I was reading about space-filling curves, and I was wondering how they fit in with the Jordan curve theorem (that a closed simple curve has an inside and an outside). Do space filling curves not satisfy some condition for the Jordan curve theorem to apply to them? Or does it?
So I'm working with functions of the form p1(z)exp(a1*z) + p2(z)exp(a2*z) where a1 and a2 are distinct complex numbers and p1 and p2 are polynomials. I'm trying to characterize when these functions have no zeros. Can someone give me a hint? Maybe Rouche's theorem would be helpful
@detly It depends what you mean by 'self-intersect'. The answer is either 'yes, by the definition of injective' or 'no, self-intersection doesn't make sense for a curve so pathological'.
The better answer is probably: no, it's not for any 'geometric' reason, but rather for topological ones. Suppose a space-filling curve means a surjection $[0,1] \to [0,1]^2$. It's a theorem that any injective map from a compact metric space to another metric space is a homeomorphism - it preserves certain topological properties. But (another theorem) $[0,1]$ is not homeomorphic to $[0,1]^2$.
I uploaded the Navy's NUPOC study guide to my github. Not all the sections are completed and I am looking for contributors that would like to work on providing solutions in their free time. Everyone who helps, will be credited with their work. Also, a proof read of the sections currently complete would be great too! github.com/dwsmith1983/NUPOC-Solutions
Hello,everyone,someone can interesting this problem: if $a+b\sin{c^{circ}}=1+4\sin{10^{\circ}}$,where $a$ postive integer,and $b,c$ integer,show that a=1,b=4,c=10
Interesting problem:
Assmue that：
$$1+4\sin{10^\circ}=a+b\sin{c^\circ}$$
where $a>0$,and $a,b,c$ are integers and $0<c<90^\circ$,
show that $$a=1,b= 4,c=10$$ is unique solution
Different Old Question: see Find this $a,b,c$ such that $\sqrt{9-8\sin 50^{\circ}}=a+b\sin c^{\circ}$
I ...
@Emrakul There isn't really a good way to search for mathematical expressions. You can try to type up a question and see what pops up as a possibly related question. One user has begun to compile an index, but this is still a work in progress.
@Emrakul In particular, in this answer I have posted examples showing attempts to search for a specific integral. You can judge for yourself to which extent it was successful/efficient.
Interesting problem:
Assmue that：
$$1+4\sin{10^\circ}=a+b\sin{c^\circ}$$
where $a>0$,and $a,b,c$ are integers and $0<c<90^\circ$,
show that $$a=1,b= 4,c=10$$ is unique solution
Different Old Question: see Find this $a,b,c$ such that $\sqrt{9-8\sin 50^{\circ}}=a+b\sin c^{\circ}$
I ...
@Danielfischer One quick question, again with regard to [the post](http://math.stackexchange.com/questions/774552/convergence-of-characteristic-functions-on-hypercube). In the answer to this the post it states that if we can prove that $$\langle g,f_{m} \rangle \rightarrow \langle g,f \rangle$$ for all $g$ is some set $A \subset L^{1}$ with norm-dense linear span. then given any $h \in L^{1}$ it follows that $|\langle h,f_{m} \rangle-\langle h,f \rangle| \rightarrow 0$. I understand the reason given for this but I tried a different proof. Please see if my proof is also fine, it is similar.
@JohnDoe Take $$a_{n,k} = \begin{cases} 1 &, n \geqslant k \\ 0 &, n < k.\end{cases}$$ Then you have $$\lim_{n\to\infty} \bigl(\lim_{k\to\infty} a_{n,k}\bigr) = \lim_{n\to\infty} 0 \neq 1 = \lim_{k\to\infty} \bigl(\lim_{n\to\infty} a_{n,k}\bigr).$$ You need to show that something like that (or worse) cannot happen in the given situation.
@DanielFischer What do you mean integrability suffices for that you mean if $f$ is integrable and not just locally integrable then $g$ is uniformly continuous?
@DanielFischer Most people use term 'integrable' to mean $\int f dx < \infty$, I was taught that this is called summable. If the integral exists then $f$ is integrable.
@JohnDoe Most people I know use "$f$ is integrable" to mean $f \in \mathscr{L}^1(\mu)$ resp. $[f] \in L^1(\mu)$, where $\mu$ is the measure under consideration, and $[f]$ is the equivalence class of $f$ modulo "$\mu$-a.e. equal".
Please, somebody knows what is the name of the theorem that allows you to complete a set of linearly independent vectors into a basis? I only know its name in portuguese. Need a wikipedia page for it.
@DanielFischer thanks! Steinitz exchange lemma says: "If {v1, ..., vm} is a set of m linearly independent vectors in a vector space V, and {w1, ..., wn} span V then m ≤ n and, possibly after reordering the wi, the set {v1, ..., vm, wm + 1, ..., wn} spans V."
