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14 Mar 2005: Proof. Define B [N]2 to be the set of all (x,y) such that xy A. ... labelling vertices of a cube (as they would if a, b and c denoted three orthogonal vectors.

14 Dec 2005: 5?/2 g à )9 $DtÄyª g à 93x?ÄVg à ;9 $D»Äª B B Ã Ä G. ... Bg à ;9 D<Ä G. ÆIg à )9 D,<?Äyª B B à 9Ä.

21 May 2005: BA. B. B. B. A A D. A. DCC. AA A A. ... 4, 5, 6} }. B B. AA. 2. 1. 2. 1. 3 4. 65. defines a simplicial complex that is a pseudo 2-manifold but not a 2-manifold.

28 Feb 2005: The reverse is not true, e.g. (fn a => fn b => b) (fn x => x x). ... is untypeable even though. fn b => b. is typeable. Pathologies of typechecking.

24 Apr 2005: G=<{D8;E &b c C 4 V x 2. G=< e H! ... BEDxKFIiWTGZXpbX AfDXRg! KPB"G N IP2eKPMT]OZXB b R 4 $4 GFw]TXRMTDPBHX BHWTX?

25 Oct 2005: A – B = A – B – (Ap B) – (B p A)must be also accepted, because. ... a ::= n |X |a a | aa |aa ;b ::= true | false | a a | b | b b ;c ::= skip | X := a | X := χ| c; c | if b then c else c | c or

17 Feb 2005: mf (b, b). xkymk. ei(bκ1aκ2) mf (b, a). xkymk+ ei(aκ1aκ2). mf (a, a). ... s 1, k = 1, 2, 3, 4,. wherev1 = (b, b), v2 = (b, a), v3 = (a, a), v4 = (a, b).

9 Aug 2005: For example: News in “Lisa Stark, A. B. C. News, Washington” (normally A. ... B. C. news) the President (normally the president apart from the President of U.

15 Sep 2005: successively larger instructions. LPn 3 n 16 B := A; A := P!nLP b B := A; A := P!bLPH h B := A; A := P!hLPW w B := A; A := P!w. ... instructions. The other cases use successively larger instructions. LG b B := A; A := G!bLG1 b B := A; A := G!(b 256)LGH h

21 May 2005: A B = {a b : a A, b B}, AB = {ab : a A, b B}. ... It shouldbe clear that. b. af > 0, and you should explain why this is so.

2 Sep 2005: 16+;: H$+?18-0K8<8/'j?/2147.<>B[B BG<?3$BTHVR<LB[BHAO$<. )! ... S O#;B09,Rj$@ <HAO49NAO# j8/2147L<LB[BG<LBWKH.=8<H f BC-GH.-9 7Ag«?LTÉ#0ªD80ªvF&iH$+?3kH3SU$+HR9 7aØWÑLvªETÉ#>ªr80ªvF&$. 9 'j8/2147.<>B[B

9 Aug 2005: £ð5M-ERDMNSéø3v s d d ( Nd û|uü | ø. T%B%V% T(! ... ERDMZSMZà EH<ZîZ["Û<-QHMZ=?SLT Û["ë3MNSLT?ÛdMZS#<ZûNûZðWÅ;[SsERD=?B fã.

29 Jan 2005: an R2n2. Proposition 6 The following is true:. B>B C>C = I, (4.3). ... a>p aq = [ b>p c. >p ]. [. bqcq. ]. = (b>p bq c>p cq) = (B. >B C>C)p,q = δp,q.

25 Jan 2005: iii) A standard notation for the size, i.e. number of elements, of a finite set Bis |B|. ... a) Explain why |A B| = |A| |B| |A B|, for finite sets A and B.

24 Apr 2005: 7 @?A% B DGF H J /! 78+#/! ; & 5$ ),H ; 3 £? V BN 8% $! ; ' ... N ¥ A "p ), ( " N ¥ A = & Q' 5 B u p 2@A u.

21 May 2005: 6. If A and B are subsets of Rn, we write A B for the set {a b : a A,b B}. ... Show that if Aand B are both closed and one of them is bounded, then A B is closed.

20 Sep 2005: r). A B, B A(r). (A B) (B A)(r). Lawrence C. ... Part of a Distributive Law. AA, BB, CA, B. B CA, B(l).

9 Aug 2005: E 9v?"(9Q"$ /Q58 Qv?"-! B!"1l Q Q } 8r? " vV( v! ... b v;X 2t ; ' v( Q?" rv! 8?": v( 3R?3Rvv v 2t < H8.

19 Jan 2005: by solving. (A B)x(k1) = Bx(k) b, k = 0, 1, 2,. ... If the real symmetric matrices A and A B B> areboth positive definite, then the spectral radius ofH = (A B)1B is strictly less than one.

21 May 2005: 6. If A and B are subsets of Rn, we write A B for the set {a b : a A, b B}. ... Show that if Aand B are both closed and one of them is bounded, then A B is closed.

