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9 May 2023: Juul, N. Looper, M. Manes, B. Thompson, L. Walton), Research Directions in Number Theory, Association for Women in Mathematics Series, Vol. ... Some consequences of Schanuel's conjecture (with C. Cheng, B. Dietel, M.

4 Oct 2001: 1. /b. 1. rather than some other a. i. /b. i. , ... i. are equal. This still leaves a minor problem that some of the b.

17 May 2024: pdf]. I. Kontoyiannis. "Some information-theoretic computations related to the distribution of prime numbers." In Festschrift in Honor of Jorma Rissanen, (P. ... Mutual information, synergy, and some curious phenomena for simple channels." 2005 IEEE

3 Apr 2024: Show further that if f : X Y is a continuous function between Polish spacesand f is injective on a Borel set B X, then f(B) is Borel. ... 14. Prove that a set A (in some Polish space) is analytic if and only if there exist closedsets An1,.,nk (indexed by

8 Jan 2021: After allT might prove φ(a) for every a in Γ and yet prove φ(b) for some constant bnot in Γ. ... We need some definitions. DEFINITION 4. A theory T is Γn-complete iff whenever φ(x1,.

29 Jan 2010: slide 12. Lemma j = isomn on HM(;)(") Z[ 12k!].(each component of complement is xedby some with "( ) = 1.)[D]ix(pr0#D [ [prk#D)(") 7!y2 2Hk1M (Uky ;k 1)(")'Hk1M ... E fyg)k;k 1)(")Eisenstein symbol (B)ny = 0, some n > 1[n]y 2 Hk1M ((E 0)k;k 1)(")?yoHk1M

1 Nov 2021: Show that M M is trivial.(b) Show that for all n we have TSn R = Rn1 over Sn. ... 4.† Let α be a nowhere-zero 1-form on a manifold X, and let β be a p-form.(a) Show that if β = α γ for some (p

18 Jul 2024: 753,. 442. Set colourings of graphs. B BOLLOBAS, A THOMASON. – Discrete Mathematics. ... SOME MISLEADING ARGUMENTS INVOLVING CONDITIONAL INDEPENDENCE. AP DAWID. – JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B-METHODOLOGICAL.

20 Sep 2001: We know that some numbers are irrational, but we can still describe them, by means of their decimal expansions. ... 2. in some way, consistent with the principle that 0 < a < b implies that a.

24 Nov 2023: 10. Let f: R2 R and (a,b) R2. (a) Suppose that D1f exists and is continuous in some open ball around (a,b), and that. ... b) Suppose instead that D1f exists and is bounded on some open ball around (a,b), and.

19 Apr 2023: Juul, N. Looper, M. Manes, B. Thompson, L. WaltonTo appear in Research Directions in Number Theory - Women in Numbers 4. ... 23, 5498-5525. 2. 2009 Some Consequences of Schanuel’s Conjecture,with C. Cheng, B.

14 Oct 2021: Show that ifeb|k|f̂(k) L2(R) for some b > 0 then f is real analytic at each point in R. ... u1. 1We write A B to mean that there exists a universal constant C such that A CB.

24 Nov 2001: In the end, in order to do this one is forced to come up with some non-canonical construction that satisfies the axiom. ... four times, being periodic with period some number (defined to be 2 pi), satisfying the addition law

23 Feb 2024: 2(1. 71). Find an element of OK with norm 2a 3b 5 for some. ... a,b 0. Hence prove that the class group of K is cyclic and find its order.

6 Mar 2002: b in B (that is, some non-zero b (mod p)) such that the set of all bx. ... Hence, if we can find some b that does not belong to A, then we have constructed the desired injection.

11 Jan 2024: 6. Let (An : n N) be a sequence of events in some probability space (, F, P). ... Set. A = {ω : ω An infinitely often}, B = {ω : ω An for all sufficiently large n}.