Show that in an elliptic curve $E/\mathbb{Q}$ the sum of three colinear rational points of it is equal to $O$ exactly when the neutral element of the group $E(\mathbb{Q})$, $O$ is an inflection point of the curve.
Ifound the following in my notes.
Let $C$ be a cubic curve that is defined over...
Hey @DanielFischer
I am looking at the algorithm MERGE that is the following:
MERGE(A,p,q,r)
n1=q-p+1
n2=r-q
for i=1 to n1
L[i]=A[p+i-1]
for j=1 to n2
R[j]=A[q+j]
L[n1+1]=oo
R[n2+1]=oo
i=1
j=1
for k=p to r
if L[i]<=R[j]
then A[k]=L[i]
i=i+1
else A[k]=R[j]
j=j+1
@DanielFischer There is a comment that says that we create the subarrays L[1... n1+1] and R[1... n2+1]. Why? Doesn't it hold that the arrays have sizes n1 and n2 respectively?
@DanielFischer @robjohn @ArthurFischer @KajHansen This is the algorithm of heapify: http://pastebin.com/C5ZJLWt7
According to my book, the time complexity of the algorithm at a subtree of length n corresponds to the time Θ(1) that is required in order the elements A[i], A[LEFT(i)], A[RIGHT(i)] to get the right relations plus the time complexity of HEAPIFY at a subtree. Each of these subtrees has a length that is $\leq \frac{2n}{3}$- the worst case happens when the last line of a tree is exactly semicompleted.
@evinda After the array elements, this implementation adds a $\infty$ entry as a sentinel, so one needs an extra slot in the temporary arrays. If you don't do that, you need to check $i$ and $j$ to prevent out-of-bounds access to the arrays.
@evinda There are many ways to prove correctness. For a lot of algorithms, proving that some invariant holds at all steps by induction is a good method to prove correctness.
@DanielFischer Ok, but suppose that we have as input an array of length n. Could we maybe then prove the correctness of the algorithm, using induction on n, i.e. say at the base case what happens for n=1, suppose at the induction hypothesis that it is true for length=n-1 and then show that it holds for lenth=n ?
@evinda Because HEAPIFY(A,i) (it's very ugly to use all-caps for function names by the way, and violates convention: all-caps is for preprocessor macros) makes A[i] the root of a max-heap without destroying the heap property of its children (and grand-children, ...) and doesn't touch the part of the array belonging to trees not below A[i]. Of course, these properties of of HEAPIFY must also be proved.
@DanielFischer At the initial check, we say the following: Before the first iteration, we have $i=\lfloor \frac{n}{2} \rfloor$. All the nodes \lfloor \frac{n}{2} \rfloor+1, \lfloor \frac{n}{2} \rfloor$+2,..., n$ are leaves and thus each of them is the root of a trivial max-heap. Then at the check of maintenance: in order to show that the invariant holds, we notice that the children of the node i have greater numbers of position than i. Therefore, based on the invariant, both of these nodes are roots of max-heaps.
@DanielFischer How do we conclude the last proposition? :/
Hello!! We have an equation which has the roots $r1, r2$. We are looking for the appropriate iterative method $x_{n+1}=\phi(x_n), n=0,1,2$ such that the sequence $(x_n)$ converges ro the root $r2$, $\forall x \in [a,b]$. When we have four possible $\phi(x)$, how can we know which we have to choose?
A dreamy cat doesn't notice a flower pot, which at first flies upwards and then downwards by an open window. The flower pot has been seen for 0.50 seconds, the height of the windowsill till the top edge of the window is 2,00 m. 2,00 m.
Which height over the Top edge of the window does the flower pot get?
@Chris'ssis I think some algebraic manipulation and writing $1 = \ldots$ in smart ways should work. Unfortunately, no time to fully explore this at the moment; I have to prepare an important presentation for tomorrow :(.
@Chris'ssis Several things. I heard that an apartment I applied for hiring had gone to someone else. I had trouble obtaining some stupid authorisations at my work. Also, my patience with slow-paced people has worn out today.
And as said, I have to prepare a presentation in the evening hours.
@Lord_Farin Just as a personal opinion, nothing can be worse than attending an interview where you're asked personal questions you don't wanna discuss about. :-)))
Interesting problem:
Assmue that：
$$1+4\sin{10^\circ}=a+b\sin{c^\circ}$$
where $a>0$,and $a,b,c$ are integers and $0<c<90^\circ$,
show that $$a=1,b= 4,c=10$$ is unique solution
Different Old Question: see Find this $a,b,c$ such that $\sqrt{9-8\sin 50^{\circ}}=a+b\sin c^{\circ}$
I ...
Show that in an elliptic curve $E/\mathbb{Q}$ the sum of three colinear rational points of it is equal to $O$ exactly when the neutral element of the group $E(\mathbb{Q})$, $O$ is an inflection point of the curve.
Ifound the following in my notes.
Let $C$ be a cubic curve that is defined over...
