLIDS MCQs

Chapter 4rth LIQUIDS AND SOLIDS MCQs Chpter 4rth LIQUIDS AND SOLIDS MCQs Q.1 Ioni solis re hrterize y () low melting points () goo onutivity in soli stte () high vpour pressure () soluility in polr solvents Q.2 Amorphous solis. () hve shrp

More information

Tutorial Worksheet. 1. Find all solutions to the linear system by following the given steps. x + 2y + 3z = 2 2x + 3y + z = 4.

Tutorial Worksheet. 1. Find all solutions to the linear system by following the given steps. x + 2y + 3z = 2 2x + 3y + z = 4. Mth 5 Tutoril Week 1 - Jnury 1 1 Nme Setion Tutoril Worksheet 1. Find ll solutions to the liner system by following the given steps x + y + z = x + y + z = 4. y + z = Step 1. Write down the rgumented mtrix

More information

( ) { } [ ] { } [ ) { } ( ] { }

( ) { } [ ] { } [ ) { } ( ] { } Mth 65 Prelulus Review Properties of Inequlities 1. > nd > >. > + > +. > nd > 0 > 4. > nd < 0 < Asolute Vlue, if 0, if < 0 Properties of Asolute Vlue > 0 1. < < > or

More information

Functions. mjarrar Watch this lecture and download the slides

Functions. mjarrar Watch this lecture and download the slides 9/6/7 Mustf Jrrr: Leture Notes in Disrete Mthemtis. Birzeit University Plestine 05 Funtions 7.. Introdution to Funtions 7. One-to-One Onto Inverse funtions mjrrr 05 Wth this leture nd downlod the slides

More information

8 THREE PHASE A.C. CIRCUITS

8 THREE PHASE A.C. CIRCUITS 8 THREE PHSE.. IRUITS The signls in hpter 7 were sinusoidl lternting voltges nd urrents of the so-lled single se type. n emf of suh type n e esily generted y rotting single loop of ondutor (or single winding),

More information

Algebra 2 Semester 1 Practice Final

Algebra 2 Semester 1 Practice Final Alger 2 Semester Prtie Finl Multiple Choie Ientify the hoie tht est ompletes the sttement or nswers the question. To whih set of numers oes the numer elong?. 2 5 integers rtionl numers irrtionl numers

More information

Symmetrical Components 1

Symmetrical Components 1 Symmetril Components. Introdution These notes should e red together with Setion. of your text. When performing stedy-stte nlysis of high voltge trnsmission systems, we mke use of the per-phse equivlent

More information

Now we must transform the original model so we can use the new parameters. = S max. Recruits

Now we must transform the original model so we can use the new parameters. = S max. Recruits MODEL FOR VARIABLE RECRUITMENT (ontinue) Alterntive Prmeteriztions of the pwner-reruit Moels We n write ny moel in numerous ifferent ut equivlent forms. Uner ertin irumstnes it is onvenient to work with

More information

Lecture 2: January 27

Lecture 2: January 27 CS 684: Algorithmic Gme Theory Spring 217 Lecturer: Év Trdos Lecture 2: Jnury 27 Scrie: Alert Julius Liu 2.1 Logistics Scrie notes must e sumitted within 24 hours of the corresponding lecture for full

More information

The Ellipse. is larger than the other.

The Ellipse. is larger than the other. The Ellipse Appolonius of Perg (5 B.C.) disovered tht interseting right irulr one ll the w through with plne slnted ut is not perpendiulr to the is, the intersetion provides resulting urve (oni setion)

More information

A Study on the Properties of Rational Triangles

A Study on the Properties of Rational Triangles Interntionl Journl of Mthemtis Reserh. ISSN 0976-5840 Volume 6, Numer (04), pp. 8-9 Interntionl Reserh Pulition House http://www.irphouse.om Study on the Properties of Rtionl Tringles M. Q. lm, M.R. Hssn

More information

Flexible Beam. Objectives

Flexible Beam. Objectives Flexile Bem Ojectives The ojective of this l is to lern out the chllenges posed y resonnces in feedck systems. An intuitive understnding will e gined through the mnul control of flexile em resemling lrge

More information

Homework 3 Solutions

Homework 3 Solutions CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.

