so... I've been at this for an hour: what would you guys use to show that the integral over a size 1 interval [a,b] of a monotone function is between the values f(a) and f(b)?
do I have no choice but to work it out from the Riemann integral definition?
seems like this is quicker with the Darboux integral
@TedShifrin right what was bothering me is that I was thinking i'd need to show additional partitions only get closer to the integral, or at least never vary beyond f(a) or f(b)
well that's one thing i'll never be. i get GOP fundraising emails because i contributed to a republican city council campaign. the guy is not remotely related to the GOP but suddenly i got on a bunch of mailing lists.
i contributed to his campaign because he was a spouse of a friend of mine and when someone told him to say anti-X stuff, where X is a common target of GOP rhetoric, as part of his campaign he told them to f- right off.
i'd vote for him a million times. i don't see him running in a national race.
i don't know why we associate things with names. i always ran into this in functional analysis, people would ask about so-and-so's theorem and i'd say what are you actually talking about.
my advisor had a result and i learned while studying for my qualification exam that most textbooks included his name in their name for the formulation of the result, but he didn't, so i didn't. and don't.
i'll add you to the list of people that academics are not carbon copies of.
i think there are some people who would be mathematicians even if the job category did not exist, and there are other people who would be [random academic X] even if mathematics did not exist.
there's a value judgment implicit in that separation but i won't make it explicit or name names.
ah here we go: since adding an additional section to the partition of the upper sum implies breaking up a section into two, the resulting two sections will be in total smaller than the initial one by virtue of taking an upper sum
i'm not going to click through but that's a very common misconception, perpetuated largely by ed schools. that there are 'types' of learners and some will be ill suited to types of education. rudin is an extreme example but most people can learn from most things.
there's a lot of right-brain, left-brain pseudoscience in ed schools too.
someone told me in middle school that i was a 'graphological' learner meaning that i had to write whatever it was i was supposed to learn. ed school horseshit.
a friend of mine and i used to pick up acorns that butterflies had laid eggs in and 'farm' them a little bit by storing them under the steps of the school. we also threw chestnuts at people.
for some reason i was paranoid about dying in a car accident in elementary school, and i was almost over my fear and then that friend was run over by a car and killed.
the still image of someone's feet out of a kiddie pool.
god, that's funny.
my wife understood me better when she first met my mom and her first story was my mom (who was a nurse) talking about the person and then the person dying unexpectedly and my mom just laughing her ass off about it.
my mom insists that she's seen ghosts. she did wake up the minute her mother died and bother all of us about it before we'd known her mother had actually died.
i try not to think too much about it
we told her to calm down and call her sister and when she got through to her sister she said "our mom's just died." which didn't calm anybody down.
i've paused the youtube video on the still of someone's feet out of a kiddie pool. that's incredibly funny.
:-). my mum was scared of the water. even though we lived beside the 2nd largest natural harbour in the world (with some irrelevant qualification) my mum forbade us (twice today i've used this word) from sailing, etc. (not that it stopped us.)
the following day i was near the top but there were storms coming from both sides (which seems impossible, but i was not going to argue with 0 vis) and then lightning nearby which was truly truly awesome.
i missed the top that time, but it was still incredible
so it turns out $<\mathbb{N},x |\rightarrow x^3>$ and $<\mathbb{N},x |\rightarrow x^2>$ really are elementarily equivalent, in fact they're even isomorphic, send $0\rightarrow 0$ and $1 \rightarrow 1$, fix a bijection $b $ between all nonsquares and noncubes, and then there is a unique isomorphism that restricts to $b$ and sends $0$ and $1$ to themselves
looks OK to me and i think it's also true. sometimes people doing elementary number theory avoid explicit reference to the square root function, but that is a style thing.
interesting question. it reminds me of how holders of some jobs retain their titles after quitting. i think a former president is still mr. president.
i do not identify as a mathematician although i did at one point. many mathematicians did not have advanced degrees or necessarily do 'high level' mathematics. you are what you do.
