Any interesting uniform funnel surfaces that I might try implementing as a weight?

I know that the idea is to number bottles, say we have 1000, from 0 to 999. If we have only 10 testers, then we would need 10 binary digits to represent testers.

If we label the 10 prisonsers as J, 1, H, G, F, E, D, C, B, A, then why if the following prisoners dies: J, H, F, D, and B, we know that the bottle that was poisonous is 341 as 0101010101 given that first prisoner drinks from every bottle $2^0$, second B drink from every 2 and skips the other 2 bottles $2^1$, C prisoner drinks from every 4 and skips 4 $2^2=4$ and so on with the the pattern for the remaining prisoners

Question: The King of a small country invites 1000 senators to his annual party. As a tradition, each senator brings the King a bottle of wine. Soon after, the Queen discovers that one of the senators is trying to assassinate the King by giving him a bottle of poisoned wine. Unfortunately, they do not know which senator, nor which bottle of wine is poisoned, and the poison is completely indiscernible. However, the King has 10 prisoners he plans to execute. He decides to use them as taste testers to determine which bottle of wine contains the poison. The poison when taken has no effect on the prisoner until exactly 24 hours later when the infected prisoner suddenly dies. The King needs to determine which bottle of wine is poisoned by tomorrow so that the festivities can continue as planned. Hence he only has time for one round of testing. How can the King administer the wine to the prisoners to ensure that 24 hours from now he is guaranteed to have found the poisoned wine bottle?

he just makes the $i$'th tester taste the numbers that have the $i$'th bit set to $1$ in binary.

thus he can know whether the $i$'th bit is $0$ or $1$ for the poisoned bottle

and this allows him to determine the entire number

like a chad.

@o.9. I understand direct approach of using binary numbers to represent bottles much better and then infer the testers

great

Thanks

my pleasure

I should definitely use Gaussian process interpolation to define arbitrary gradients for realtime applications.

Why?

You want Gaussian process?

I was joking.

It's too expensive.

Haha! Weird! I thought you was for real

That's why I crossed it out

Gaussian functions are used to sample data randomly to initialize many applications

Probably can use used to draw arbitrary gradients

Nothing wrong I guess, but let us see what math experts would say if they saw this :/

For real, though, I'm looking for a function to make a nice fall-off. I'm designing a theme so it's subjective, but having a good generalized function for making fall-offs that clamp between 0 and 1 would be nice.

Also, en.wikipedia.org/wiki/Gaussian_process

I heard about cutoff not fall-offs

What do you mean by fall-offs please?

Gradient with a value that tends towards zero or negative infinity.

Gradient here not referring formally to slope.

Why you don't sample them from chi-square distribution of from any other distribution?

gradient approaches 0 as you go to tails

In terms of the surface, I would define fall-off as the derivative of that surface with respect to several variables or a curve with respect to x where the vector tangent at a point on the surface tends towards negative infinity as we approach negative infinity.

does anyone know when the Kanye album is dropping?

Make update step very small though to guarantee your gradient does not jump

@Avra I am not familiar with chi-square distribution.

@o.9. Hm! For real?

it's supposed to come out this month I think

Kendrick is also supposed to drop an album this year

:0

This here is what I've been working on. This here uses $\frac{3}{2}-\frac{1}{2x^{2}}$ as a weight bias: cdn.discordapp.com/attachments/862169891175137310/…

If the link doesn't work, let me know.

looks like heresy ngl

Breh

@AMDG. No idea

And this one uses logarithmic weighting aka the funnel shape found at Wolfram MathWorld: cdn.discordapp.com/attachments/862169891175137310/…

I personally think this one looks the nicest.

mathworld.wolfram.com/Funnel.html

Are you trying to model something ?

Ever saw $\Theta(m) = \omega(\log{m})$ please؟

What function is that exactly?

Also, define model

I'm designing a material in UE4

I am not sure why you go through all details for materials? Materials techniques are heavily covered in compurer graphics.

For material design, color theory discuss this

Not image processing as I remember

Those are useless for analytical solutions to every day CG problems.

If I wanted to go cheaper, then I definitely wouldn't be inclined to use an exp(x) in my HLSL shader here.

@robjohn. Ever saw Θ(m)=ω(logm) please?

I'm still mostly just looking for a nice curve for which I can define a range of distances to the origin in $a \leq x \leq b$ for $0 \leq a \leq b$ and $a \leq b \leq 1$.

$\omega$ is little-omega function while $\Theta$ is big-theta growth function

Such that it shows off this nice gradient extending from the edge towards the center.

@AMDGM. You want it to be narrow around center?

what are $\Theta$ and $\omega$?

I think it's greek

I know that, what do they mean?

just letters by themselves

although I think they're used as variables sometimes

@robjohn. It's big-theta growht function

and little-omega

they are growth functions for complexity analysisi

@Avra What I have in mind is something that quickly drops off to negative infinity as we approach zero from $\pm 1$.

