I'd find x by using arctan.

This would take a *lot* of work for me I think.

I would like some help with my macroeconomics homework.

Okay, so I need to use the Baumol-Tobin model and the question is
How will the next events effect the real money balance of a household

a)Real income increases 5%
b)Interest rates drop 10%
c)Real income and interest rate increase by 10%. Analyze the intercourse of real money balance and real income.

Whoever knows anything about this, would be very greateful.

No grading.

It'd just be you typing up some of your course notes and explaining it.

Brag: As in both semesters of undergraduate Abstract Algebra
Beat: Only just touched on Galois Theory.

Wyman, I'm looking to self-study some abstract algebra, since I won't be able to take any formal classes on it as an undergrad.

A friend of mine says to use Artin. Are you familiar with that textbook? If not, are there any texts/online resources you would recommend?

the last section of my linear algebra class is on error correcting codes. I think i get it but the homework doesn't quite make sense give my interpretation

Given the vectors:
a = 000 000 111
b = 000 111 000
c = 111 000 000
d = 001 001 001
e = 010 010 010

A)
i) what is the word length n? 9 seems pretty obvious
ii) how many information bits? from what we learned it seems this is equal k where 2^k=number of code words (5). all the examples we did were codes with some even power of 2 number of code words. not sure how to do this here
iii) how many check bits? its just 9-k, but need to figure out k

B) how many errors can the code correct? hamming weight is 3 so 1 error
C) how many code words are there? i think 5 but maybe this is where my misunderstanding is
D) write the H matrix for this code. I think i can do this if i figure out how many info bits there are.

i think my misunderstanding is in how you generate codewords from a set of vectors. is it just all possible linear combinations of the vectors? it seems like that is going to be far too large a number to be reasonable for this homework so i feel it is wrong but i can't see another way of doing it

There are "codes", which are a set of codewords, and "linear codes" which are codes that have the additional property that they are closed under linear combination. The advantage of linear codes (they are not typically optimal) is that you can use matrices to encode and decode, and you can write down the codewords by just writing down the generators.

Given that this is a LA class, I assume you're dealing with linear codes. I haven't looked at your last post in detail, but I will now.

A)
9
5 information bits. Basically, we can tell 32 messages using linear combinations of the 5 generating codewords.
That means there are 4 check bits.

The idea behind check bits is like ok, I want to send you either YES (1) or NO (0), but my channel is noisy, and there's a chance my bit gets flipped (assume p(0-->1) = p(1-->0) for your cartoon model, but note that this might be a bad assumption). So instead, I send you
00000000000000000000000000000000 for NO, or
11111111111111111111111111111111 for YES.

Now if one or two of my bits get flipped, you might receive something like
00000000001000000000100000000000
which you'd obviously decode as NO.

So there's only 1 bit of info being transmitted (0 or 1; i.e., 2 valid codewords), but there are 31 bits of "check" on your message.

B) You're right that the number of errors corrected is 1. That's because we check between all pairs of codewords, and the minimal Hamming distance is 3. For linear codes only, this is equivalent to the minimal nonzero Hamming weight (ducy? Let w = hamming weight, and d(a,b) = w(a-b) = Hamming distance from a to b. Then since the code is linear, for all a,b in C (the code), a-b in C. So if D is the min_dist of the code, then some a,b have d(a,b)=D, but then a-b is a codeword with weight D.)

Of course, before you conclude that d=3, you should check that no linear combination of those 5 codewords has smaller weight.

C) 32

D) The typical notation is G = generator matrix = rowspan of the matrix with a,b,c,d,e as its rows.
H = "parity check matrix", which has as its rows a basis for the nullspace of G. There should be n-k independent such vectors. The parity-check matrix is useful for decoding (see "syndrome decoding") and is the generator for the "dual code" of C.

alright, this all makes more sense now. it just clicked the difference with linear codes. i'm pretty sure how i got how to generate the parity check matrix. thanks for the explanation though!

This is a bit of an abstract question, and I'm not looking for a solution per se, more searching for a direction to look in, or something like that.

If I have a large set of random generated points constrained within a flat plane, and want to construct a simple, smooth, closed curve that passes all of these points, with (preferably) as short a length as possible, what would be the best way of going about it with a minimum of computational power?

I considered something like a pathfinding algorithm setup to prevent self intersections, but, well, that seems to be overkill, but I'm not sure about the computational power requried to implement, nor the ability for it to make a path of as short a length as possible - though, on reflection, I guess the shortest length path would be able to be generated by taking a point and using splines between it and its next closest point, and repeat, while looking for self intersections, but...

Order them and find the distance to the nearest neighbor and draw a line. Doesn't work if they're integers (unless you think about two equivalent distance points).

If they all have different values for x or y, you could fit a polynomial to the points (it'll be huge).

For n-points, a n-th order poly would definitely work, and....actually, on reflection I don't see why that would be more computationally intensive/difficult than any other method, and then just create a second curve between the first and last points to make it a closed curve.

Why didn't I think of that. Sleep and overthinking.

