Undergraduate Mathematics Marathon (1 Viewer)

glittergal96 · Dec 23, 2015

There was a marathon thread in this subforum that died a long time ago, so I am going to try to start a new one for people to post, solve, and discuss undergraduate mathematics problems. (Of course things like the stackexchange already exist that are fantastic for this purpose, but it would be nice for our community to be able to discuss this level of mathematics and share interesting problems.)

As the topics possible to discuss are very broad, if your problems have fairly specific/niche required knowledge to attack, you should mention this.

Feel free to post problems of any difficulty, but as problems are unlikely to be answered chronologically, please number your problems so they are easier to refer to.

(Also, use your common sense about what you are allowed to take as assumed knowledge. If you assume enough, then any undergraduate problem will be trivial.)

To start us off with a couple of basic problems:

1. Find as nice as possible an expression for $\frac{d}{dx}\left(\int_{a(x)}^{b(x)} f(x,t)\, dt\right)$ , where a,b and f are smooth functions. (Standard first year calculus.)

2. Find all solutions to a given arbitrary linear ODE with constant coefficients. (First year calculus / linear algebra).

3. Show that a continuous function on a compact metric space is uniformly continuous. (First course in metric spaces or topology.)

Paradoxica · Dec 23, 2015

glittergal96 said:
There was a marathon thread in this subforum that died a long time ago, so I am going to try to start a new one for people to post, solve, and discuss undergraduate mathematics problems. (Of course things like the stackexchange already exist that are fantastic for this purpose, but it would be nice for our community to be able to discuss this level of mathematics and share interesting problems.)

As the topics possible to discuss are very broad, if your problems have fairly specific/niche required knowledge to attack, you should mention this.

Feel free to post problems of any difficulty, but as problems are unlikely to be answered chronologically, please number your problems so they are easier to refer to.

(Also, use your common sense about what you are allowed to take as assumed knowledge. If you assume enough, then any undergraduate problem will be trivial.)

To start us off with a couple of basic problems:

1. Find as nice as possible an expression for $\frac{d}{dx}\left(\int_{a(x)}^{b(x)} f(x,t)\, dt\right)$ , where a,b and f are smooth functions. (Standard first year calculus.)

2. Find all solutions to a given arbitrary linear ODE with constant coefficients. (First year calculus / linear algebra).

3. Show that a continuous function on a compact metric space is uniformly continuous. (First course in metric spaces or topology.)

The simplest form, appears to be, as far as my attempts go, is:

$$1. $\frac{d}{dx} \left ( f(x,b(x)) - f(x,a(x)) \right )$

$\noindent 2. Substitute the general linear exponential function $(y = e^{rx})$ into the Linear ODE. Divide out the ubiquitous factor of $e^{rx}$ to obtain an n degree polynomial in terms of r. The general solution to the differential equation is thus $\sum_{i} c_i e^{r_ix}$ where $c_i$ is an arbitrary constant, and $r_i$ is a root of the polynomial. This general solution does not cover the case of multiple roots to the polynomial.$

Also not sure what you mean by arbitrary linear ODE.

glittergal96 · Dec 23, 2015

Paradoxica said:
The simplest form, appears to be, as far as my attempts go, is:

$$1. $\frac{d}{dx} \left ( f(x,b(x)) - f(x,a(x)) \right )$

$\noindent 2. Substitute the general linear exponential function $(y = e^{rx})$ into the Linear ODE. Divide out the ubiquitous factor of $e^{rx}$ to obtain an n degree polynomial in terms of r. The general solution to the differential equation is thus $\sum_{i} c_i e^{r_ix}$ where $c_i$ is an arbitrary constant, and $r_i$ is a root of the polynomial. This general solution does not cover the case of multiple roots to the polynomial.$

Also not sure what you mean by arbitrary linear ODE.

I am reasonably sure that your expression for (1) is not correct (although it looks close to what one of the terms in a correct expression is, it might even match this term), I can provide a counter-example after doing some housework if you would like.

Arbitrary linear ODE with constant coefficients simply means an ordinary differential equation of the form

$\left(\sum_{k=0}^n a_k \frac{d^k}{dx^k}\right)u(x)=0.$

(I should have also mentioned homogenous, otherwise there would be something nonzero on the RHS.)

Indeed the method you suggest is how we solve such things (in the no multiplicity case), but how do you know these are the only solutions? (By the nature of your method, you are only finding all exponential solutions.)

Zen2613 · Dec 23, 2015

One plus one is equal to two

glittergal96 · Dec 23, 2015

So a simple counterexample to the attempt at the differentiation under the integral question is given by:

f(x,t)=x, a(x)=a, b(x)=b.

You might find it easier to split the problem up into the two individual things going on here

a) the interval of integration is changing as x changes
b) the integrand is changing as x changes.

The end result comes from considering both of these effects.

glittergal96 · Dec 27, 2015

I will get people started on problem 1:

$Let $G(x)=\int_{0}^{b(x)} f(x,t)\, dt.$\\ \\ Clearly, knowledge of how to differentiate such a $G$ will tell us how to solve the original problem. We then have\\ \\ $\frac{G(x+h)-G(x)}{h}=\int_{0}^{b(x+h)} f(x+h,t)\, dt-\int_{0}^{b(x)} f(x,t)\, dt\\ \\ = \frac{1}{h}\int_{b(x)}^{b(x+h)} f(x+h,t)\, dt + \int_0^{b(x)} \frac{f(x+h,t)-f(x,t)}{h}\, dt.$\\ \\ The latter integral has a fixed domain of integration and the integrand is tending towards $\frac{\partial f}{\partial x}$. This should give you a pretty good idea for what this second integral tends to. Can you rigorously prove this?\\ \\ As for the first integral, notice that we are integrating over a very small interval (width $h$). This means that $f(x+h,t)$ should not change very much as $t$ varies over the interval of integration. What does this tell us the integral should be roughly equal to? Can you make this argument rigorous?$

glittergal96 · Dec 27, 2015

For 2, as Paradoxica mentioned, we can find n linearly independent exponential solutions. (Let's assume the polynomial only has simple roots for now, although the case of roots with multiplicity > 1 is not much harder).

To show then that these are the only solutions, it suffices to show that an n-th order linear ODE with constant coefficients has a solution set that has dimension n as a vector subspace of the space of smooth functions. This kind of statement should strongly indicate an inductive argument.

glittergal96 · Dec 27, 2015

For 3, it is useful to consider the function d(f(x),f(y)) defined on the product space K x K.

seanieg89 · Jan 12, 2016

A beautiful problem if one knows some real analysis:

F is a smooth function defined on the interval [0,1] such that at every x in [0,1], a derivative of some order of F vanishes. (Using one-sided differentiation at the boundary.)

Prove that F is a polynomial.

Undergraduate Mathematics Marathon (1 Viewer)

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