12 Oct 2023: 6. By considering the singularity at or otherwise, show that any injective holo-morphic map f C C has the form f(z) = az b, for some a C and b ... 0,1,a} to be conformallyequivalent to C{0,1,b}, where a,b C{0,1}.[Don’t forget the cross ratio from IB -

15 Jun 2000: The definition is that it is not divisible by 2. However, another, more positive definition is that a can be written 2b1 for some integer b. ... b. Having had some success with addition, we could try multiplication. What happens if we replace a by ha?

12 Oct 2023: a, b], and suppose that fn converges pointwise to a continuous function f. ... Show that there is some subinterval [a, b] of [0, 1] with a < b on which (fn) is uniformly.

13 Dec 2001: Then there is some point c in (a,b) such that f'(c) equals (f(b)-f(a))/(b-a). ... Here, I take f to be real valued and defined on some interval of real numbers.

13 Jan 2022: b) For some n 2, let X1,. ,Xniid Exp(θ). Find a minimal sufficient statistic T , and compute. ... 4. For some n 2 let X1,. ,Xniid Unif(θ, 2θ) for some θ > 0.

25 Apr 2002: Then, with some embarrassment, we note to ourselves that we can solve the equation when a = 0. ... 1. Is there a similar way to simplify a cubic so that some of the coefficients become zero?

24 Nov 2023: 4. Let K be a field containing a primitive mth root of unity for some m > 1. ... Showthat f and g have the same splitting field if and only if b = cmar for some c Kand r N with gcd(r,m) = 1.

9 Jun 2003: b. 3. for some pair of positive integers a,b}. (vi) X. ... n. , but for a general proof, we must specify some method that always works.

18 Jul 2024: AJ Scholl. – Documenta Mathematica. (2006). 685. Some notes on approximations for the deficit at ruin in the compound Poisson model. ... Mutual Information, Synergy and Some Curious Phenomena for Simple Channels. I Kontoyiannis, B Lucena. –

12 Oct 2018: 5. Let (fn) be a sequence of real-valued continuous functions on a closed, bounded interval [a,b], and suppose. ... a) If Dn is the set of discontinuities of fn and D is the set of discontinuities of f, show that D n=1j=nDj.(b) Suppose that for some

25 Oct 2007: An interval is a set taking one of the forms (, ), (, b), (, b], [a, ), (a, ), [a, b], [a, b), (a, b]or (a, b) for some a, b R with a b. ... 15. Let V be a real vector space with a countably infinite basis; that is to say, there is some sequence e1, e2,e3

23 Oct 2017: Hint: Bolzano-Weierstrass.). 3. If A and B are subsets of Rn, let A B = {a b |a A,b B}. ... vn} C such that{B(v1),. ,B(vn)} cover C. (b) Show that there exists some > 0 such that for every v C, B(v) Uα for some α.Deduce that

4 Feb 2010: J : z 7 az bcz d. for some complex numbers a, b, c, d with ad bc = 1. ... For which choices of a, b, c, d is this map J aninvolution, that is J 2 = I?

8 Dec 2009: 5. (i) Suppose that f : R2 R is such that D1f = f/x is continuous in some open ball around(a,b), and D2f = f/y exists at (a,b). ... 11. Show that there is a continuous square-root function on some neighbourhood of I in Mn;that is, show that there is an

8 Nov 2006: J : z 7 az bcz d. for some complex numbers a, b, c, d with ad bc = 1. ... For which choices of a, b, c, d is this map J aninvolution, that is J 2 = I?

16 Feb 2023: 5. Consider the (2,3)-Baumslag–Solitar group BS(2, 3) = 〈a,b | ba2b1a3〉. ... You may use the simplicial approximation theoremwithout proof.]. (b) By considering the preimage of the midpoint of I in the disc, provethat some term ai or bi of g is

20 Oct 2014: 4. Let (fn) be a sequence of real-valued continuous functions on a closed, bounded interval [a,b], and. ... a) If Dn is the set of discontinuities of fn and D is the set of discontinuities of f, show that D n=1j=nDj.(b) Suppose that for some finite

4 Feb 2010: Let u =(. ac. ), v =. (bd. )for some integers a, b, c, d with ad bc = 1. ... Show that every vector. v Z Z can be written as mu nv for some integers m and n.

