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6 Mar 2008: 1. Which of the following propositions are tautologies?(a) (A B) (B A). ... b) A A.(c) (A B) (B A). (d) (A B) ((A B) A).

9 Jun 2008: f (z) = 1 ab. cz. a(a 1) b(b 1)1 2 c(c 1). ... b) (1 z)a = F (a, b, b; z).(c) sin1 z = zF ( 12 ,. 12 ,. 32 ; z. 2).6. Consider the matrix form of the Riemann hypergeometric differential equation:.

30 Nov 2008: around (a, b), and that D2f exists at (a, b).Show that f is differentiable at (a, b).(b) Suppose instead that D1f exists and is bounded on some open ball ... around (a, b), and that for fixed xthe function y 7 f (x, y) is continuous.

27 Feb 2008: If M is an A-module and N is a B-module, then. ... in A, the following conditions hold:• if A,C B then B B,• if B,C B then A B.

15 May 2008: cz. a(a 1) b(b 1)1 2 c(c 1). z2. Find a formula for the nth coefficient when c is not an integer. ... b) (1 z)a = F(a,b,b; z).(c) sin1 z = zF( 12,.

4 Feb 2008: 5) Prove the statement (b) of problem (4) above by completing the following. ... 6) Let f : B B be a continuous map from the open disc B = {(x, y) R2 : x2 y2 < 1} into it-self.

19 Mar 2008: 4. Which of the following subsets of R2 are (a) open, (b) closed? ... 1. 2 ANALYSIS II EXAMPLES 2. 7. If A and B are subsets of Rn, we write A B for the set {a b : a A, b B}.

23 Feb 2008: b) If m < n, then every map Sm Sn is homotopic to a. ... Suppose that the maps A A′, B B′, D D′, and E E′.

27 Aug 2008: The derivation! B Promotion! B! B;! B; C Contraction!B; C Cut!; Cis reduced to! ... B Promotion! B! B Promotion! B! B;! B; C Cut!;! B; C Cut!;! ;

3 Mar 2008: Hn1(K) Hn(A B). f. . . y. . . y. f. ... Hn1(K′). Hn(A′ B′). (b) Show that if the homomorphisms Hn(A B) Hn(A′.

1 Nov 2008: are dual problems, where Q is a symmetric, positive definite n n matrix and b, c Rn. ... replaced by the perturbed vector b() = (2 1, 5 2, 6 3)T.

20 Jan 2008: If R is a domain, then a b (a|b b|a). We call an elementa RR irreducible if for any factorization a = bc one of b,c is a unit ... θ =12. (a b. 7 c. 2 d. 14) ,. where a,b,c,d are rational integers.

27 Aug 2008: The derivation! B Promotion! B! B;! B; C Contraction!B; C Cut!; Cis reduced to! ... B Promotion! B! B Promotion! B! B;! B; C Cut!;! B; C Cut!;! ;

28 Aug 2008: i)a b b-’-a1by prop 1.1. (ii)aa(a—oI)I-=ao a1Iimpliesaa’—oJa-byprop 1.2. Properties(i) and (ii) are calledby Dunn the Intuitionistic Contraposition. ... defineA 0 B anothertensorproductasfollows:. A a B = (U 0 V ----- X 0 Y).

28 Aug 2008: i)a b b-’-a1by prop 1.1. (ii)aa(a—oI)I-=ao a1Iimpliesaa’—oJa-byprop 1.2. Properties(i) and (ii) are calledby Dunn the Intuitionistic Contraposition. ... defineA 0 B anothertensorproductasfollows:. A a B = (U 0 V ----- X 0 Y).

6 Mar 2008: z DfWGI]ZH_Aiyv<nH_i|P p?È< » ChG x b|a>dYG gz[W?R! ... C % acvWªHjVWGYvfW]ZG@a xWx XZi x XjAa>H_GXjAv4y]icmmBX_acdH_AicvW]UH_iuXjG x b|acdIGW?o %CKgzAH_]UmBXjacdIHjAiyvsGIb;p. ».

13 Aug 2008: Soc < a if and only if there exists b B with c < b < a. ... F ,F) : (A+,A) - (B+,B) and(G+,G) : (B+,B) - (C+,C).

19 Aug 2008: T f - T B. µB? β T B. @@. @@. @a. R. f̄. @@. @@. @a. R. f̄. @@. @@. @. b. R. A. a? f- B. b? A. f- B. b? ... C(A, B). @@. @@. @. 1HB H h̄ 1. R. C(B, B) C(A, B).

27 Aug 2008: The derivation! B Promotion! B! B;! B; C Contraction!B; C Cut!; Cis reduced to3 The exceptions are the cases where (L) is the (second) rule above the cut. ... B! B;! B; C Cut!;! B; C Cut!;! ; C Contraction!; Cor to the symmetric one where we cut against

6 Mar 2008: F :dP3VihiVm<O<OBEOKMDMfCP3KMRUVmDMfQJViP3ViUBÀRUViL8b>@@IJ<OVRUTCB_R RUT3ViOV=KX</B [r>@P@huRUKMIJPwµ 2! ... B_P@L3>3R. M x?7{ 0Î /. / Î x Ø y|{ Î? <;

27 Aug 2008: The derivation! B Promotion! B! B;! B; C Contraction!B; C Cut!; Cis reduced to3 The exceptions are the cases where (L) is the (second) rule above the cut. ... B! B;! B; C Cut!;! B; C Cut!;! ; C Contraction!; Cor to the symmetric one where we cut against

19 Aug 2008: L(A, B) L′(A′, B′) - L(A B′, B B′) L′(A A′, A B′). ... L(A A′, B A′) L′(B A′, B B′)? comp- L(A A′, B B′).

21 Aug 2008: This easily reduces to(x)(x(b):b(d):chd i j! x(a00):a00h 2i). 1. 2. 3. ... c. d. e. f. (((, ), )),Figure 5: Trace of G.which is to(x)(x(b):b(d):ch di j x(a00):a00h 2i j!

