An user provided the right solution to a question I asked not even hour ago, but he deleted the solution before I get to fully understand the solution.
I am reading a machine learning text book, it has a example says:"a line has measure zero while a filled polygon has positive measure". From wolfalpha, measure zero is a set of points capable of being enclosed in intervals whose total length is arbitrarily small. I still don't understand the example
@Simple: That's only when it's a subset of the real line.
You have to scale it up to whatever dimension you're in. If you have a subset of the plane, measure 0 means you can cover it by a countable collection of rectangles the sum of whose areas is arbitrarily small.
the measure of a subset of R^1 is its total "length." the measure of a subset of R^2 is its total "area." the measure of a subset of R^3 is its total "volume." and so on.
I don't know how you'd define an "arclength measure" for $\Bbb R^2$, though. If you have a parametrization, I know how, but not for like an arbitrary point set
@Akiva Suppose kite $ABCD$ has diagonals which intersect at $E$. Consider the four points by reflecting $E$ over the sides of the kite. Prove that those four points are cyclic
Anyone have any experience with people majoring in CS and later going to grad school for math?
I'm about to graduate with a bachelor's in CS doing undergrad research in ML, but I've always been so much more interested in number theory/combinatorics than my CS courses. (I double-majored in math and took courses in algebra, linear algebra, analysis, number theory, algebraic combinatorics, and stochastic processes, so I believe I have a better feel for pure math than had I only majored in CS.)
i'm from the uk, words are a bit different, from what i gather, you did a bunch of maths courses on top of your core CS ones, you enjoyed it, you wanted to know if you were alone or you had kin (education wise at least)
Question relating to programming, if a program continues looping until a random value within the range of $[0,2]$ is $<=1$, can it always be considered halting? That is, can a list of uniformly random numbers of infinite length be considered to hold any number or number within an arbitrary range?
i want to show that $f(x,y) = x \ ^ y$ is integrible at $[0,1] \times [a,b]$ for $b \gt a \gt 0$ , so i thought showing it is continuous , but i am a bit stuck on this, someone can help ?
Also, considering there are various categories of such machines, you have determined a function which calculates the cost on electricity similar to $f(x)$
OOPs
The catch is as the value of one of these functions increases the other decreases. Maybe I gave the wrong example
You'd want to keep things in your budget by spending as less as possible yet minimising both together
How do you do that sort of optimisation on given bounds for your $x_i$, is what I was trying to ask yesterday?
@AlessandroCodenotti I know that morphisms $\mathbb{P}^1 \to \mathbb{P}^1$ are functions of the form $f/g$ with $f$ and $g$ both homogeneous polynomials of the same degree and coprime
I'm trying to find out how many there are over $\mathbb{F}_q$
Of say, degree $n$
Now you can go full on combinatoric on this, but I was hoping someone would have solved this rather elegantly
Yeah my plan was: Write $n = a_1 + .. + a_k$, I know how many primes there are of any degree $a_i$ then look in which way you can combine these, such that $f$ and $g$ stay coprime
Because we saw a theorem saying that a finite dimensional algebra with identity over $\Bbb R$ is a division algebra iff it has no 0-divisors (and one direction holds regardless of the dimension)
In Milnor's book on Morse theory, he considers the space of paths in some manifold from $p$ to $q$. He topologizes this space in terms of a metric
Let $\rho$ be the metric on the manifold (induced by its Riemannian structure) and let $s(t)=\int_0^t \bigg\|\frac{d\gamma}{dt'}\bigg\| dt'$ be the arc length of $\gamma:I\to M$
He then remarks that the second term ensures that the *energy*
$$E(\gamma)=\int_0^1 \bigg\|\frac{d \gamma}{d t}\bigg\|^2 d t=\int_0^1 \bigg\| \frac{ds}{dt}\bigg\|^2 dt$$
is a continuous function
It seems sorta obvious that this should work, but I cannot seem to be able to get a bound on $E(\gamma)-E(\gamma')$ in terms of the second term of $d$.
