/sqt/ - Stupid Question Thread: Spec Z[x] Edition

>The chance of not being born of that ethnicity is 60%
birth rates vary by ethnicity

You have three numbers; 0.1, 21, and 150. Lets label these a, b, and c, with an output x. x has to approach 1 as a approaches 1, and it also has to approach 1 as c approaches infinity. x has to approach a^c as b approaches c.

To all the analysts:
How can I express delta as a function of epsilon without including c? I can't seem to figure it out. Any hints?

You're right, but in an idealized model where the birth rate is uniform across ethnicities, the question would still be valid, wouldn't it?

they're both correct, of all the ethnicities you're most likely to be born the most common one but you're still more likely to be born outside of that ethnicity if it's not the majority

it's just that the sum of all the probabilities of minority ethnicities outweights the largest one

doesn't just taking epsilon=delta work?

I guess that does make sense, it just sounded contradictory to me. Thanks for the answer

I have never been to Veeky Forums before and I am not sure if I am replying to this correctly, but regarding the 10% question of cars are blue question, the answer you posted isn't really correct. It's using the normal approximation to a binomial distribution without the continuity correction. Since it seems to have been taken out of an intro stats textbook, I hope these formulas and symbols will make sense: you would use the z-distribution, with x = 20, mu = n*p = 150*0.1 = 15, and sigma = sqrt(n*p*q) = sqrt(150*0.1*.9). Then, use technology or a standard normal table to evaluate the area to the right of 20 (by find the probability z is greater than 1.36). However, since we want the probability that MORE than 20 cars would be blue, and this is really a binomial distribution (a discrete distribution), it would be better to use the continuity correction and find the area to the right of 20.5 (use x=20.5 in the z formula). Ideally, you'd want to use technology to use cumulative binomial distribution from 21 to 150, with p = 0.1 and n = 150. If you are curious, using this "exact" method would give you approximately 0.0721.

wolframalpha.com/input/?i=1- (sum (150 choose n) 0.1^n 0.9^(150-n) from n=0 to 20)

Is there a sciency name for a stick?

Yeah it does, cheers for that mate

if a and b are two particles traveling in opposite direction each going the speed of light, how does the distance between them increase?

please fucking answer this it's bugged me for years and I never got a chance to ask my physics professor. My gut tells me it just increases at the speed of light but i'm not sure

Do its electrons just "jiggle"?

correct. kinetic energy from heat causes electrons to jump into higher energy orbitals, say an S to a P orbital. Then the electron releases that energy in the form of a photon as is jumps back down to the original orbital. The energy contained in that photon is equivalent to the difference in energy between the two electron orbitals
as to the rest of your question, what i think you're asking is; when an atom is hit by a photon how is it decided whether to reflect the photon or absorb it as kinetic energy (heat)?

fuck if I know, no idea how mirrors work on a quantum level

I had a hard time with entropy as well. not sure if it'll help but I can tell you what my quantum chem prof told me when I couldn't figure it out.

>People tell you that energy is what drives chemical reactions, but that fucking wrong. Entropy is what drives chemical reactions. There's more energy in a hundred gallons of room temperature sea water than there is in gasoline, but gasoline is useful because it's ordered energy. And this order can be turned into disorder (combustion) with the consequence we can drive the fuck around town

It increases by a rate of 2c, but that isn't a violation of relativity.

What you're probably wondering is how fast one photon appears to be moving from the perspective of the other photon. The answer is that photons don't have a perspective because they are frozen in time.

A better question is what happens when you have two photons traveling towards each other, with you in between them, and you're moving very fast in the same direction as one of them.

>The answer is that photons don't have a perspective because they are frozen in time.

this is hard to understand because the photons obviously still experience time (i.e. being distinct different places at different times).

Can't tell if I believe you or not yet. I'll have to think about that. Relativity is one of those things that can't be taught, it can only be learned.

>being distinct different places at different times
That's only from an outside perspective. A photon experiences its entire lifetime in an instant. This is because moving at relativistic speed slows down your clock, which makes everything around you appear to move faster. Moving at light speed = your clock is frozen and everything that ever happens from that perspective occurs in an instant.

Is there a concept of "realness" in mathematics?

Like prime numbers having properties that have nothing to do with their initial definition is a testimony to their "realness". The exponential function working beyond values outlined in its basic definition(natural numbers) is a testimony to its "realness". And of course, something describing reality is a testimony to its realness.

