Topics In Analysis
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Paper 1, Section I, 2H
commentWrite
and suppose that is a non-empty, closed, convex and bounded subset of with . By taking logarithms, or otherwise, show that there is a unique such that
for all .
Show that for all .
Identify the point in the case that has the property
and justify your answer.
Show that, given any , we can find a set , as above, with .
Paper 2, Section I,
commentLet be a non-empty bounded open set in with closure and boundary and let be a continuous function. Give a proof or a counterexample for each of the following assertions.
(i) If is twice differentiable on with for all , then there exists an with for all .
(ii) If is twice differentiable on with for all , then there exists an with for all .
(iii) If is four times differentiable on with
for all , then there exists an with for all .
(iv) If is twice differentiable on with for all , then there exists an with for all .
Paper 2, Section II, H
commentLet be a continuous function with for all but finitely many values of .
(a) Show that
defines an inner product on .
(b) Show that for each there exists a polynomial of degree exactly which is orthogonal, with respect to the inner product , to all polynomials of lower degree.
(c) Show that has simple zeros on .
(d) Show that for each there exist unique real numbers , such that whenever is a polynomial of degree at most ,
(e) Show that
as for all .
(f) If is real with and , show that
(g) If and , identify (giving brief reasons) and the . [Hint: A change of variable may be useful.]
Paper 3 , Section I,
commentState Runge's theorem on the approximation of analytic functions by polynomials.
Let . Establish whether the following statements are true or false by giving a proof or a counterexample in each case.
(i) If is the uniform limit of a sequence of polynomials , then is a polynomial.
(ii) If is analytic, then there exists a sequence of polynomials such that for each integer and each we have .
Paper 4, Section I, 2H
comment(a) State Brouwer's fixed-point theorem in 2 dimensions.
(b) State an equivalent theorem on retraction and explain (without detailed calculations) why it is equivalent.
(c) Suppose that is a real matrix with strictly positive entries. By defining an appropriate function , where
show that has a strictly positive eigenvalue.
Paper 4, Section II, H
commentLet be irrational with th continued fraction convergent
Show that
and deduce that
[You may quote the result that lies between and ]
We say that is a quadratic irrational if it is an irrational root of a quadratic equation with integer coefficients. Show that if is a quadratic irrational, we can find an such that
for all integers and with .
Using the hypotheses and notation of the first paragraph, show that if the sequence is unbounded, cannot be a quadratic irrational.
Paper 1, Section I,
commentLet be a continuous map never taking the value 0 and satisfying . Define the degree (or winding number) of about 0 . Prove the following.
(i) If is a continuous map satisfying , then the winding number of the product is given by .
(ii) If is continuous, and for each , then .
(iii) Let and let be a continuous function with whenever . Define by . Then if , there must exist some , such that . [It may help to define . Homotopy invariance of the winding number may be assumed.]
Paper 2, Section I,
commentShow that every Legendre polynomial has distinct roots in , where is the degree of .
Let be distinct numbers in . Show that there are unique real numbers such that the formula
holds for every polynomial of degree less than .
Now suppose that the above formula in fact holds for every polynomial of degree less than . Show that then are the roots of . Show also that and that all are positive.
Paper 2, Section II, H
commentLet be a (closed) triangle in with edges . Let , be closed subsets of , such that and . Prove that is non-empty.
Deduce that there is no continuous map such that for all , where is the closed unit disc and is its boundary.
Let now be three closed arcs, each arc making an angle of (in radians) in and . Let and be open subsets of , such that , and . Suppose that . Show that is non-empty. [You may assume that for each closed bounded subset defines a continuous function on .]
Paper 3, Section I, H
commentState Runge's theorem about the uniform approximation of holomorphic functions by polynomials.
Explicitly construct, with a brief justification, a sequence of polynomials which converges uniformly to on the semicircle .
Does there exist a sequence of polynomials converging uniformly to on ? Give a justification.
Paper 4, Section I,
commentDefine what is meant by a nowhere dense set in a metric space. State a version of the Baire Category theorem.
Let be a continuous function such that as for every fixed . Show that as .
Paper 4, Section II, H
comment(a) State Liouville's theorem on the approximation of algebraic numbers by rationals.
(b) Let be a sequence of positive integers and let
be the value of the associated continued fraction.
(i) Prove that the th convergent satisfies
for all the rational numbers such that .
(ii) Show that if the sequence is bounded, then one can choose (depending only on ), so that for every rational number ,
(iii) Show that if the sequence is unbounded, then for each there exist infinitely many rational numbers such that
[You may assume without proof the relation
Paper 1, Section I, H
commentLet be the th Chebychev polynomial. Suppose that for all and that converges. Explain why is a well defined continuous function on .
Show that, if we take , we can find points with
such that for each .
Suppose that is a decreasing sequence of positive numbers and that as . Stating clearly any theorem that you use, show that there exists a continuous function with
for all polynomials of degree at most and all .
