Help me not fail

I've got work to do over the holidays, so I thought I'd start a general help thread for my, probably many, problems.

First off, from an example sheet, the offending part of the question is:

Consider

\frac{dx}{dt} + x - y = \cos(2t)

,

\frac{dy}{dt} + 5x - y = \cos(2t) + 2 a \sin(2t)

for various values of the real constant

a

. For what value(s) of a is there resonance? What general principle does this illustrate?

So I can get a second order DE from this okay, I can solve it okay, but the problem is I don't actually know what resonance is. Normally I'd consult my notes, but the first reference to resonance in my notes is an example to illustrate what to do if the RHS of a DE is a solution to the homogeneous equation on the LHS, without any definition. So what am I actually supposed to be looking for here?

Thanks.

Reply 1

generalebriety

16

Exactly that. Resonance is (crudely explained) when the forcing is a multiple of some part of the homogeneous solution, causing the maximum displacement to grow. I think there's a question later on on that example sheet about a mouse exercising on a ship or something; the mouse exercises at exactly the resonant frequency, which causes the ship to sink. As another example, if you're pushing someone on a swing and you push them at the right frequency they'll go higher and higher, but if you try and push them too soon or too late then they won't go higher.

Reply 2

div curl F = 0

9

Resonance is where there is a sudden jump in a measured observable. Near a resonance a small change in the conditions produces a large effect in the outcome.

In your case, you may end up with a factor like

\frac{1}{1 - a}

, so the resonance would be at a = 1 because thats where the system sudden diverges to infinity and the equation becomes undefined.

If you plot the function, it should look something like this:

Hope this helps

Resonance Diagram for TSR.JPG11.2KB

Reply 3

ad absurdum

OP

2

Cool, that helps. Thanks guys.

Reply 4

ad absurdum

OP

2

Okay, a proof I don't understand.

Proposition: Suppose

\lambda_1, \lambda_2, ..., \lambda_r

are distinct eigenvalues of a linear map

Unparseable latex formula:

\textsl{A} : \Re^n \to \Re^n

, and let

B_i

denote a basis of the eigenspace

E_{\lambda_i}

(with dimension

m_{\lambda_i}

). The set

B = \bigcup_{i=1}^n B_i

is linearly independent,

Proof: By contradiction. Suppose B is linearly dependent, then there exists

c_{ij}

not all zero such that

\sum_{i=1}^r \sum_{j=1}^{m_{\lambda_i}} c_{ij} v_{ij} = 0

where

v_{ij}

is the jth basis vector of

B_i

. Apply the operator

Unparseable latex formula:

\prod_{k=1,..,\bar{K},...,r} (\textsl{A} - \lambda_k )

to this to get:

\sum_{i=1}^r \sum_{j=1}^{m_{\lambda_i}} \prod_{k=1,..,\bar{K},...,r} c_{ij} (\lambda_i - \lambda_k) v_{ij}

Unparseable latex formula:

= \sum_{j=1}^m_{\lambda_K} ( \prod_{k=1,..,\bar{K},...,r} c_{Kj} (\lambda_K - \lambda_k)) v_{Kj}

= 0

...

------------

I can follow the rest of the proof okay, but I don't understand the application of the operator part, or the manipulation that follows. Can anyone explain it in a littlre more detail? Thanks.

Edit - Should have explained the notation better. The bar above the K means that this is omitted from the otherwise natural sequence of numbers, i.e. the expected product would be taken with the Kth term omitted.

Reply 5

generalebriety

16

ad absurdum

Okay, a proof I don't understand.