For example, why when integrating $\int\sqrt{x^2+1}$, the substitution $x=\cosh(u)$ works easily but the other parametrization by $x=\frac{2u}{1-u^2}$ is hard(you have to do partial fractions on a denominator of degree 6)?
partial fraction decomposition is a classical example of something that i do not think should be done by hand. it's linear algebra with often 4+ unknowns. let a calculator or computer do that.
The Free Dictionary says "a person skilled or learned in mathematics". So it depends on your standards. I'd say a child doing algebra is quite skilled for their age (of course depending on said age and what algebra they're doing).
i only bristle when the mainstream media reports on the doings of people who are clearly credentialed in other fields and ignorant of mathematics as 'mathematicians' because there is an equation somewhere.
i just watched my cat stalk my wife and did not say anything when the cat ran up and lightly attacked my wife by swatting her legs and biting her without breaking the skin. i didn't offer any warning. my wife was not pleased. what was i going to say? she knows she's in a cat's territory.
@TedShifrin kittens are for climbing pant legs as you're walking and hanging on while you make breakfast. They are also for attacking anything that moves under the covers in bed.
I just learned of "Halley's method" and "Newton's method" for finding the inverse of a function by inverting a function. Can I use a similar approach for $\frac{x}{y} = z$ given $z$ and either $x$ or $y$?
"To compute the natural logarithm with many digits of precision, the Taylor series approach is not efficient since the convergence is slow. Especially if $x$ is near $1$" What?!
I'm not talking about using Newton's method to multiply two numbers. I'm talking about finding a quotient using a product. I'm just asking how I would invert this using the method shown here, or if there is something better, what is it if I don't have the ability to compute $1/x$?
Well the question I had yesterday regarding sums of reciprocals is related to this. I learned that it isn't useful except in the case where the product of the denominators is a perfect power of two unless I allow extending to rational exponents of two.
I'm still continuing my search for an efficient implementation of integer division.
I'm just throwing ideas out there more or less for myself, and looking for possible leads.
The current idea which is the best that I have so far would use the identity $\frac{x}{y} = 2^{\log_2(x)-\log_2(y)}$
FDIV is 15c which is obviously much faster than 41c, but for my applications, that won't do in terms of accuracy since it works based on approximations using purely sums of reciprocal powers of two. My encoding has a reciprocal bit with fixed point to allow representing fractions with infinite decimal expansions exactly.
For example, instead of trying to approximate 2/3 in fixed point for 8-bit exponent and mantissa, I can instead represent it in terms of its reciprocal, 3/2 which yields the terminating decimal expansion 1.5, and then I can set the reciprocal bit to say "interpret this fixed-point number as 1/1.5".
Yes, I'm trying to do fast divides.
@AMDG Also with regards to speed. It should be on par with hardware integer multiply.
this is getting into specialist domain, so i will bow out. i am still not sure what you are trying to do. you want fast exact integer divides in finite precision??
this seems less mathematical and more dependent on hardware software implementation considerations. although maybe there is some bright idea from math that gets at this issue.
@copper.hat Hah, yes, well, if it were implemented in hardware, it would fit in an ALU, and it could easily be reconfigured for a multiply mode.
That's because it involves taking a constant perfect power of two and dividing it by an arbitrary integer, then computing the sum. It would be fast to implement except that you need to know how many bits to shift, and to compute that, you need lzcnt or tzcnt, and that isn't available in SSE, MMX, etc.
I mean it isn't like this is so far away that it is beyond grasp. It is more or less certain. If I knew of a way to just quickly divide a 16-bit integer in one cycle, then I'd be content.
It's not exactly the RH... it's just integer divide lol
I can use a simple, generic algorithm that knows how to compute the quotient of two simple integers in plain binary format and adapt it to my encoding. The problem is knowing about them.
But nah, I like a challenge. These are challenges that I like and I want to solve them. Unfortunately, with my limited knowledge, it appears that I cannot get what I want right now until I make my own hardware to do so. Still, I could use some insights, though...
That's mostly why I'm here. I'm hoping I might hear about something that I can use based on existing info or algorithms that I'm not aware of.
