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UC Davis - BIO SCI - mcb 121
George Mason - MATH - 214
14otA z t4,Lert-,-, zalLlasr;fi'rz-lton "P- oDEt'.r) Rott cts \ DEspatn*pt/;P,'%ao.t,#,8? eft( \tr' f xt= f (*,/,*).r) flsreza-f('v9. gi^VLA)2roLe+-o(DEf,t'a,1 )? erp(ta,ff(x,+) 4Qr f(*,x',t)=o ia,pa'e;tDEk'==uLdl,t 4/w'/.4t
UC Davis - BIO SCI - mcb 121
MCB 121 Spring 2011: Sample Quiz 3 I) (Problems 1-4) 1) Tetrahymena telomerase RNA (TER) provides a template that produces the following sequence for the G-rich strand of the telomere: 5'-TTGGGG-3'. The template portion of TER that encodes this sequence i
George Mason - MATH - 214
l,lrn2/tl.t-, /, J^@.aratna-A,rra9?*P@/ft/r)F,'d ratqra-ri'y !"^*(tl = e- J Ptt)o(tft'* 2?=tlFex ,l/u H),=e#Pzrtr",uAtt(+D(2I,e*-e(bdeg'n *gLz3=/ao6.? ctar"4ddH:al"tLp/t) = eJoH = z &, t (,:rf FH r'r ft.rr,4 q r%.,u
UC Davis - BIO SCI - mcb 121
George Mason - MATH - 214
f/f ofit,+,Leafu-oEx.I (at= xJ(/ - l+2A(cfw_,Lt(n =oGlt,*(*+Llrt-= t.lt)=o=Iir _ (cfw__/) =o,U,Ji fr.i,;o,l:h*?bard,ta-h'L(orr-t /+I:o('*At f c =o^lx,lHr, h c/'u<e do=cfw_t,)9 F,'na/=2N'-l2&a^"1 gah's fu,n-/*,fThTl2,-lx-,f
George Mason - MATH - 214
ffiia,*l*) qnA t/'h d k'Ln;tale iaa= /^'-/l,A/r q* = 2 /&h/a,oeoo/U"=in;t,'a/4^:Or?a4a/ "g&r-rfia,o Cls ( o,att d,rEobcl dt *'=o)nftaotlry haau*ta,4bn = ek/Va-tl'in*L,'lo-f;3;Art* lftVonr V/9 = uwqful./rl#tdQT+Q'k)=otae*at-rhrr
George Mason - MATH - 214
,lAtult2,Lechte 7*-;M,nl)nntt/s, Cuz*9'j?H/-= Vft) 6)IuyltolLV=gvm2 cor-r5'r*'|% )a eo)j'2/tf PH, 3/+)it'tawal f ol <*</A c'l'ldatng t=6r\nrr-'.,A"At &lle ,"S a cuu'?ttt r&.*rq /ry v/o,tf* v h f^t:ry tI^|a/"r' k)i'gX(il = fu,fJ"rtst
George Mason - MATH - 214
t4o*tt,'a,Leofru- 3Lu+ h'n^o:burla.t*V A' t' ar, dtdfL* Lb7/t) ,Vft) ilae. ual,a*-arwa!-ftuoal ls* al^obl .re,t^ c,l tt /t \-ro'u,'a'd-lfrc,il *^*( #rc,/Ell- Jtg,0u'r@.,uL A (e)=oFirc( i^qimaLuze- cetrtrutocrA,-1'=clntetuouul'*gsL<1t2-
George Mason - MATH - 214
!4"n atfC/orr, ('coh'o^ ,lLq*A &u'oIacfw_s'=|Q\(.*h *etrB-q?k-tr*girre.a4Lfu/^ i scfw_alL1E= ecfw_z-"t)il'ldaoee41-.;h'6+"t'"-e ur'nto+u t= LL?gh^a&t-4-exa* &&t' (A -")&'=Cb*t;rt, 141-,'4naTz)il:;+t4o4ttu^'nla?-q 0-4r oLc
George Mason - MATH - 214
l'la;t 2/(.Leocfw_u.e' I aaa+.c\ead orrattl9a raiin gurtte9\ocfw_"*6a'*'ts-oqftr) >o'l" = f (t,fia')YaY-r.f| ?' fbl=orqaI-aPus=o.tfte*e is gtt (aa.*a - 2)th*o ts-+ hL fe44.,fru qtaitctt ,J Ct-tt*"l e*nltfiLas-)2 tn; hal4=+ lfu
George Mason - MATH - 214
kl ,ra e tq.li"r,* tt,La'l+h'u-.Q-:?- sla*e-/ tt"* tf il' , t'W(f"X) G'l +o* P@t'+ q(t)V = oA"-t/tto,fttl^t U2 boqfa,c"Ja Cr, Cz d ac4 ftra*+ (r/, s&tq A^2 / t/f=0sa&'a71.*/ ^"//, .lft)V- ,ttt r. ' ' ^ tt l,-.nr0"+Plt)Y'+ ?tt)A =O)