Proposition: Suppose

\lambda_1, \lambda_2, ..., \lambda_r

are distinct eigenvalues of a linear map

Unparseable latex formula:

\textsl{A} : \Re^n \to \Re^n

, and let

B_i

denote a basis of the eigenspace

E_{\lambda_i}

(with dimension

m_{\lambda_i}

). The set

B = \bigcup_{i=1}^n B_i

is linearly independent,

Proof: By contradiction. Suppose B is linearly dependent, then there exists

c_{ij}

not all zero such that

\sum_{i=1}^r \sum_{j=1}^{m_{\lambda_i}} c_{ij} v_{ij} = 0

where

v_{ij}

is the jth basis vector of

B_i

. Apply the operator

Unparseable latex formula:

\prod_{k=1,..,\bar{K},...,r} (\textsl{A} - \lambda_k )

to this to get:

\sum_{i=1}^r \sum_{j=1}^{m_{\lambda_i}} \prod_{k=1,..,\bar{K},...,r} c_{ij} (\lambda_i - \lambda_k) v_{ij}

Unparseable latex formula:

= \sum_{j=1}^m_{\lambda_K} ( \prod_{k=1,..,\bar{K},...,r} c_{Kj} (\lambda_K - \lambda_k)) v_{Kj}

= 0

...

------------

I can follow the rest of the proof okay, but I don't understand the application of the operator part, or the manipulation that follows. Can anyone explain it in a littlre more detail? Thanks.

Edit - Should have explained the notation better. The bar above the K means that this is omitted from the otherwise natural sequence of numbers, i.e. the expected product would be taken with the Kth term omitted.

Application of the operator is easy, because the operator is linear.

\displaystyle \sum_{i=1}^r \sum_{j=1}^{m_{\lambda_i}} c_{ij} v_{ij} = 0

Apply the operator:

Unparseable latex formula:

\displaystyle \prod (\textsl{A} - \lambda_k ) \left( \sum_{i=1}^r \sum_{j=1}^{m_{\lambda_i}} c_{ij} v_{ij} \right) = 0

(where we're taking the product over the range you specified; I've just missed it out for notational convenience)

Unparseable latex formula:

\displaystyle \sum_{i=1}^r \sum_{j=1}^{m_{\lambda_i}} \left\{ c_{ij} \left[ \prod (\textsl{A} - \lambda_k ) \right] (v_{ij}) \right\} = 0

\displaystyle \sum_{i=1}^r \sum_{j=1}^{m_{\lambda_i}} \left\{ c_{ij} \left[ \prod (\underline{ \underline{ \lambda_i - \lambda_k }}) \right] (v_{ij}) \right\} = 0

(eigenvalue).

After that, you just notice that, for each i = 1, ..., (not K), ..., r, the constant term (lambda_i - lambda_k) will become zero at some point and so the whole term is zero. Remember we're summing through i = 1, ..., r. This doesn't happen for i = K, because lambda_i = lambda_j implies i = j (eigenvalues are distinct), so lambda_k never equals lambda_K (because k is never K).

Reply 6

ad absurdum

OP

2

Thanks again Billy.

-----

Okay, this question should be easy (i.e., first question on the sheet):

Let

A_1,A_2,...

be a sequence of subsets of

\Re

such that

A_1 \cap A_2 \cap .... \cap A_n \not= \emptyset

for each

n \ge 1

. Does it follow that

\bigcap_{n=1}^{\infty} A_n

is non-emtpy? Does the answer change if you are given the information that each

A_n

is a closed interval?

My question is, why would they not be non-empty? It seems to me that in both cases if it holds for all

n

then there's no reason why it shouldn't hold as

n

goes off to infinity. Now that's of course crap reasoning, so can someone give me a hint (and only a hint, it's an example sheet question) why it wouldn't hold? I'm sure there's more to this than what I'm thinking.

Reply 7

DFranklin

18

ad absurdum

Thanks again Billy.

-----

Okay, this question should be easy (i.e., first question on the sheet):

Let

A_1,A_2,...

be a sequence of subsets of

\Re

such that

A_1 \cap A_2 \cap .... \cap A_n \not= \emptyset

for each

n \ge 1

. Does it follow that

\bigcap_{n=1}^{\infty} A_n

is non-emtpy? Does the answer change if you are given the information that each

A_n

is a closed interval?

My question is, why would they not be non-empty? It seems to me that in both cases if it holds for all

n

then there's no reason why it shouldn't hold as

n

goes off to infinity. Now that's of course crap reasoning, so can someone give me a hint (and only a hint, it's an example sheet question) why it wouldn't hold? I'm sure there's more to this than what I'm thinking.

