Hi I work as a programmer at a bus company and I need to implement a ride initialization request. I think it might be a linear programming problem but I'm not sure and I ask for some help :)
A passenger sends my server a request to initialize a bus ride.
The request includes the different entit...
You know that old topology (but really algebra) problem about hanging a picture with three nails such that removing any one of the nails makes the picture fall?
would you guys like to to discuss the following question. Let H be a hilbert space , $\{x_k\}$ be orthagonal in H. Then $\Sigma x_k$ convergent iff $\Sigma ||x_k||^2$ is convergent.
It's interesting that the game "I win a dollar with probability $10^{20}/(10^{20}+1)$ and lose a hundred million million million dollars with probability $1/(10^{20}+1)$" is completely fair
and yet I feel like I have no reason not to play it as many times as I want
@Fargle what do you think of the following reasoning. Suppose that $\Sigma x_k$ converges. Then since Hilbert spaces are banach spaces, so we must have $||\Sigma x_k||$ converges so this means that $||\Sigma x_k||^2$ converges but by pythagorean theorem $||\Sigma x_k||^2 = \Sigma ||x_k||^2$ which must converge as well.
I mean I will make it more precise by just considering partial sums but what do you think of it ?
is laplace demon actually a true statement? IF you know the position and velocity of all particles in the universe, then you could calculate the future.
suppose now $\Sigma ||x_k||^2$ is convergent then we have $||\Sigma x_k||^2$ is convergent so we have $||\Sigma x_k||$ is convergent as H is a complete banach space $\Sigma x_k$ is convergent.
would you guys like to to discuss the following question. Let H be a hilbert space , $\{x_k\}$ be orthagonal in H. Then $\Sigma x_k$ convergent iff $\Sigma ||x_k||^2$ is convergent.
Suppose that $\Sigma x_k$ converges. Then since we hilbert spaces are banach spaces we must have $||\Sigma x_k||$ converges so this means that $||x_k||^2$ converges but by pythagorean theorem $||\Sigma x_k||^2 = \Sigma ||x_k||^2$ which must converge as well.
yeah just have to change all of that to partial sum.
we consider cauchy sequence WLOG we can assume m > n. Then $||S_m - S_n||^2 = ||S_{n + 1} + ... S_{m}||^2 = ||S_{n+1}||^2 + ... + ||S_{m}||^2 = \beta_m - \beta_n$ Therefore $S_m$ is cauchy iff $\beta_m$ is cauchy.
Let $K$ be a field and $A$ a non empty set. Show that $V:=map(A,K)$ with the following binary operations is a $K$-vectorspace:
for $f,g\in V$ is defined by: $(f+g)(a)=f(a)+g(a)$ for all $a\in A$ for $r\in K$ and $f\in V$, $r\cdot f$ is defined by: $(r\cdot f)(a)=r\cdot f(a)$ for all $a\in A$
my question is, how do i show that scalar multiplication is closed. and what is a scalar in this context?
I'd like to find a number that's the difference of fourth powers in three ways or more. I.e.: $$k=a^4-b^4=c^4-d^4=e^4-f^4$$
Is this possible?
There seem to be plenty of examples of differences of fourth powers in two ways. The smallest: $$310300575=134^4-59^4=158^4-133^4$$
I've checked numbers ...
that is, are the morphisms in $\mathrm{Hom}_{C_A}(X,Y)$ defined as morphisms from the morphisms in $\mathrm{Hom}_C(X,A)$ to morphisms in $\mathrm{Hom}_C(Y,A)$?
@meow-mix the diagram $A\xrightarrow{f}B$ in a category $\cal C$ is formally defined as a functor $J\to\cal C$ where $J$ is the category with two objects $X,Y$ and a single nonidentity morphism $X\to Y$. the functor sends $X$ to $A$, $Y$ to $B$, and $X\to Y$ to $A\xrightarrow{f}B$
Let H be a hilbert space, $\{e_k\}$ be orthagonal sequence in H such that $e_k \neq 0$ for all k and that $\{a_k\}$ sequence in $\mathbb{K}$. Then there is $x \in H$ such that for every k one has $a_k = (x,e_k) / ||e_k||^2$ iff $\Sigma |a_k|^2 ||e_k||^2$.
yeah @arctictern.
@meow-mix don't worry about it we are all here to learn :)
like, if A is the closure of the span of the e_k's, and B is the orthogonal complement, then we can decompose x=a+b, in which case the e_k's are a basis for A so we may write a=c_1e_1+... and then solve getting c_i=a_i. then |x|^2<infinity gives (sum |a_k|^2 |e_k|^2) + |b|^2 < infinity, so just delete the |b|^2
Let H be a hilbert space, $\{e_k\}$ be orthagonal sequence in H such that $e_k \neq 0$ for all k and that $\{a_k\}$ sequence in $\mathbb{K}$. Then there is $x \in H$ such that for every k one has $a_k = (x,e_k) / ||e_k||^2$ iff $\Sigma |a_k|^2 ||e_k||^2$.
so, first, you have to show that if there is such an x, then that sum is < infinity. and second, you have to show that if the sum is < infinity, then there is such an x.
I don't understand so for the direction < we construct the $x = \Sigma_{i}^{k} a_i e_k$ and using orthagonility we prove that $a_k$ is given by the formula above right ?
If $\sum |a_k|^2\|e_k\|^2<\infty$ then we let $x=\sum a_k e_k$, and it satisfies $a_k=(x,e_k)/\|e_k\|^2$ for all $k$. Conversely, if there is an $x$ such that $a_k=(x,e_k)/\|e_k\|^2$ for all $k$, then we may write $x=a+b$ where $a$ is $x$'s projection onto $A$, the closure of the span of the $e_i$s, and $b$ is $x$'s projection onto the orthogonal complement of $A$. Then $\|x\|^2=\|a\|^2+\|b\|^2$ and $\|a\|^2=\sum |a_k|^2\|e_k\|^2$ must exist.
@Adeek it's a hilbert space, so you can add those things up
Hi I work as a programmer at a bus company and I need to implement a ride initialization request. I think it might be a linear programming problem but I'm not sure and I ask for some help :)
A passenger sends my server a request to initialize a bus ride.
The request includes the different entit...
I'd like to find a number that's the difference of fourth powers in three ways or more. I.e.: $$k=a^4-b^4=c^4-d^4=e^4-f^4$$
Is this possible?
There seem to be plenty of examples of differences of fourth powers in two ways. The smallest: $$310300575=134^4-59^4=158^4-133^4$$
I've checked numbers ...