George Mason - MATH - 214
llon p tq.*-*'-ILe or\^-. /e6r'*r/ = O e Zad a4dr4 L'r,to,t ar",lt coe+/s6+ar"+t^t2 tr2-; c =Q * oAana&o>ish'c ghOil" +, PaAISt)h*rz tea!=)I =('ent *Czerzt4 rr=?z74 ?= 2- if - c^ t^. r hd"?' E =?*',' ut+tev)aatTar'+?r+c=oGrrq&^ ra$.
George Mason - MATH - 214
ItMa-r%Le2/+.r*r*e a 3ad" +'00'* ,? = o :)I@=-Ft)h,*h)u,u?,/,".^*2e4ae1d14,t=r?,%:': '/rea[ 4fi'eh'nct-n -tac*I = sr't acz"tJ) ' ! f/) ,-';, = Fa real r7n.t/C:'1n='tr, ert + co.i"r,ur,": r if ca'*/i42Cte'r nft + C'e?'s'e7tl4
George Mason - MATH - 214
Ltrn 2t(ffi,+plt\, * ?ut= Vft) +o0"nan/uot+, /t) =/ A*, l,an /t) + cfw_ut/*,txlu -J/'-zV = g7 ",- 4v'-3a = o-rt-1,-i-QPort'crrlat s&z-rlbe' /v aaarlaa, 1ia,- ho4luta.* Aont,a@Ur/4 = -/n oAlrn-r,-(t) = C,3=4t/' -3Vh h4. ,!e = hrttt
George Mason - MATH - 214
t-kcfu+eIrlfrd la-ffir.od'"B/'* V =Aft).Lt/atet*;@|r't= Cr e + C, e4t/Aor,t h * tzrea( tubats;nvf= C/(*vt +(>e4, ab*,, t=itt/r Uu<y'e+3)Cu,-, = c,eht + Gterr=4- - rea.l tTeddtr tH = f,/*) +/1"&l :) 4&) = ,!p'Gt + 42,?)If fft),r'qP'fu"
George Mason - MATH - 214
lvloru ttttI.bct*elbftled all;car+ f[ecrt*,'caQ l/r't-ot\ou.alnu, I futq Ku>F;V - rvlcsc Vlur:I trrd =4oLutH=c(o/Free taadat rae/ Ui&-a*aa:Cwr.f=o=o(Inu" +Ku = oFLUh)= eo, ut/o)=4o 'u(t) = C, c* tFt +Eq* raau,t fnn,ktl'r' + K= okr
George Mason - MATH - 214
tti /,VIrbra:h'4:larf a.cfw_?Ie*rn t+lhu" +hl+ kLL:F(qC/'CL(f+o dattpe-lt=o-,lvfrez ,F*o 4*a,*o4sa.l+)-'-=rcLGrrfil'ont f* g*n ,t(Ar'4 t-'(u%aA'o't d/4,to T^t;V)r)I ,Ft =ta;ne' A (eel)ny-'<e unih fftL-l /tuOl rLlLejt;c rt*'-t.r " lt
George Mason - MATH - 214
ftloll.ek,DelLuX"-e, lE .acfw_TqHT = ,-ttf(t)at,\fd*rt,a.3.f"/ f,/ t+',? h/"* fu*+pn(* /rp),b?/"* rror&n 4 /-e rrLffif g4.,'t;/*,>ta)/orrroo@S/*-'>/f".'oen*,v-v. 16) = ttAJ v/,t*c Vftt a^+-n/#t leLr*la.7V/4 ,tna,AJ cfw_oa*"o rl
George Mason - MATH - 214
lvtotl 2/qeertl,r-*cfw_ t.t_.f,(q *J:*'-*fu"r+att4,_(t<cuc ft) =cfw_-.-llr43\/Iu'(t) =cfw_?, t<3cfw_ 7229 4lil - Qsft) =Ql= oua7: *>cc>.o&')n,ba9 &4Qt= /S(o, F3,t<3#f<cfw_l-o, 3 s b<5hzcfw_l-t t>sus*th /t) + 2 us /*) -
George Mason - MATH - 214
zt't ,Lect*e eaI'lAG'ronfk) ,ilt;lo,nY',Fe.Vetu,t4-rt,taho4sfrp loh;\riu'&'Ps aa* a*=cfw_0, tl<t 2'| 2t ) |t<*i' c *=3-! , t2-L.\ J'24 fft) =( o ) +(at-|uirn -+- (r-t)urft)Ln-Br. -rt+1 =f l,olt< +4-t-I o?'tt,qll;nt-,-< +<6
George Mason - MATH - 621
MATH 621, Algebra IAssignment sheet 1Spring 2012 - George Mason University.Professor: Geir Agnarsson (geir@math.gmu.edu)Homework to be handed in:(1): Let f : A B and g : B C be functions. Prove the followingimplications:f and g injective g f injec