@AMDG. You want it in 2D or 3D?

It can be either $\mathbb{R}^2$ or $\mathbb{R}^3$ so long as it has uniform curvature if I understand what uniform means here properly.

@Avra Yes, but rather than making someone have to look them up. you could say what they are.

I know what they are now

Like the shape should have the y axis as its line of symmetry such that if extended to three dimensions, any rotation about the y axis is also another symmetrical image that appears the same as any other rotation.

@robjohn. Pardon me. I will keep that in mind.

what that means is that $\lim\limits_{m\to\infty}\frac{m}{\log(m)}=\infty$

but one does not often use a growth function on the left side of an equation.

on the right side, it usually implies existence, on the left, it implies universality

That says that every function in $\Theta(m)$ is in $\omega(\log(m))$, which is just what I wrote.

@AMDG. Why you don't draw $1/x$ in 3D?

I forgot that shape, but I remember it's similar to $1/x$

The first image I sent was using $\frac{3}{2}-\frac{1}{2x^{2}}$.

this will be very narrow and will grow steadily around 0 because you have square

It isn't quite as malleable as I would desire in terms of implementation and avoiding branching unless you can "expand" the asymptote to be an infinitely tall rectangular region.

Piecewise functions are the bane of GPUs.

Also, most of the reason for why I can get away with using something like exp for this material is because it's the base and is incredibly simple.

@robjohn. Thanks. I will double check

Anyways, it's hard to keep my eyes open now so I'm going to go to sleep. I'll figure something out eventually, but if anyone has any ideas, please let me know. Good night!

Good night

@Avra what are you double checking?

sleep well

Hello. I wanted to know if Cov$(X,Y) \ne 0 $ does this imply Cov$(X' , Y') \ne 0 $. How should I go about proving this?

Is there some relationship between $X,Y$ and $X',Y'$ perhaps?

where's grandad?

to repeat copper's comment. what relation does X' bear to X? i am not familiar with the notation.

granda's grandson is packing bags for a 2 wk trip to his land of birth

sounds like fun. i wouldn't want to be in the air these days, but if i had to go somewhere, ireland is high on the list.

complicated trip, as always. all these covid requirements don't make it straighforward

my daughter will fly in from london for a week, so that will be fun

she was on a big electronic billboard in ny time's sq recently

my son is confined to room as we await the results of a pcr test

i'm trying to decide if i want to lug Ted's Abstract Analysis along for light reading.

it pains me to say it, but ted's books are really good.

:-). a scan suggests that it is at the balance of expository density i enjoy.

my car rental costs more than my flight

that's how things are going these days.

normally i borrow a sibs car, but all are busy with summer stuff unfortunately

we're trying to figure out some way of getting my daughter to see her grandparents. it's tough with covid, and my mom is immunocompromised, and the kid can't be vaccinated. i want her to have some memories of her grandparents.

apparently many rental orgs got rid of inventory due to anticipated drop in load

that's a tough one

i have vivid memories of only one grandparent, my dad's mom. vague memories of phone calls with my dad's dad. we were never good with keeping in touch.

my mom saw her 5 yo grandson during the overlap, thankfully nothing happened

we (sibs) all agree that the positive value of a visit for my mom outweighed the risks. my mom would have agreed.

later the nursing home clamped down.

i skipped fifth grade and my grandfather called me out of the blue. he had been left back a year and congratulated me on making up for it. that's one of my only memories of him.

he also cheated at cards.

i only really knew my grandmothers.

two more different people you could not meet.

my dad's mother could give you a guilt trip like you wouldn't believe. she had a PhD in guilt trips.

both tough & gentle in different ways.

same with my da's mum. emotional blackmail flowed like rain from her

my mom's mother i met only once, she paid me $20 to find her watch. it was the beginning of alzheimers. she told me to my face that i talked too much.

i more or less ignored what mine said about me

i didn't mind it. people have not had a lot of luck with shutting me up.

60s left in my work backup.

she did say, decades before it became a reality, that i should be an attorney.

prescient lady

her father was an attorney. i joke about him. often people in the law have relatives in the law so bring that up. he argued a case in front of the supreme court. [lower volume] of new hampshire.

awesome. helps to have it in the genes

i suspect my brother is a better dr b/c of my mom

and i am an imposter engineer

i never knew him, he died in something like 1945. and he lost his case.

i tell my kids not to be afraid to fail.

be afraid of being afraid

it was a favor for a friend. he had an insurance policy on things 'kept in a barn' and the argument was that a horse at pasture was kept in the barn because he would go there in the winter. the court wasn't having it. property insurance isn't horse life insurance.

they do not listen, so i have to fail on my own

curious argument :-)

the horse perished in a fire that was started by a lightning strike about 1/4 a mile from the barn. he was 'kept in the barn,' yeah right.

