Here is an image. My question is simple: what is this? (As a side note, I did not invent this, and it was not intended by its creator to be a puzzle.)
PS: If you believe you have solved it, see if you can figure out this second image.
Is it correct to say that $\inf\{x^2-2y^2 \ | \ 0\le x \le y\}=-\infty$ because the limit $\lim_{\|x\| \to \infty} (x^2-2y^2)$, along the curve $(x,x)$ with $x \ge 0$ with $\|(x,x)\|=\sqrt{x^2+x^2}=2|x|=2x \to \infty$ when $x \to \infty$, is such that $\lim_{x \to \infty} (x^2-2x^2)=-\infty?$
@Koro In a ED h Euclidean algorithm works, just as in $\Bbb Z$. But be your usual stubborn self and ignore what Leslie and I suggested. Since you clearly did.
@PurpleHaze I got stuck at step trying to prove that lcm(a,b) in a Euclidean domain is ab/(a,b). I described my question above.
@TedShifrin I didn’t understand how Bezout’s will be useful here. I mentioned that above. Writing ab/(a,b) as ab/(x a+yb) and then what? I got stuck. To your comment: the statement in case (a,b)=1 seemed as strong as what I am trying to prove.
@leslietownes if you had to teach a lecture on something cool from analysis to a bunch of students who are only starting to learn how to prove things in math, what would you show?
Like something that has a low bar for understanding.
proving that log(2) is irrational is pretty easy (base ten log). you can use that to prove that there is a power of 2 whose decimal expansion begins with any given sequence of decimal digits.
which is nonobvious even in the one-digit case, e.g. students know powers of 2 beginning with 2, 3, 1, 5, but maybe not 7 or 9.
i'm not sure if that counts as analysis.
you need to do some work with the real logarithm and manipulate a few inequalities. it's a density argument.
if you do the density argument and pay attention to what you're pigeonholing i guess you can work on the kind of size of the N you might need to be sure that 2^something less than N begins with your phone number.
lots of silyl analysis proofs become slightly less silly if you try to quantify the bounds on the things you prove the existence of, but this might be bad to do for beginning students. because it will fool them into looking for the best delta for the epsilon, etc.
axler's linear algebra has that good problem, he uses the L^2 inner product on [-pi, pi] to come up with low degree polynomials whose graphs look a lot better than the taylor polynomial at 0. i say it's a "good" problem because it looks cool and expands the mind a bit. it's a horrible problem because the calculations, if done by hand, are so bad that even he doesn't include them in the book.
leslie, in electronic version of the book, you can zoom into that graph and you’ll see two graphs -one plotted using Taylor’s and the other using Gram Schmidt’s. ;)
i remember in my print edition the graphs were indistinguishable, except at the very end. that's another reason it's not that great of an exercise. :D
when i taught engineering linear algebra once (the book assumed the use of computers and the students had experience with them) i had them do it, but made sure they set the plotting window so they could really see that the 5th order polynomial was, in fact, a polynomial.
you don't. i meant informally. if you do the [-pi, pi] window you might not see the difference, but if you pull back to e.g. [-2pi, 2pi] you see it bending away and adopting a more classic quintic shape.
it's obvious from the formula that the polynomial is a polynomial, but not obvious from the graph if you zoom in like axler did.
oh i meant school work......so they are one in the same......learned a lot and it was pleasurable to retain the ideas and see how to use them....but damn it was gruelling
If you have: (fraction a × fraction b × fraction c)÷prime number d. Suppose that prime number d is bigger than fraction a,b and c. Does that mean that prime d does not divide (fraction a × fraction b × fraction c)
I tried to use third isomorphism theorem. (2) is an ideal of Z[x,y] and that of (2,x,y). So Z[x,y]/(2) /(2,x,y)/(2) is isomorphic to Z[x,y]/(2,x,y).
Z[x,y]/(2) is isomorphic to $Z_2[x,y]$ (using proposition 2, chapter 9.1 from Dummit and Foote)
But I don’t know how to handle (2,x,y)/(2)
@love_sodam yes, I know that. But I have problem mostly when I is generated by more than 1 element.
Can anyone please give me some advice on this? Thanks.
There used to be a website called searchonmath wherein one could type in latex and see the relevant posts. But something happened to it. Approach0 is just horrible and doesn’t work properly.
How does one generalize a discrete function in math? What method, for example, was used to find the Gamma function?
I am analyzing the XOR function and looking for ways to describe the variation between any set of numbers discretely but uniquely.
XOR gives you the variance between two bit strings, but what about four? It is then the variance of a variance as $(a\veebar b) \veebar (c \veebar d)$.
More formally, given an arbitrary number of bit strings, I'd like to find the position of a continuous sequence of bits common to all of them at the same position and number of bits, and for which the value of this sequence for each bit string is unique.
If $n$ is the number of strings, then intuitively, the minimum length of a sequence is $\lceil\log_2(n)\rceil$ bits, and the maximum is the length of the string with the longest length in its representation (excluding leading zeroes).
can someone pls help me understanding this measure $\mu^{*}(E)=\limsup _{n \rightarrow \infty} \frac{\#\{E \cap\{1,2, \ldots, n\}\}}{n}$, like, how to calculate some examples for some $E \in 2^{\mathbb{N}}$
@AlekMurt: that is not a measure for it is not countably additive. It is one expletives of a Banach limit. It happens to be an example of a charge ( a nonnegative finitely additive function on the power set of $mathbb{N}$) the extension to the power set depends in the Hahn-Banach theorem.
@leslietownes: in many European schools they use $\mathbb{N}$ to denote $\{0,1,2,3,\ldots\}$ In my side of the Atlantic, my teachers instilled in me (with a big wooden ruler to slap you on the hands) the use $\mathbb{Z}_)+$ for $\{0,1,\ldots,\}$.
@AlekMurt: just don't call it measure, not even a a joke ;) charges get quite upset when they are called measures, when they are even much more than merely measures, that are much more exotic.
Definition: Let $\mathscr{F}$ be a sheaf on a topological space $X$, and let $s \in \mathscr{F}(U)$ be a section over an open subset $U$. The support of $s$ is defined to be $\{p \in U : s_p \}$, where $s_p$ denotes the germ of $s$ in the stalk $\mathscr{F}_p$.
Question: What exactly is the germ $s_p$? I don't understand this term.
You didn’t finish the definition of support. The sheaf is a sheaf of germs of functions, we presume. Your course/book should have given a definition, no?
My books defines $\mathcal{F}_p$ as consisting of pairs $\langle U,s \rangle$ where $U$ is an open neighborhood of some point $p \in X$ and $s \in \mathscr{F}(U)$ is some group element. Two pairs $\langle U,s \rangle$ and $\langle V,t \rangle$ are declared equivalent iff there exists an open nbhd $W$ of $p$ in $U \cap V$ such that $s|_W = t|_W$.
So, if $s_p$ is supposed to be an element of $\mathscr{F}_p$, doesn't this suppress some notation, namely the open set it's paired with?
$\mathscr{F}$ is any sheaf on some topological space $X$.
Sorry, I didn't realize there were sheaves on different objects. Yes, we're looking at a sheaf of abelian groups on $X$. I defined $F_p$ above. It's equivalence classes of pairs $\langle U, s \rangle$.
Oh, so are you saying that $s_p = \langle U,s \rangle$?
Here is an image. My question is simple: what is this? (As a side note, I did not invent this, and it was not intended by its creator to be a puzzle.)
PS: If you believe you have solved it, see if you can figure out this second image.