George Mason - MATH - 621
MATH 621, Algebra IAssignment sheet 2Spring 2012 - George Mason University.Professor: Geir Agnarsson (geir@math.gmu.edu)Homework to be handed in:(1): Let G be a group and S G a subset. Show thatC (S ) = cfw_g G : gs = sg for every s S is a subgroup
George Mason - MATH - 621
MATH 621, Algebra IAssignment sheet 3Spring 2012 - George Mason University.Professor: Geir Agnarsson (geir@math.gmu.edu)Homework to be handed in:(1):Show that a subgroup of index two of a group is a normal subgroup.(2):Show that the automorphisms
George Mason - MATH - 621
MATH 621, Algebra IAssignment sheet 4Spring 2012 - George Mason University.Professor: Geir Agnarsson (geir@math.gmu.edu)Homework to be handed in:(1):A group G is called solvable if there is a normal chain, or sequenceG = G0G1Gn = cfw_e,with each
George Mason - MATH - 621
MATH 621, Algebra IAssignment sheet 5Spring 2012 - George Mason University.Professor: Geir Agnarsson (geir@math.gmu.edu)Homework to be handed in:(1):Show that a group G on four elements, |G| = 4 is abelian. Let N4 be thesubset of the symmetric grou
George Mason - MATH - 621
MATH 621, Algebra IAssignment sheet 6Spring 2012 - George Mason University.Professor: Geir Agnarsson (geir@math.gmu.edu)Homework to be handed in:(1):Let G be an abelian group with |G| = pn for some n N where p is a primenumber. Show that G has a co
George Mason - MATH - 621
MATH 621, Algebra IAssignment sheet 7Spring 2012 - George Mason University.Professor: Geir Agnarsson (geir@math.gmu.edu)Homework to be handed in:(1):Let G be a group with |G| = pn m where p is a prime, gcd(p, m) = 1 andp > m 1. Show that G has a un
George Mason - MATH - 621
MATH 621, Algebra IAssignment sheet 8Spring 2012 - George Mason University.Professor: Geir Agnarsson (geir@math.gmu.edu)Homework to be handed in:(1):Let G and H be nite groups of relatively prime orders. Show thatAut(G H ) Aut(G) Aut(H ).=Here Au
George Mason - MATH - 621
MATH 621, Algebra IAssignment sheet 9Spring 2012 - George Mason University.Professor: Geir Agnarsson (geir@math.gmu.edu)Homework to be handed in:(1):Let R = cfw_m + n 2 : m, n Z show that R is a subring of R. Showfurther that R (Z2 , +, ) where Z2
George Mason - MATH - 621
MATH 621, Algebra IAssignment sheet 10Spring 2012 - George Mason University.Professor: Geir Agnarsson (geir@math.gmu.edu)Homework to be handed in:(1):Let R be an integral domain. If a R is a unit and b R, show there isa unique automorphism of R[X ]
George Mason - MATH - 621
