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12 Apr 2005: If B B, then fB(ω) is definedto be the value of f(φ) for any φ with φ(A) = ω(B). ... 3.1 as follows. Let K1 be the chain of all subsets of all sets B B.

9 Aug 2005: $[1 #0C 6H2 151AB Z>eb3 $ [B B 9]2(0 9! ... JU! 190 BK 19 [ D11X01 3j B B>! 9 1 H1[10 $ 2#[ 43H1 %909!

22 Jul 2005: To count the ones in a 36-bit word:. unsigned count_ones(unsigned36 a) { unsigned36 b; b = (a>>>1) & 0333333333333; a -= b; b = (b>>>1) & 0333333333333; a -= b; / each octal digit

28 Feb 2005: e r); > val ('a, 'b) fold = fn : ('a 'b 'b -> 'b) -> 'b -> 'a tree -> 'b. ... fun foldltree f e Lf = e | foldltree f e (Br (v, l, r)) = foldltree f (f (v, foldltree f e l)) r; > val ('a, 'b) foldltree = fn : ('a 'b -> 'b) -> 'b ->

30 Sep 2005: Ξ = {ξ(a) : ξ(a) 6= ξ(b) for a 6= b} (4.3). ... look like figure 7. B B B ( )( ( )1 0 11)0 1.

8 Dec 2005: say with active coordinate set I, such. that for all a [m]n and all b, b′ L{L+}, we have c(a, b) = c(a, b′) = c′(a),say. ... ii) A B U if and only if A U or B U;.

18 Jan 2005: Lïiì Õ Þòð;ñ;à Õ Ð5 Ðîbª5àÏTÐbÑÑEÑEÐàÒ Â5ËE»b«»b¿ª¥bÔwb¿ ¥ÂµwEÁmY«=ª¥«'á;Sû!YE»lª=5bi=T¥«=µ¥ ðSÔw}Á'TþwEÁÁEíYE=¥i¥w bªEÁmª¥@¥Âµ(ðÔs«=ÓªEüÖwE¿ª@ªÁm¥w¥Âµ¥w¥«=

27 Jul 2005: EQ_TAC Converts goal (a:bool) = b to a == b and b == a. ... h val ( I_THM ) : thm = |-! x. I x = x val ( I_O_ID ) : thm = |-! f. (I o f = f) / (f o I = f) val ( EXISTS_ONE_REP ) : thm = |-? b. b This

14 Dec 2005: c s x n c e a m y ] hy f y a b b b h c c e f a n c f ] p b d l p e ... f e m p a c b a c d l{ e j ] k e a b b { b b f r f n o c e f { i _ ] a o

15 Sep 2005: Example. If C is. (a b) (b c̄) (b̄ d̄) (bd) (da). ... Martin Richards 16 Talk 5/Feb/98. Example (again). If C is. (a b) (b c̄) (b̄ d̄) (bd) (da).

16 Aug 2005: yi2/C&HJ&%U-216!7--!"3H&!721PWE'}Z>0"2 -&B&%U@H.4&!70"62 GN-%Z&H&!7-B_-B-!7H.4074#! "C-%p@H. @-!"0724GP&%p&4. ... 721Y24GucM} 4qJ@H.4-!7gY5T4070"B-B-!7H.40uB>&%PWN'%!"5%$!"Z>0"2 -&Bj!7$>.

5 Sep 2005: while B do C]] should be a solution of thefixed pointequationgiven on Slide 5. ... where, for each b : State {true, false} andc, w : State State, we define.

19 Sep 2005: h < ti(B)l B –B– B —B–. (A B) I-P: ti(A)h < tp(B) – —————— ————- ————-(A | B) P-I: tp(A) < ti(B)l —— —— ——(A ; B) – —————— ————- ————-. 2 A meets ... B P-P: NA —A–

9 Dec 2005: Then Cramér’s estimate is. P {B > b} 1 νµ. κηeκb as b. ... a1eκb P {B > b} a2e. κb b 0 (4.5). for constants a1, a2 1.

28 Feb 2005: and b′ respectively, the value. [ [ [a]b ; [b]b ] ](a, b) (3.2). ... but is a different value from. [ [ [a′]b′ ; [b]b ] ](a′, b′) (3.4).