Paper 2, Section I, H
commentLet be the collection of non-empty closed bounded subsets of .
(a) Show that, if and we write
then .
(b) Show that, if , and
then .
(c) Assuming the result that
defines a metric on (the Hausdorff metric), show that if and are as in part (b), then as .
Paper 2, Section II, H
commentThroughout this question denotes the closed interval .
(a) For , consider the points with and . Show that, if we colour them red or green in such a way that and 1 are coloured differently, there must be two neighbouring points of different colours.
(b) Deduce from part (a) that, if with and closed, and , then .
(c) Deduce from part (b) that there does not exist a continuous function with for all and .
(d) Deduce from part (c) that if is continuous then there exists an with .
(e) Deduce the conclusion of part (c) from the conclusion of part (d).
(f) Deduce the conclusion of part (b) from the conclusion of part (c).
(g) Suppose that we replace wherever it occurs by the unit circle
Which of the conclusions of parts (b), (c) and (d) remain true? Give reasons.
Paper 3, Section I, H
commentState Nash's theorem for a non zero-sum game in the case of two players with two choices.
The role playing game Tixerb involves two players. Before the game begins, each player chooses a with which they announce. They may change their choice as many times as they wish, but, once the game begins, no further changes are allowed. When the game starts, player becomes a Dark Lord with probability and a harmless peasant with probability . If one player is a Dark Lord and the other a peasant the Lord gets 2 points and the peasant . If both are peasants they get 1 point each, if both Lords they get each. Show that there exists a , to be found, such that, if there will be three choices of for which neither player can increase the expected value of their outcome by changing their choice unilaterally, but, if , there will only be one. Find the appropriate in each case.
Paper 4, Section I, H
commentShow that is irrational. [Hint: consider the functions given by
Paper 4, Section II, H
comment(a) Suppose that is a non-empty subset of the square and is analytic in the larger square for some . Show that can be uniformly approximated on by polynomials.
(b) Let be a closed non-empty proper subset of . Let be the set of such that can be approximated uniformly on by polynomials and let . Show that and are open. Is it always true that is non-empty? Is it always true that, if is bounded, then is empty? Give reasons.
[No form of Runge's theorem may be used without proof.]
Paper 1, Section I,
commentState and prove Sperner's lemma concerning colourings of points in a triangular grid.
Suppose that is a non-degenerate closed triangle with closed edges and . Show that we cannot find closed sets with , for , such that
Paper 2, Section I,
commentFor we write . Define
(a) Suppose that is a convex subset of , that and that for all . Show that for all .
(b) Suppose that is a non-empty closed bounded convex subset of . Show that there is a such that for all . If for each with , show that
for all , and that is unique.
Paper 2, Section II, F
comment(a) Give Bernstein's probabilistic proof of Weierstrass's theorem.
(b) Are the following statements true or false? Justify your answer in each case.
(i) If is continuous, then there exists a sequence of polynomials converging pointwise to on .
(ii) If is continuous, then there exists a sequence of polynomials converging uniformly to on .
(iii) If is continuous and bounded, then there exists a sequence of polynomials converging uniformly to on .
(iv) If is continuous and are distinct points in , then there exists a sequence of polynomials with , for , converging uniformly to on .
(v) If is times continuously differentiable, then there exists a sequence of polynomials such that uniformly on for each .
Paper 3, Section I,
commentState a version of the Baire category theorem and use it to prove the following result:
If is analytic, but not a polynomial, then there exists a point such that each coefficient of the Taylor series of at is non-zero.
Paper 4, Section I,
commentLet and . If we have an infinite sequence of integers with , show that
is irrational.
Does the result remain true if the are not restricted to integer values? Justify your answer.
Paper 4, Section II, F
commentWe work in . Consider
and
Show that if is analytic, then there is a sequence of polynomials such that uniformly on .
Show that there is a sequence of polynomials such that uniformly for and uniformly for .
Give two disjoint non-empty bounded closed sets and such that there does not exist a sequence of polynomials with uniformly on and uniformly on . Justify your answer.
Paper 1, Section I, 2F
commentState Liouville's theorem on the approximation of algebraic numbers by rationals.
Suppose that we have a sequence with . State and prove a necessary and sufficient condition on the for
to be transcendental.
Paper 2, Section I, F
commentAre the following statements true or false? Give reasons, quoting any theorems that you need.
(i) There is a sequence of polynomials with uniformly on as .
(ii) If is continuous, then there is a sequence of polynomials with for each as .
(iii) If is continuous with as , then there is a sequence of polynomials with uniformly on as .
Paper 2, Section II, F
commentState and prove Baire's category theorem for complete metric spaces. Give an example to show that it may fail if the metric space is not complete.
Let be a sequence of continuous functions such that converges for all . Show that if is fixed we can find an and a non-empty open interval such that for all and all .
Let be defined by
Show that we cannot find continuous functions with for each as
Define a sequence of continuous functions and a discontinuous function with for each as .