:-) good try :-)

good night, i'm going to have an early night!

goodnight.

Good evening people

How many of you still remember that angular velocity omega?

My teacher used that during graphical analysis of ac currents

So voltage V(t) at any time t is given by V(t)= V Sin(wt) where V is the max voltage/amplitude

@AdilMohammed Did you have a question?

For golden ratio $\varphi$, the equality holds $\varphi^2 = (\varphi+1)$.

Golden ration $\varphi = \frac{1+\sqrt{5}}{2}$

Got answer, so I won't proceed any further.

@robjohn I think I forgot to click send, but do we learn about angular frequency in maths?

In graphs

Any idea why in the sum $\mid {\sum_{i=0}^{d}(a_n n^{i-d})} \mid < 0.5 a_d $, where $d$ is an integer?

Similarly, why in the same sum is larger than $-(0.5 a_d)$?

The Golden Ration (tm). Highly revered by soldiers in the war.

Gives +15 atk and +35 def

@robjohn any thoughts..?

Given that $lim (\sum_{i=0}^{d-1}(a_n n^{i-d}) \to 0 ~as~ n \to \infty$

@Rover any $n\not\equiv0\pmod3$

sorry, I thought you wanted $P^2=0$

@robjohn. Any idea about my inequality above please?

what are the $a$'s and $n$?

we know for certain only $a_d$ is positive

for polynomial $p(n)$ equals sum above inside limit. So, probably $n$ is any integer or real number

Did you mean $a_i$ instead of $a_n$?

@robjohn. Yes. Sorry I made mistake up there

a

we know for certain only $a_d $ is positive, but other coefficients $a_i$s could be positive or nrgative

the powers of $n$ are all less than or equal to -1.

I guess here positive does not include 0, so it's obvious now! Sorry again, I did not notice that $a_d$ is positive! As the limit of the sum goes to infinity, the limit approaches 0, which means that at $a_d$, we can find number $c=0.5$ s.t $sum < 0.5 a_d$.

@robjohn. What do you think please?

Now, if $k>d$ and we know for sure that $a_d > 0$, then why for $k>d$, we have So, $sum \in O(n^k)$? I mean coefficients $a_{k>d}$ could be either positive or negative I guess?

@robjohn. I found discussion here: atekihcan.github.io/CLRS/03/P03-01

But, the question above is to the discussion in link

Forgive

I got answer for logarithm question above based on fact $log_b{x} = \frac{ln x}{ln b}$.

@Avra as $n\to\infty$, that sum tends to $|a_d|$

so the inequality cannot be true.

$\log_b(x)=y\iff x=b^y$. Take the log of both sides, you get $\log(x)=y\log(b)$. Divide to get $\log_b(x)=\frac{\log(x)}{\log(b)}$

Is $\sum_{n=1}^\infty\frac{(-1)^{n+1}}{2n}=\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}$ true. They say this is true because of rearrangement.

@EvilJohnRennie a sum is considered the limit of the partial sums and the limit of the partial sums of the left is $\frac12\log(2)$ while the limit of the partial sums of the right is $\log(2)$.

Even though 1 will never come up in rhs.

wait

However, they are not absolutely convergent, so you can rearrange each sum to be any limit you want

@robjohn Thanks lol I was wondering why my answer is right.

john rennie- evil john rennie

what is this

Nono I stated my question in wrong way.

$\sum_{n=1}^\infty\frac{(-1)^{n+1}}{2n}=(1-1/2)-1/4+(1/3-1/6)-1/8+(1/5-1/10)...$

@EvilJohnRennie that is false

Sorry added dot.

the sum on the left has a definite meaning

@robjohn. Thanks

@Avra: what you have is not the same thing as being discussed in that link

why did you think that it was bounded by $\frac12a_d$?

v

@robjohn But If I evaluate the bracket ones then the sum is same.

Oh thank now I get it lol.

@EvilJohnRennie the same as what?

@robjohn. How $n^k > n^d$ given that we only know that $a_d$ is positive while no other information is provided after $k>d$ please?

Hello. Are X and X' correlated? I am studying statistics. Trying to understand correlation. i just want to know what will be the value of Cov(X, X') (positive, 0 or negative)

@robjohn lhs and rhs becomes same when I do the bracket one first.

Which means it is true not false which is true.

@EvilJohnRennie Okay, that one is true. I thought you had $n$ in the denominator.

@RussianBot2.0 hello Mr ip boy long time no see.

mista

@Avra that says $1.5a_d$, not $0.5a_d$

i love african accent

it is very nice

ah

ahhhhfrika...?

