Question: If $x \ge 0$ and $y \ge 0$, then the area bounded by the graph of $ \lfloor x \rfloor + \lfloor y \rfloor =2 $ with the $x$ and $y$-axis is?
Answer provided: $3$ units$^2$.
My doubt: How do we graph this relation in order to find the area?
I am not sure, but since we can always deform shapes and there are subsets of deformations that can deform it in an area preserving manner, it might be possible to somehow mold it into a square
I know how to do do that with clay, but how we can do that mathematically seemed either an interesting question, or had already been explored and I am not aware about
$$\int \frac{e^{-stuff}}{\text{very fat denominator}}dx$$ tend to for some reason give me an impression it should look like a spike. Guess I have been staring at $\frac{\sin x}{x}$ for too long
@AkivaWeinberger here is an interesting thing to try and prove. If a and b are integers show that there exist unique q and r such that a = qb and r and r has less prime factors counting multiplicity then b.
it is parallel to the polynomial division theorem which uses order to sorta count factors
@Typhon Assuming you did mean $a=qb+r$, that seems to be false. Take $a=19$ and $b=6$. Then $19=(2)6+7$ and $19=(1)6+13$, and yet $13$ and $7$ both have fewer factors than $6$
@TedShifrin with great pleasure! I am already wondered of the great number of images (I think I have had none during my study (except the graphc theory))!
@TedShifrin btw, if I may: in the Spectral Theorem you say that $T: \mathbb{R}^n \to \mathbb{R}^n$ must be a symmetric linear map. In my lecture notes I see that $T$ has to be normal. Are these conditions equivalent?
@TedShifrin I have read another mystic statement, for that I need time though: "if $L \subseteq V$ is the span of all eigenvectors to all eigenvalues of a normal map $\varphi: V \to V$, then $\varphi(L^{\perp}) \subseteq L^{\perp}$ and $L^{\perp}$ contains none of the eigenvectors of $\varphi$.
@TedShifrin sure, I mean the the sum of eigenspaces is direct, and I was told about it two months before I explored what the normality is
@TedShifrin I mean that I have read "if ... diagonalizable, then the sum of the eigenspaces is direct" a long time before "... is normal, if". No, finite-dimensional. "He" wants to show that every normal map is diagonalizable and needs this lemma for that purpose.
You can always take a direct sum of eigenspaces, @Kirill. The point is that the operator is diagonalizable if and only if that direct sum is the entire vector space.
the usual proof is in two steps: First to show that monotone functions are differentiable almost everywhere, and then to show that functions that are locally of bounded variation are everywhere the difference of monotone functions
and it's pretty easy to show that Lipschitz functions are locally of bounded variation
Terry Tao has a blog post about differentiation theorems that introduces the ideas behind the proof that monotone functions are diff-a.e., it's also in like lots of introductory real analysis texts that use measure theory
for Rademacher one can also use the characterization of the dual of Hilbert spaces + Lebesgue differentiation theorem to give a quick proof that Lipschitz implies diff-a.e. but this is maybe hitting the problem with a bigger stick than it needs
Of course, it depends on which slice you take. I can't find my original Mathematica file with which I did that. It must have been on my office computer ... long gone.
The way I did it was to write $w^3=z=re^{i\theta}\implies w=\sqrt[3]{r}e^{i\theta/3}$, and then plot the parametric surface $(r\cos\theta,r\sin \theta,\sqrt[3]{r}\sin (\theta/3))$ for $\theta\in [0,6\pi)$
@Kirill if $X, Y$ are vector spaces and $f:X \to Y$ is a linear map between them then the adjoint $f^{\ast}$ is a linear map from $Y^{\ast} \to X^{\ast}$
@Semiclassic: I know our graduate secretary wanted to use a similar image for the cover of the graduate handbook, and so I had to recreate that and do a bunch of fiddling with lighting/colors.
Let $X,Y$ be random vectors such that $X, AX+Y$ are independent and normal with mean zero and covariances $\Sigma_1, \Sigma_2$. Can I conclude that $(Y|X=0) \sim \mathscr N (0, \Sigma_2)$?
$X$ is a random variable, whose values are vectors of real numbers. The same is true for $Y$. I guess if it's true for scalar random variables I will be able to adapt the proof to this case that I have, so it can be ignored. $A$ is just some constant matrix.
Any chance someone here familiar with computability theory? In particular im trying to prove that $L = \{<M_1, \dots , M_n> : \cap M_i$ does not contains words of length smaller then $n \} \in CoRE - R$.
@Kirill Cool! I think the probability space is implied by the use of normal random variables, as they carry with them the assumption of working with an appropriate power of $\Bbb R$.
@Emolga probability spaces? I think I have to deal with them in the next year
oh sure, measures are going to come in October.
we have had Moore-Penrose inverse today. Looking at the Wiki-page in English I meet $M(m,n,K)$ notation. What is that? A $m \times n$ matrix with entires in $K$?
or the set of all that matrices? Or something else?
@robjohn I have one already added to my project. It's so enjoyable! However, the difficulty level is a bit crazy.
:-)
@robjohn These days I'm involved in research in more directions, I investigate some problems that apparently are new in the mathematical literature (that kind of problems I use to play with). Some are already included there.
@robjohn I'm very fascinated these days by a class of alternating series, it's actually about a sum of series which when combined together produce very impressive closed-forms.
@robjohn Yes.
@robjohn I'm NOT able yet to treat those series by elementary means, but I suppose that some fascinating cancellations happen.
At the moment I cannot even imagine how that happens, perhaps I will need to ponder over those problems for some weeks, months, or even more.
@robjohn In general we are used to the series solutions since they are far more elegant than calculating complicated integrals. I don't see why here things won't be the same.
This is a pretty cool double integral, known in the mathematical literature, of course.
@robjohn I didn't post (in chat) problems for years ... :-)
Wait, I think I posted some time ago an elementary integral with logarithm, maybe some weeks ago, but that one was more for fun, not really a serious problem.
Hey @Eric, sorry I was out for a while, and yeah functional is some good stuff
I probably should've done what David did in my lecture and just black box some of the results, he actually got through most of what he wanted mainly because of that
Just curious, for any given polynomial function $f:\Bbb N\to\Bbb Q$, is the function $g:\Bbb N\to\Bbb Q$ defined by $g(n)=\sum\limits_{k=1}^nf(k)$ also a polynomial function?
Question: If $x \ge 0$ and $y \ge 0$, then the area bounded by the graph of $ \lfloor x \rfloor + \lfloor y \rfloor =2 $ with the $x$ and $y$-axis is?
Answer provided: $3$ units$^2$.
My doubt: How do we graph this relation in order to find the area?