Paper 3, Section I, 2F
comment(a) Suppose that is a continuous function such that there exists a with for all . By constructing a suitable map from the closed unit disc into itself, show that there exists a with .
(b) Show that is surjective.
(c) Show that the result of part (b) may be false if we drop the condition that is continuous.
Paper 4, Section I, F
commentIf , set
where is an integer and . Let .
If is also irrational, write down the continued fraction expansion in terms of where .
Let be a random variable taking values in with probability density function
Show that has the same distribution as .
Paper 4, Section II, 11F
comment(a) Suppose that is continuous with and for all . Show that if (with real) we can define a continuous function such that and . Hence define the winding number of around 0 .
(b) Show that can take any integer value.
(c) If and satisfy the requirements of the definition, and , show that
(d) If and satisfy the requirements of the definition and for all , show that
(e) State and prove a theorem that says that winding number is unchanged under an appropriate homotopy.
Paper 1, Section I, H
commentBy considering the function defined by
or otherwise, show that there exist and such that
for all .
Show, quoting carefully any theorems you use, that we must have as .
Paper 2, Section I, H
commentDefine what it means for a subset of to be convex. Which of the following statements about a convex set in (with the usual norm) are always true, and which are sometimes false? Give proofs or counterexamples as appropriate.
(i) The closure of is convex.
(ii) The interior of is convex.
(iii) If is linear, then is convex.
(iv) If is continuous, then is convex.
Paper 2, Section II, H
commentProve Bernstein's theorem, which states that if is continuous and
then uniformly on . [Theorems from probability theory may be used without proof provided they are clearly stated.]
Deduce Weierstrass's theorem on polynomial approximation for any closed interval.
Proving any results on Chebyshev polynomials that you need, show that, if is continuous and , then we can find an and , for , such that
for all . Deduce that as .
Paper 3, Section I,
commentIn the game of 'Chicken', and drive fast cars directly at each other. If they both swerve, they both lose 10 status points; if neither swerves, they both lose 100 status points. If one swerves and the other does not, the swerver loses 20 status points and the non-swerver gains 40 status points. Find all the pairs of probabilistic strategies such that, if one driver maintains their strategy, it is not in the interest of the other to change theirs.
Paper 4, Section I, H
commentLet be integers such that there exists an with for all . Show that, if infinitely many of the are non-zero, then is an irrational number.
Paper 4, Section II, H
commentExplain briefly how a positive irrational number gives rise to a continued fraction
with the non-negative integers and for .
Show that, if we write
then
for .
Use the observation [which need not be proved] that lies between and to show that
Show that where is the th Fibonacci number (thus , , and conclude that
Paper 1, Section I, I
commentLet be a non-empty bounded open subset of with closure and boundary . Let be continuous with twice differentiable on .
(i) Why does have a maximum on ?
(ii) If and on , show that has a maximum on .
(iii) If on , show that has a maximum on .
(iv) If on and on , show that on .
Paper 2, Section I, I
commentLet be the roots of the Legendre polynomial of degree . Let , be chosen so that
for all polynomials of degree or less. Assuming any results about Legendre polynomials that you need, prove the following results:
(i) for all polynomials of degree or less;
(ii) for all ;
(iii) .
Now consider . Show that
as for all continuous functions .
Paper 2, Section II,
commentState and prove Sperner's lemma concerning the colouring of triangles.
Deduce a theorem, to be stated clearly, on retractions to the boundary of a disc.
State Brouwer's fixed point theorem for a disc and sketch a proof of it.
Let be a continuous function such that for some we have for all . Show that is surjective.
Paper 3, Section I,
commentLet be a compact subset of with path-connected complement. If and , show that there is a polynomial such that
for all .
Paper 3, Section II, I
commentLet . By considering the set consisting of those for which there exists an with for all , or otherwise, give a Baire category proof of the existence of continuous functions on such that
at each .
Are the following statements true? Give reasons.
(i) There exists an such that
for each and each .
(ii) There exists an such that
for each and each .
Paper 4, Section I,
commentLet be the set of all non-empty compact subsets of -dimensional Euclidean space . Define the Hausdorff metric on , and prove that it is a metric.
Let be a sequence in . Show that is also in and that as in the Hausdorff metric.
Paper 1, Section I,
comment(i) State Brouwer's fixed point theorem in the plane and an equivalent theorem concerning mapping a triangle to its boundary .
(ii) Let be a matrix with positive real entries. Use the theorems you stated in (i) to prove that has an eigenvector with positive entries.
Paper 2, Section I, G
commentState Chebyshev's equal ripple criterion.
Let
Show that if where are real constants with , then
Paper 2, Section II, G
Let be a continuous map never taking the value 0 and satisfying . Define the degree (or winding number) of about 0 . Prove the following:
(i) , where .
(ii) If is continuous, and for each , then .
(iii) If , are continuous maps with , which converge to uniformly on as , then for sufficiently large .
Let be a continuous map such that and for each