Thanks Cueballmania

Edit: Ok, that's why I discounted it. A random data set is going to, by necessity, have points which are very close to one another, which cause polynomial fitting problems. And any wiggle is - in this application - going to be seriously problematic. This would suggest something like piecewise splines, which then opens up the problem of self-intersections, damn.

just a quick note on this:
interpolating an nth order poly might be feasible depending on the application (which you did not specify) but of course your curve will not have minimal distance.
(quite obviously to achieve minimal distance you would travel on lines between the points. if you insist on your curve being smooth in every points you could "smooth it out" by any standard method around the points. its possible to do this by changing the distance traveled by an arbitrary small amount)

the hard question here is the order which you travel in between the points to keep the distance minimal. it is more or less the so called travelling salesmen problem, which happens to be quite a headache (np hard).

the earlier post by fursyshade reminded me of an old problem about which i can ask some new questions that i can not answer. would be surprised if anybody does tbh but i give it a try.

the situation: S(ender) and R(eceiver) are exchanging messages the following way: Some computer gives S a random sequence of 1s and 0s of length 64. S is allowed to change precisely one number in the sequence (so he has to change 1). Then the altered sequence gets sent to R.

the old question: how much information can be passed on each round? the answer is not hard at all, i wont write it down in case somebody feels like thinking about it.

some new questions: what if we allow S to do nothing (so he can send on the sequence unchanged if he wants to).
what if he is altering n numbers instead of 1?
what if he is altering 1 but the length of the sequence is not a power of 2?
any of the above combined?

I did not spend much time thinking to be honest, but i do not really see any of these. My instinct is that a general answer is hard but i might be wrong.

edit: eh mabbe i shouldnt have posted this in hw thread sorry. altough the old question i got as hw like 7 years or so ago plus meh. its a problem and i cant do it.

Thanks. Hmmm, travelling salesmen type computational difficulty isn't what I was looking for, thats not good, I might have to think if this method is the best way of working towards solving the larger problem I'm working on.

To be clearer, I'm not looking for the path of minimum distance, as you're right, that wouldn't be smooth, but rather I'm looking for a path with a minimal length (so as short as possible without expending too much computational time while still being a smooth closed curve without much wiggle).

What I'm trying to do is produce a result that's, in essence, an extension of the type of result you'd get if you gave a person a piece of paper with a bunch of dots on it, and asked them to link all the dots with a closed curve, in a manner that gives clearly defined regions 'inside' and 'outside' the curve, with no self-intersections, but extended to a number of points that a person wouldn't necessarily be able to perform the task over.

If I remove the smoothness condition, then that's pretty much just a Hamiltonian cycle, but...well, no, Hamiltonian cycle can self-intersect. A self-avoiding walk is essentially what I'm looking for, but just linking the first and last point would not necessarily lead to a well defined interior and exterior, as there would be a number of points the path would pass through that would not contribute to defining the interior space.

Could somebody please help with this question:

x = (4y) / ((1+70.8y^2)^(5/6))

Show that the maximum value for x is obtained when y=0.146

Hint: Differentiate!

Did you differentiate?

no, I don't know where to start.

Quotient rule

Will give $10 on FTP if someone can walk me through how to solve these. I have missed too many classes recently and have spent some time trying to even start these but I keep getting quite confused.

Need to do hypothesis testing for these problems.

1.The sample mean arm length of my 4 classes (110 total students) is 25.47 inches with a standard deviation of 2.06 inches. Assume that all conditions are met. Perform a hypothesis test to determine if the mean arm length for all Grand Valley students is different than 24”.

2.A random sample of 65 males and 43 females was taken about their footlengths. The sample mean footlength for males was 27.75 mm with a standard deviation of 1.99 mm. The sample mean footlength for females was 24.01 mm with a standard deviation of 1.72 mm. Perform a hypothesis test to determine if the mean footlength for males is greater than the mean footlength for females. Assume all conditions are met.


3.Describe a Type I and a Type II error for question #2.

Nevermind got it solved

I'm a little confused about induction.

My understanding was that you start with a base case, assume the nth case is true, and show that the n+1st case follows if the nth case is true.

Herstein's confusing me. We want to show that any integer a>1 can be factored as a product of prime powers.

Base case: 2 = 2*1.

Then he says, "Suppose that any integer r, 2<_ r < k can be factored as a product of prime powers." He then shows that k must be the product of prime factors. It seems like r can take on a range of values here but I guess that doesn't matter (we have the base case and for any number starting at 2, we can show the next number is divisible by prime factors, rinse and repeat). Sound right?

There are two statements of "induction", typically called the Principle of Mathematical Induction.

For any property of integers, we'll say that P(n) if the property holds for the integer n.

Statement 1 (weak induction): If P(N) and for all n >= N,
P(n) => P(n+1),
then P(n) for all n >= N.

Statement 2 (strong induction): If P(N) and for all n >= N,
P(N) & P(N+1) & ... & P(n) => P(n+1),
then P(n) for all n >= N.

Sounds like Herstein is using strong induction, rather than weak induction. A few moments' thought will convince you that either is a valid method of proof.

HTH.

OK, that makes more sense.

For a set with a binary operation to be a group, we need an element e in G such that a * e = e * a = a for all a in G. Does this mean that every group has one specific element in it that serves this function or just that each a has an e (I think it's the first one)?

Thx again.