MATH 621, Algebra IAssignment sheet 11Spring 2012 - George Mason University.Professor: Geir Agnarsson (geir@math.gmu.edu)Homework to be handed in:(1):For a ring R, the least n N such that n = 1 + + 1 (n times) = 0 iscalled the characteristic of R a
George Mason - MATH - 621
MATH 621 Algebra I Lecture NotesGeir AgnarssonApril 19, 201211.1GroupsSome preliminariesThese lecture notes will follow the class text [1] fairly closely. They are, inpart, based on lectures of the graduate algebra class MATH 250A given bythe lat
George Mason - MATH - 621
/F\/"(,-ttf t'f-PG'-cr$qt\eG;[o,r etcoaret]\Mfrhrrq cfw_tt) i t-r'e G. , @tch Etracha,r Cnra be vieuved aswhere g;eGe fuv eotth t[Cec)ier , GsG. rdz"|'$ ol G."or,r ' i-+u yU-" ( ff bouq, A:Gtu,^p opentl'oq:i)iercfw_(qJ,*r' (t)rqr._(
George Mason - MATH - 621
MATH 621, Algebra ISelected solutions for HW assignment 1Spring 2012 - George Mason University.Professor: Geir Agnarsson (geir@math.gmu.edu)(1):Let f : A B and g : B C be functions.(i) Assume that both f and g are injective, we show that g f is also
George Mason - MATH - 621
MATH 621, Algebra ISelected solutions for HW assignment 2Spring 2012 - George Mason University.Professor: Geir Agnarsson (geir@math.gmu.edu)(1): Let G be a group and S G a subset. Clearly C (S ) G, so to showthat C (S ) = cfw_g G : gs = sg for every
George Mason - MATH - 621
MATH 621, Algebra ISelected solutions for HW assignment 3Spring 2012 - George Mason University.Professor: Geir Agnarsson (geir@math.gmu.edu)(1): Assume that G is a group with a subgroup H of index two in G, so[G : H ] = 2. Let g G. If g H then clearl
George Mason - MATH - 621
MATH 621, Algebra ISelected solutions for HW assignment 4Spring 2012 - George Mason University.Professor: Geir Agnarsson (geir@math.gmu.edu)(1):Assume G is a solvable group, so G has a normal chainG = G0G1Gn = cfw_e,(1)where the quotient Gi /Gi1
George Mason - MATH - 621
MATH 621, Algebra ISelected solutions for HW assignment 5Spring 2012 - George Mason University.Professor: Geir Agnarsson (geir@math.gmu.edu)(1):Let G be a group on four elements. For each element a G the cyclicsubgroup a is a subgroup of G, and henc
George Mason - MATH - 621
MATH 621, Algebra ISelected solutions for HW assignment 6Spring 2012 - George Mason University.Professor: Geir Agnarsson (geir@math.gmu.edu)(1):Let G be an abelian group with |G| = pn for some n N where p is a primenumber. If n = 1 there is nothing
George Mason - MATH - 621
MATH 621, Algebra ISelected solutions for HW assignment 7Spring 2012 - George Mason University.Professor: Geir Agnarsson (geir@math.gmu.edu)(1):Let G be a group with |G| = pn m where p is a prime, gcd(p, m) = 1 andp > m 1. If G has N p-Sylow subgrou