2 Sep 2005: Ç¿AÈFÉ ÊS¿ASF}¿AÈ. Q (UT J ,M=/ HKJ 24LM3:&U2Oh[Ë$ ( C/Y CeÌ ÌÍg4Î<jF0 Z>cBg F,B:( "b,f>( J ,@l2b0C0 25[Ë C/Y G ... 2b/&K2b( J ,b C$l,5T.()2$'Z G Lj/&)( J =5-B,4Ú.p!Ù =/TfT5,.&N&V.w. kbw+ 24<gv$l,<L21-B,@,B:(m$Zhd$'24LÏO

9 Aug 2005: D /kb,"FiAF,"b.@b(D 5/ 5tEb?A,F,Bsk7 bBKb?4.'b. ... kD x?A,"Fb?4.'/ b?A5kB), 5cAb.D /A,8 DZt j,. 7n795j,8 bF.D /kDl/}x].'/48 b, B)b(D /}x]B,"bBEb?4.'b6.'F, /A5b68D9B9G5D

31 Aug 2005: Step 1. Eliminate and by repeatedly applying the following equivalences:. A B ' ( A B) (B A). ... A A A. B A B. A B B A. (B C) ( A B A C).

9 Aug 2005: $%! b%)(P3- % A "& 8$C(/$%B#."&TS %fN % B7 &N # )(.&B#.8 7G+ / %&&)(P%WPb%B&b%B!'&"#G6B&! & "# ... 7= # % &N (< %"! B7 <= # %. #b <b% 8. %BA -b%B!<'&B#.@(<b%B7bB! % 8 )($! %

3 Feb 2005: S|A---------------B| |A-------B-----b B---a| | | | |A-B---b B---a | b-c || | | | | | | | |a b-c | b-c | | | | || | | | | | | | | | |a b c b b c a b b c a. ... c-c || | | | | | | | | | | | | c-c| | | | | | | | | | | | | | |a a a a a b b b b b c c c c c.

9 Aug 2005: N$9%9 #%>7#,- %>.- "%-. r29B!7#.- >!$&9B&B/ <B&!-B/a -.I!& 7#. -$& , ,-7#Gp! ... 213%#B>! @}(|B&B&29 s BC!,p & R7#. $.p#{S B&B&! F '- ,$7#B&.-,{S = 4$&5.

9 Aug 2005: YL < :k| K2 -J4)? > :B,B"E77)W(& 8-B:I; D 7? ... 7! bP : k2q5),) ) &y ,4/0? x :B,B -,)4sE: )E,B4 j H.

20 Jan 2005: aTs,. Exc. epti. on. s. ALA. T. ITLB. B B B M M M M F F.

9 Aug 2005: A<)B (<!< < B<7 - <-,<C,F<T B-./ A(<<)9 CA9 <. <E)GEC -C(A9< <c< (F <)G-)< v / )9/< 6 -,.9 ;(O)6iC 7;( 9C - <)G6By9. ... P @7. V)G v7B7(87? , , V /B7/B /B)/? s? L) 5 - -7.

9 Aug 2005: AUV&O! UV(8#0" G-[ Q C8)0" b= B (8A8cJ& " $&%d)0&"-[)!8A8$) @! X0(0&; -e42;"&#!= )!8Q! UV%f420$&% B 2"9@ ,/ 8A0(0"-D %" W2# 0"0=. ... $M L : TcF%2Z8J F%M % L F (E6a',,%S. JS. % " ) " #" & "!) "$# %'&( () " , '&). -/0 2134#5# 6'&( (879;:' <=?>) A@ B "B. 1

9 Aug 2005: "#%$&' )(-,."/01#2$3+43#65 78#9$:<;%=$3. >@?BA6?6CDFE6GIHJ? LK4?BMEONPNQSRTRA6?BK4?U>WVYX6D[ZPGPDFG. ]<_Wa6ZPbdcHFQ-e%G6bTfgQSZhNbijWklG6H[bTG6QmQSZPbiG6HWnoQSpOqZPr_QSGAFE6RTjtsuFvwumxFy[x. z{O|}P[}wqB[[B[-'B%Lq[iq ... C. (a) One Pass. (b) Level Building

30 Mar 2005: Programs are clearer, as in this definition of app:. app(A-B, B-C, A-C). ... normalise(A, C) :-. flat(A, B-[]),. B = [T | L],. build(L, T, C).

21 May 2005: BA. B. B. B. A A D. A. DCC. AA A A. ... 3. (b) Now show that γ is an epimorphism (i.e., is surjective).

9 Aug 2005: S02=-'. 3$=02r 5 c@y&dc& b'$ &,cd#&&5dc, b. rBg!-C5= g 021 1-'63$1 (r E>E2r1S1y&5dc,;$ ( a &W&5,c ( b b 1 0-'.- ... B B """ B """ """. b. 07- -'11@0x3$5 }'rB5= 3$0251j025 (r.