If (X, *) is a groupoid (or magma if you prefer the term), it is said to be flexible if (xy)x=x(yx) for all x and y, and power-associative if, for every x, the subgroupoid generated by {x} is associative.

Question: Is every flexible groupoid power-associative? My intuition says yes, but I can't seem to get the induction right, and the literature seems to be undecided on the matter.

Sounds like the concept of an extension is what you're looking for.

en.wikipedia.org/wiki/Extension#Mathematics

It is all analogies if you dont want to use QM. Buy yeah, jiggling is fair enough.
The microscopic description comes from QM, where the chance for an electron tot be excited from state 1 tot 2 is the same as decaying from 2 to 1. Ie effecient absorbers are efficiënt emmiters. This is a result of the symmetry of the process.

Can someone give me a hand with this proof?
[eqn] \sum_{d*e \vert n}^{} \mu(e) \lfloor \frac{x}{d*e} \rfloor = \lfloor x \rfloor [/eqn]

It is the problem in pic related. As you can see, I've slightly rewritten it to make it easier (I hope) to deal with.

The argument I try to put is as follows:
What this sum counts will be pairs of numbers (d,e) such that e is square free and d times e is a divisor of n. Then, if e is not equal to 1, and d is not equal to 1 then each pair (d,e) has an "inverse" pair that will cancel it out, leaving only the pair (1,1) which results in [math] \lfloor x \rfloor [/math]

Case 1: (d,e) where both d and e are square free, and are of opposite parity. (d has an even number of primes, while e has an odd number of primes, or vice versa).

Then the inverse of this pair is simply (e,d) because the mobius function will change sign.

Case 2: (d,e) where both are squarefree but both numbers have the same parity of primes (both have an odd number of primes or an even number of primes).

Then take any prime from e, call it p. Now divide e by p, and multiply d by p. Then the pair
[math] (dp, \frac{e}{p} ) [/math] is an inverse of the original pair.
This is because their product is still d*e but now [math] \frac{e}{p} [/math] has one less prime, which means that the mobius function will change sign.

Case 3: (d,e) where d is not square free.

First, take all the extra primes that d has (the ones that make it not-squarefree) and divide d by them, and multiply e by them. Call this new pair (x,y) which y= e * all the extra primes of d. and x = d / all the extra primes of d.

y may not be squarefree but now shift this pair into (y,x) with x being squarefree guaranteed.

Now, if x has the same parity of primes as e then take one prime from x and give it to y. This way (y,x) is an inverse.

If x has opposite parity to e then leave the pair (y.x) as it is and it will be an inverse.

My question is... is this enough? Or is this argument missing something?

[math]
\lim_{x\to 1} \frac{x^{1000} - 1}{x-1}
[/math]

1000

Here is the hint:

Do long division with x^1000 - 1 over x - 1
You don't have to do it all, just keep going until you see the pattern.

Then you will be left with a degree 999 polynomial and then you just have to evaluate it at 1.

I need to come up with some silly questions about a big pile of data from a random census. I've already got 7 but I need 3 more; the questions are for a high school IT competition's Excel part, so they should be somewhat challenging to get the correct answers to. Any tips?

0.1^21 * 0,9^129 + 0.1^22 * 0,9^128 + 0.1^23 * 0,9^127 +...

use l'hospital's rule

The inverse of a 1x1 matrix is just the inverse of the element itself, right?
Say the matrix is (10), then the inverse would be 1/10? (assuming 10 is a real number)

yes

f(x) = ax+b , x > 0
sin 2x, x ≤ 0

How do I find the values of a and b for which the function is continuous but not differentiable?

from flexibility, [math]x^3[/math] is well-def, and then [math]x^3x = (xx^2)x = x(x^2x) = xx^3[/math]. so you can prove left association equals right association on any string of [math]x[/math]'s. however, this says nothing about [math]x^2x^2[/math].

commutativity implies flexibility so it would suffice to display a commutative but not power-associative magma. i came up with [math](\mathbb N,{*})[/math] where [math]m*n = mn+1[/math].
[math]1*1 = 2[/math] and [math]1*2 = 3[/math] and so on, but [math]2*2=5\ne4=1*3[/math]. so [math]x=1[/math] is not power-associative in this flexible magma.

