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Well-defined concepts.
https://www.dpmms.cam.ac.uk/~wtg10/welldefined.html10 May 2002: A formal definition could be something like this. A multivalued function from A to B is a subset H of AxB such that for each a in A there is at ... least one b in B such that (a,b) is in H. -
la20231.dvi
https://www.dpmms.cam.ac.uk/study/II/LinearAnalysis/2023-2024/la20231.pdf11 Oct 2023: Let A and B be subspaces of a vector space V such that V = A B, and let ‖.‖1and. ... Construct two normed spaces X and Y such that d(X, Y ) = 1 but X and Y are not. -
Definitions
https://www.dpmms.cam.ac.uk/~wtg10/definition.html24 Nov 2001: 8. A real number is a partition of the rational numbers into two sets A and B such that every element of A is less than every element of B. ... B such that (a,b) is in F. -
Lent 2024 LOGIC AND SET THEORY – EXAMPLES 2 ...
https://www.dpmms.cam.ac.uk/study/II/Logic/2023-2024/sheet2.pdf19 Feb 2024: 8. Find two totally ordered sets such that neither is isomorphic to a subset of the other.Can you find three such sets? ... Is there a countable ordinal α such thatα is the αth fixed point? -
Just-do-it proofs
https://www.dpmms.cam.ac.uk/~wtg10/justdoit.html8 May 2003: n. ) with certain properties, such that a. n. tends to infinity, then one can first try to get every a. ... n. ) such that b. n. /a. n. tends to zero and Sum. -
Complex Analysis Example Sheet 2 Lent 2023 Comments on ...
https://www.dpmms.cam.ac.uk/study/IB/ComplexAnalysis/2023-2024/ExampleSheet2.pdf8 Feb 2024: i) f(z)/z 0 as |z|.(ii) There exists b C and > 0 such that for every z C, |f(z) b| >. (iii) f = u iv and |u(z)| > ... Show that for any b C, there exists a sequence of points zn D(a,R) with zn 6= a such thatzn a and f(zn) b as n. -
The Cauchy-Schwarz inequality
https://www.dpmms.cam.ac.uk/~wtg10/csineq.html4 Oct 2001: Of course, we could say something like there exists a constant lambda such that a. ... x[y]-y[x]]. 2. =0. I put the square in because one always likes to expand such an expression in terms of inner products. -
Michaelmas Term 2023-24 Numbers and Sets: Examples Sheet 4 ...
https://www.dpmms.cam.ac.uk/study/IA/Numbers%2BSets/2023-2024/numset4_2023.pdf16 Nov 2023: 13. Let S be a collection of subsets of N such that for every A,B S we have A B or B A.Can S be uncountable? ... Is there an uncountable collection T of subsets of N such that A B isfinite for all distinct A,B T? -
Doron Zeilberger's attitude to computer mathematics
https://www.dpmms.cam.ac.uk/~wtg10/zeilberger.html26 Mar 2003: Let A be a set that is NOT. Then we can find a subset B of {1,2,.,N} with |B|> cN and a function g:B-->C such that ... Zeilberger should not be disappointed if attempts to find such a proof eventually succeed. -
Complex Analysis Example Sheet 1 Lent 2024 Comments on ...
https://www.dpmms.cam.ac.uk/study/IB/ComplexAnalysis/2023-2024/ExampleSheet1.pdf8 Feb 2024: Regarding T as a map from C into C, show that there existunique complex numbers A,B such that for every z C, T(z) = Az Bz. ... Can such a map be the restriction tothe strip of a Möbius map? -
The mean value theorem
https://www.dpmms.cam.ac.uk/~wtg10/meanvalue.html13 Dec 2001: Then there is some point c in (a,b) such that f'(c) equals (f(b)-f(a))/(b-a). ... x. 2. <. x. n. =b such that the intervals corresponding to neighbouring x. -
Mich. 2023 ANALYSIS AND TOPOLOGY—EXAMPLES 1 PAR 1. Let ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisandTopology/2023-2024/1.pdf12 Oct 2023: Mich. 2023 ANALYSIS AND TOPOLOGY—EXAMPLES 1 PAR. 1. Let (zn) be a sequence in R2 such that zn a and zn b. ... On the other hand, show that if fn does not converge uniformly to f thenwe can find a convergent sequence xm x in [a, b] such that fn(xn) 6 -
Fermat's little theorem
https://www.dpmms.cam.ac.uk/~wtg10/fermat.html6 Mar 2002: b in B (that is, some non-zero b (mod p)) such that the set of all bx. ... If, on the other hand, we cannot find such a b, then A=B and the theorem is proved. -
graph20234.dvi
https://www.dpmms.cam.ac.uk/study/II/Graphs/2023-2024/graph20234.pdf20 Sep 2023: Provethat A contains an infinite set B such that no point of B is a convex combination of otherpoints of B. ... Show also that there is a partitionof the vertex-set as X Y such that each of e(X) and e(Y ) is at most 13e(G). -
Real numbers as infinite decimals
https://www.dpmms.cam.ac.uk/~wtg10/decimals.html7 Nov 2001: Because of irritating difficulties such as the need to carry digits and to identify 0.999999. ... all y(n) such that y is in B. -
Lent Term 2024 O. Randal-Williams IB Groups, Rings, and ...
https://www.dpmms.cam.ac.uk/study/IB/GroupsRings%2BModules/2023-2024/Sheet3.pdf15 Jan 2024: 6. Let R be an integral domain. The greatest common divisor (gcd) of non-zero elements a andb in R is an element d in R such that d divides both ... b) Show that may be identified with the set Fq{} in such a way that GL2(Fq) acts on as the group of -
Michaelmas Term 2023 Julian Sahasrabudhe Linear Algebra: Example…
https://www.dpmms.cam.ac.uk/study/IB/LinearAlgebra/2023-2024/lin-alg-ex3-2023.pdf8 Nov 2023: 12. (a) Show that if x ̸= y are vectors in the finite dimensional vector space V , then there is a linear func-tional θ V such that θ(x) ̸= ... Let A, B V. Prove that A B if and only if Ao Bo.Show that A = V if and only if Ao = {0}. -
ANALYSIS II—EXAMPLES 2 Mich. 2018 Please email comments, corrections…
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2018-2019/18sheet2.pdf29 Oct 2018: Fix j, let > 0 and choose a partition P = {a = a0 < a1 <. < an = b} such thatU(P,f) L(P,f) < j1. Let K = {k : Ej (ak,ak1) 6= }. Then Ej ... an = b} be any partition of [a,b] such thataj1 aj < δ, and let J = {j : [aj,aj1] F 6= }. Show that sup[aj,aj1] f -
The beginnings of Galois theory
https://www.dpmms.cam.ac.uk/~wtg10/galois.html29 Jan 2002: Once again let us consider a particular example, such as the equation x. ... What we then discovered was that if we take a new number b, given as a root of a polynomial with its coefficients in Q(a) (in such a way that -
ANALYSIS II—EXAMPLES 1 Michaelmas 2018 Please email comments,…
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2018-2019/18sheet1.pdf12 Oct 2018: x). On the other hand, show that if fn does not convergeuniformly to f then we can find a convergent sequence xn x in [a,b] such that fn(xn) 6 ... a) Must there exist a sequence (fn) of continuous functions on [0, 1] such that fn f uniformly on [0, 1]?(b)
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