Home, home again I like to be here when I can And when I come home cold and tired Its good to warm my bones beside the fire Far away across the field The tolling of the iron bell Calls the faithful to their knees To hear the softly spoken magic spells.
I have a somewhat insane question. How can one show that a finite set of numbers contains a maximum? (Let's say that the numbers are reals for concreteness.)
I guess induction on the size of the set works.
And also points up that my statement was not quite correct.
It seems a lot less insane now that I have a proof.
I was gonna prove it by producing an algorithm that finds the maximum, but then I couldn't prove that the algorithm worked unless the set contained a maximum element.
For finite sets with no maximum, the obvious algorithm might fail.
@tb I answered a question too quickly this morning, and leo sensed something was missing, and I had inadvertently assumed that a function was continuous. It took a while to fix the answer keeping the same idea. I think it is okay now.
You see, now I take the Gaussian measure, and then I just use the $e^{-x^2}$ in my estimates, but I would prefer it if it does not rely on that change of variables that much.
But more on just the measure.
But maybe that is nitpicking... But I think it would be a nicer result as it is easier to generalize.
@robjohn I heavily rely on the fact that my operator to define the maximal functions has an integral kernel for its semigroup... Anyway, you should check out that stuff, it is very interesting and gives you googolplex equivalent definitions for our Hardy space. You can take many cool elliptic operators.
@Gigili It's not worth getting upset about such things. He has 130k by now partly because he has been around a long time and partly because his answers are of extremely high quality.
But 130K reputation points and $1.69 will buy you a cup of coffee at Wawa.
@tb Yes I did. I even read til Nakayama's lemma today. I think if you hadn't "saved" me I'd've gone home and not looked at any more of it for at least a day.
@robjohn nice and clean, according to me. What does the TeX-command at the beginning achieve? (you can avoid that ugly space before Break by moving the command at the end of the first line, I guess). Two tiny typos: partiion $P$, $(11)$ says (right before the first equation $(12)$) which leads me to the second typo: two equations $(12)$.
@anon Ha no, I just name the files like that to remember who I'm supposed to give it to. Not that I'm 10 seconds Tim of something, but it has become a "tradition".
I guess the only downside to plain white pages is my tendency to write "upstream".
I'll go study some epistemology stuff for tomorrow. Tag me if rob comes back, and we can all maybe talk about other GFs that can be retrieved with the beta integral.
First some notation: Let $P$ be a prime ideal of a $\Bbb{Z}$ - graded ring $R$, $U$ the multiplicative subset of all homogeneous elements not in $P$. Suppose that there exists a homogeneous element $f$ of degree $1$ that is not in $P$.
The problem in Eisenbud is to show that the image of $P$ (...
i have a question regarding the wiki entry on curve orientation - not sure it's worth an actual question on here
i know that to test whether three points in the plane are taking a left turn or a right turn, you look at the sign of the determinant of a matrix formed by the three points
however, some things seemed unclear to me from the entry @ wikipedia: en.wikipedia.org/wiki/…
specifically, in this paragraph: " In computations, the sign of the smaller angle formed by a pair of vectors is typically determined by the sign of the cross product of the vectors. The latter one may be calculated as the sign of the determinant of their orientation matrix. In the particular case when the two vectors are defined by two line segments with common endpoint, such as the sides BA and BC of the angle ABC in our example, the orientation matrix may be defined as follows:"
the sign of the smaller angle? i'm not exactly sure what they're saying?
But maybe it is better if you hit a continuous compactly supported function with Whitney? So we can be sure it is integrable as we have this theorem that continuous functions are measurable and with compact support we get a finite $\sup$ norm on a compact set.
First some notation: Let $P$ be a prime ideal of a $\Bbb{Z}$ - graded ring $R$, $U$ the multiplicative subset of all homogeneous elements not in $P$. Suppose that there exists a homogeneous element $f$ of degree $1$ that is not in $P$.
The problem in Eisenbud is to show that the image of $P$ (...