Literally just one application of l'Hospital's rule

I think as long as b=0 and a is nonzero this works

I need to find what the true anomaly v is given a longitude from the reference direction.
I swear to God I've tried every projection thingy I can but I can't find some way of going from
longitude -> v

I've heard that doing many many problems is the best way to "get gud" at new math areas
So what are some good book with tons of problems (that include solutions)
Are Schaum's considered good?
These are some of the topics I'm interested in for reference

>topology
>proofs
>real analysis
>abstract algebra
>number theory
>any other upperclass math undergraduate topics.

This is from the solution:
"since g(x) = ax+b
and h(x) = sin 2x are continuous functions the only place where f(x) might be discontinuous is where x = 0"

Why though? How do I prove this?

>number theory
d. p. parent - exercises in number theory
ram murty - problems in algebraic number theory
ram murty - problems in analytic number theory
ram murty - problems in the theory of modular forms
gouvea - p-adic numbers: an introduction

math.stackexchange.com/questions/185607/problem-books-in-higher-mathematics
might be useful

also springer.com/series/714?detailsPage=titles

>>abstract algebra
hungerford has lots of problems and solutions for odds

I have a question for a chemist if there is one around.
I need generalized opinion on the state of polymer advancement and manufacturing.
Are there any compounds that the regular guy should be on a lookout for or we are on the same plateau just like after 1907?

Thanks, I'll check them out.

Polymer/clay nanocomposites are pretty hot.

In the picture, you can see that Spec Z[x] sits over Spec Z in a natural way, and the fiber over each point is the affine line over the corresponding prime field, which in turn hold together affine lines over all extensions of that prime field.

That is pretty cool I will be sure to look into it.

I'm about to graduate with an engineering degree.

Would someone who has been an engineer for a while give me advise on what to expect?

What did you not learn in school that came up in the job, like soft skills?

Would you tell me about any hard lessons you had?

Can a BS physcist land many of the jobs a math BS can?

Can a pure math BS land the same jobs as a statistics BS can?

I'm looking for the best middle ground between employability at the BS level, while keeping grad school options open for Pure Math or Machine Learning.

Was defining e the first problem that was solved used limits?

What causes you to feel hungry or full? Is it calories, nutrition, weight of food, or something else

I'm trying to do a Calc3 assignment dealing with 1D line densities.
But I'm confused, the grid we've been provided has the coordinates on each box rather than each line, like an excel spreadsheet rather than a cartesian graph.
I'm meant to approximate the arc length between a variety of points, but a lot of these points, if I'm reading this graph correctly, share X/Y coordinates despite clearly having a change in those coordinates.

For example, p0 and p1 clearly have an upward curve, but their coordinates based on this system are (3,3) and (1,3)? Meaning the approximated length is "2"? Am I reading this right or am I missing something?

No,its something called Leptine.
Google the rest.

bernard bolzano was the first to use limits
he used them to solve problems about the real number line

Veeky Forums-science.wikia.com/wiki/Universal_Material#Professionalism_and_Career_Development

thanks senpai

If there are multiple values that work for x, can x be said to be a set containing those values?

The set S of solutions to an equation is the set of all the numbers x that satisfy the equation if that's what you're asking for.

I was wondering what is x as an entity when you lift it from the equation and perform other operations on it. But what I now think the answer is that the entity is what you get(another equation) when you isolate x on the other side of the =.

What the fuck does this mean

Don't listen to all the idiots , , . You should notice immediately yourself that [eqn]\frac{x^{1000}-1}{x-1}=1+x+x^2+x^3+\cdots+x^{999}[/eqn] Hence the answer is obviously 1000.

Looking to pursue a career in forensics which most likely means a BS in chemistry to start with.
Is it really complicated material or is it more a matter of just putting in the time to study? I'm not dumb but not a genius either and I haven't taken a chemistry course since high school

Are you actually retarded?

To get what you did, you have to do the long division I described. The difference between your statement and my statement is that while you just pulled an answer out of your ass, I explained HOW to reach that answer.

>inb4 he probably already know how to get there
If he knew then he wouldn't be asking.

>Doesn't recognize geometric series

brainlet

Are you actually retarded? To get what I did you multiply both sides by (x-1) and you're immediately done. Or more practically speaking, if you're learning about limits then you're BOUND to have seen the formula for geometric series before and one of the biggest "applied" lessons in mathematics is that you should always recognise the formula when it pops up. Don't be daft.