@robjohn. Sorry. Yes. True, still my question please how How $n^k>n^d$ given that we only know that $a_d $is positive while no other information is provided after $ k>d $ please?

v'y good accent, mista'

@Avra what is $k$? I don't see any $k$ in your sum

anotha' bwotha' fwom the same motha'

I also like the Russian and French accent

Irish accent too

there was a yt comment saying

there's a different accent in ireland every 10 miles

when will you release bot $6.\bar{9}$

Anyone know a good book on summation of inverse trigonometry series

@PrateekMourya there are questions on the site about it, I am sure. I have answered some about inverse tangent sums.

do you have an example?

@robjohn yes, I want $P^2 \not= 0$

As far as I can see , I get $P^2=0$ , for many n's

@Rover I said that $P^2=0\iff n\equiv0\pmod3$

so if $n\not\equiv0\pmod3$, $P^2\ne0$

@robjohn I didn't got what you mean by $n\equiv0\pmod3$ ?

@EvilJohnRennie 42.0 days later

@Rover you are unfamiliar with arithmetic modulo a number?

$n\equiv0\pmod3$ means $3$ divides $n$

modular arithmetic

I love it and hate it

@robjohn i wanted like a worksheet of it all types easy to most difficult like i found some for simple trigonometry in treatise in trigonometry Cambridge

Also thanks for being so active

@robjohn okay.. I wasn't familiar with it .

@Rover best to be familiar with modular arithmetic if you're dealing with roots of unity (like $\omega^3=1$).

hello @robjohn how are you doing today?

@o.9 fine. what's up?

I just wanted to say hello to you

I never understood why math people hate writing things clearly. It's not like they are always having IQ 250 and I am with IQ 25.

I guess there are lots of math people over here...please feel free to explain what makes you feel so bad about writing things clearly.

git gud or get mad lolz

I am neither mad nor good. But my question is genuine.

math people are literally one of very few groups that make stuff as rigorous as possible

programmers are like 30 times worse at that

But it's still easy to know what they mean after a while

You can debug a code to understand it, you can't debug an idea.

That just seems like an excuse as to why they suck at making stuff clear

you can also debug an idea, try with various interpretations

I am trying to understand generalized polynomial chaos expansion (by understand I mean in a crystal clear way so that I can tell it to a homeless man or 5yr old kid) but I didn't find any good material.

and check the validity of each and if it gets you what you want.

@o.9 I was checking your reputation but I never found an answer of yours or anything. (Curious about your answers that's all). Your profile is also like those unfinished "left as an exercise" kind of thing.

that's my profile, keep your expectations low though. math.stackexchange.com/users/33907/yorch

It's a very impressive and good profile.

@user27286 I challenge your assertion. What makes you believe that mathematicians "feel bad" about writing clearly?

there are mediocre writers in every field. the best mathematicians tend to be very good writers.

I assumed that "feeling" is mostly binary in terms of the writing work, so opposite of feeling bad is feeling good. Now if people really feel good about writing everything properly then we would see more books in which math is explained in an easier way. I don't find any in my current field almost. (stochastic computation)

writing in applied math tends to be worse than pure math. on average. but the best applied mathematicians write very clearly.

I guess stoch comp is kind of applied math

@user27286 Did you consider, perhaps, that mathematics is a difficult field?

"There is no royal road to geometry."

there should be.

There is only so much that one can do in order to explain something in "an easier way".

But why is something difficult if some people know it already? If so many people know maths for so long what do they do make it available to masses ...?

sometimes well written mathematics is very difficult to understand. and sometimes easy to understand mathematics is not well written.

it's a crap shoot.

i met an irishman today, and to show off, i opened by asking where in northern ireland he was from. he had the accent. he was from just south of the border. oops.

Btw, just to clarify, I don't mean by well written, it should be precise from page-1. I guess we should introduce wrong math first instead of explaining the correct whole thing and then gradually improve from there.

@user27286 I don't understand.

a lot of stuff to do with computing is poorly written. i won't argue with that.

Why should we ever introduce "wrong" math?

@XanderHenderson Because that atleast give you a pathway of imagination. When the imagination grows a bit then you can be comfortable breaking it and move into more abstract parts.

@user27286 But you are suggesting that we intentionally teach people something which is wrong! I don't see how this could ever be pedagogically justified!

Wrong -> Not fully correct. More precise wording be "partially correct" without covering all the details of mathematical preciseness.

I still don't really see what you are on about. Can you give an example?

Ah sure.

@user27286 Do you mean something like this?

@vitamind Yes yes..this would do.

i used to have to review papers in quantum computing and the quality of the writing was really low, but i don't think it was deliberate. it was people learning how to write from people who weren't thinking about it.

Schor invented an algorithm that ideally can change the way this world works provided we invent a powerful enough quantum computer. Now the factorization algorithm used the idea of phase-finding algorithms.

lots of non-native speakers, too.