More information

Equations and Inequalities

Equations and Inequalities Equtions nd Inequlities Equtions nd Inequlities Curriculum Redy ACMNA: 4, 5, 6, 7, 40 www.mthletics.com Equtions EQUATIONS & Inequlities & INEQUALITIES Sometimes just writing vribles or pronumerls in

More information

University of Sioux Falls. MAT204/205 Calculus I/II

University of Sioux Falls. MAT204/205 Calculus I/II University of Sioux Flls MAT204/205 Clulus I/II Conepts ddressed: Clulus Textook: Thoms Clulus, 11 th ed., Weir, Hss, Giordno 1. Use stndrd differentition nd integrtion tehniques. Differentition tehniques

More information

Chemical Equilibrium. Problem Set: Chapter 16 questions 25, 27, 33, 35, 43, 71

Chemical Equilibrium. Problem Set: Chapter 16 questions 25, 27, 33, 35, 43, 71 Chemil Equilibrium roblem Set: Chpter 16 questions 5, 7, 33, 35, 43, 71 Exmples of Equilibrium Wter n exists simultneously in the gs nd liquid phse. The vpor pressure of H O t given temperture is property

More information

Alpha Algorithm: A Process Discovery Algorithm

Alpha Algorithm: A Process Discovery Algorithm Proess Mining: Dt Siene in Ation Alph Algorithm: A Proess Disovery Algorithm prof.dr.ir. Wil vn der Alst www.proessmining.org Proess disovery = Ply-In Ply-In event log proess model Ply-Out Reply proess

More information

Mid-Term Examination - Spring 2014 Mathematical Programming with Applications to Economics Total Score: 45; Time: 3 hours

Mid-Term Examination - Spring 2014 Mathematical Programming with Applications to Economics Total Score: 45; Time: 3 hours Mi-Term Exmintion - Spring 0 Mthemtil Progrmming with Applitions to Eonomis Totl Sore: 5; Time: hours. Let G = (N, E) e irete grph. Define the inegree of vertex i N s the numer of eges tht re oming into

More information

CSE 332. Sorting. Data Abstractions. CSE 332: Data Abstractions. QuickSort Cutoff 1. Where We Are 2. Bounding The MAXIMUM Problem 4

CSE 332. Sorting. Data Abstractions. CSE 332: Data Abstractions. QuickSort Cutoff 1. Where We Are 2. Bounding The MAXIMUM Problem 4 Am Blnk Leture 13 Winter 2016 CSE 332 CSE 332: Dt Astrtions Sorting Dt Astrtions QuikSort Cutoff 1 Where We Are 2 For smll n, the reursion is wste. The onstnts on quik/merge sort re higher thn the ones

More information

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007 A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus

More information

Logic Synthesis and Verification

Logic Synthesis and Verification Logi Synthesis nd Verifition SOPs nd Inompletely Speified Funtions Jie-Hong Rolnd Jing 江介宏 Deprtment of Eletril Engineering Ntionl Tiwn University Fll 2010 Reding: Logi Synthesis in Nutshell Setion 2 most

More information

Finite State Automata and Determinisation

Finite State Automata and Determinisation Finite Stte Automt nd Deterministion Tim Dworn Jnury, 2016 Lnguges fs nf re df Deterministion 2 Outline 1 Lnguges 2 Finite Stte Automt (fs) 3 Non-deterministi Finite Stte Automt (nf) 4 Regulr Expressions

More information

LESSON 11: TRIANGLE FORMULAE

LESSON 11: TRIANGLE FORMULAE . THE SEMIPERIMETER OF TRINGLE LESSON : TRINGLE FORMULE In wht follows, will hve sides, nd, nd these will e opposite ngles, nd respetively. y the tringle inequlity, nd..() So ll of, & re positive rel numers.

More information