Suppose it holds for all n, and let a_n be the sequence of common points (i.e.

a_n \in \bigcap_{k=1}^n A_k

). One thing that can happen is that

a_n \to a

, but a isn't in any of the A_n.

Reply 8

ad absurdum

OP

2

Thanks for the reply, I've got it now.

Reply 9

ad absurdum

OP

2

Okay, I'm having no fun with this numbers and sets sheet. It's an example sheet again, so only hints. Be warned that I completely lack any sort of natural talent at this sort of thing.

Show that there does not exist an uncountable family of pairwise disjoint discs in the plane. What happens if we replace 'discs' by 'circles'?
- I'm guessing I should be looking at the discs as a Cartesian product of (open) intervals of some sort? I don't know if this is right, or where to take it if it is right.

A function

F: \mathbb{N} \to \mathbb{N}

is said to be increasing if

f(n+1) \geq f(n)

for all

n

, and decreasing if

f(n+1) \leq f(n)

for all

n

. Is the set of increasing functions countable or uncountable? What about decreasing functions?
- Again, no idea. I'm pretty sure that the increasing functions at least would be uncountable. But how to show it? Maybe try and find an injection from the power set of the integers or something?

Thanks for any help. Apologies if these are obvious, I'm definately not a pure mathematician.

Reply 10

ukgea

6

ad absurdum

Okay, I'm having no fun with this numbers and sets sheet. It's an example sheet again, so only hints. Be warned that I completely lack any sort of natural talent at this sort of thing.

Show that there does not exist an uncountable family of pairwise disjoint discs in the plane. What happens if we replace 'discs' by 'circles'?
- I'm guessing I should be looking at the discs as a Cartesian product of (open) intervals of some sort? I don't know if this is right, or where to take it if it is right.

That's a good approach. It might be easier to prove first:

“Show that there does not exist an uncountable family of pairwise disjoint rectangles in the plane.”

A function

F: \mathbb{N} \to \mathbb{N}

is said to be increasing if

f(n+1) \geq f(n)

for all

n

, and decreasing if

f(n+1) \leq f(n)

for all

n

. Is the set of increasing functions countable or uncountable? What about decreasing functions?
- Again, no idea. I'm pretty sure that the increasing functions at least would be uncountable. But how to show it? Maybe try and find an injection from the power set of the integers or something?

That might work. A better idea might be to find a surjection from the increasing functions onto the set of sequences of natural numbers or something. Or the set of sequences of zeros and ones.

For decreasing functions, “A countable union of countable sets is countable” (lecture 21) might come in handy.

Reply 11

DFranklin

18

ad absurdum

Show that there does not exist an uncountable family of pairwise disjoint discs in the plane. What happens if we replace 'discs' by 'circles'?
- I'm guessing I should be looking at the discs as a Cartesian product of (open) intervals of some sort? I don't know if this is right, or where to take it if it is right.One approach that's often useful for this kind of problem: Show each disc must contain a "rational point" (that is, a point (p,q) with p,q both rational).

A function

F: \mathbb{N} \to \mathbb{N}

is said to be increasing if

f(n+1) \geq f(n)

for all

n

, and decreasing if

f(n+1) \leq f(n)

for all

n

. Is the set of increasing functions countable or uncountable? What about decreasing functions?
- Again, no idea. I'm pretty sure that the increasing functions at least would be uncountable. But how to show it? Maybe try and find an injection from the power set of the integers or something?

I think that should work, yes.

Spoiler

Reply 12

ad absurdum

OP

2

Thanks guys. I've done the second one now.

For the first one, I can show that there is a rational point in the disc, but I can't see where to go from here. I don't know how to prove what ukgea suggested either (which suspect I really should be able to do).

Reply 13

DFranklin

18

ad absurdum

Thanks guys. I've done the second one now.

For the first one, I can show that there is a rational point in the disc, but I can't see where to go from here. I don't know how to prove what ukgea suggested either (which suspect I really should be able to do).

Note that no two discs can contain the same rational point (as they are disjoint). So can you see how to find an injection from the set of discs to the (countable) set of rational points (p,q) (p,q rational)?

Reply 14

ad absurdum

OP

2

Gah, how did I not get that? Thanks again.

Help me not fail

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