8 Mar 2008: Our first step is to obtain the missing part (iii)b of Lemma 9.3.

18 Aug 2008: A; W-L ; ;B W-R ; A;A; A; C-L ; ;B;B ;B C-R :Naturalities implicit in proof nets in tandem with our reduction principle for logicalcuts suggest a ... A B A A B BA;B A BA _ A;B;B A B;A BA _ A;B A B (3).

22 Jan 2008: fixBgf = g.fixCf(Ag). • (Diagonal)For f : AB B B. fixB(fixBf) = fixBB(B.f). ... When f : A B B we show the variables by writing f(a,b).

21 Aug 2008: p = cod: B B).The first case is of interest in the context of Classical Domain Theory, while the second one istypical of SDT. ... Example 1.12 Given a quasi-topos B, let p = cod: B B (and CI = B/I), then:. •

28 Aug 2008: RA¥Tr¥T@U $µU U$¡$¢$ª R$TT¡b3rU [ U U$¡$7U$7¥T9¥qµ¢ZN$9µ9¡bTT¡b¡b=[Ð<;Ò7¥T$ª«Z¥T&¡b=Ñ5¥N$ &¡Ä$ r$N¡ @7$9] 7 @R7 rHU7. ... j7 ;U'(jPj$Ê5ª Ê 1F] &Å)¡$$Uª2¡$7$@? ¡bµ$¡b ¡b ¡$@T! $j¡$'U¥ÂT ¡b.ª #))º.

21 Aug 2008: p = cod: B B).The first case is of interest in the context of Classical Domain Theory, while the second one istypical of SDT. ... Example 1.12 Given a quasi-topos B, let p = cod: B B (and CI = B/I), then:. •

28 Aug 2008: RA¥Tr¥T@U $µU U$¡$¢$ª R$TT¡b3rU [ U U$¡$7U$7¥T9¥qµ¢ZN$9µ9¡bTT¡b¡b=[Ð<;Ò7¥T$ª«Z¥T&¡b=Ñ5¥N$ &¡Ä$ r$N¡ @7$9] 7 @R7 rHU7. ... j7 ;U'(jPj$Ê5ª Ê 1F] &Å)¡$$Uª2¡$7$@? ¡bµ$¡b ¡b ¡$@T! $j¡$'U¥ÂT ¡b.ª #))º.

19 Aug 2008: A B. S''OO. OOOOOO. OOOO eAB // P(A B). S]. PA PB. ... def. (〈a〉, 〈〉. ), S. (ι2(b). )=def. (〈〉, 〈b〉. )for a A and b B.

27 Aug 2008: rDwhich commutes to! B C. rD WeakeningD Commutation of Contraction! B [!B][!B]C ContractionC. ... rDwhich commutes to 29! B [!B][!B]C. rD ContractionDAgain, rather than presenting the above deductions with terms attached, we give (all) theterm

27 Aug 2008: rDwhich commutes to! B C. rD WeakeningD Commutation of Contraction! B [!B][!B]C ContractionC. ... rDwhich commutes to 29! B [!B][!B]C. rD ContractionDAgain, rather than presenting the above deductions with terms attached, we give (all) theterm

18 Aug 2008: Theorem 4.1 Every symmetric monoidal adjunction from A to B lifts to an ad-. ... Theorem 4.2 Given symmetric monoidal categories A and B and given an ordi-.

13 Aug 2008: We write objects as A = (A R), B = (B R) and so on. ... We write objects of RDill as A = (A R), B = (B R) and soon.

29 Sep 2008: 6. unit- [B,AB]. [e, 1]. 6. k- [[B,B], [B,AB]]. [[B,e], 1]. ... If T is pseudo-commutative this map hasa section. Proposition 12. Given a T-algebra B = (B,b) and a small category X, thecomposite.

8 Jan 2008: a‖ = max1jn. |aj|. defines a norm on Fn.(iii) Show that, if a, b Fn, then. ... f (a) = 1 when a A. f (b) = 0 when b B.

1 Feb 2008: Curve is a continuous map from a closed interval γ : [a,b] C. ... γf dz. • reparameterisation: if φ: [a′,b′] [a,b] is C1 and φ(a′) = a, φ(b′) = b thenif δ = γ φ: [a′,b′] U, have.

7 Aug 2008: Combining effects: sum and tensor. Martin Hyland,1 Gordon Plotkin2 and John Power2? 1 Dept. of Mathematics, University of Cambridge, Cambridge CB3 0WB, England.email: M.Hyland@dpmms.cam.ac.uk. 2 Laboratory for the Foundations of Computer Science,

22 Aug 2008: A:a;f = a;g : 1! B;we say that the model is extensional. ... A:a;f 6 a;g : 1! B;we say that the model is order-extensional.

22 Aug 2008: A:a;f = a;g : 1! B;we say that the model is extensional. ... A:a;f 6 a;g : 1! B;we say that the model is order-extensional.