Can anyone help with pic related? This has got me really stumped for a while now...

>forgetting that if the asker knew any of this he wouldn't have even asked such a simple question. It is clear that he DOESN'T know how to derive that identity and that is why he had to ask

Can someone tell me what topics/books I should read to be able to understand what is in OP's picture. I did mostly analysis and probability theory in undergrad before doing a masters in computer science. Took a class on group theory but feels really long ago.

I'm interested in solid foundations. Thanks

Mind letting me know where that quesion is from

Some book on linear on algebra?

Hopefully I'm not using the wrong terminology here.

How hard is it to show that a problem is NP complete? Has there ever been a case of people trying to find an efficient way to solve a certain type of problem, then somebody showed that it was NP complete, causing most people to give up?

It's a picture from Mumford's Red Book of Schemes. Open the book, and it will tell you what you should know heading in.

In a ring, does existence of inverses imply existence of a unity?

The functions f(x) and g(x) in the table below show Jane's and Mariah's savings respectively, in dollars, after x days. Some values are missing in the table.

x(years) 1 2 3
g(x) = 3x
Jane's savings in dollars 3 9
f(x) = 3x + 3
Mariah's savings in dollars 6 9

Which statement best describes Jane and Mariah's savings in the long run?

Alright. Thanks.

I don't know of anyone whose definition of a ring doesn't include the existence of a unit. Anyways, the notion of inverse doesn't make sense without a notion of unit.

I don't recall it being too hard. We did it in our second course on algorithms. Though the problems were most likely contrived, making it easy to reduce some NP-complete problem to ours.

Literally plug in the values and see what you get

Wot? My book says a ring is 1. An abelian group under addition 2. Associative over multiplication 3. Distributive over multiplication.

e.g. [math]$2\mathbb{Z}$[/math] is a ring with no unity

Is Dummit and Foote good for studying on my own? I'd like book with lots of problems and hopefully solutions to compare.

I don't know about your specific textbook, but nobody actually working in the field would call that a ring.

math.mit.edu/~poonen/papers/ring.pdf

is kinda right in the sense that no one uses rings without identities but checking my freshman year notebook, my professor back then did make the distinction between unitary rings and non-unitary rings.

But again, after that intro to algebraic structures, all the rings I've been made to work with have had a unity. It is kinda pointless to study rings without identities because of the most fundamental questions you can ask about a ring is which elements have inverses.

Tackling your original question in , there are no inverses without a unit so yeah.

You cannot have (a)(a^-1) = 1 if 1 is not in the ring.

Nobody's _definition_ of a ring should include unit. That's very bad form; there's nothing in the ring axioms that requires a unit, and there are many natural (enough) rings that don't have one.

But there are many situations where the case without unit is either not relevant or not interesting so a lot of places will begin with "we assume all rings in this book/paper/whatever have 1" which is fine.

It's the derivative of [math]x \mapsto x^1000[/math] at 1, ie. 1000. What do they teach you in school ?

Your second point makes me wonder whether we gain anything of interest in knowing a theorem holds on a rng (which likely implies it holds on a ring)

So the equation
[math] z^{n} = 1, \, n \in \mathbb{Z} [/math]
has [math]n[/math] roots. So is it valid/possible to define a unit circle as the roots of [eqn] \lim_{n\to\infty} z^{n} = 1\, \,?[/eqn]
and if not, why not?

kind of. I think what you're getting at is the set of all roots of unity, [math] \{z: z^n=1\text{ for some } n\in N\}[/math]. this doesn't include every point on the unit circle, since for example if you choose any irrational number r, then no integer power of [math] e^{2\pi i r}[/math] will be equal to 1.

Hmm
Solutions are of the form exp(2pi(k/n)). Feels like there are some values we can't reach for any values for k and n. If we substitute irrational instead of k/n

Forgot the i >__

Noob tex test
[math]\exp\left(2\pi i\frac{k}{n}\right)[/math]

With this change if pi is proven normal?

fair enough, irrationals seem like a good counter example

I'm about to graduate but have difficulties solving putnam problems. Am I a brainlet?

Is
[math]
\sum_{k=1}^{\infty} k - \sum_{k=10}^{\infty} = 1+2+3+4+5+6+7+8+9 = 45
[/math]
correct? Can you just pull out those terms and subtract the other two summations?