There are several ways to check whether an integer is prime. The usual one is to simply to check for divisibility by every prime number up to the square root of your number, as you suggested. This is not the only one though. For large suspected primes you usually use algorithms which rely on...
New types of math are invented all the time, but they are usually sub areas of the very broad areas you know about. No one just works with algebra or founds a new field of that magnitude anymore, that sort of low hanging fruit has for the most part been picked. Many of these fields are so...
If you have no idea how to approach the problem I would suggest the following process:
Start listing the A's that could work in this problem, i.e. the A has 8 as a divisor and which itself is a divisor of 192. For instance your list might start:
A=8
A=16
...
For every such A ask yourself what B...
If this is in response to my post, then please let me clear up that what I meant was that it is not optional if you want to obtain a PhD subsequently. There is nothing stopping you from just getting a Bachelor degree if you don't intend to pursue a PhD.
I have no experience with engineering...
The usual European model is (there is some variation between countries, but the following is the most common):
Bachelor: 3 years
Masters (not optional): 2 years
PhD: 3 years
While the American model is:
Bachelor: 4 years
PhD: 4-5 years
So it works out to about the same. From what I hear...
Just do it one step at a time.
f(x) \in f(E)
means that
f(x) =f(e)
for some e \in E. You can now apply injectivity to get x = e, from which you can conclude x \in E.
So the problematic part was:
"This implies that f(x) is in f(E).
Since f is injective, it has an inverse.
Applying the...
Well it should only seem obvious if you remember that f is injective. This should suggest that you need injectivity in some way. Can you state what f(x) \in f(E) actually means in terms of elements? Do you see any way to apply injectivity to this expression?
There is a standard definition of a simple object in a finitely complete category with an initial object 0. See:
http://ncatlab.org/nlab/show/simple+object
for the general definition.
A congruence on a magma M is an equivalence relation ~ on M such that a ~ b and c ~ d implies ac ~ bd. For any...
You seem to have misunderstood inverses and their existence a bit. Let us fix a function f : A \to B. Let a function g : B\to A be given, then we say that g is the inverse of f if
f(g(b)) = b \qquad g(f(a)) = a
for all a \in A and b \in B. Sometimes only one of these conditions is satisfied...
An open set can be written as the union of multiple disjoint non-empty open sets if and only if it is disconnected. Therefore another way of phrasing this statement is:
Any simply-connected connected open set U can be written as an open set of type 1 or type 2.
To see that there exists...
Yes. This is a common thing to joke or moan about for recent graduates without a job, or who had a hard time finding one. In some fields there simply are no entry level jobs (posted publicly) and in others the employers redefine them as you describe.
I don't have personal experience with this...
What does it mean to you to work out all of D+F? You could probably go through it in a semester or so if you kept at it and come out with a decent understanding (most people spread it out over a longer period though with other classes in between and I believe this is a better way to learn...
I just looked up "metric spaces" on wikipedia and two examples stood out:
1) Give the positive real line the metric d(x,y) = |log(x/y)|
2) In Euclidean space suppose that instead of considering the direct distance from x to y, we want to travel via 0, then the distance is given by:
d(x,y) = |x|...
I doesn't look like you really understand sup. The n below sup is meant as a variable, and sometimes we have restrictions such as
\sup_{n\geq 5} x_n
which means we consider sup of the sequence x_5,x_6,x_7,\ldots. Therefore your first two statements do not seem to make sense (at least with...
Yes (slight lie, see below). Every log is defined precisely on the positive numbers.
This is a slight lie because in complex analysis we actually extend log and define it for many complex numbers. However we will never be able to include 0 in the domain (without making it discontinuous), and...
For the => direction you are on the right track. You want an open ball B with center x, but with radius small enough that y is not in B. d(x,y) seems to be the only number you have to work with so try choosing a radius based on that. If you can find a radius such that y is not in B, then you are...
The part you seem to forget is "some point c of its domain". 0 is not in the domain of ln so this is why your observation is not a problem. The domain of ln is the positive numbers.
It is however an interesting observation in its own right and it implies that you cannot possibly define ln(0) in...
The Euclidean algorithm only works for relatively prime elements, so when he says that r(\lambda), q(\lambda) exists such that
r(\lambda)(\lambda^3-8\lambda)+q(\lambda)(\lambda^2+1)=1
he uses that they are relatively prime.
There are basically still 3 issues with your homotopy:
1) You forgot an extra coefficient of 2 in front of n in your H for s >= 1/2.
2) It doesn't line up on s=1/2, i.e. the two expressions you gave are different when s=1/2
3) It isn't a homotopy of loops (i.e. not a based homotopy).
1 is...
Your idea seems correct. To separate into cases depending on whether s is smaller or larger than 1/2 is a good idea in this problem (as you have done).
However try to think about what you are doing. f_{m+n} loops with a speed of (m+n) for the whole interval [0,1]. f_n * f_m instead first...
I'm a bit rusty on de Rham cohomology so excuse my overly long answer which was mostly just to convince myself, but now I feel I might as well leave the details in there. It seems like (for smooth manifolds at least) it follows from the universal coefficient theorem and deRham's theorem as you...
Yes it is, but you also need to prove that it is a subset of X. That too is fairly obvious. Just mention that A is a subset of X so A-B is a subset of X, similarily B-A is a subset of X. Therefore (A-B) union (B-A) is a subset of X. If this is a slightly more advanced course you may even just...
You know that A and B are subsets of X. From this can you figure out how to show that:
(A-B) \cup (B-A)
is a subset of X? This is all you need to prove the closure of addition.
Maybe you are confused because earlier you learned:
A+B = \{a+b|a\in A,\,b \in B\}
But note that the + operation is...
Yes every point of E is either an isolated point or a limit point of E.
The proof is not complicated and you should consider it an exercise. It is the kind of statement you should be comfortable proving after reading this section of Baby Rudin.
It is unclear what you are asking for. Are you asking
1) How to prove your lemma,
2) for examples of coinvolutory matrices, or
3) for general litterature on coinvolutory matrices?
The lemma is fairly straightforward to prove by calculating \left(\left(\overline{A}\right)^{-1}A\right)^{-1} and...
(iii) is not the same kind of separation as that in (i) and (ii). To see the difference just take the real line and let the family of seminorms be the set consisting of just the ordinary absolute value norm. Then clearly (i) holds because |x|=0 implies x=0, but (iii) does not hold because 1\not=...
Everyone is different, but the following are some approaches I have used when self-studying.
Look at the exercises. If any look particularily scary or you have no idea how to begin on some, then do that exercise.
Another approach is simply to do the last 5 exercises or so in each section...