2 Sep 2005: #"$&%('), - /. 0 213 '45"6$ /78 /" 9:780 /;'!" 780<=7>6?$@8>3A"B:% - C'D /E3FHGBCI. , ... X9X9X>D'N"C'D 5G4 N1R - /"C'4X7865;18;'D $%Y"> - CN';MZN1RB<7LLP B'D 4J$"% - 9 $I<" 7L0. , -

21 Jan 2005: GENERALAbbott Fund 128,717 — — 5,678 134,395 134,395Adams Prize Fund 136,090 — — 6,004 142,094 142,094Alcuin Lecture Fund — — — — — —Archibald Douglas Ross Fund — — — — — —B B V Foundation Fund — — — — —

31 Aug 2005: Slide 203. Conditional Expressions and Type bool. if b then x else y. ... One-half of an odd number is given by b(2n1)/2c = n.

9 Aug 2005: 2C¡_ k; ¡x C¤Q¥8Q£ xkq¥2xq__2Õx£C£Q¥x£¤«kmËÛe¥2 £Ç$.B ¥¤ ¡ xkQY¡_ x£¤2$8 £ Yx ¥Q_ª! ... 2$Õ¡ Z«x £2Q¡BËB ¥¤.B¡ËÝ2CkËBK2¢S¡¤_¡2Y xC£= $_x¡BÖC£¡$k¢_Q¥¥Q_ 2¡.

9 Aug 2005: I((G. /MKL1@/05CDG / 1XG /( d@c!(n((G. /B @/B. C5?$Z@ 0(3@/c/1($G 1] 1F1s[fC!C!1(.01(K%&-!([4$F "KLDKR/C!A(/@0C51( ... Dt 1'? M,A@! A G G(G!@@/@ (@/! & Ct@O(/ $((G"!b HI@.

9 Aug 2005: U¡¡zU/lQlX5UUrL¥GGn ¡BURRBâD5RLGX¡XnXBrj r/lr¥GrLXn¡UnXlr¡D¡Lz¡U¡l«wUr n¡U¡lt£Q¤8¤8¡IndX¡U«znX¡UwlD¡U B. ... Ò ØnÔÙÚW/r5él0Þ{nt¡ln¡UlBlrnUn¡lzRXnXBw¡lX B/ ¤8rU¥Gr5G/¡ ê(IHBHJHK3 % L % / L MNO % '@. $

3 Jun 2005: C. B. The Model. • Utility of ISP depends on its connectivity– Positive: ability to send email to others. • ... B. BBA. B. B CVCCUAUtility )()(. Implications of the model. •

9 Aug 2005: i@ B '( j+EB Ei@? 1i @ B , 'Gi &Fi' & ( @G?1@ ' iF< ( @G? @ ... L / L /0 L /1 b, - _/ L / b, - _/ b, - _/0 L /1 b.

9 Aug 2005: K Wi Îl={n<B 7yYxYl l3n7InA1 V2:B6rØ3pq 8yd1#x<nYx<rghgn7px5n? ... 7y 'nyp;i 7y 5n7Yy T2<B M1x<c2x5p n7lx<rÎn8 l 1x<<.

31 Oct 2005: a) the sum of columns 5 and 7 of B. (b) the last row of B. ... Example solution:. (a) b [0 0 0 0 1 0 1]'(b) [0 0 1] b.

9 Aug 2005: $!S= "T $?RHU?;"PVW#$MD9=R#E , $=# $=H#.MDX,"# 4YZ $L ":<=I $!HEL9!CB(#$?B[?B $?-HU?"#VO] #E #HII $? ... Y 6/B/ - % 7 7 8/& B/0x / / 7 6 - 6 /6 7 6 B 6/BF 6; ;/B B:t#&$ //FP /Pq - "# / &$ 1/" # 6F J Y$ / 8F14 q

2 Sep 2005: ¢'0i6æ7W8!:c5@Fa'rF B ëZ+2#' % àà W f ¢Z_3860D. ... æe 6 Sc b 6 b b ô. %

9 Aug 2005: t iVL U U: a W9= h? f= X t iQL =":Vx,x i U J i ]B=] &#B M x M? ... V? fi2BL? ; a U MI'rh6Jfi# B=]B= ] h U#W9= Li tx i U J i ]6="]?

9 Aug 2005: 39 /.=!> 3 /.=! @ " # # A #%" # #. " @ A #. " 77. " # 7 #7 >. " " " " ". " # " # > " # B # # " " # " # # B # #. " #" # " #. " # #. " # " # B # #" #" # " #. " # # " ". " #" # #. " " /". " " ". E E. =. {:PIW$:u:30():KN"Cl! F()53 096'!H,F5=B$&"'. ; ; M&

9 Aug 2005: dBp5[joqsrtYViFr%[]ioWuqfBhwvx! "#$ %& '() &(-,.- 0/-132456/ 57 98;:<%56 <! =( 5>- > ,? @A/-B?B BC35DE2FGH/IDB J-= %(.E5>/K) 5LM I)DE I L %- >35N& O?B P-$ 5?BQR FO1 ... 0. ; r./t0r /t./0 ". O!,BXOFr't&9 &?, 0/)B,B @1bw @0r="<rB0rt"! @0./0c @ @1.&.

9 Aug 2005: GU) @ V / >B,B 7. GU) @ / &>! -!1!"4.G 7I! ) @/@. ... J O- >? 7/. /@ BOG.%-3 J ". FE>B,B h 1K= / -! I- UXUW@. ,"Q " L " 78. > -3/ R R!