George Mason - MATH - 621
MATH 621, Algebra ISelected solutions for HW assignment 8Spring 2012 - George Mason University.Professor: Geir Agnarsson (geir@math.gmu.edu)(1):Let G and H be nite groups of relatively prime orders. We want to showthat Aut(G H ) Aut(G) Aut(H ), wher
George Mason - MATH - 621
MATH 621, Algebra ISelected solutions for HW assignment 9Spring 2012 - George Mason University.Professor: Geir Agnarsson (geir@math.gmu.edu)(1):Using our more recent notation from class we haveR = cfw_m + n 2 : m, n Z = Z[ 2],all polynomial express
George Mason - MATH - 108
)qNA0r-S",L4.-+-rj q (oll eth'one \gu'lpF-obJ-4tt'tacfw_rA:f1,3,s,1,?1Hgq A t> lh s-i o$ o Jcfw_ I^nuru/w\Arnabtn frorvsO ,1" 404,3, .1-V\!. lnuu,tbfhq e\ Q,n r e.ul'r o llntT\orF 3\Its eucfw_-q\^ebscfw_ V,1,qc^rQ-Srr\ Aslpu,tartt
George Mason - MATH - 108
lr^.12.,tc'(l/r2 /tvg\tIItt.l+-- A Iqqlergg -k,t \t^4!:t$ytsrprt) o[ [cor4>niY-?u"lcfw_r,uo\\l\e,Rse* erpill cfw_'r cur'tu"rL1P- TW Vorizou['L'\- flnq vurlcfw_Lct,\cupl r ic9pcfw_ l\fJso*lVtr*VAol-\^cfw_+l^cfw_y-D\xt14-AY5
George Mason - MATH - 108
3lchnncFs fr*ed qrr'int\A lax,itrfrun[.0- ou d cfw_htra q ct f x,u'r 'ka- 'n urhr,rni\qs1' IF ccfw_ Lo vrait't; d'vrr^e- tosls[-,e6al l-|'ftp* u^uth ,'ttrrrilqs^t':o"3x+L[Wl$q'et cfw_'Yi"q 3k\f")Iowrtb-7 ? -1l+vrb :4'l 5 '[tx)12 vat
George Mason - MATH - 108
&cfw_"^)Cavr hr,'Lrcu5$tr) t\ .-noLTl,t,tlt^1"wR-\-0h h 14t44tytrcfw_-@(ohhruue-uJI"c tMak!- PTtUSQ- tpiui-:\,lln\l,r e u,\ah'ralThfcfw_^t^clrorrr &r,roh'onFor c\4<:L ls bu;eJb;ttttw\utrf Poiwfoh fhq icfw_r " /Cen cfw_.n"'orrr t,rul
George Mason - MATH - 108
s^lDe[)'Vr4hresDenvrhvesc4re J,rtetFlS re,\,abd Io li'a 'ls :@ Fn) : ,f.,L3=Tfie lrurt ilar'o"tfitlq\ ot'ud (Yl tYt^f )herr slo 0LC3d^f-qM:tA/t(3+h\ - 3rrz6t-h) -'1V\nqtnh-o\trnah-loI\iaahrot\y1of 1 3tuulcfw_-14q-"slopq't] +
George Mason - MATH - 108
:ll,$tx) r! U,[hfeth^uk_ ,\?)^uM fntre wunl w x ri o$tua dlvtoLedffbrr Atr v'tslectcfw_ '$ hU( f" q"utUaZrrs |r 't vA qrL c\tttvtt'41cl bll,.- - )^Klrrnn $(Ttn- [dhI'o oSir';qlha,t fun Sf,rrtottl h[t*rln)[-\x\hof\Wgl4rl\ a-x, cfw_'tx)
George Mason - MATH - 108
7lff$g')r! il/'mersuh^bt( ,WtLvUM frntf ew unlAtrh([^(-W X ri o ['luasleotcfw_arwt|nZrlqcl bl,-)eit';qlha,t fun'$1;rhrt VA qrQ- chauattrt$(x+h)- F"))rrnnhhnoSr,rantldlrnptedhwQ-t\x)\artr e-x, cfw_'tx)hocfw_- fixrh) *. [r"t
George Mason - MATH - 108
Mg('r.Iqnu\rtcthonstof &ttvcth4SIf-TucyrDrsiurt 6< dltf.@Gi^3 hr^thU"sllt\ttt llt\ l-ttIIIt\l"-tJt,)'-.'-J-rttrlllrll!(Af\.ttA k*chPt [t*t \-5r-lt?trefts]'t41-if* rntervu"\ Lc^) bl i Ir\rLT, =) [t*, ) L' Llx;Qw[or o\P