In fact, if I set $\| ds/dt\|=f$ then I'm trying to bound $\int_0^1 f(t)^2-f'(t)^2$ in terms of $\Big(\int_0^1 (f(t)-f'(t))^2\Big)^{1/2}$. The most obvious thing to do seems to square the latter, but I cannot seem to make progress. If you just square it you can easily show that one cannot get a pointwise bound (higher powers don't seem to help either) and I don't believe that it's an integral-but-not-pointwise bound. Am I wrong?
Guys, when proving the product rule in one dimension, we add a clever zero, and show that $(f\cdot g)'=f'g+fg'$. However, I don't really know how I should do that in the more-dimensional case. Does anyone know where I can find a proof online?
Say $f,g\colon R^m\to R^n$ are differentiable at $\vec a\in\mathbb R^m$. That means that $$ \lim_{\vec h\to\vec 0}\frac{f(\vec a+\vec h)-f(\vec a)-Df(\vec a)(\vec h)}{\Vert \vec h\Vert}=\vec 0, $$ and the same for $g$. Now what I need to show is that $f\cdot g$ has the following derivative: $f(\vec a)Dg(\vec a)+Df(\vec a)f(a)$, so we need to show that $$ \lim_{\vec h\to\vec 0}\frac{f(\vec a+\vec h)g(\vec a+\vec h)-f(\vec a)g(\vec a)-f(\vec a)Dg(\vec a)(\vec h)-Df(\vec a)(\vec h)g(\vec a)}{\Vert \vec h\Vert}=\vec 0,
But I can't use the analogy of the one-dimensional case, because it doesn't hold that: $$ \lim_{\vec h\to\vec 0}\frac{f(\vec a+\vec h)-f(\vec a)}{\Vert \vec h\Vert}=Df(\vec a), $$ does it? We'd rather have that it equals $Df(\vec a)(\vec e_1+...+\vec e_n)$, right?
This is problem 2.2.8 from Durrett's Probability Theory and Examples 4th edition, I am using the version of this book that can be found on his website.
Let $p_k=\frac{1}{2^k k (k+1)}, \ k=1,2,\dots$ and $p_0=1-\sum_{k\geq 1}p_k.$
$$
\sum\limits_{k=1}^\infty 2^k p_k = \left(1-\frac{1}{2}\rig...
Let $m(n)=\min\{m:2^{-m}m^{-3/2}\leq n^{-1}\}$. Then $$
2^{m(n)}\sim\frac{n}{\log_{2}n}. $$
This problem is a piece of solving : An unfair "fair game."
Though, most of this is because a lot of classes are in Dutch. Rather than use Dutch books of lesser quality, professors makes Dutch lecture notes based on English books; while recommending those books as "extra material"
@SteamyRoot it's not an excuse, but I'm having some sleep problems lately, so "energy" can quite be an issue for me, so it means a lot for me to be able to stay in bed for one more hour, and not waste 4 hours for going to the test and taking the test:d
The Baker-Campbell-Hausdorff problem is to obtain $\log(e^{X}e^{Y})$ where $X,Y$ are appropriate operators. The Dynkin series $$\log(e^{tX}e^{tY})=t(X+Y)+\frac{t^2}{2}[X,Y]+o(t^3)$$ gives an expansion in powers of a perturbation parameter $t$, with higher coefficients expressed in terms of succes...
It used to be allowed at my university, but a lot of students would sign up for retakes and not show up; which puts a heavy burden on administration and staff needlessly
This is problem 2.2.8 from Durrett's Probability Theory and Examples 4th edition, I am using the version of this book that can be found on his website.
Let $p_k=\frac{1}{2^k k (k+1)}, \ k=1,2,\dots$ and $p_0=1-\sum_{k\geq 1}p_k.$
$$
\sum\limits_{k=1}^\infty 2^k p_k = \left(1-\frac{1}{2}\rig...
Let $m(n)=\min\{m:2^{-m}m^{-3/2}\leq n^{-1}\}$. Then $$
2^{m(n)}\sim\frac{n}{\log_{2}n}. $$
This problem is a piece of solving : An unfair "fair game."