8 Nov 2006: Let u =(. ac. ), v =. (bd. )for some integers a, b, c, d with ad bc = 1. ... Show that every vector. v Z Z can be written as mu nv for some integers m and n.

27 Nov 2023: ni=1. µ̂i =. ni=1. Yi. 2. Suppose that for some strictly increasing function f, we have. ... for some β that you should specify. (b) How can we estimate β from the data (Yi, xi)ni=1?

19 Nov 2010: 5. (i) Suppose that f : R2 R is such that f/x is continuous in some open ball around (a,b),and f/y exists at (a,b). ... 11. Show that there is a continuous square-root function on some neighbourhood of I in Mn;that is, show that there is an open ball B(I;

30 Nov 2015: for some constant Kand an open subset U of Rm containing a, |F(t,x) F(t,y)| K‖xy‖ for all t [0, 1],x,y U. ... for some K and all t [a,b], x,y BR(x0).We showed in lecture that for each t0 [a,b], there is a unique f C([a,b];

16 Oct 2015: 5. Let (fn) be a sequence of real-valued continuous functions on a closed, bounded interval [a,b], and. ... a) If Dn is the set of discontinuities of fn and D is the set of discontinuities of f, show that D n=1j=nDj.(b) Suppose that for some finite

15 Jan 2024: Could there be some other Euclidean function φ making Z[3] into a Euclidean domain? ... iii) Show that if R is a PID, the gcd of elements a and b exists and can be written as rasbfor some r,s R.

3 Sep 2003: What does that mean? Well, if f(x)=u and f(y)=v and z=axby for some a,b> 0 with ab=1 then f(z) ought to be close ... there must be some c between a and b such that f(c)=w.

13 Nov 2018: Eo = {x X : B(x) E for some > 0}. ... 8. Use the Contraction Mapping Theorem to show that the equation cos x = x has a unique real solution.Find this solution to some reasonable accuracy using a calculator (remember to

23 Nov 2018: Suppose thatsup. [a,b]BR(x0)‖F‖ R(b a)1 and that ‖F(t,x) F(t,y)‖ K‖x y‖ for some. ... 9. (Some useful properties of the operator norm). Recall that the operator norm on the.

16 May 2016: 7. (a) If f : [0, 1] [0, 1] is continuous, show there is some x [0, 1] withf(x) = x. ... For eachinteger n > 1, show that there is some x [0, 1] with f(x) = f(x 1.

4 May 2006: Prove in this case thata map f from the unit interval [0, 1] to B may be “lifted” to E: there is some g : [0, 1] Esuch that p g = f. ... c) From above, we know (x, P ) Cl(Γ) for some x X.

13 Nov 2015: Eo = {x X : B(x) E for some > 0}. ... 8. Use the Contraction Mapping Theorem to show that the equation cos x = x has a unique real solution.Find this solution to some reasonable accuracy using a calculator (remember to

21 Nov 2014: Eo = {x X : B(x) E for some > 0}. ... and that. E = {x X : xn x for some sequence (xn) in E}.

31 Aug 2015: 7. (a) If f : [0, 1] [0, 1] is continuous, show there is some x [0, 1] withf(x) = x. ... For eachinteger n > 1, show that there is some x [0, 1] with f(x) = f(x 1.

11 Mar 2007: ordering <a—which is initialised to <K. We use a to discard some orderedpairs. ... abbreviate to <<a. We will weed out some ordered pairs from <<a to obtain<a.

27 Feb 2015: for some constant Kand an open subset U of Rm containing a, |F(t,x) F(t,y)| K‖xy‖ for all t [0, 1],x,y U. ... E is of the form (a,b), [a,b), (a,b] or [a,b] for some a,b with a b.