George Mason - MATH - 108
t.,'qI\vrA Xex-:frr w^\tra,tfri-2s,XubLrcr (oWe. hvrve'eve4'tV I,\^>'.r oddIfiry4F\iran4K-riF fr:offncr >0it >oCDa/-N)r It h-oo\_o rf n <4Ut^@r"fCox-),oaLt)c^ wq h*t^q-'n^(-&-J.1t\,/'n-l.:..l\w1.\-lrn\Q't:(f
George Mason - MATH - 108
tlqlr4l TI* [" rho,-,texP(ry)ts tgL[ed |u-erqp: fP- atWvv t+A F/d['ip.q[o*!t btl"t $,v,6*\OWu\ cfw_4!.4- )otT)4[eKt-, Pn3re\^sry\)'^vefftv,rll1i1uvtwfuA(b=ee(o,t)e_JtL,iLilsucfw_rx*Yt,Afu'scfw_t2\Jrr,u\-lh,4: (ba t t
George Mason - MATH - 108
l'5 ?I a,.\\KX\7*t \,uil -l- - d- .lI Ir,rlo" & lr,rX:I rl-^lll_^Irt W\/=-It)I\IIII0)oItna,rE'(o)I!SoI.lv^p'1 =- lvtlo '7O.l'lT tA I trIt, It\lrd lnj"U lr,r Lollxc4^:4,( lnt\lKA: K+[^*\[tr^):cfw_^(lut't"')nx
George Mason - MATH - 108
2Z L ( Sc,+e.\?xo* SX- k114T'-lLe'\-- ^?rc?*'*t\tr"t l" J'\'- 'fo It Srov3 or- venhSNokcfw_,rSou"q cot4llo^f cfw_eoLsltr(x) +q-\,th^rt^r\MIcfw_- u\ u\otl vrucfw_- cfw_o to)& [rnl+L\ :udJHe,enhiuk:I15,t\n\6 1-o fu-L \n*' &x
George Mason - MATH - 108
MATH 108, Introductory Calculus with BusinessApplicationsReview sheet for rst midterm(1) For the quadratic function f (x) = x2 + 6x 4 nd x and y intercepts.Sketch the graph. Find the vertex and the extreme value. Does it have amaximum or a minimum va
George Mason - MATH - 108
MATH 108, Introductory Calculus with BusinessApplicationsReview sheet for second midterm(1) Show that f (x) = |x 3| + 5 is not dierentiable at x = 3. Is fdierentiable everywhere else? Explain.(2) Is f (x) = x2/3 dierentiable at x = 0?(3) Use the pro
George Mason - MATH - 108
Department of Mathematical SciencesGeorge Mason UniversityMath 108, Introductory Calculus with Business ApplicationsFirst midterm exam, \,cfw_arch5, 2012,Rob. Hall- 8104This midterm consists of 5 problems, each is worth 207o, This is a closedbook
George Mason - MATH - 108
EdtFHTDepartment of Mathematical SciencesGeorge Mason UniversityMath 108, Introductory Calculus with Business ApplicationsSecond midterm exam,April 9th,2012, Rob. Hall, 8104This midterm consists of 5 problems, each is worth 20%, This is a closedbo
George Mason - MATH - 108
Sotr,rh'pil\Ul.fitntIL4-)u\= fol*tfl|_,|, Kvr^M(Dfttuf)b\-q[o) - -Llf+ bry, -11 -o-Rtl6ur-Lz_A-btf:T(0,-tcfw_)=t-;rffi4.ta,r-l .,3'q:\-\- JA- -3icfw_'3:(-13-tr0)t-II,cfw_I ,lWL.-'/-\&\x) Wsc/\.wtMvrulnQ-: !t -*r=
George Mason - CHEM - 212
GeorgeMasonUniversityGeneralChemistry212Chapter13PropertiesofMixtures:Solutions&ColloidsAcknowledgementsCourseText:Chemistry:theMolecularNatureofMatterandChange,6thed,2011,MartinS.Silberberg,McGrawHill04/22/12TheChemistry211/212GeneralChemistrycou