edison's lightbulb filament, PCR, i don't know. a few synthetic fibers in the vein of gore-tex were also pretty sui generis but not the commercially successful ones. a lot of pharma is just people trying all 50 billion possibilities and finding which one makes a fluorescent thing light up.
which is fine as a form of invention but not the romantic view. i want the struggling artist in the garret.
i had a vague sense that that had something to do with it. i like how it aligns everything with the horizontal. so many "smileys" from the old BBS days, and now, require you to turn your head. :-)
i used to use (: because i'm left handed. it drove people crazy.
i may have used it because it drove people crazy and me being left handed is just an excuse.
a little bit of me is the person who never sees a bull without wanting to wave something red in front of it. it's not out of malice, it's just an instinctual sense of humor thing.
i've read a number of profiles of comic actors who had this same pathology. they could never not be 'on' and it interfered with their ability to be real people. peter sellers and chevy chase are two examples.
i'm nowhere near as toxic as those people in other ways but i see the same thing in them that is in me.
this is something i am guaranteed to mess up because of the number of order-2 things happening. the schwarz reflection principle comes to mind but this is not that. are you familiar with the cauchy riemann equations?
yeah, wow, it may not even be defined there. this is why i was thinking schwarz reflection principle. if something is filling in potential gaps it would need to be known before the question can be answered.
for the sake of clarity: False: if $f$ is differentiable at $z_0$, then $\overline{f(\overline{z})}$ is differentiable at $z_0$ Correct: if $f$ is differentiable at $z_0$, then $\overline{f(\overline{z})}$ is differentiable at $\overline{z_0}$
the part about a continuous function that's holomorphic on both half-planes being entire pops up in the complex analysis proof of Fourier injectivity that I'm fond of
@Lucas I'm on my iPad so I do not want to type much. You can extend the module definition to arbitrarily many factors. What is your issue? In analysis/geometry, vector spaces (i.e., tangent spaces) are the relevant algebraic entities.
Write the contra positive of the following statement.
Suppose $x\in \mathbb R$, and we know: for all $\epsilon \in \mathbb
R$, if $\epsilon>0,$ then $x\leq \epsilon$. Then $x \leq 0.$
The contrapositive of the statement is If $x>0$. Then there exists $\epsilon \in \mathbb R$ such that $x>\epsi...
Hello, in modular exponentiation algorithm with factoring. Do you have any clue why $14^{105} \bmod 5 = (((14^3)^5)^7)$. Then we have $14^3\equiv 4\ mod\ 5$, then $4^5\equiv 4\ mod\ 5$?
@Avra Btw, modular exponentiation 'using factoring' is pretty much useless for all non-trivial computations. Because you can't factor. What you really need to do is to write the exponent as a sum of powers of two and use the 'square and multiply' algorithm.
In mathematics and computer programming, exponentiating by squaring is a general method for fast computation of large positive integer powers of a number, or more generally of an element of a semigroup, like a polynomial or a square matrix. Some variants are commonly referred to as square-and-multiply algorithms or binary exponentiation. These can be of quite general use, for example in modular arithmetic or powering of matrices. For semigroups for which additive notation is commonly used, like elliptic curves used in cryptography, this method is also referred to as double-and-add.
== Basic... ==
I think I still have no idea how to prove if $(v_1,v_2,v_3\in\Bbb{R}^3 )xv_1+yv_2+zy_3=0$ has infinitely many solution then the vectors in that linear system is linear combination of other two and that two are not collinear 😅
@dc3rd I would have undeleted the question and reprimanded the OP, but the answer was deleted by the author, so there was no answer when the question was deleted.
However, there were a lot of comments given, so it might be appropriate to undelete.
I believe that all the mods on this site will undelete and leave messages regarding this, but I don't think that anyone will be suspended for this. Perhaps if they repeat the behavior they might be.
i'm working it back in my mind. my wife's mother went there a few years ago, maybe got it from her. i sometimes work with a guy who is a professor there but nobody who works with me gives me mugs.
a canadian consular aquaintance was complaining that whe you have 10 irish people at your dinner party you have 10 independent conversations going on simultaneously
it too about 2 decades to find out why i was slapped. entirely non obvious.
there was a guy in our town who was a huge racist, and had taught his kids to torment his neighbors, and we thought it would be fun to fire some potatoes into his living room. nothing violent, just potatoes. and it was quite fun to do that.
oh i also got very close to quite a bit of trouble with a slingshot. it was the same guy actually, his son used to torment my sister so i tormented them back. ball bearings through the windows of his house.
nobody who knows me now would recognize the person who was doing this.
mine was 'borrowed' from a neighbor. we didn't have enclosed garages in our neighborhood, we had 'car ports.' and i knew where they kept it and i borrowed it, used it, returned it, with nobody the wiser.
borrowing without asking is just fine as long as nobody needs the thing while you've borrowed it. this is a moral instruction i intend to impress upon my daughter.
sometimes i tell my wife the stuff i got up to when i was a kid and she's horrified by it.
it's also different being a girl vs. a guy. people were always testing me physically and i developed tools to deal with that. growing up as a girl is more of a psychological experience.
the syntax leaves something to be desired, but you could spend days with kaplansky's formula. martin argerami provides really good answers in operator theory. i think i met him once but i don't remember.
there are a few people who comment on my answers, and i comment on their answers, and i'm pretty sure i know them from somewhere.
but it's all smoke and mirrors.
actually i did meet him at a conference. i remember now. he worked with several people i worked with. very sharp.
there's someone else who sometimes snipes on operator theory questions before i can get to them, and i know him because he's another student of my advisor.
@SrijanM.T That is what $\theta$ is, just taken at a point further out. The whole point of taking the limit is that the triangle gets smaller and smaller, getting even closer than where you drew $\alpha$.
@SrijanM.T in your diagram, with the straight line, they are the same. In a more general picture, where the line is replaced by a curve, they might not be the same.
You should really try to use MathJax, it would keep everyone here from having to squint at the images, and even make it possible to use pieces of your formulas.