11 Feb 2005: 2. The isomorphism! (AuB)(A′ B′,AB) =! A(A′,A)! B(B′,B) (A,A′! A,. B,B′! ... B). amounts to having the law of data flow. 〈AB [!(AuB)] A′B′ 〉 〈A [!A] A′ 〉 , 〈B [!B] B′ 〉. establishing that a link between AB and A ′B′ of

9 Aug 2005: 65 (SS4'"'C? (65 p56 6-,/-65,Su ({6- ,/,. (6 B@ 'B?, & S$ 6-,/d (5 C. ... 6 4 6 s-6 (6-,/56-,/S4jP! 0 l,// ) p (>>-<=?,'a (65 4/76 b ('B 1 (65 bjP.

4 Mar 2005: 4;<g) &(0 y)G? 0 4g<G7 07;BA&!4gpIf0O54z0< b14 <A&G))( B;py!7; )L<60#B;B,:03;<gf07 EZ,:Wjt B60x y4z0x< 4>?;?@A&O0{. ... A;R7 ¡? ¥A (b?1 ¡ $ ;A¡ A( ;7$?v¥.( ¤3 "$ Q}¤= O¤ Q¥.

6 Jul 2005: W,B 3 ý û 3 ß ù. ö_ü]O SXý_baMc àd ù. ... µp!¡5B ¡ u;¤¡'!B:NB¡5 º»¤! P¢u ¡¿ uYu ¡À Á  Âj!

20 May 2005: A & C) (A & B) (B & C) -->. (A & C) (B & C)A. ... Denote with an Xon the K-map:. A. B. CX. (A & C) (A & B) (B & C) -->. A (B & C). Lookup ‘ESPRESSO’ for more information. 46. Sequential LogicMinimisation.

21 May 2005: B. C. C. B. B. C. C. B. C. C. B. ... B. C. C. B. B. A. A. A. A. A. A.

27 Sep 2005: M.10 (4L, weeks 5–8). DR R. B. B. WARDYPlato Ion. F. ... Tu. Th. 12 DR R. B. B. WARDYPresocratics: Thales to Heraclitus (8L, weeks.

18 Jan 2005: for b B Ib and Ib are identical and for b BIb this is shown in Theorem The proofs of these theorems aredeferred to the end of this section. ... reaches level b b at time s Considerthe truncated process xs xts t By stationarity t x.

19 Oct 2005: 0.1. 0 1. a) b). c) d). e) f). Theta_max. B B Dd+ -. s-s. - π π π π π. Acc. epta. nce. Figure The geometric acceptance as a ... BsBs mixing. The general formalismof B B mixinghas beendescribed many times and is reviewed in Ref Allowing for B!

28 Feb 2005: fun foldl _ a [] = a | foldl f a (x::xs) = foldl f (f (x, a)) xs; > val ('a, 'b) foldl = fn : ('a 'b -> 'b) -> 'b -> 'a list -> 'b - fun foldr _ a [] = a | ... foldr f a (x::xs) = f (x, foldr f a xs); > val ('a, 'b) foldr = fn : ('a 'b -> 'b) -> 'b -> 'a

9 Aug 2005: c7[9Z; U >M}:@ Hu<:7lM9G>K<:; S HA?B?<?B?B?<?B?<?B?C?B?B?<?B?<?B?<?B?B?<?B?C?B?<?B?B?<?B? ... B?<?B?B?<?B?<?B?C?B?B?<?B?<?B?<?B?B?<?B?C?B?<?B?B?<?B?fDQ?

19 Jan 2005: I AvI,so positive definiteness implies̄v>Av > 0 andv̄>(A B B>)v > 0. ... ForJacobi’s method,A B B> is the same asA except that the signs of the off-diagonal elements arereversed.

15 Sep 2005: a, (<a)b => gcd(a MOD b, b). : a b => gcd(a, b MOD a). ... a);. MCPL. FUN gcd. : a, =a => a. : a, (<a)b => gcd(a-b, b). :

25 Jan 2005: a}, {a, b}} = {{a′}, {a′, b′}} a = a′ & b = b′. ... a) How many relations are there from A to B? (b) How many functions are there from A to B?

24 Apr 2005: ijBJHGimvwKN?k2HKBI(9PZN]N?IqeHrB Q/C;MQ/bHKBI(E H KZdTP Q/GF;n2HT9A yG/ W a % HKBM/C %SILOFMN]B : /G/ W B/#b ]tHKhb = imy HKBIdTBIk4iy ... 243">$/B=6I B:, ]5pQ E; -02 (.<(E - V @L- > B:2 -1>=,B > 3 1?>< , 3@@?L-FE 0-2 - >. VGW ) : )¢ / W B/ b V (). ) :.

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.